*Article* **Adaptive Fixed-Time Control of Strict-Feedback High-Order Nonlinear Systems**

**Yang Li 1,\* , Jianhua Zhang <sup>1</sup> , Xiaoyun Ye <sup>1</sup> and Cheng Siong Chin <sup>2</sup>**


**Abstract:** This paper examines the adaptive control of high-order nonlinear systems with strictfeedback form. An adaptive fixed-time control scheme is designed for nonlinear systems with unknown uncertainties. In the design process of a backstepping controller, the Lyapunov function, an effective controller, and adaptive law are constructed. Combined with the fixed-time Lyapunov stability criterion, it is proved that the proposed control scheme can ensure the stability of the error system in finite time, and the convergence time is independent of the initial condition. Finally, simulation results verify the effectiveness of the proposed control strategy.

**Keywords:** adaptive fixed-time control; neural network control; strict-feedback high-order nonlinear systems

#### **1. Introduction**

Recently, the adaptive trajectory tracking control of uncertain nonlinear systems has made a significant breakthrough [1–3]. In addition, neural network adaptive control has become a popular method in the past decades [4–6]. Many remarkable results have extended to strict-feedback systems, pure-feedback systems, and Brunovsky systems, and neural networks are combined with various techniques, such as the backstepping technique, the adaptive technique, and the sliding mode control method [7–9]. The neural network is used to identify the nonlinear term of the uncertain system, which combines the advantages of adaptive control. Many excellent articles and monographs have been published. In the design of these control systems, the neural network is used as a general approximator to the uncertain nonlinear term of the systems [10–12]. In these systems, the unknown nonlinear systems are approximate by neural networks, which are valid only within a compact set, and the neural network controller is designed. Based on Lyapunov uniformly bounded (UUB) theory, the closed-loop error systems are bounded [13–15]. In order to overcome the problem of uncertainty or disturbance that does not meet the specific matching conditions, the adaptive controller is usually constructed by combining backstepping control technology with the adaptive neural network. The high-order system is divided into multiple subsystems. The virtual controller of the low-order subsystem is designed first. Then, the recursive design is used until the final design of the neural network adaptive controller to achieve stability of the system, allowing it to possess the desired performance indicators.

In practical engineering applications, the research of high-order nonlinear systems has attracted much attention, and their application is also extensive, for example, as financial systems, communication systems, biological systems, and machine systems [16–18]. Some results regarding high-order system control have been obtained following the development of adding a power integrator [19]. The problems studied in recent years involve robust control [20,21], adaptive global stabilization [17], global asymptotic stabilization [22],

**Citation:** Li, Y.; Zhang, J.; Ye, X.; Chin, C.S. Adaptive Fixed-Time Control of Strict-Feedback High-Order Nonlinear Systems. *Entropy* **2021**, *23*, 963. https:// doi.org/10.3390/e23080963

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 2 July 2021 Accepted: 26 July 2021 Published: 27 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

output feedback stabilization [23], and state feedback output tracking [16]. Many methods have been proposed, such as backstepping technology, adaptive technology, sliding mode control, neural network control, and fuzzy control. However, the above results need to be precise with some unknown coefficients in the system model. In [20], the unknown function in the system is described by the mathematical model of an online neural network. In addition to this pioneering result, high-order system control based on neural networks has been widely developed and applied [24–26].

In the actual industrial process, such as in missile systems, aircraft attitude control systems, robot control systems and other industrial control systems, the purpose of controller design is to achieve stability of the controlled system and maintain it for a limited time. However, the control method without considering the convergence time cannot achieve this objective. Compared with the traditional Lyapunov stability theory, the finite-time Lyapunov stability theory has attracted the attention of many researchers because it can make the controlled system stable near the equilibrium state in finite time [27–29].

Many researchers combine finite-time control with neural network adaptive control for nonlinear systems with nonlinear functions and dynamic uncertainties based on backstepping and propose many related adaptive finite-time control schemes [30–32]. However, there are still many problems to be solved in these existing control strategies. For finite-time control, the convergence time is dependent on the initial condition. However, the ideal weights of NNs are unknown, and it is difficult to obtain a convergence time. Therefore, to solve this issue, fixed-time neural network control is an appropriate selection of the control method.

The high-order systems' neural network control problem is discussed in the articles [33–35]. The fixed-time neural network adaptive controller is present for nonlinear high-order systems. Based on the fixed-time adaptive technology, the strict-feedback high-order system has fixed-time Lyapunov stability based on Lyapunov stability theory [36–38]. The convergence time of the system can be accurately calculated, and the settling time does not rely on the initial situation. The main contributions of this article are as follows:


This article consists of the following sections: in Section 2, a strict-feedback high-order nonlinear mathematical description of the problem is presented; in Section 3, the adaptive fixed-time neural network control scheme for the strict-feedback high-order nonlinear system is designed; in Section 4, simulation results show the effectiveness of the proposed control strategy; in Section 5, the conclusion of the article is presented.

#### **2. Problem Formation and Preliminaries**

Consider the following strict-feedback high-order nonlinear system:

$$\begin{array}{l} \dot{\mathbf{x}}\_{i} = \mathbf{g}\_{i}\mathbf{x}\_{i+1}^{\eta\_{i}} + f\_{i}(\overline{\mathbf{x}}\_{i})\\ \dot{\mathbf{x}}\_{n} = \mathbf{g}\_{n}\mathbf{u}^{\eta\_{n}} + f\_{n}(\overline{\mathbf{x}}\_{n})\\ y = \mathbf{x}\_{1} \end{array} \tag{1}$$

where *x<sup>i</sup>* ∈ *R* is the state of the system; *x<sup>i</sup>* = [*x*1, . . . , *x<sup>i</sup>* ] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup> i* is the state vector of the system; *fi*(*xi*) : *R <sup>i</sup>* <sup>→</sup> *<sup>R</sup>* is the unknown smooth function; *<sup>y</sup>* <sup>∈</sup> *<sup>R</sup>* is the output of the system; *u* ∈ *R* is the corresponding control input of the system; *η<sup>i</sup>* is the order of the system; *g<sup>i</sup>* is the unknown control gain parameter and satisfies 0 < *g i* ≤ *g<sup>i</sup>* ≤ *g<sup>i</sup>* , where *g i* , *g<sup>i</sup>* are known parameters; and the desired trajectory *y<sup>d</sup>* and its derivative are continuous and bounded.

**Lemma 1.** *For positive real numbers p*, *q*, *p* ∈ (0, 1), *q* ∈ (1, ∞) *with a denominator and numerator, both are odd numbers and positive real numbers ρ*, *σ*, *ρ*1, *ρ*2, *σ*1, *σ*2; *then, the following inequalities hold*:

$$\begin{aligned} -\rho \widetilde{\theta} \widehat{\theta}^p &\le -\rho\_1 \widetilde{\theta}^{p+1} + \rho\_2 \theta^{p+1} \\ -\sigma \widetilde{\theta} \widehat{\theta}^q &\le -\sigma\_1 \widetilde{\theta}^{q+1} + \sigma\_2 \theta^{q+1} \end{aligned} \tag{2}$$

*where ρ*1, *ρ*2, *σ*1, *σ*<sup>2</sup> *are determined by p*, *q*, *ρ*, *σ* [39].

**Lemma 2.** *For any constant where x*, *y* ∈ *R and p*, *q are odd, the following inequality holds*:

$$\mathbf{x}^{\mathfrak{J}} - y^{\mathfrak{J}} \le \mathfrak{z}|\mathbf{x} - y| \left( \mathbf{x}^{\mathfrak{J}-1} + y^{\mathfrak{J}-1} \right) \le \mathfrak{z}|\mathbf{x} - y| \left( (\mathbf{x} - y)^{\mathfrak{J}-1} + y^{\mathfrak{J}-1} \right) \tag{3}$$
  $\text{and } a > u > 1, \ \mathfrak{z} - \mathfrak{z} \land \mathfrak{z}^{\mathfrak{J}-2} \ \mathfrak{z} \ \mathfrak{z}$ 

*where ξ* = *q p and q* > *p* > 1, *ζ* = *ξ* 2 *<sup>ξ</sup>*−<sup>2</sup> + 2 .

**Proof.** Assuming *x* ≥ *y*, for any constant, the following equation holds:

$$\frac{x^{\tilde{\varsigma}} - y^{\tilde{\varsigma}}}{x - y} = \tilde{\varsigma} c^{\tilde{\varsigma} - 1} \tag{4}$$

where *c* is an existent constant and satisfies *y* ≤ *c* ≤ *x*; therefore,

$$\begin{aligned} \mathbf{x}^{\mathfrak{E}} - \mathbf{y}^{\mathfrak{E}} &= \mathfrak{F} \mathbf{c}^{\mathfrak{E} - 1} (\mathfrak{x} - \mathfrak{y}) \\ &\le \mathfrak{F} \mathbf{c}^{\mathfrak{E} - 1} |\mathfrak{x} - \mathfrak{y}| \end{aligned} \tag{5}$$

because *y* ≤ *c* ≤ *x*, then *c <sup>ξ</sup>*−<sup>1</sup> <sup>≤</sup> max *x ξ*−1 , *y ξ*−1 ≤ *x <sup>ξ</sup>*−<sup>1</sup> + *y ξ*−1 ; therefore,

$$|\mathbf{x}^{\xi} - \mathbf{y}^{\xi}| \le \xi |\mathbf{x} - \mathbf{y}| \left(\mathbf{x}^{\xi - 1} + \mathbf{y}^{\xi - 1}\right) \tag{6}$$

On the other hand, based on *ξ* > 1, for *x ξ*−1 ,(*x* − *y*) *ξ*−1 , *y ξ*−1 , we have

$$\mathbf{x}^{\mathfrak{z}^{\mathfrak{z}}-1} \le \begin{cases} (\mathfrak{x}-\mathfrak{y})^{\mathfrak{z}^{\mathfrak{z}}-1} + \mathfrak{y}^{\mathfrak{z}-1}, \mathbf{1} < \mathfrak{z} < \mathbf{2} \\\ 2^{\mathfrak{z}} \mathbf{2} \left( (\mathfrak{x}-\mathfrak{y})^{\mathfrak{z}^{\mathfrak{z}}-1} + \mathfrak{y}^{\mathfrak{z}-1} \right), \mathfrak{z} \ge \mathbf{2} \end{cases} \tag{7}$$

then, we choose

$$\alpha^{\tilde{\varsigma}-1} \le \left(2^{\tilde{\varsigma}-2} + 1\right) \left(\left(\chi - y\right)^{\tilde{\varsigma}-1} + y^{\tilde{\varsigma}-1}\right) \tag{8}$$

therefore,

$$x^{\sharp -1} + y^{\sharp -1} \le \left(2^{\sharp -2} + 2\right) \left( (x - y)^{\sharp -1} + y^{\sharp -1} \right) \tag{9}$$

Then,

$$|\mathcal{J}|\mathbf{x} - y| \left(\mathbf{x}^{\tilde{\xi}-1} + y^{\tilde{\xi}-1}\right) \le \mathcal{J}|\mathbf{x} - y| \left((\mathbf{x} - y)^{\tilde{\xi}-1} + y^{\tilde{\xi}-1}\right) \tag{10}$$

where *ζ* = *ξ* 2 *<sup>ξ</sup>*−<sup>2</sup> + 2 .

#### **3. Main Results**

In this section, for the strict-feedback high-order nonlinear system, the neural network is used to identify the nonlinear system, and an adaptive algorithm is used to adjust the weight coefficient of the neural network. Based on fixed-time Lyapunov stability theory, a neural network adaptive tracker based on backstepping control strategy is designed so that the system state can track the preset trajectory. Theoretical proof and a numerical simulation are given.

The design block diagram of the closed-loop system is shown in Figure 1. For highorder nonlinear systems with strict-feedback form, a neural network adaptive controller is

designed to make the system track a given target signal in finite time. The convergence time is independent of the initial condition to achieve fixed-time Lyapunov stability of the closed-loop error system. The controller design can be divided into the following N steps: Step 1: First, for the system, the following variables are selected:

$$z\_1 = x\_1 - y\_d \tag{11}$$

the dynamics of *z*<sup>1</sup> can be obtained as

$$
\dot{z}\_1 = \mathbf{g}\_1 \mathbf{x}\_2^{\eta\_1} + f\_1(\mathbf{x}\_1) - \dot{y}\_d \tag{12}
$$

Moreover, we have

$$f\_1(\mathbf{x}\_1) = \mathcal{W}\_1^T \mathbf{Y}(Z\_1) + \varepsilon\_1(Z\_1) \tag{13}$$

where |*ε*1(*x*1)| ≤ *ε*1, we have

$$|z\_1 f\_1(x\_1) \le \theta\_1 |z\_1| \|\Psi(Z\_1)\| + |z\_1| \varepsilon\_1 \tag{14}$$

*θ*<sup>1</sup> = k*W*1k is defined, and the Lyapunov candidate functional is chosen as

$$V\_1 = \frac{1}{2}z\_1^2 + \frac{1}{2\mu\_1}\tilde{\theta}\_1^2\tag{15}$$

where *µ*<sup>1</sup> > 0 is positive constant, and *θ*e <sup>1</sup> <sup>=</sup> <sup>ˆ</sup>*θ*<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*1. Differentiating *<sup>V</sup>*<sup>1</sup> with respect to time *t* yields

$$\dot{V}\_1 \le g\_1 z\_1 \mathbf{x}\_2^{\eta\_1} + \theta\_1 |z\_1| \|\mathbf{Y}(Z\_1)\| + |z\_1| \varepsilon\_1 - z\_1 \dot{y}\_d + \frac{1}{\mu\_1} \widetilde{\theta}\_1 \dot{\theta}\_1 \tag{16}$$

The virtual control signal *α*<sup>1</sup> is selected as

$$\mathfrak{a}\_{1} = -\underline{\mathfrak{g}}\_{1}^{-\frac{1}{\eta\_{1}}} \left( \operatorname{sign}(z\_{1}) \widehat{\theta}\_{1} || \mathbb{1} (Z\_{1}) || + \frac{z\_{1} \varepsilon\_{1}^{2}}{|z\_{1}| \varepsilon\_{1} + \eta\_{1}} + \operatorname{sign}(z\_{1}) |\dot{y}\_{d}| + \kappa\_{1} z\_{1}^{p} + \iota\_{1} z\_{1}^{q} \right)^{\frac{1}{\theta\_{1}^{2}}} \tag{17}$$

Then, based on Lemma 2, we have

$$\dot{V}\_1 \le c\_1 \overline{\varrho}\_1 |z\_1| \left( \left| z\_2^{\eta\_1} \right| + |z\_2| z\_2^{\eta\_1 - 1} \right) - |z\_1| \widetilde{\theta}\_1 \|\Psi(Z\_1)\| + \delta\_1 + \frac{1}{\mu\_1} \widetilde{\theta}\_1 \dot{\theta}\_1 - \kappa\_1 z\_1^{p+1} - \iota\_1 z\_1^{q+1} \tag{18}$$

where

$$z\_2 = x\_2 - x\_1 \tag{19}$$

then the adaptive law design as

$$\dot{\theta}\_1 = \mu\_1 \left( |z\_1| |\Psi\_1| - \rho\_1 \theta\_1^p - \sigma\_1 \theta\_1^q \right) \tag{20}$$

based on Equation (18), we have

$$\begin{split} \dot{V}\_1 &\leq c\_1 \overline{\mathbf{g}}\_1 |z\_1| \left( \left| z\_2^{\eta\_1} \right| + |z\_2| \mathbf{x}\_2^{\eta\_1 - 1} \right) - |z\_1| \widetilde{\theta}\_1 ||\mathbf{Y}(Z\_1)|| + \delta\_1 + \widetilde{\theta}\_1 |z\_1| |\Psi\_1| \\ &- \rho\_1 \widetilde{\theta}\_1 \theta\_1^p - \sigma\_1 \widetilde{\theta}\_1 \theta\_1^q - \kappa\_1 z\_1^{p+1} - \iota\_1 z\_1^{q+1} \end{split} \tag{21}$$

based on Lemma 1, we have

$$\begin{aligned} -\rho\_1 \widetilde{\theta}\_1 \theta\_1^p &\le -\varsigma\_1 \widetilde{\theta}\_1^{p+1} + \upsilon\_1 \theta\_1^{p+1} \\ -\sigma\_1 \widetilde{\theta}\_1 \theta\_1^q &\le -\omega\_1 \widetilde{\theta}\_1^{q+1} + \vartheta\_1 \theta\_1^{q+1} \end{aligned} \tag{22}$$

then

$$\dot{V}\_1 \le c\_1 \overline{\varrho}\_1 |z\_1| \left( \left| z\_2^{\theta\_1} \right| + |z\_2| \mathbf{x}\_2^{\theta\_1 - 1} \right) + \delta\_1 - \zeta\_1 \overline{\theta}\_1^{p+1} + v\_1 \theta\_1^{p+1} - \omega\_1 \overline{\theta}\_1^{q+1} + \theta\_1 \theta\_1^{q+1} - \kappa\_1 z\_1^{p+1} - \iota\_1 z\_1^{q+1} \tag{23}$$

Step i: the tracking error can be described as

$$z\_i = x\_i - \mathfrak{a}\_{i-1} \tag{24}$$

Based on dynamics and tracking error, the dynamics of *z<sup>i</sup>* can be obtained as

$$
\dot{z}\_i = g\_i \mathbf{x}\_{i+1}^{\theta\_i} + f\_i(\mathbf{x}\_i) - \dot{a}\_{i-1} \tag{25}
$$

Moreover, we have

$$f\_i(\mathbf{x}\_i) - \dot{\alpha}\_{i-1} = \mathcal{W}\_i^T \Psi(Z\_i) + \varepsilon\_i(Z\_i) \tag{26}$$

where |*εi*(*xi*)| ≤ *ε<sup>i</sup>* ; we have

$$z\_i(f\_i(\overline{x}\_i) - \dot{a}\_{i-1}) \le \theta\_i |z\_i| \|\Psi(Z\_i)\| + |z\_i|\varepsilon\_i \tag{27}$$

*θ<sup>i</sup>* = k*Wi*k is defined, and the Lyapunov candidate functional is chosen as

$$V\_i = \frac{1}{2}z\_i^2 + \frac{1}{2\mu\_i}\hat{\theta}\_i^2\tag{28}$$

where *µ<sup>i</sup>* > 0 is positive constant and *θ*e *<sup>i</sup>* <sup>=</sup> <sup>ˆ</sup>*θ<sup>i</sup>* <sup>−</sup> *<sup>θ</sup><sup>i</sup>* . Differentiating *V<sup>i</sup>* with respect to time *t*, yields

$$\dot{V}\_{i} \le g\_{i} z\_{i} \mathbf{x}\_{i+1}^{\theta\_{i}} + \theta\_{i} |z\_{i}| \|\mathbf{Y}(Z\_{i})\| + |z\_{i}| \varepsilon\_{i} + \frac{1}{\mu\_{i}} \widetilde{\theta}\_{i} \dot{\theta}\_{i} \tag{29}$$

The virtual control signal *α<sup>i</sup>* is designed as

$$\mathfrak{a}\_{i} = -\underline{\underline{\mathcal{G}}}\_{i}^{-\frac{1}{\eta\_{i}}} \begin{pmatrix} \mathfrak{c}\_{i-1}\operatorname{sign}(z\_{i})\overline{\mathfrak{g}}\_{i-1}|z\_{i-1}|\left(\boldsymbol{z}\_{i}^{\eta\_{i-1}-1} + \boldsymbol{x}\_{i}^{\eta\_{i-1}-1}\right) + \operatorname{sign}(z\_{i})\boldsymbol{\theta}\_{i}||\mathbb{1}(\boldsymbol{Z}\_{i})|| \\\\ + \frac{\boldsymbol{z}\_{i}\boldsymbol{z}\_{i}^{2}}{\overline{\boldsymbol{z}\_{i}}|\boldsymbol{\varepsilon}\_{i} + \boldsymbol{\delta}\_{i}^{\*}} + \boldsymbol{\kappa}\_{i}\boldsymbol{z}\_{i}^{p} + \boldsymbol{\iota}\_{i}\boldsymbol{z}\_{i}^{q} \end{pmatrix}^{\frac{1}{\boldsymbol{\theta}\_{i}}} \tag{30}$$

then, based on Lemma 2, we have

$$\begin{split} \dot{V}\_{i} &\leq -c\_{i-1}\overline{\mathfrak{g}}\_{i-1}|z\_{i-1}|\left( \left| z\_{i}^{\eta\_{i-1}} \right| + |z\_{i}|\mathbf{x}\_{i}^{\eta\_{i-1}-1} \right) + c\_{i}\overline{\mathfrak{g}}\_{i}|z\_{i}|\left( \left| z\_{i+1}^{\eta\_{i}} \right| + |z\_{i+1}|\mathbf{x}\_{i+1}^{\eta\_{i}-1} \right) \\ &- |z\_{i}|\widetilde{\theta}\_{i}|\|\mathbf{Y}(Z\_{i})\| + \eta\_{i} + \frac{1}{\mu\_{i}}\widetilde{\theta}\_{i}\dot{\theta}\_{i} - \kappa\_{i}z\_{i}^{p+1} - \iota\_{i}z\_{i}^{q+1} \end{split} \tag{31}$$

where

$$z\_{i+1} = \mathfrak{x}\_{i+1} - \mathfrak{a}\_i \tag{32}$$

Then, the adaptive law design as

$$\theta\_i = \mu\_i \left( |z\_i| |\Psi\_i| - \rho\_i \theta\_i^p - \sigma\_i \theta\_i^q \right) \tag{33}$$

based on Equation (31), we have

$$\begin{split} \dot{V}\_{i} &\leq -\varepsilon\_{i-1}\overline{\mathfrak{z}}\_{i-1}|z\_{i-1}|\left( \left| z\_{i}^{\eta\_{i-1}} \right| + |z\_{i}|\mathbf{x}\_{i}^{\eta\_{i-1}-1} \right) + c\_{i}\overline{\mathfrak{z}}\_{i}|z\_{i}|\left( \left| z\_{i+1}^{\eta\_{i}} \right| + |z\_{i+1}|\mathbf{x}\_{i+1}^{\eta\_{i}-1} \right) - |z\_{i}|\overline{\theta}\_{i}|\mathbf{V}(Z\_{i})|| + \eta\_{i} \\ &+ \overline{\theta}\_{i}|z\_{i}|\left| \mathbf{Y}\_{i} \right| - \rho\_{i}\overline{\theta}\_{i}\theta\_{i}^{p} - \sigma\_{i}\overline{\theta}\_{i}\theta\_{i}^{q} - \kappa\_{i}z\_{i}^{p+1} - \iota\_{i}z\_{i}^{q+1} \end{split} \tag{34}$$

based on Lemma 1, we have

$$\begin{aligned} -\rho\_l \widetilde{\theta}\_l \theta\_i^p &\le -\varrho\_l \widetilde{\theta}\_i^{p+1} + \upsilon\_l \theta\_i^{p+1} \\ -\sigma\_l \widetilde{\theta}\_l \theta\_i^q &\le -\omega\_l \widetilde{\theta}\_i^{q+1} + \vartheta\_l \theta\_i^{q+1} \end{aligned} \tag{35}$$

then

$$\begin{split} \dot{V}\_{i} &\leq -\varepsilon\_{i-1} \overline{\mathfrak{z}}\_{i-1} |z\_{i-1}| \left( \left| z\_{i}^{\eta\_{i-1}} \right| + |z\_{i}| \mathbf{x}\_{i}^{\eta\_{i-1}-1} \right) + \varepsilon\_{i} \overline{\mathfrak{z}}\_{i} |z\_{i}| \left( \left| z\_{i+1}^{\eta\_{i}} \right| + |z\_{i+1}| \mathbf{x}\_{i+1}^{\eta\_{i}-1} \right) + \delta\_{i} - \varsigma\_{i} \overline{\mathfrak{z}}\_{i}^{p+1} + \upsilon\_{i} \theta\_{i}^{p+1} \\ &- \omega\_{i} \overline{\theta}\_{i}^{p+1} + \vartheta\_{i} \theta\_{i}^{q+1} - \kappa\_{i} z\_{i}^{p+1} - \iota z\_{i}^{q+1} \end{split} \tag{36}$$

Step *n*: the time derivative of *z<sup>n</sup>* can be described as

$$z\_n = \mathfrak{x}\_n - \mathfrak{a}\_{n-1} \tag{37}$$

Based on dynamics and tracking error, the dynamics of *z<sup>n</sup>* can be obtained as

$$
\dot{z}\_n = g\_n \mu^{\eta\_n} + f\_n(\overline{x}\_n) - \dot{\alpha}\_{n-1} \tag{38}
$$

Moreover, we have

$$f\_n(\overline{x}\_n) - \dot{a}\_{n-1} = \mathcal{W}\_n^T \Psi(Z\_n) + \varepsilon\_n(Z\_n) \tag{39}$$

where |*εn*(*xn*)| ≤ *εn*, we have

$$z\_n(f\_n(\overline{x}\_n) - \dot{\mathfrak{a}}\_{n-1}) \le \theta\_n |z\_n| \|\Psi(Z\_n)\| + |z\_n|\varepsilon\_n \tag{40}$$

*θ<sup>n</sup>* = k*Wn*k is defined, and the Lyapunov candidate functional is chosen as

$$V\_n = \frac{1}{2}z\_n^2 + \frac{1}{2\mu\_n}\widetilde{\theta}\_n^2\tag{41}$$

where *<sup>µ</sup><sup>n</sup>* <sup>&</sup>gt; <sup>0</sup> is positive constant and *<sup>θ</sup>*e*<sup>n</sup>* <sup>=</sup> <sup>ˆ</sup>*θ<sup>n</sup>* <sup>−</sup> *<sup>θ</sup>n*. Differentiating *<sup>V</sup><sup>n</sup>* with respect to time *t* yields .

$$\dot{V}\_n \le g\_n z\_n u^{\theta\_n} + \theta\_n |z\_n| \|\Psi(Z\_n)\| + |z\_n| \varepsilon\_n + \frac{1}{\mu\_n} \tilde{\theta}\_n \dot{\theta}\_n \tag{42}$$

The actual control is designed as

$$u = -\underline{\mathbf{g}}\_{n}^{-\frac{1}{\eta\_{n}}} \begin{pmatrix} c\_{n-1} \operatorname{sign}(z\_{n}) \overline{\mathbf{g}}\_{n-1} |z\_{n-1}| \left( \mathbf{z}\_{n}^{\theta\_{n-1}-1} + \mathbf{x}\_{n}^{\theta\_{n-1}-1} \right) + \operatorname{sign}(z\_{n}) \boldsymbol{\theta}\_{n} \|\mathbf{Y}(Z\_{n})\| \\ + \frac{\underline{\mathbf{z}}\_{n} \mathbf{z}\_{n}^{2}}{|z\_{n}| \varepsilon\_{n} + \delta\_{n}} + \kappa\_{n} \boldsymbol{\varepsilon}\_{n}^{p} + \iota\_{n} \boldsymbol{\varepsilon}\_{n}^{q} \end{pmatrix} \tag{43}$$

then, based on Lemma 2, we have

$$\begin{split} \dot{V}\_{n} &\leq -c\_{n-1}\overline{\mathfrak{g}}\_{n-1}|z\_{n-1}|\left(\left(\left|z\_{n}^{\eta\_{n-1}}\right| + |z\_{n}|\chi\_{n}^{\eta\_{n-1}-1}\right)\right) - |z\_{n}|\widetilde{\theta}\_{n}||\mathbb{1}(\mathbb{Z}\_{n})|| + \eta\_{n} \\ &+ \frac{1}{\mu\_{n}}\widetilde{\theta}\_{n}\dot{\widehat{\theta}}\_{n} - \kappa\_{n}z\_{n}^{p+1} - \iota\_{n}z\_{n}^{q+1} \end{split} \tag{44}$$

then, the adaptive law design as

$$\boldsymbol{\theta}\_{n} = \mu\_{n} \left( |z\_{n}| |\Psi\_{n}| - \rho\_{n} \boldsymbol{\theta}\_{n}^{p} - \sigma\_{n} \boldsymbol{\theta}\_{n}^{q} \right) \tag{45}$$

based on Equation (20), we have

$$\dot{V}\_{n} \leq -\varepsilon\_{n-1}\overline{\mathfrak{z}}\_{n-1}|z\_{n-1}|\left(\left(\left|z\_{n}^{\eta\_{n-1}}\right| + |z\_{n}|\mathbf{x}\_{n}^{\eta\_{n-1}-1}\right)\right) + \eta\_{n} - \kappa\_{n}z\_{n}^{p+1} - \iota\_{n}z\_{n}^{q+1} - \rho\_{n}\widetilde{\theta}\_{n}\widehat{\theta}\_{n}^{p} - \sigma\_{n}\widetilde{\theta}\_{n}\widehat{\theta}\_{n}^{q} \tag{46}$$

based on Lemma 1, we have

$$\begin{aligned} -\rho\_n \widetilde{\theta}\_n \widetilde{\theta}\_n^p &\le -\varsigma\_n \widetilde{\theta}\_n^{p+1} + \upsilon\_n \theta\_n^{p+1} \\ -\sigma\_n \widetilde{\theta}\_n \widetilde{\theta}\_n^q &\le -\omega\_n \widetilde{\theta}\_n^{q+1} + \vartheta\_n \theta\_n^{q+1} \end{aligned} \tag{47}$$

**Figure 1.** Block diagram of the closed-loop system.

**Theorem 1.** *For the strict-feedback high-order nonlinear system with unknown nonlinearity (1), based on the feasible virtual control signal (17), (30), actual controller (43), and adaptive function (20) (33), the error system is fixed-time Lyapunov stable, and the convergence time is independent of the initial condition*.

**Proof.** Based on Lyapunov candidate functional (15), (28), (41), the Lyapunov candidate functional is chosen.

$$V = \sum\_{j=1}^{n} V\_{\bar{i}} \tag{49}$$

The virtual control signal is chosen as (17), (30), and the fixed-time adaptive function is chosen as (20), (33); the controller is designed as (43) according to the fixed-time Lyapunov stability theory. Based on fixed-time adaptive neural network control and backstepping technology, and taking the trajectory along the system, we have

$$\begin{split} \dot{V} &\leq -\sum\_{j=1}^{n} \kappa\_{j} \boldsymbol{z}\_{j}^{p+1} - \sum\_{j=1}^{n} \iota\_{j} \boldsymbol{z}\_{j}^{q+1} - \sum\_{j=1}^{n} \varsigma\_{j} \hat{\theta}\_{j}^{p+1} - \sum\_{j=1}^{n} \omega\_{j} \tilde{\theta}\_{j}^{q+1} \\ &+ \sum\_{j=1}^{n} \eta\_{j} + \sum\_{j=1}^{n} \upsilon\_{j} \theta\_{j}^{p+1} + \sum\_{j=1}^{n} \vartheta\_{j} \theta\_{j}^{q+1} \\ &\leq -aV^{\frac{p+1}{2}} - bV^{\frac{q+1}{2}} + c \end{split} \tag{50}$$

where

$$\begin{aligned} a &= \frac{\min\left\{\kappa\_{j\in N}\mathfrak{L}\_{j\in N}\right\}}{\left(\max\left\{\frac{1}{2}, \frac{1}{2\mu\_{j\in N}}\right\}\right)^{\frac{p+1}{2}}}, b = \frac{(2n)^{\frac{1-q}{2}}\min\left\{\iota\_{j\in N}, \omega\_{j\in N}\right\}}{\left(\max\left\{\frac{1}{2}, \frac{1}{2\mu\_{j\in N}}\right\}\right)^{\frac{q+1}{2}}} \\ c &= \sum\_{j=1}^{n} \delta\_{j} + \sum\_{j=1}^{n} \nu\_{j}\theta\_{j}^{p+1} + \sum\_{j=1}^{n} \vartheta\_{j}\theta\_{j}^{q+1} \end{aligned} \tag{51}$$

Therefore, according to the lemma in [39], all closed-loop signals possess fixed-time Lyapunov stability.

The design details are summarized in Figure 2 to show the procedure of the control process.

**Figure 2.** Design procedure.

#### **4. Numerical Examples**

In this paper, the feasibility and effectiveness of the algorithm are verified by numerical simulation. A strict-feedback high-order system is considered as follows:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = g\_1 \mathbf{x}\_2^{\eta\_1} + f\_1(\mathbf{x}\_1) \\
\dot{\mathbf{x}}\_2 = g\_2 \mathbf{u}^{\eta\_2} + f\_2(\mathbf{x}\_1, \mathbf{x}\_2) \\
y = \mathbf{x}\_1
\end{cases} \tag{52}$$

where the function *f*1(*x*1) = *x*1(*t*) + sin(0.1*x*1(*t*)), *f*2(*x*1, *x*2) = *x*2(*t*), *g*<sup>1</sup> = 1, *g*<sup>2</sup> = 1, *η*<sup>1</sup> = <sup>5</sup> 3 , *η*<sup>2</sup> = <sup>7</sup> 5 , and the control input under the adaptive law is designed

$$\dot{\theta}\_1 = 0.01 \left( |z\_1| |\Psi\_1| - 0.1 \dot{\theta}\_1^{\frac{5}{3}} - 0.1 \dot{\theta}\_1^{\frac{1}{3}} \right) \tag{53}$$

$$\dot{\hat{\theta}}\_2 = 0.01 \left( |z\_2| |\Psi\_2| - 0.1 \hat{\theta}\_2^{\frac{5}{3}} - 0.1 \hat{\theta}\_2^{\frac{1}{3}} \right) \tag{54}$$

the control input is designed as

$$u(t) = \begin{pmatrix} -5\text{sign}(z\_2(t)) \ast |z\_1(t)| \ast \left( |z\_2(t)| + |x\_2(t)|^{\frac{2}{5}} \right) + \text{sign}(z\_2(t))\dot{\theta}\_2 \mathbf{V}\_2(Z\_2) \\\ + \frac{0.01z\_2}{0.1|z\_2(t)| + 0.1} + \text{5}z\_2(t)^{\frac{1}{5}} + \text{5}z\_2(t)^{\frac{5}{5}} \end{pmatrix}^{\frac{5}{7}} \tag{55}$$

The desired reference signal is *y<sup>d</sup>* = sin(*t*). The initial condition is selected as *x*1(0) = 1, *x*2(0) = 1, ˆ*θ*1(0) = 1, ˆ*θ*2(0) = 1. The neural network consists of seven nodes, centers *c* = [−3, −2, −1, 0, 1, 2, 3], and widths *b* = 1.

Figure 3 shows that under the action of the neural network adaptive controller, the state of the controlled system state can track the preset trajectory in finite time. Figure 4 shows the state trajectory of the error system. It can be seen from the figure that under the action of the controller, the error system achieves fixed-time Lyapunov stability. The adaptive function curve is shown in Figure 5. For fixed-time control, *α*<sup>1</sup> is designed by *f*1(*x*1), which is approximated by NNs, but its derivative is not easy to approximate; therefore, *f*2(*x*2) − . *α*<sup>1</sup> is not easy to approximate, and the amplitude is inevitable. Figure 6 shows that the system's controllers are bounded. It can be seen from the figures that the designed method is effective.

**Figure 3.** Trajectories of *x*<sup>1</sup> and *y<sup>d</sup>* of a strict-feedback high-order nonlinear system.

**Figure 4.** Trajectories of error states of a strict-feedback high-order nonlinear system.

**Figure 5.** Trajectories of adaptive functions of a strict-feedback high-order nonlinear system.

**Figure 6.** Trajectories of system input of a strict-feedback high-order nonlinear system.

#### **5. Conclusions**

In this paper, based on backstepping adaptive control technology, a neural network is used to approximate some unknown signals in a system. Combined with Lyapunov stability theorem and fixed time stability, an effective adaptive control scheme is designed. A class of strict-feedback high-order systems is further studied. The main contributions of this paper are as follows: the fixed-time control problem of strict-feedback high-order nonlinear systems is solved; the Lyapunov function is designed for each subsystem; at the same time, combined with adaptive backstepping technology, an adaptive neural network fixed-time controller is designed. The tracking error converges in finite time through stability analysis, and the convergence time does not relay on the initial condition. The most popular controller is designed in a linear control strategy, which controls the state's exponential stability. At present, the adaptive neural network control method based on backstepping has some limitations, and many problems need to be further studied and solved. In the finite-time adaptive control method for the multi-agent system, the finite time obtained by most finite-time control strategies often depends on the initial conditions of the system. Therefore, a finite-time control scheme independent of the initial value for a multi-agent system must be designed.

**Author Contributions:** Conceptualization, Y.L. and J.Z.; methodology, Y.L.; software, X.Y.; validation, Y.L., J.Z. and X.Y.; formal analysis, J.Z.; investigation, Y.L.; resources, C.S.C.; data curation, C.S.C.; writing—original draft preparation, Y.L.; writing—review and editing, Y.L.; visualization, Y.L.; supervision, Y.L.; project administration, Y.L.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Yuntian Zhang <sup>1</sup> , Aiping Pang 1,2,\*, Hui Zhu <sup>1</sup> and Huan Feng <sup>1</sup>**

<sup>1</sup> College of Electrical Engineering, Guizhou University, Guiyang 550025, China;

ee.ytzhang18@gzu.edu.cn (Y.Z.); gs.huizhu19@gzu.edu.cn (H.Z.); gs.hfeng20@gzu.edu.cn (H.F.)

<sup>2</sup> Guizhou Provincial Key Laboratory of Internet + Intelligent Manufacturing, Guiyang 550025, China

**\*** Correspondence: appang@gzu.edu.cn

**Abstract:** Spacecraft with large flexible appendages are characterized by multiple system modes. They suffer from inherent low-frequency disturbances in the operating environment that consequently result in considerable interference in the operational performance of the system. It is required that the control design ensures the system's high pointing precision, and it is also necessary to suppress low-frequency resonant interference as well as take into account multiple performance criteria such as attitude stability and bandwidth constraints. Aiming at the comprehensive control problem of this kind of flexible spacecraft, we propose a control strategy using a structured H-infinity controller with low complexity that was designed to meet the multiple performance requirements, so as to reduce the project cost and implementation difficulty. According to the specific resonant mode of the system, the design strategy of adding an internal mode controller, a trap filter, and a series PID controller to the structured controller is proposed, so as to achieve the comprehensive control goals through cooperative control of multiple control modules. A spacecraft with flexible appendages (solar array) is presented as an illustrative example in which a weighted function was designed for each performance requirement of the system (namely robustness, stability, bandwidth limit, etc.), and a structured comprehensive performance matrix with multiple performance weights and decoupled outputs was constructed. A structured H-infinity controller meeting the comprehensive performance requirements is given, which provides a structured integrated control method with low complexity for large flexible systems that is convenient for engineering practice, and provides a theoretical basis and reference examples for structured H-infinity control. The simulation results show that the proposed controller gives better control performance compared with the traditional H-infinity one, and can successfully suppress the vibration of large flexible appendages at 0.12 Hz and 0.66 Hz.

**Keywords:** structured control; flexible spacecraft; prevent oscillations

### **1. Introduction**

With the rapid development of the aerospace industry and of composite material technology, along with its broad application in aerospace, the structure of spacecrafts is becoming larger and more flexible, featuring multi-system modalities. The resonant mode of a flexible system of this type leads to tremendous changes in amplitude features. Meanwhile, the inherent low-frequency interference caused by the complex launch environment and the high-altitude environment during orbit operation, as well as the flexible mode of the system, greatly limit the choice of bandwidths, i.e., robust stability. The flexible modes and low-frequency disturbances inherent in the high-altitude environment impair the stability and performance of the spacecraft, which will cause performance degradation and failure to meet mission requirements, or result in unstable control or even failure of the spacecraft.

With the increasing diversification of spacecraft missions, the requirements for pointing accuracy of large spacecraft have become increasingly stringent, which makes control research more complicated and difficult to delve into. The difficulties of the control design of such large flexible systems are as follows: to suppress the external interference caused

**Citation:** Zhang, Y.; Pang, A.; Zhu, H.; Feng, H. Structured H∞ Control for Spacecraft with Flexible Appendages. *Entropy* **2021**, *23*, 930. https://doi.org/10.3390/e23080930

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 22 June 2021 Accepted: 19 July 2021 Published: 22 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

by the complex space environment and the inherent low-frequency resonance interference of flexible spacecraft; to meet "high-precision" performance requirements; and to ensure attitude stability and bandwidth amount (robustness requirement).

Previous studies show that in terms of a synthesis control over these spacecraft with multi-performance requirements, it is difficult to apply classic analysis methods to balance the requirements. Traditional control design schemes, in general, fail to simultaneously meet the requirements of pointing accuracy and robustness [1–5]. All of them [6–8] used some fuzzy/neural control scheme to deal with at least two of these undesirable aspects: presence of inertia uncertainties, misalignment, unknown or external disturbance, vibration, actuator saturation, and faults, to ensure spacecraft stability. H∞ control theory is a comprehensive control theory that can take multiple performance requirements into consideration in the design and is suitable for such comprehensive control problems with multiple performance requirements. Currently, robust adaptive control, robust H∞ control, and µ synthesis control are mainly adopted to realize vibration control during the stable operation of flexible spacecraft [9–13].

Despite its synthesis advantages, H∞ design has engineering application limitations [14] mainly due to its high-order and complicated controller. Apart from high costs, it is a tremendous challenge to decompose a high-order and complicated controller into multiple low-complexity control structures based on experience in engineering practices [15,16], thus leading to a low feasibility of the practical use of traditional H∞ controllers. A new H∞ control method combining the advantages of the traditional H∞ control [15–17] was proposed by Apkarian in recent years, which factors into system performances in all aspects and overcomes the infeasibility of engineering applications of traditional H∞ controllers due to non-transparency and high complexity. The new method has attracted much attention and been widely applied since it was proposed [18–20]. The logic of the new structured H∞ control method is as follows: the structure of a controller is first designed according to actual needs and control objectives; on this basis, appropriate weighting functions are selected as per the specific performance requirements of the control object in order to form an H∞ performance matrix with multi-dimensional performance output; and finally, the structured H∞ controller with optimal parameters can be obtained through parameter optimization of the controller with a fixed structure [21–23].

In view of problems in integrated control of spacecraft with large flexible solar panels, based on the structured H∞ control design strategy and the specific resonance mode of the system, this paper proposes incorporating into the controller structure an internal mode controller, a notch filter, and a serial PID controller, which can achieve integrated control through multi-control module collaboration. A control solution that satisfies the comprehensive performance requirements is provided, thus reducing the project cost and implementation difficulty. Apart from the introduction, this paper deals with system modeling in the second part, structured H∞ controller design in the third part, performance simulation and analysis in the fourth part, and conclusions in the last part.

#### **2. System Model Case and Control Analysis**

The spacecraft with large flexible appendages is shown in Figure 1. It mainly comprises solar arrays that provide energy, velocity gyroscopes, precision guidance sensors, and star trackers that provide spacecraft attitude data, as well as a reaction wheel and an electromagnetic torque device for momentum management, and a digital computer.

Only the pitch axis model that is most jitter-prone and the most important in the entire system is considered when constructing the simulation model. Official data show that the flexible spacecraft's pitch axis model is composed of a rigid body model and sev-Only the pitch axis model that is most jitter-prone and the most important in the entire system is considered when constructing the simulation model. Official data show that the flexible spacecraft's pitch axis model is composed of a rigid body model and several flexible modules, as shown in the following formula:

$$\frac{\theta(s)}{u(s)} = \frac{1}{IS^2} + \sum\_{i=1} \frac{K\_i/I}{s^2 + 2\zeta w\_i s + w\_i^2} \tag{1}$$
 
$$\text{In the formula } \theta \text{ is the number error of the critical axis affected by the } \overline{\omega} \text{-inter of order } \theta$$

ω<sup>i</sup> is

( ) 22 2 1 2 *i i i u s IS s w s w* = ζ In the formula, θ is the angular error of the pitch axis affected by the jitter of solar arrays; u is the given input of the pitch axis torque; s is the Laplace operator; I is the spacecraft pitch inertia constant with the value of 77,076 kg· <sup>2</sup> m ; ξ = 0.005 is the passive damp-In the formula, *θ* is the angular error of the pitch axis affected by the jitter of solar arrays; *u* is the given input of the pitch axis torque; *s* is the Laplace operator; *I* is the spacecraft pitch inertia constant with the value of 77,076 kg·m<sup>2</sup> ; *ξ* = 0.005 is the passive damping ratio constant of the system; *Ki* is the flexibility gain of the flexible module; and *ωi* is the flexible frequency of the flexible module. The data are shown in Table 1.


ing ratio constant of the system; Ki is the flexibility gain of the flexible module; and **Table 1.** Flexible spacecraft system model parameters.

−1.341 13.201 −1.387 14.068 −0.806 14.285 The Bode plot of the system without a controller is shown in Figure 2. The Bode plot shows that the cut-off frequency *ω*<sup>c</sup> is only 0.16 Hz and the bandwidths of the system are very small, indicating poor interference suppression.

−0.134 15.264

ω

The Bode plot of the system without a controller is shown in Figure 2. The Bode plot

<sup>c</sup> is only 0.16 Hz and the bandwidths of the system

are very small, indicating poor interference suppression.

shows that the cut-off frequency

Since they were put into operation, flexible spacecraft, as high-precision spacecraft, have never produced an output error of pitch axis exceeding 0.007 arcsec, which requires good control performance of the system. An effective controller should be good at disturbance suppression of solar panels and inherent flexibility suppression of the system, Since they were put into operation, flexible spacecraft, as high-precision spacecraft, have never produced an output error of pitch axis exceeding 0.007 arcsec, which requires good control performance of the system. An effective controller should be good at disturbance suppression of solar panels and inherent flexibility suppression of the system, with a certain number of bandwidths.

with a certain number of bandwidths. The control redesign requirements can be stated as follows [24]:


#### with respect to the original design. **3. Design of Structured H∞ Control**

4. Maintain the bandwidth (the open-loop gain crossover frequency) close to 1.5 Hz. **3. Design of Structured H∞ Control** In general, the complete design of structured control is divided into three steps: firstly, to design a structured controller according to design requirements and control ob-In general, the complete design of structured control is divided into three steps: firstly, to design a structured controller according to design requirements and control objectives; secondly, to select and design proper weighting functions based on control objectives and performance requirements; and finally, to obtain a desired structured controller according to performance requirements and selected weighting functions.

jectives; secondly, to select and design proper weighting functions based on control objectives and performance requirements; and finally, to obtain a desired structured control-In Section 3.1, the author briefly analyzes the control objectives and performance requirements in the control design of the large flexible spacecraft in this case and presents controller structure design in terms of the flexible modes and disturbance model of the system.

#### ler according to performance requirements and selected weighting functions. In section 3.1, the author briefly analyzes the control objectives and performance re-*3.1. Structured Controller Setting*

quirements in the control design of the large flexible spacecraft in this case and presents controller structure design in terms of the flexible modes and disturbance model of the system. *3.1. Structured Controller Setting* The main problems confronted in the control of large flexible spacecraft are as follows: first, the multiple low-frequency resonance modes of the system result in huge changes in amplitude characteristics; second, the inherent low-frequency interference caused by the complex launch environment and the high-altitude environment during orbit operation, as well as the flexible mode of the system, greatly limit the choice of bandwidths. The controller is designed to suppress the inherent low-frequency resonance disturbance of the

The main problems confronted in the control of large flexible spacecraft are as follows: first, the multiple low-frequency resonance modes of the system result in huge changes in amplitude characteristics; second, the inherent low-frequency interference

orbit operation, as well as the flexible mode of the system, greatly limit the choice of bandwidths. The controller is designed to suppress the inherent low-frequency resonance disturbance of the system and achieve high pointing accuracy, while ensuring the bandwidth

amount of the flexible system (robustness) and stability.

system and achieve high pointing accuracy, while ensuring the bandwidth amount of the flexible system (robustness) and stability. widths. The controller is designed to suppress the inherent low-frequency resonance disturbance of the system and achieve high pointing accuracy, while ensuring the bandwidth amount of the flexible system (robustness) and stability.

To meet the above control target, based on the flexible mode and inherent disturbance frequency of the system in this case, the authors design a structured integrated controller as shown in Figure 3. To meet the above control target, based on the flexible mode and inherent disturbance frequency of the system in this case, the authors design a structured integrated controller as shown in Figure 3.

*Entropy* **2021**, *23*, x FOR PEER REVIEW 5 of 12

**Figure 3.** Control system block diagram. **Figure 3.** Control system block diagram.

ance.

The dotted line in Figure 3 shows the structured controller of the system. The red modules are for adjustable parameters, and the blue modules are fixed parameters. The dotted line in Figure 3 shows the structured controller of the system. The red modules are for adjustable parameters, and the blue modules are fixed parameters.

The structured H∞ controller consists of three parts. The first part is an internal mode controller S(s) designed for the system's resonance modes to suppress resonance disturb-The structured H∞ controller consists of three parts. The first part is an internal mode controller *S*(*s*) designed for the system's resonance modes to suppress resonance disturbance.

$$S(s) = \frac{\left[\left(\frac{s}{87}\right)^2 + \frac{0.002s}{87} + 1\right]}{\left[\left(\frac{s}{45}\right)^2 + \frac{1.4s}{45} + 1\right]}\tag{2}$$
 
$$\text{As are shown in Figure 4 and its purpose is to reduce system.}$$
 
$$\text{imping of the system for system vibration at 14 Hz.}$$

2 ( ) 1.4 <sup>1</sup> 45 45 *S s s s* (2) Its frequency characteristics are shown in Figure 4 and its purpose is to reduce system vibration by increasing the damping of the system for system vibration at 14 Hz. *Entropy* **2021**, *23*, x FOR PEER REVIEW 6 of 13

The second part is a notch filter R(s) set by the disturbance characteristics of the space

2 2

0.77872 0.77872 3.84336 3.84336

*s s s s*

0.7536 4.1448

The third part is an adjustable parameter PID controller set to ensure the stability of the system, whose rate path is supplemented with an FIR filter to provide gain suppres-

*s s*

2 2

0.728 0.254 1 1

 + + + + = + + 

1 1

(3)

**Figure 4.** Internal mode controller S(s). **Figure 4.** Internal mode controller *S*(*s*).

tem. The Bode plot is shown in Figure 5.

sion.

**Figure 5.** Notch filter R(s).

(s)

*R*

The second part is a notch filter *R*(*s*) set by the disturbance characteristics of the space environment to ensure sufficient suppression of the disturbance of solar panels at 0.12 Hz and 0.66 Hz on the premise of not affecting the stability of the frequency band in the system. The Bode plot is shown in Figure 5. tem. The Bode plot is shown in Figure 5. 2 2 2 2 0.728 0.254 1 1 0.77872 0.77872 3.84336 3.84336 (s) 1 1 *s s s s R s s* + + + + = + +

*Entropy* **2021**, *23*, x FOR PEER REVIEW 6 of 13

donde plot is shown in Figure 5.

$$R(s) = \frac{\left[\left(\frac{s}{0.77872}\right)^2 + \frac{0.728s}{0.77872} + 1\right]}{\left[\left(\frac{s}{0.7785}\right)^2 + 1\right]} \times \frac{\left[\left(\frac{s}{3.84336}\right)^2 + \frac{0.254s}{3.84336} + 1\right]}{\left[\left(\frac{s}{4.1448}\right)^2 + 1\right]}\tag{3}$$

The second part is a notch filter R(s) set by the disturbance characteristics of the space environment to ensure sufficient suppression of the disturbance of solar panels at 0.12 Hz and 0.66 Hz on the premise of not affecting the stability of the frequency band in the sys-

(3)

The third part is an adjustable parameter PID controller set to ensure the stability of the system, whose rate path is supplemented with an FIR filter to provide gain suppression. the system, whose rate path is supplemented with an FIR filter to provide gain suppression.

**Figure 5. Figure 5.** Notch filter Notch filter *R* (*s*).

#### *3.2. Selection of Performance Weighting Functions*

R(s).

**Figure 4.** Internal mode controller S(s).

The following indicators need to be factored into the selection of weighting functions: the requirement of pointing accuracy; the stability and sensitivity of the system after the internal model controller of disturbance suppression is added; and the bandwidth limitations (robustness) of the flexible system.

By setting *T*<sup>1</sup> as the transfer function of *r* − *e*, the stability of the system is the distance from the transfer function *T*<sup>1</sup> to the critical operating point, which is also the upper limit of the gain of tracking performance, requiring:

$$\|\|W\_1(s)T\_1(s)\|\|\_{\infty} \le \gamma \tag{4}$$

where *γ* is the norm index, and *W*1(*s*) is the weighting function. The upper limit of the design stability margin is 1, *W*1(*s*) = 1.

*T*<sup>2</sup> is set as the transfer function from *d* to *θ*, which is the robust stability requirement of the system:

$$\|\|W\_2(s)T\_2(s)\|\|\_{\infty} \le \gamma \tag{5}$$

where the weighting function *W*2(*s*) = 0.8.

By setting *T*<sup>3</sup> as the transfer function from *r* to *θ*, the bandwidth requirement of the system is: selected in the high-pass filter form as follows: The Bode diagram for *W s* <sup>3</sup> ( ) Figure 6.

In order to limit the bandwidth of the system, the weighting function

*Entropy* **2021**, *23*, x FOR PEER REVIEW 7 of 13

as the transfer function of

*W s* <sup>1</sup>

*T*1

( ) *W s* <sup>1</sup>

( ) = 1 .

*W s* <sup>2</sup>

The following indicators need to be factored into the selection of weighting functions: the requirement of pointing accuracy; the stability and sensitivity of the system after the internal model controller of disturbance suppression is added; and the bandwidth limita-

*W s T s* 1 1 ( ) ( )

*W s T s* 2 2 ( ) ( )

*W s T s* 3 3 ( ) ( )

( ) = 0.8 .

as the transfer function from r to θ, the bandwidth requirement of the

 

is set as the transfer function from d to θ, which is the robust stability requirement

 

 

*r e* − , the stability of the system is the dis-

(4)

(5)

(6)

(7)

can be

is shown in

*W s* <sup>3</sup> ( )

to the critical operating point, which is also the upper

is the weighting function. The upper limit of the

*3.2. Selection of Performance Weighting Functions*

limit of the gain of tracking performance, requiring:

is the norm index, and

tions (robustness) of the flexible system.

*T*1

tance from the transfer function

design stability margin is 1,

where the weighting function

*T*3

By setting

By setting

where

*T*2

system is:

of the system:

$$\|\|\mathcal{W}\_{\mathfrak{d}}(s)T\_{\mathfrak{d}}(s)\|\|\_{\infty} \leq \gamma \tag{6}$$

In order to limit the bandwidth of the system, the weighting function *W*3(*s*) can be selected in the high-pass filter form as follows: The Bode diagram for *W*3(*s*) is shown in Figure 6. 3 ( ) 2 220 *s W s s* = +

$$\mathcal{W}\_{\mathfrak{Z}}(\mathbf{s}) = \frac{2s}{s + 220} \tag{7}$$

#### *3.3. Parameter Optimization of Structured Controller*

( ) .

Based on comprehensive considerations of the performance requirements of the system and the implementation cost of the controller, the controller was designed as shown in Formula (8).

$$K(\mathbf{s}) = K\_P(1 + K\_I/\mathbf{s} + K\_D \mathbf{s}) \times R(\mathbf{s}) \times S(\mathbf{s}) \tag{8}$$

*KP*, *K<sup>I</sup>* , *K<sup>D</sup>* is the to-be-optimized parameter.

Given the above analysis, and for the structured *H*<sup>∞</sup> optimization of the designed controller structure and the weighting function, the control performance requirements of the flexible system was considered comprehensively, and the minimum *KP*, *K<sup>I</sup>* , *KD*, and the minimum value satisfying Formula (9) can be obtained by optimizing the adjustable parameters.

$$\|H\|\_{\infty} \le \gamma \tag{9}$$

In formula *H* = *diag W*1*T*<sup>1</sup> *W*2*T*<sup>2</sup> *W*3*T*<sup>3</sup> , the adjustable parameter yielded is the optimal one for the system controller.

When seeking the optimal parameter of the structured controller *H*∞, the linear fractional transformation (LFT) [18–24] was employed with *T*1, *T*2, and *T*<sup>3</sup> in Formulas (4)–(6), and the structured controller *C* with parameters was extracted and expressed in the following linear fraction forms to optimize the parameters:

$$\begin{cases} \begin{array}{ll} T\_1 = \mathrm{F}\_I(\ \ P\_1 \ \ \ \text{C} \\ T\_2 = \mathrm{F}\_I(\ \ P\_2 \ \ \text{C} \\ T\_3 = \mathrm{F}\_I(\ \ P\_3 \ \ \text{C} \end{array}) \\\end{cases} \tag{10}$$

In this case, the optimal parameters of the structured controller in Figure 1 yielded through repeated iterative calculations are as follows:

$$K\_P = 8, K\_I = 0.5, K\_D = 0.95$$

The final controller is:

$$K(s) = 8(1 + 0.5/s + 0.95s) \times R(s) \times S(s) \tag{11}$$

#### **4. Simulation Performance Analysis**

Figure 7 shows the Bode plot of the unified open-loop frequency domain. According to Figure 7, the gain margin of the system is 5.17 dB; the phase margin is 22.5◦ ; and the cut-off frequency is approximately 1.8 Hz, exceeding the required cut-off frequency of 1.5 Hz, which indicates that the stability and bandwidth requirements of the system have been met. For the system flexibility at 13 to 14 Hz, the controller provides about −15 to −55 dB gain suppression, which basically meets the suppression requirements for the inherent flexibility of the system. *Entropy* **2021**, *23*, x FOR PEER REVIEW 9 of 12

**Figure 7.** Open-loop Bode plot of the system. **Figure 7.** Open-loop Bode plot of the system.

Figure 8 is a closed-loop Bode plot of the system. Figure 9 is the time-domain response of the system. The figure shows that the proposed controller provides gain attenuation far higher than 20 dB for the jitter at 0.12 Hz or 0.66 Hz, which is also proved by Figure 8 is a closed-loop Bode plot of the system. Figure 9 is the time-domain response of the system. The figure shows that the proposed controller provides gain attenuation far higher than 20 dB for the jitter at 0.12 Hz or 0.66 Hz, which is also proved by the time-domain response of the system shown in Figure 7.

.

**Figure 8.** Closed-loop Bode plot of the system.

the time-domain response of the system shown in Figure 7.

the time-domain response of the system shown in Figure 7.

.

Figure 8 is a closed-loop Bode plot of the system. Figure 9 is the time-domain response of the system. The figure shows that the proposed controller provides gain attenuation far higher than 20 dB for the jitter at 0.12 Hz or 0.66 Hz, which is also proved by

**Figure 8. Figure 8.** Closed-loop Bode plot of the system. Closed-loop Bode plot of the system.

**Figure 7.** Open-loop Bode plot of the system.

With the same weight function, the obtained H-inf controller is as follows: 7 2 62 22 2 6 3.492 10 ( 0.6402)( 0.002009 1.609 10 )( 0.0003605 0.002128) ( 0.01) ( 0.00022 0.002118)( 3357 3.763 10 ) ( ) *<sup>h</sup> ss s s s ss s <sup>K</sup> s s <sup>s</sup>* − − × ×+ + + × + + + + ++ <sup>=</sup> + × (12) With the same weight function, the obtained H-inf controller is as follows: *Kh* (*s*) = 3.492 <sup>×</sup> <sup>10</sup><sup>7</sup> <sup>×</sup> (*<sup>s</sup>* <sup>+</sup> 0.6402)(*<sup>s</sup>* <sup>2</sup> <sup>+</sup> 0.002009*<sup>s</sup>* <sup>+</sup> 1.609 <sup>×</sup> <sup>10</sup>−<sup>6</sup> )(*s* <sup>2</sup> + 0.0003605*s* + 0.002128) (*s* + 0.01) 2 (*s* <sup>2</sup> + 0.00022*s* + 0.002118)(*s* <sup>2</sup> + <sup>3357</sup>*<sup>s</sup>* + 3.763 × <sup>10</sup>−6)

The open-loop Bode plot of the H-inf controller is shown in Figure 10.

The open-loop Bode plot of the H-inf controller is shown in Figure 10.

(12)

The frequency domain response shown in Figure 11 shows that the structured integrated controller is excellent in vibration suppression for the flexible structure at 0.12 Hz and 0.66 Hz. The time-domain response shown in Figure 12 shows that under the same weight function, both controllers (structured integrated controller and H-inf controller) can completely suppress solar panel disturbance within a certain period, meet the control requirements, and present stability. For the structured integrated controller, the overshoot of the system is 2.101%; the peak time is 0.711 s; and the adjustment time is 13.562 s. For the traditional H-inf controller, the overshoot is 2.31%; the peak time is 0.876 s; and the

**Figure 10.** Open-loop Bode plot of H-inf controller.

With the same weight function, the obtained H-inf controller is as follows:

+ + ++ <sup>=</sup> + × (12)

−

−

22 2 6 3.492 10 ( 0.6402)( 0.002009 1.609 10 )( 0.0003605 0.002128)

The open-loop Bode plot of the H-inf controller is shown in Figure 10.

*ss s s s*

× ×+ + + × + +

**Figure 10.** Open-loop Bode plot of H-inf controller. **Figure 10.** Open-loop Bode plot of H-inf controller.

**Figure 9.** System time-domain response.

*ss s <sup>K</sup> s s <sup>s</sup>*

7 2 62

( 0.01) ( 0.00022 0.002118)( 3357 3.763 10 ) ( ) *<sup>h</sup>*

The frequency domain response shown in Figure 11 shows that the structured integrated controller is excellent in vibration suppression for the flexible structure at 0.12 Hz and 0.66 Hz. The time-domain response shown in Figure 12 shows that under the same weight function, both controllers (structured integrated controller and H-inf controller) can completely suppress solar panel disturbance within a certain period, meet the control requirements, and present stability. For the structured integrated controller, the overshoot of the system is 2.101%; the peak time is 0.711 s; and the adjustment time is 13.562 s. For the traditional H-inf controller, the overshoot is 2.31%; the peak time is 0.876 s; and the The frequency domain response shown in Figure 11 shows that the structured integrated controller is excellent in vibration suppression for the flexible structure at 0.12 Hz and 0.66 Hz. The time-domain response shown in Figure 12 shows that under the same weight function, both controllers (structured integrated controller and H-inf controller) can completely suppress solar panel disturbance within a certain period, meet the control requirements, and present stability. For the structured integrated controller, the overshoot of the system is 2.101%; the peak time is 0.711 s; and the adjustment time is 13.562 s. For the traditional H-inf controller, the overshoot is 2.31%; the peak time is 0.876 s; and the adjustment time is 14.227 s. The data clearly show that the structured integrated controller has significantly better dynamic performance than the H-inf controller and the structured integrated controller has a lower order. *Entropy* **2021**, *23*, x FOR PEER REVIEW 11 of 12 adjustment time is 14.227 s. The data clearly show that the structured integrated controller has significantly better dynamic performance than the H-inf controller and the structured integrated controller has a lower order.

**Figure 11.** Open-loop simulation comparison between the structured integrated controller and traditional H-inf controller. **Figure 11.** Open-loop simulation comparison between the structured integrated controller and traditional H-inf controller.

**Figure 12.** Simulation comparison of time-domain response between the structured integrated con-

To solve the comprehensive control problems of spacecraft with large flexible appendages such as insufficient bandwidths, low system directivity accuracy, and flexible structure vibration, this paper proposes a structured integrated controller that satisfies control requirements by selecting appropriate weight functions. The simulation results indicate that the proposed controller can effectively suppress the vibration of large flexible appendages at 0.12 Hz and 0.66 Hz. While ensuring high pointing accuracy, the structured integrated controller can meet the requirements of attitude stability and bandwidths. Compared with the traditional H-inf controller, the proposed controller has the advantages of lower complexity and system order as well as lower engineering costs and

**Author Contributions:** Conceptualization, A.P. and Y.Z.; Methodology, A.P. and Y.Z.; Validation, Y.Z. and H.Z.; Data Curation, Y.Z.; Formal analysis Y.Z. and H.F.; Funding acquisition, A.P.; Writing—Original Draft Preparation, Y.Z.; Writing—Review and Editing, Y.Z. and H.F.; Visualization,

Y.Z. All authors have read and agreed to the published version of the manuscript.

implementation difficulty, with better control performance.

troller and traditional H-inf controller.

**5. Conclusions** 

**Figure 11.** Open-loop simulation comparison between the structured integrated controller and tra-

adjustment time is 14.227 s. The data clearly show that the structured integrated controller has significantly better dynamic performance than the H-inf controller and the structured

integrated controller has a lower order.

**Figure 12.** Simulation comparison of time-domain response between the structured integrated controller and traditional H-inf controller. **Figure 12.** Simulation comparison of time-domain response between the structured integrated controller and traditional H-inf controller.

#### **5. Conclusions 5. Conclusions**

ditional H-inf controller.

To solve the comprehensive control problems of spacecraft with large flexible appendages such as insufficient bandwidths, low system directivity accuracy, and flexible structure vibration, this paper proposes a structured integrated controller that satisfies control requirements by selecting appropriate weight functions. The simulation results indicate that the proposed controller can effectively suppress the vibration of large flexible appendages at 0.12 Hz and 0.66 Hz. While ensuring high pointing accuracy, the structured integrated controller can meet the requirements of attitude stability and bandwidths. Compared with the traditional H-inf controller, the proposed controller has the advantages of lower complexity and system order as well as lower engineering costs and To solve the comprehensive control problems of spacecraft with large flexible appendages such as insufficient bandwidths, low system directivity accuracy, and flexible structure vibration, this paper proposes a structured integrated controller that satisfies control requirements by selecting appropriate weight functions. The simulation results indicate that the proposed controller can effectively suppress the vibration of large flexible appendages at 0.12 Hz and 0.66 Hz. While ensuring high pointing accuracy, the structured integrated controller can meet the requirements of attitude stability and bandwidths. Compared with the traditional H-inf controller, the proposed controller has the advantages of lower complexity and system order as well as lower engineering costs and implementation difficulty, with better control performance.

implementation difficulty, with better control performance. **Author Contributions:** Conceptualization, A.P. and Y.Z.; Methodology, A.P. and Y.Z.; Validation, Y.Z. and H.Z.; Data Curation, Y.Z.; Formal analysis Y.Z. and H.F.; Funding acquisition, A.P.; Writ-**Author Contributions:** Conceptualization, A.P. and Y.Z.; Methodology, A.P. and Y.Z.; Validation, Y.Z. and H.Z.; Data Curation, Y.Z.; Formal analysis Y.Z. and H.F.; Funding acquisition, A.P.; Writing— Original Draft Preparation, Y.Z.; Writing—Review and Editing, Y.Z. and H.F.; Visualization, Y.Z. All authors have read and agreed to the published version of the manuscript.

ing—Original Draft Preparation, Y.Z.; Writing—Review and Editing, Y.Z. and H.F.; Visualization, Y.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This work was funded in part by the Science and Technology Foundation of Guizhou Province (QKH [2020]1Y273, [2018]5781, [2018]5615), Guizhou Provincial Department of Education, Youth Talent Development Project, Qianke [2021] 100, Research on Robust Integrated Control of Flexible System, and the National Natural Science Foundation (NNSF) of China under Grant 51867005.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


### *Article* **Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration**

**Aiping Pang 1,2, Hongbo Zhou 1,2, Wenjie Cai <sup>3</sup> and Jing Zhang 1,\***


**Abstract:** The continuous development of spacecraft with large flexible structures has resulted in an increase in the mass and aspect ratio of launch vehicles, while the wide application of lightweight materials in the aerospace field has increased the flexible modes of launch vehicles. In order to solve the problem of deviation from the nominal control or even destabilization of the system caused by uncertainties such as unknown or unmodelled dynamics, frequency perturbation of the flexible mode, changes in its own parameters, and external environmental disturbances during the flight of such large-scale flexible launch vehicles with simultaneous structural deformation, rigid-elastic coupling and multimodal vibrations, an improved adaptive augmentation control method based on model reference adaption, and spectral damping is proposed in this paper, including a basic PD controller, a reference model, and an adaptive gain adjustment based on spectral damping. The baseline PD controller was used for flight attitude control in the nominal state. In the non-nominal state, the spectral dampers in the adaptive gain adjustment law extracted and processed the high-frequency signal from the tracking error and control-command error between the reference model and the actual system to generate the adaptive gain. The adjustment gain was multiplied by the baseline controller gain to increase/decrease the overall gain of the system to improve the system's performance and robust stability, so that the system had the ability to return to the nominal state when it was affected by various uncertainties and deviated from the nominal state, or even destabilized.

**Keywords:** multiplicative adaptation; gain adjustment; spectral damping; robust stability

### **1. Introduction**

As the exploration of the space environment progresses, the missions of spacecraft exploration become more and more diversified, and as the application of polymer materials in the space field progresses, the structure of spacecraft is gradually developing towards large and flexible structures. In order to carry these large and flexible-structure spacecraft, launch vehicles with a large carrying capacity have become an inevitable requirement of space-development strategies [1]. At the same time, the lightweight polymer material used in the body of launch vehicles has increased the flexible mode of the vehicles, leading to the presence of structural deformation, rigid body-elastic vibration coupling, multi-modal vibration, and other characteristics of the body at the same time. These factors make the attitude control of launch vehicles subject to oscillations and difficult to attenuate, or even lead to system instability, which poses a new challenge to the reliability and robustness of launch vehicles [2].

For high-risk aerospace applications, both government and industry rely heavily on classical control theory, and gain-scheduling PID control is still the mainstream control method for current launch vehicles, due to the advantages of its simple structure, good anti-interference, and ease of analysis in the time domain (or frequency domain). Typical applications include the Saturn V and Space Shuttle of the United States, the Ariane of

**Citation:** Pang, A.; Zhou, H.; Cai, W.; Zhang, J. Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration. *Entropy* **2021**, *23*, 1058. https:// doi.org/10.3390/e23081058

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 16 June 2021 Accepted: 10 August 2021 Published: 16 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Europe, and the Long March series of launch vehicles of China [1]. The application of classical control theory in launch vehicles has matured, and is well verified with few failures. Although the classical control methods can usually meet the flight requirements, the traditional gain-scheduling PID control is no longer able to meet the control requirements of launch vehicles due to their increasing mass and aspect ratio, the increase in the flexible structure of their components, and the obvious influence of the flexible mode and elastic vibration of large launch vehicles, and is unable to cope with the control instability problems caused by the excessive interference and modal uncertainty during the flight. Due to the unknown and unmodeled dynamics, external perturbations of the flight environment, and changes in its own parameters, the attitude of a large flexible launch vehicle will inevitably generate errors during flight [3]. Traditional controllers are usually designed with high gain to suppress attitude errors, but excessive gain during full flight will easily cause the control commands to vibrate, and the flexible mode will also generate a series of vibration signals, which will affect the control effect. There are two main solutions to the problem of elastic-vibration suppression in the design of launch vehicle control systems: one is to design controllers for different characteristics using robust control theory, and the second is to suppress the elastic vibration signal by designing a notch filter. Unlike the robust controller design, the notch filter does not require significant changes to the original rocket control system to deal with elastic vibration. The zero point of the notch filter is used to eliminate the high-frequency pole of the elastic rocket system and determine the frequency center according to the system requirements of the filter. The method of using a notch filter to suppress the elastic vibration of a projectile is widely used in the design of the launcher attitude-control system. The determination of the parameters of the notch filter is the key to the design, and the location of the zero point of the notch filter; i.e., the frequency center, can be determined after the transfer function of the elastic launch vehicle is determined with sufficient accuracy in the model [4,5]. In order to solve the low-frequency, dense-frequency elastic vibration modes appearing in the launch vehicle, some scholars adopted the method of attitude control of the flexible launch vehicle by adaptive control of the adaptive notch, and the adaptive controller of the adaptive notch filter successfully stabilized the uncertain and time-varying equations of the launch vehicle model dynamics through thrust vector control [6]. Another scholar designed a bending mode filter for the whole system, which had a better filtering function for low-frequency, dense-frequency modes, and achieved good control results [7].

In response to the limitations of the classical approach, in order to increase the robust stability of the launch vehicle attitude-control system, many scholars began to work on advanced control methods, and since 1990, NASA has developed a variety of launch vehicle control techniques in the Advanced Guidance Control program, including trajectory linearization control methods, neural network adaptive control methods, and higher-order sliding-mode control methods; the development of such advanced controls has the potential to improve system performance and increase robustness [8–10]. Classical adaptive-control concepts were proposed for attitude-control systems applied to rockets [11–13]. However, many adaptive-control concepts are not feasible when applied to high-risk aerospace systems due to the stringent flight environment. In addition, many adaptive techniques appearing in the literature are not applicable to conditionally stable systems with complex flexible dynamics. Therefore, researchers have optimized model-referenced adaptive control for these situations and proposed an Adaptive Augmentation Control (AAC) [14], widely used in launch-vehicle and missile-longitudinal control in recent years. AAC a gain-adjustment method based on a model-referenced adaptive-control design that generates adaptively adjusted gain from the generalized error between the reference model and the actual system as a supplement to the nominal controller. Orr et al. introduced a scheme for adaptive control of multiplicity applicable to rockets [3], and then improved the adaptive-control scheme [15] to improve the performance of the original method with higher sensitivity to external inputs. The method was developed by the NASA Marshall Space Center (MSFC) and became a major part of the U.S. Space Launch System (SLS) to

adapt to unpredictable external environmental disturbances and a variety of flight dynamics characteristics (elastic vibration of flexible modes, control structure coupling, servo delay, etc.) and to reduce the probability of flight destabilization [15,16]. NASA included the method in the development of the flight control system for the SLS program in early 2013, and tested the designed method in the F/18-A to verify the resilience of the control system in adverse flight conditions [17,18]. Brinda et al. performed an adaptive gainadjustment controller design for the longitudinal channel of a two-stage launch vehicle using a Chebyshev high-pass filter to improve the problem of insufficient amplitude of the low-frequency part of the control signal in the original adaptive gain-adjustment structure of the low-order high-pass filter [19,20]; however, there were equal-amplitude ripples in the passband of the Chebyshev filter. Zhang applied a fault-tolerant control method and adaptive vibration frequency recognition method to AAC, and designed a corresponding correction network based on an SMM algorithm to identify each order vibration frequency to improve system control performance and stability [21]. Cui Naigang et al. applied an interference compensation control loop and active load reduction control loop based on the dilated state observer to the adaptive gain adjustment structure, and performed a simulation analysis of the pitch channel control [22]. To enhance the robustness to changes in elastic modal parameters, Domenico Trotta integrated the AAC control architecture with adaptive notch filters and proposed two novel and effective tuning methods for adaptively enhanced control systems, which were optimized by robust design and solved by genetic algorithms to achieve continuous improvement in the performance and robustness of standard launch vehicle single-axis attitude controllers in atmospheric flight [6,23]. Diego Navarro designed two adaptive augmentation control laws using a robust control design (structured H∞ control) as a baseline controller to improve the robust performance of AAC control, while analyzing the effect of the adaptive action on the classical stability margin, and validated this analysis using nonlinear time-domain stability margin evaluation techniques [24]. However, if the expansion state observer is not properly selected, the observer is easily affected by the noise signal, or even diverges when there is additional measurement noise caused by elastic vibration and other additional dynamics. However, advanced nonlinear stabilization techniques to reduce the error by increasing the control gain are not feasible for aerospace systems with high complexity; meanwhile, the abovementioned adaptive gain-adjustment control scheme features complicated algorithms, costly computation, and challenging implementation, which introduce unknown risks to the actual system.

In this paper, we aim to establish a rigid-bullet coupling model of a large flexible spacecraft with second-order vibration signals and design an improved adaptive augmentation control method based on the reference model adaptive control method to address additional dynamics issues such as increased attitude-tracking errors and flexible-mode elastic vibrations caused by uncertainty (modeling uncertainty, frequency perturbation of the flexible mode) and external environmental interference during ascent of a large launch vehicle with a flexible mode. The scheme first determines the adjustment threshold of forward gain on the basis of the baseline PD controller, and takes the tracking error and control command error signals as the input of the two channels of the adaptive control law. The spectral dampers in the two channels (tracking error and control command elastic vibration) process the error signals (the high-pass filter extracts the high-frequency signal of a specific frequency from the error signal, and the low-frequency filter lowers the frequency to reduce the influence of the high-frequency signal on the actuator) to produce the corresponding suppression gain (error-suppression gain and elastic-suppression gain) to form the overall gain of the AAC, and increase or decrease the forward gain of the system to improve the control performance of the system. The AAC controller will not affect the PD controller when the basic PD controller is able to handle the control tasks better [3,15], whereas the AAC controller will adjust the adaptive gain to achieve the overall gain of the system when the impact of external perturbations and uncertainties is significant, so as to recover the system performance when the system deviates severely from the nominal state

and meet the performance requirements of attitude control and robust stability of the large flexible launch vehicle in the flight process.

This paper is organized as follows. Section 2 introduces the rigid-bullet coupling model of a large flexible launch vehicle with a second-order elastic vibration signal. Section 3, on controller design, details the improved adaptive augmented control scheme: (1) based on the rigid-bullet coupling model of a large flexible launch vehicle, the base-line PD controller is given and the maximum critical value of the forward gain tunable is determined after analyzing the flexible mode of the system in the frequency domain; (2) Two error signals are selected between the reference model and the actual system as input signals for the two channels of the adaptive control law, thereby increasing/decreasing the overall forward gain of the system; and (3) we introduce the role of the spectrum damper and parameter selection (mainly the extraction of high-frequency signal and low-frequency output for the error input signal of two channels). Section 4 presents a simulation analysis of the improved adaptive augmentation control designed in this paper, and the simulation results of several launch vehicle runaway scenarios are presented and discussed. By comparing the traditional PD control with the adaptive augmentation controller containing the baseline PD controller, we observed that the adaptive augmentation control improved the performance and robust stability of the large flexible launch vehicle during flight.

#### **2. Mathematical Model of the Launch Vehicle with Second-Order Vibration Modes**

The coupling problem between the rigid body motion and elastic vibration of large launch vehicles is more prominent than that of medium-sized and small launch vehicles. The motion process is more complicated due to the large mass of large launch vehicles, the increase in the aspect ratio, and the complex interference and uncertainty during flight, so the elastic vibration mode cannot be neglected. The rigid-bullet coupling mathematical model of the launch vehicle was established based on the forces (gravity, aerodynamic, thrust, control, etc.), moments, and vibration factors during the ascent of the launch vehicle.

In the velocity coordinate system, the translational and rotational motion of the launch vehicle around the centroid (pitch channel) can be expressed as (1) and (2). Considering the plane bending vibration and torsional elastic vibration of the arrow body, the general elastic vibration equation can be obtained by using the modal superposition method and the orthogonality of the vibration pattern, as shown in (3).

$$\begin{split} mV\dot{\theta}\cos\sigma &= -mg\cos\theta + q\mathbf{S}\_{\text{u}}\mathbf{C}\_{\text{y}}^{\text{u}}\mathbf{a} - q\mathbf{S}\_{\text{u}}\sum\_{i=1}^{n} \left( q\_{\text{ij}}\int\_{\mathbf{x}\_{0}}^{\mathbf{x}\_{\text{I}}} \mathbf{C}\_{\text{y}}^{\text{u}}\mathbf{R}\_{\text{i}\mathbf{z}}(\mathbf{x})d\mathbf{x} \right) + \frac{q\mathbf{S}\_{\text{u}}}{V}\mathbf{C}\_{\text{y}}^{\text{u}}(\mathbf{x}\_{\text{I}} - \mathbf{x}\_{\text{z}})\omega\_{\text{z}} \\ &- \frac{q\mathbf{S}\_{\text{u}}}{\nabla} \sum\_{i=1}^{n} \left( \dot{q}\_{\text{ij}}f\_{\text{X}\_{0}}^{\text{X}}\mathbf{C}\_{\text{y}}^{\text{u}}\mathbf{L}l\_{\text{i}\mathbf{y}}(\mathbf{x})d\mathbf{x} \right) + P\sin\mathfrak{a} + \frac{p}{\mathbf{T}}\delta\_{\theta}\cos\mathfrak{a} - P\cos\mathfrak{a} \sum\_{i=1}^{n} R\_{\text{i}\mathbf{z}}(\mathbf{x}\_{\text{R}})q\_{\text{ij}} + 2m\_{\text{R}}l\_{\text{R}}\ddot{\delta}\_{\theta} + F\_{\text{y}} \end{split} \tag{1}$$

*Jz* . *ω<sup>z</sup>* + (*J<sup>y</sup>* − *Jx*)*ωxω<sup>y</sup>* = −2*mRlR*(*x<sup>R</sup>* − *xz*) .. *δ<sup>ϕ</sup>* − 2*mRl<sup>R</sup>* . *<sup>W</sup>xδ<sup>ϕ</sup>* <sup>−</sup> *<sup>P</sup>* 2 (*x<sup>R</sup>* − *xz*)*δ<sup>ϕ</sup>* − *P n* ∑ *i*=1 *Uiy*(*x*)*qiy* −*P*(*x<sup>R</sup>* − *xz*) *n* ∑ *i*=1 *Riz*(*x*)*qiy* − *qSmC α y* (*x<sup>d</sup>* − *xz*)*α* + *qS<sup>m</sup> n* ∑ *i*=1 *<sup>q</sup>iy*<sup>R</sup> *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>z*)*Riz*(*x*)*dx* − *qSm <sup>V</sup> m ωz z l* <sup>2</sup>*ω<sup>z</sup>* + *qSm V n* ∑ *i*=1 . *<sup>q</sup>iy*<sup>R</sup> *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>z*)*Uiy*(*x*)*dx* + *M<sup>z</sup>* (2)

.. *qiy* + 2*ξiω<sup>i</sup>* . *<sup>q</sup>iy* + *<sup>ω</sup>*<sup>2</sup> *i qiy* = −*P n* ∑ *j*=1 *Rjz*(*x*)*qjy* + *<sup>P</sup>* 2 *δϕ* ! *Uiy*(*xR*) + 2*mRl<sup>R</sup>* .. *δϕUiy*(*xR*) + 2*mRl<sup>R</sup>* .. *δ<sup>ϕ</sup>* + 2*mRl<sup>R</sup>* . *Wxδ<sup>ϕ</sup> Riz*(*xR*) + *qS<sup>m</sup>* R *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>Uiy*(*x*)*dx α* −*qS<sup>m</sup> n* ∑ *j*=1 *<sup>q</sup>jy*<sup>R</sup> *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>Uiy*(*x*)*Rjz*(*x*)*dx* + *qSm V* R *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>z*)*Uiy*(*x*)*dx ωz* − *qSm V n* ∑ *j*=1 . *<sup>q</sup>jy*<sup>R</sup> *<sup>x</sup><sup>l</sup> x*0 *C α <sup>y</sup> <sup>x</sup>Uiy*(*x*)*Ujy*(*x*)*dx* (3)

The identification and meaning of the parameters in the above formula are shown in Table 1.


**Table 1.** Parameters and identification.

The nominal controller of the control system of the large flexible launch vehicle designed in this paper was based on a small perturbation linearization model, so the mathematical model of the rigid-bullet coupling of the large flexible launch vehicle was simplified to a small perturbation linearization model, as in (4) [25,26]:

$$\begin{aligned} \Delta \dot{\theta} &= c\_1 \Delta a + c\_2 \Delta \theta + c\_3 \delta\_\theta + c\_3 \prime \ddot{\delta}\_\theta + \sum\_{i=1}^n c\_{1i} \dot{q}\_i + \sum\_{i=1}^n c\_{2i} q\_i + \overline{F}\_y \\ \Delta \ddot{\varphi} + b\_1 \Delta \dot{\varphi} + b\_2 \Delta a + b\_3 \delta\_\theta + b\_3 \prime \ddot{\delta}\_\theta + \sum\_{i=1}^n b\_{1i} \dot{q}\_i + \sum\_{i=1}^n b\_{2i} q\_i &= \overline{M}\_z \\ \Delta \phi &= \Delta \theta + \Delta a \\ \ddot{q}\_{iy} + 2 \xi\_i \omega\_i \dot{q}\_{iy} + \omega\_i^2 q\_{\dot{j}y} &= D\_{1i} \Delta \dot{\varphi} + D\_{2i} \Delta a + D\_{3i} \delta\_\theta + D\_{3i} \prime \ddot{\delta}\_\phi + \sum\_{j=1}^n D\_{\bar{i}\bar{j}} \dot{q}\_{\dot{j}y} + \sum\_{j=1}^n D\_{\bar{i}\bar{j}} \dot{q}\_{\dot{j}y} \end{aligned} \tag{4}$$

In addition, the period of the arrow body centroid motion is much longer than the period of the pitch attitude angle motion, so the impact of the arrow body centroid motion can be ignored in the study of the arrow body attitude angle motion, and when also ignoring the influence of each oscillation pattern in the elastic vibration equation, then (4) can be further simplified as follows:

$$\begin{aligned} \Delta \ddot{\boldsymbol{\varrho}} + b\_1 \Delta \dot{\boldsymbol{\varrho}} + b\_2 \Delta \boldsymbol{\varrho} + b\_3 \delta\_\varphi + \sum\_{i=1}^n b\_{1i} \dot{\boldsymbol{q}}\_i + \sum\_{i=1}^n b\_{2i} q\_i &= \overline{M}\_z \\ \Delta \boldsymbol{\varrho} = \Delta \boldsymbol{a} \\ \ddot{\boldsymbol{q}}\_{\dot{\mathbf{i}}\dot{\boldsymbol{y}}} + 2\xi\_i \omega\_i \dot{\boldsymbol{q}}\_{\dot{\mathbf{i}}\dot{\boldsymbol{y}}} + \omega\_i^2 q\_{\dot{\mathbf{i}}\dot{\boldsymbol{y}}} &= D\_{1i} \Delta \dot{\boldsymbol{\varrho}} + D\_{2i} \Delta \boldsymbol{\varrho} + D\_{3i} \delta\_\varphi + \overline{Q}\_{\dot{\mathbf{i}}\dot{\boldsymbol{y}}} \end{aligned} \tag{5}$$

where *Qiy* is the generalized disturbance force on the elastic vibration of the *i*th order.

When the attitude angle and angular rate signals of the arrow body are obtained through the attitude measuring element, there is an additional elastic vibration signal in the obtained measurement signal influenced by the elastic vibration of the arrow body, and the actual measurement signal is as follows (6):

$$\begin{aligned} \Delta \boldsymbol{\varrho}\_{T} &= \Delta \boldsymbol{\varrho} - \sum\_{i} \boldsymbol{\mathcal{W}}\_{i}^{\prime}(\boldsymbol{\chi}\_{T}) \boldsymbol{q}\_{i} \\ \Delta \dot{\boldsymbol{\varrho}}\_{\mathcal{S}^{T}} &= \Delta \dot{\boldsymbol{\varrho}} - \sum\_{i} \boldsymbol{\mathcal{W}}\_{i}^{\prime}(\boldsymbol{\chi}\_{\mathcal{S}^{T}}) \dot{\boldsymbol{q}}\_{i} \end{aligned} \tag{6}$$

where *W*0 *<sup>i</sup>*(*xT*) is the slope of the *i*th-order oscillation pattern of the attitude angular measuring element at the mounting position *x<sup>s</sup>* , and *W*0 *<sup>i</sup>*(*xgT*) is the slope of the *i*th order vibration pattern at the installation of the attitude angular rate measurement element.

In this paper, we considered the design of a nominal controller for a second-order elastic vibration model of a launch vehicle, with the input, state variable, and output defined as:

$$\boldsymbol{\mu} = \boldsymbol{\delta}\_{\boldsymbol{\varrho} \boldsymbol{\nu}} \ \boldsymbol{y} = \begin{bmatrix} \ \Delta \boldsymbol{\varrho} & \Delta \dot{\boldsymbol{\varrho}} \ \end{bmatrix}^T \ \boldsymbol{\varepsilon} = \begin{bmatrix} \ \Delta \boldsymbol{\varrho} & \Delta \dot{\boldsymbol{\varrho}} & \boldsymbol{q}\_1 & \boldsymbol{q}\_2 & \dot{\boldsymbol{q}}\_1 & \dot{\boldsymbol{q}}\_2 \ \end{bmatrix}^T$$

Here we first ignore the effect of external perturbations and build the standard state space model. External perturbations will be added to the control input signal and explained in Section 4. The state space of the system is described by (7):

$$\begin{array}{l}\dot{\mathbf{x}} = A\mathbf{x} + Bu\\y = \mathbf{C}\mathbf{x} + Du\end{array} \tag{7}$$

where the matrix *A*, *B*, *C*, *D* is given by (1)–(5),

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ -b\_2 & -b\_1 & -b\_{21} & -b\_{22} & -b\_{11} & -b\_{12} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ D\_{21} & D\_{11} & -\omega\_1^2 & 0 & -2\xi\_1\omega\_1 & 0 \\ D\_{22} & D\_{12} & 0 & -\omega\_2^2 & 0 & -2\xi\_2\omega\_2 \\ \end{bmatrix} \\ B = \begin{bmatrix} 1 & 0 & -W\_1(\mathcal{X}\_T) & -W\_2(\mathcal{X}\_T) & 0 & 0 \\ 0 & 1 & 0 & 0 & -W\_1(\mathcal{X}\_{\mathcal{S}T}) & -W\_2(\mathcal{X}\_{\mathcal{S}T}) \\ \end{bmatrix}$$

The data selected in this paper are shown in Table 2.

**Table 2.** Parameters and values.


#### **3. Adaptive Augmentation Controller Design**

The adaptive augmentation controller combines the adaptive controller with a classically designed linear control system using a multiplicative forward gain that enhances the system by adjusting the total loop gain in real time based on the error between the actual output and the output of the reference model. When the baseline controller performs well, the adaptive augmentation controller produces little enhancement. When the baseline controller is unable to effectively meet the performance requirements, the adaptive augmentation controller adjusts the total gain of the system by increasing/decreasing the

adaptive gain to improve the performance of the control system; when the system is in a high degree of uncertainty or deviates from the nominal system, the adaptive controller can compensate the PD baseline controller to a greater extent to avoid system instability.

Figure 1 shows the adaptive augmentation control block diagram, which mainly consists of two parts: the PD-based baseline controller and the adaptive controller composed of the reference model and the adaptive law.

**Figure 1.** The adaptive augmentation control system.

#### *3.1. Reference Model*

The reference model was used to simulate the controlled motion of the rigid body of the launch vehicle in the nominal state, which produced the nominal response to the guidance instruction by adjusting the control parameters, and then the gap with the actual response of the launch vehicle was used as the input of the adaptive control law to adjust the gain of the PD controller. In adaptive gain control, a typical second-order system is used as the reference model [27], and the reference model used in this paper was obtained by neglecting the elastic vibrations in (5); the model's state space is given in (8):

$$\begin{aligned} \dot{\mathbf{x}}\_{r} &= A\_{r}\mathbf{x}\_{r} + B\_{r}\boldsymbol{\mu}\_{r} \\ \mathbf{y}\_{r} &= \mathbf{C}\_{r}\mathbf{x}\_{r} + D\_{r}\boldsymbol{\mu}\_{r} \end{aligned} \tag{8}$$
 
$$\text{where } A\_{r} = \begin{bmatrix} 0 & 1 \\ -b\_{2} & -b\_{1} \end{bmatrix}, B\_{r} = \begin{bmatrix} 0 & -b\_{3} \end{bmatrix}^{T}, \mathbf{C}\_{r} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, D\_{r} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.$$

#### *3.2. Baseline PD Controller*

In the stability of elastic vibration, there is a difference between amplitude stability and phase stability. The so-called amplitude stability refers to the amplitude *Gain*(*ω*) < 0, when the phase-frequency curve crosses ±(2n + 1)π; the so-called phase stability refers to the amplitude *Gain*(*ω*) > 0, when the phase-frequency curve does not cross ±(2n + 1)π The magnitude of stability of the essence of the engine oscillation control force generated by the excitation is less than the elastic vibration in the inherent damping under the role of attenuation, so the magnitude of stability depends on the inherent damping of elastic vibration and control system on the elastic vibration signal of the sufficient attenuation. The essence of phase stability is to take the elastic vibration signal as part of the control signal through the correction network to obtain the correct phase; for the elastic vibration to produce additional damping to achieve the purpose of stability, the phase stability does not depend on the inherent damping of elastic vibration, but the phase-frequency characteristics of the correction network put forward strict requirements.

Figure 2 shows an open-loop Nichols plot of the pitch channel, in which the open-loop frequency response of the angle (Phi) satisfies the amplitude-stability and phase-stability conditions, while the angle rate (Omega) does not satisfy the amplitude-stability condition (amplitude *Gain*(*ω*) > 0 when the phase frequency curve traverses the 180◦ curve) or the phase-stability condition (the phase frequency curve traverses the 180◦ line at frequency

7.694 rad/s, while amplitude *Gain*(*ω*) > 0). In adaptive augmentation control, the PD controller provides the basic control gain for the launch vehicle and is the basic controller in the AAC control framework. In this paper, we directly selected the PD controller parameters *K<sup>p</sup>* = −9.2 and *K<sup>d</sup>* = −3.8. For the rigid–flexible coupling model at the 30 s moment, the following notch filter was established as in (9):

$$\mathcal{W}(\mathbf{s}) = \left(\frac{s^2 + 2 \times 0.005 \times 7.69 \mathbf{s} + 7.69^2}{s^2 + 2 \times 0.6 \times 9 \mathbf{s} + 9^2}\right)^2 \left(\frac{s^2 + 2 \times 0.005 \times 7.69 \mathbf{s} + 7.69^2}{s^2 + 2 \times 0.6 \times 6.8 \mathbf{s} + 6.8^2}\right)^2 \tag{9}$$

**Figure 2.** The open-loop frequency characteristics of the system.

In a classical PID feedback control system, a higher forward gain can improve the performance and robustness of the system with a fixed ratio of proportional and differential gain. However, due to the special performance requirements and stability requirements of large flexible launch vehicles (presence of high uncertainty, elastic vibration), forward gain must be limited to a small range to provide better performance and robustness for the baseline controller, and the design requirements usually limit the allowable forward gain to a range not less than 6 dB from the critical stability value to improve the system's ability to cope with uncertainty. The closed-loop spectral characteristics of the system can be reflected to some extent by the open-loop margin of the system, and when the adaptive gain *k<sup>T</sup>* reaches a certain critical value, the closed-loop system exhibits resonance phenomena at certain frequencies (open-loop characteristics cross the *jω* axis in the complex plane), while for any *k<sup>T</sup>* + *ε*, the closed-loop system exhibits dispersion phenomena. The spectral characteristics of the closed-loop system at different forward gain *K<sup>T</sup>* are shown in Figure 3. From this, the critical value of the octave forward gain in the adaptive augmentation control can be determined.

**Figure 3.** The adjustable range of forward gain.

#### *3.3. Multiplicative Adaptive Control Law Based on Spectral Damping*

For large launch vehicle elastic vibration, the first-order elastic vibration mode has a low frequency and small phase deviation, and is usually stabilized by the phase-stabilization method, while the second-order and higher-frequency elastic vibration modes have large phase deviations, and are usually stabilized by the amplitude-stabilization method. In general, the first-order vibration mode can be phase-stabilized by selecting the mounting position of the rate gyro, the second-order vibration mode requires amplitude stabilization, and the higher-order vibration mode is amplitude-stabilized by high-frequency filtering of the correction network. Figure 4 shows a block diagram of the improved adaptive broadening control system designed in this paper, where the control command error signal *e<sup>u</sup>* and tracking error *eϕ*, . *ϕ* between the reference model and the actual model were used as inputs to the adaptive control method, and the adaptive gain *k<sup>T</sup>* was calculated through the two channels of oscillation suppression and error suppression, respectively, to adjust the baseline PD controller gain. The control command error, tracking error, and adjustment gain of adaptive augmentation control are shown below:

$$
\mathfrak{e}\_{\mathfrak{u}} = \mathfrak{u}\_r - \mathfrak{u} \tag{10}
$$

$$e\_{\varphi,\dot{\varphi}} = 0.5e\_{\varphi} + e\_{\dot{\varphi}} = 0.5(\varphi\_r - \varphi) + (\dot{\varphi}\_r - \dot{\varphi})\tag{11}$$

$$k\_T = sat\_{k\_0}^{k\_{\text{max}}} \{ k\_\ell y\_\ell - k\_s y\_s + 1 \} \tag{12}$$

where *k*max is the upper bounds of the adjustment gain, *k*<sup>0</sup> is the lower bounds of the adjustment gain, *k<sup>e</sup>* is the adjustment gain of tracking error term, *y<sup>e</sup>* is the output signal of the tracking error signal through the high- and low-pass filters, *k*<sup>s</sup> is the adjustment gain of control command error term, and *y<sup>s</sup>* is the output signal of the control command error signal through the spectrum dampener.

**Figure 4.** The adaptive control algorithm.

#### *3.4. Spectrum Dampers*

The adjustment gains *k<sup>e</sup>* and *k<sup>s</sup>* of the two spectrum dampers adjust the spectrum output signals *y<sup>e</sup>* and *y<sup>s</sup>* of the two channels, which are formed by the tracking error signal and the controller command error signal, as follows:

$$y\_{\varepsilon} = \mathbb{E}r r L\_P(s) \left( \mathbb{E}r r H\_P(s) e\_{\varphi, \dot{\varphi}} \right)^2 \tag{13}$$

$$y\_s = SDL\_P(s)(SDH\_P(s)e\_\mu)^2\tag{14}$$

In general, the DC gain of the designed high-pass filter should be as small as possible (the passband gain is usually set to 1), while the transition band should be as steep as possible (limited to 1.5 rad/s) The forms of the high- and low-pass filters are shown in (15) and (16):

$$H\_P(s) = \frac{s^2}{s^2 + 2\xi\_{hp}\omega\_{hp}s + \omega\_{hp}^2} \tag{15}$$

$$L\_P(\mathbf{s}) = \frac{\omega\_{lp}^{-2}}{s^2 + 2\xi\_{lp}\omega\_{lp}s + \omega\_{lp}^{-2}}\tag{16}$$

where *ωhp*, *ωl p* are the cutoff frequencies of the high- and low-pass filters; and *ξhp*, *ξl p* are the damping ratios, with values ranging from 0.5 to 0.8.

In the tracking error channel, the high- and low-pass filters successively process the error signal, and the AAC gain *k*<sup>T</sup> is to be enhanced by increasing the error suppression gain *k*e, thus improving the overall forward gain of the system to reduce the system tracking error and improve system performance. In this channel, we set the control frequency near the shear frequency of the rigid-body system (0.87 rad/s), considering the need to compensate the control of the system tracking error and improve the overall gain of the system [5,28]. The shear frequency of the low-pass filter is near this frequency, and the shear frequency of the high-pass filter is one octave above this frequency, so the transfer function of the high and low-pass filters is given accordingly as follows:

$$ErrH\_P(s) = \frac{s^2}{s^2 + 2 \times 0.8 \times 8.7s + 8.7^2} \tag{17}$$

$$ErrL\_P(s) = \frac{1.2^2}{s^2 + 2 \times 0.6 \times 1.2s + 1.2^2} \tag{18}$$

In the control command error channel, the spectrum damper is mainly used to process the elastic vibration signal in the control command and reduce the AAC gain *k<sup>T</sup>* by setting a specific frequency to adjust the elastic rejection gain *k<sup>s</sup>* , thus reducing the overall gain (excessive gain) of the system to suppress the elastic vibration of the system and reduce

its instability. The input of the high- and low-pass filters is the additional instruction error generated by the rigid-body controller instruction and the elastic vibration excitation, where the high-pass filter is used to obtain the elastic vibration signal from the control instruction. The analysis in the previous section showed that the system was prone to modal vibration at 7.69 rad/s, so we designed the control frequency at this frequency point. The cut-off frequency *ωhp* of the high-pass filter should be taken slightly higher than this frequency. The low-pass filter is used to eliminate the high-frequency components of the signal, which is squared before entering the low-pass filter, so the value of the cut-off frequency *ωl p* of the low-pass filter should be taken near this frequency. The corresponding parameters of the spectrum dampers in the elastic rejection channel are as follows:

$$SDH\_P(s) = \frac{s^2}{s^2 + 2 \times 0.8 \times 24.3s + 24.3^2} \tag{19}$$

$$SDL\_P(s) = \frac{7.69^2}{s^2 + 2 \times 0.6 \times 7.69s + 7.69^2} \tag{20}$$

#### **4. Simulation Results and Analysis**

In order to illustrate the role of the improved AAC control scheme in the launch vehicle system, the tracking curves and adaptive control gains of the nominal system and two different failure scenarios were presented and analysed in the simulation to verify the effectiveness of the designed algorithm. Assuming at 10 s after the launch vehicle takes off, the angle and rate commands of pitch were given as shown in Figure 5. The gain saturation function of the AAC controller in this example was taken as *K*max = 2 and *K*min = 0.5.

**Figure 5.** The pitch and angle rate commands.

If the system was in the normal state, the output of the system under PD control and AAC was consistent with the rigid-body nominal system, as shown in Figure 6a. The control commands and control rates of PD and AAC are shown in Figure 6b; there was almost no difference in the visible output curves. At the same time, the adjusted gain *k<sup>e</sup>* and *k*<sup>S</sup> of two channels in the AAC were close to 0, as shown in Figure 6c, and the overall adaptive gain was always kept at a stable value *k<sup>T</sup>* = 1, which meant that the AAC did not produce any effect in the normal state. This was in line with our original design requirement that the AAC not be involved in control activities when the baseline PD controller was able to achieve a good performance output.

**Figure 6.** Performance in the nominal state: (**a**) pitching attitude; (**b**) control commands; (**c**) gain adjustment for the tracking error and control command error; (**d**) total gain of adaptive augmentation control.

We assumed that there was a baseline controller gain loading error during operation (the gain of the PD controller was not sufficient to meet the system requirements; in this case, *K<sup>p</sup>* = −1 and *K<sup>d</sup>* = −0.8); meanwhile, the control commands of the system were perturbed by a square wave with an amplitude of 0.5 and a duration of 5 s. As shown in Figure 7a, the system was able to track the input commands better, but then there was a certain steady-state error (about 2◦ ) in the baseline PD control compared to the nominal system, and in contrast, the steady-state error was reduced to half (less than 1◦ ) under the AAC control adjustment. In addition, in Figure 7b there is a corresponding reduction in the control command error, while the control command error rate only fluctuated significantly when it was just perturbed. Among the two channels, the adjusting gain *k*<sup>s</sup> produced elastic suppression due to the disturbance in 25 s, and then the gain for suppressing elastic vibration fell back to 0; while *k*<sup>e</sup> produced error suppression mainly after 60 s due to the tracking error. The overall gain *k*<sup>T</sup> of the AAC control was less than 1 at 25 s in the elastic suppression channel *k*s, and then gradually increased (>1) due to the error suppression gain *k*<sup>e</sup> generated, and the overall AAC gain was always maintained at a saturated value due to the long-term presence of steady-state errors.

**Figure 7.** Performance in a state in which the baseline controller gain was misloaded and suffered external perturbations at *t* = 25 s: (**a**) pitching attitude; (**b**) control commands; (**c**) gain adjustment for the tracking error and control command error; (**d**) total gain of adaptive augmentation control.

Assuming that there was uncertainty in establishing the rigid spring coupling model, the elastic vibration frequency of the model was reduced by 40% while the same perturbation signals described above existed. The pitch angle and pitch angle rate signals at this time are shown Figure 8a. The system was able to follow the control commands to some extent in the early stages when the adaptive channel was closed (i.e., only the baseline PD controller was in action), but with the passage of time and accumulation, the system ended up in a divergent state. When the model parameters were changed substantially, the model was fundamentally changed, and the controlled object deviated from the nominal state. The original PD controller parameters were not suitable for this model, and the excessive forward gain aggravated the elastic vibration of the system. At the same time, the PD controller parameters were not reduced accordingly, which intensified the system oscillation and eventually could not be suppressed, leading to system dispersion. However, with the AAC controller, the system was able to suppress the system oscillation caused by the elastic modal perturbation and could track the reference input. At the same time, as shown in Figure 8b, for the control command and rate output, we can see that the control command of the baseline PD controller began to oscillate and could not be inhibited at the same time by the perturbation signal, while the AAC could well inhibit the oscillation of the control command (the control command error was less than 0.5, and the control

### command error rate was less than 1), which was beneficial for the actuator in the actual system, and had a good input signal.

**Figure 8.** Performance at an elastic vibration frequency of 40% perturbation and external perturbation at *t* = 25 s: (**a**) pitching attitude; (**b**) control commands; (**c**) gain adjustment for the tracking error and control command error; (**d**) total gain of adaptive augmentation control.

The adjusted gain of the two channels *k*e, *k*<sup>s</sup> and the overall gain *k*<sup>T</sup> of the AAC control are shown in Figure 8c,d. We observed that in the case in which the AAC was involved in the control, the baseline PD controller was not able to achieve a good tracking effect due to the ingress of the elastic mode, then the AAC controller generated a corresponding gain value *k*<sup>s</sup> (in this case mainly for the suppression of elastic vibration), and then set the AAC gain to less than 1 to reduce the overall gain of the system and meet the requirements. When the baseline controller can achieve the tracking effect better, then the value of AAC gain KT will fall back to 1. It is obvious from the above analysis that the adaptive control designed in this paper had a good robust stability to the ingress of the elastic mode, and under the adjustment of the AAC control, the launch vehicle could adjust the control gain online and in real time to set the engine swing angle of the servo to keep the rocket stable.

#### **5. Conclusions**

Adaptive augmentation control has important research significance and development potential for the control of large flexible launch vehicles, and can increase the robustness of the system, avoid oscillations and even destabilization problems caused by the estimation errors of the flexible mode, and improve the safety and reliability of rocket operation. The

improved AAC control scheme designed in this paper had good performance, and the simulation showed that during the nominal state of the system, the AAC control did not affect the baseline PD controller. When the system was subjected to external disturbances or PD controller errors (the controller parameters were loaded at values less than the set value), the AAC control could generate a multiplicative gain greater than 1 to boost the system forward gain and reduce the steady-state error of the system to half of the PD control (<1◦ ). With large regression of the flexible modal vibration frequency, the PD controller could cause the system to become unstable (uncontrollable). The AAC control could reduce the system forward gain by generating a multiplicative gain of less than 1 and limiting the control input signal to 0.5◦ to keep the system in a stable operation.

The simulation results of the above scenarios showed that the enhancement provided by the improved AAC control designed in this paper matched the expected goals and requirements, while the scheme had the same verifiability for the poorer control due to uncertainties caused by large-scale variations in thrust, mass, and atmospheric characteristics. The current simulation results showed suppression of up to 40% of the effect of flexible mode ingress, which greatly improved the robustness of the system. Future work on this program may consider sensor measurement noise and the nonlinear environment of closedloop guidance [29], which would assist in fine-tuning and improving the extraction and calculation of error signals for use in conjunction with other load-shedding/fault-tolerant controls in the future development of large flexible launch vehicles.

**Author Contributions:** Conceptualization, A.P. and J.Z.; data curation, H.Z.; funding acquisition, J.Z.; investigation, W.C.; methodology, A.P. and H.Z.; resources, A.P.; validation, H.Z. and W.C.; visualization, H.Z.; writing—original draft, H.Z.; writing—review and editing, A.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded in part by the Science and Technology Foundation of Guizhou Province (QKH [2020] 1Y273, [2018] 5781, [2018] 5615, [2016] 5133), Guizhou Provincial Department of Education, Youth Talent Development Project (Qianke [2021] 100), and the National Natural Science Foundation (NNSF) of China under Grant No. 51867005.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


## *Article* **Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method**

**Kai Liu , Fanwei Meng \* , Shengya Meng and Chonghui Wang**

College of Department of Control Science and Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China; miaomiao@stumail.neu.edu.cn (K.L.); 2001944@stu.neu.edu.cn (S.M.); 2071918@stu.neu.edu.cn (C.W.)

**\*** Correspondence: mengfanwei@neuq.edu.cn; Tel.: +86-157-3213-7808

**Abstract:** The coupling between variables in the multi-input multi-output (MIMO) systems brings difficulties to the design of the controller. Aiming at this problem, this paper combines the particle swarm optimization (PSO) with the coefficient diagram method (CDM) and proposes a robust controller design strategy for the MIMO systems. The decoupling problem is transformed into a compensator parameter optimization problem, and PSO optimizes the compensator parameters to reduce the coupling effect in the MIMO systems. For the MIMO system with measurement noise, the effectiveness of CDM in processing measurement noise is analyzed. This paper gives the control design steps of the MIMO systems. Finally, simulation experiments of four typical MIMO systems demonstrate the effectiveness of the proposed method.

**Keywords:** MIMO; coupling; PSO; CDM; measurement noise; robust controller

#### **Citation:** Liu, K.; Meng, F.; Meng, S.; Wang, C. Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method. *Entropy* **2021**, *23*, 1180. https://doi.org/10.3390/e23091180

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 3 August 2021 Accepted: 3 September 2021 Published: 8 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Multi-input multi-output (MIMO) systems, defined as systems with multiple control inputs and outputs, are widely used in industrial systems. Many common industrial control systems can be modeled as MIMO systems, such as chemical reactors, distillers, generators, and automobile transmission systems [1–5]. A consensus is that the control of the MIMO systems is more complex than the control of the single-input single-output (SISO) systems. In the MIMO systems, the outputs are affected by each input. In other words, there is a coupled interaction between the input and output variables of the MIMO systems. Due to the interaction in the MIMO systems, it is not easy to directly apply the advanced control methods based on the SISO systems.

Currently, the control strategies of the MIMO systems are mainly based on the methods of decoupling. Decoupling strategies can be divided into static decoupling and dynamic decoupling. The former achieves decoupling based on steady-state gain, which can effectively reduce the impact of model uncertainty, but the high-frequency response of MIMO systems is often not ideal [6]. The dynamic decoupling can achieve a trade-off between complexity and decoupling performance. In recent years, various dynamic decoupling strategies have been developed, such as ideal decoupling, simplified decoupling, and reverse decoupling. The ideal decoupler can provide a simple decoupling system, but the ideal decoupler is difficult to realize in practical applications. The opposite is simplified decoupling. Although simplified decoupling can obtain simple decoupling and decoupling, the decoupling system will be very complicated. In [7], Hagglund T proposed a decoupling method that approximates the sum of elements to reduce the system's complexity after decoupling. Reverse decoupling takes into account the advantages of ideal decoupling and simplified decoupling. However, when there is a time-delay element in the MIMO systems, reverse decoupling cannot guarantee the system's stability. In addition, researchers have used intelligent algorithms for MIMO systems and proposed various intelligent decoupling algorithms [8–12]. However, because the design of this kind of method is difficult to understand and the controller is complicated, it is difficult for engineers to adopt.

As an algebraic design method, the CDM proposed by S. Manabe is simple and easy to implement [13]. Compared with other control methods, CDM only requires the designer to define one parameter: the equivalent time constant [14]. At the same time, all algebraic equations in the CDM are expressed in the form of polynomials, which facilitates the elimination of poles and zeros in the design and analysis of the control systems. CDM has been proven to be a method to ensure the robustness of the control system, and its effectiveness has been proven through a series of experiments [15–17]. Therefore, with the continuous improvement of CDM, CDM has been continuously applied to existing control systems. Mohamed T. H. combined CDM with ecological optimization technology (ECO) for load frequency design in multi-regional power systems in [18]. Experimental results show that the proposed method is robust in the presence of disturbance uncertainty. Because CDM is simple, effective and robust, it is also applied in MIMO system control [19]. CDM was be used to design a PI controller with two cone-shaped official position research objects in [20]. The simulation results prove the effectiveness of CDM on disturbance suppression. In [21], CDM was used to solve the controller gain to suppress the vibration in the flexible robot system.

The first problem to be solved in the controller design of the MIMO system is how to achieve decoupling. Compared with other existing results, this article transforms the decoupling problem into the parameter optimization problem and gives an interaction measurement to evaluate the decoupling degree of the MIMO systems. The PSO algorithm is used to optimize the parameters of the compensator to achieve decoupling. After decoupling, the systems tend to have high order. The CDM considers the robustness and interference suppression performance of the system and the simplicity of design. Therefore, motivated by the advantages of CDM, this paper applies the CDM to the field of controller design for MIMO systems. At the same time, considering measurement noise can generate undesired control activity resulting in wear of actuators and reduced performance, this article analyzes the controller's suppression effect on measurement noise based on the CDM. To verify the effectiveness and universality of the proposed method, this paper gives four typical design examples of MIMO systems in the hope of providing engineers and technicians a reference.

The main innovations of this paper are as follows:


The rest of this article is organized as follows: Section 2 gives an interaction measurement of the coupling interaction and uses the PSO algorithm to design the compensator to achieve decoupling; Section 3 summarizes the design process of the CDM controller and analyzes the controller's suppression effect on measurement noise based on the CDM. Section 4 outlines a set of controller design procedures for MIMO systems; four unique objects are simulated to verify the effectiveness of the proposed method in Section 5. Finally, a conclusion is given.

#### **2. Decoupling Design**

At present, there are two solutions to the interaction of MIMO systems. One is to use modern control theory, and the other is to limit the interaction to a certain extent and treat MIMO systems as multiple SISO systems, which is called decoupling control. Generally speaking, decoupling control is simple to operate, so it is often used. This article designs a

compensator in the frequency domain to decouple. At the same time, in order to verify whether the designed compensator achieves the expected decoupling effect, this section provides the MIMO system interaction measurement.

#### *2.1. Compensator Design*

The schematic diagram of the decoupling design of the *n* × *m* MIMO system is shown in Figure 1.

**Figure 1.** Schematic diagram of the decoupling design of the *n* × *m* MIMO system.

where the model of the MIMO system is represented by the transfer function *Gp*(*s*) ∈ *R <sup>n</sup>*×*m*, which is Equation (1).

$$\mathbf{G}\_p(\mathbf{s}) = \begin{bmatrix} \mathbf{g}\_{11}(\mathbf{s}) & \cdots & \mathbf{g}\_{1m}(\mathbf{s}) \\ \dots & \ddots & \dots \\ \mathbf{g}\_{n1}(\mathbf{s}) & \cdots & \mathbf{g}\_{nm}(\mathbf{s}) \end{bmatrix} . \tag{1}$$

where *gij*(*s*), *i* = 1, 2, · · · , *n*; *j* = 1, 2, · · · , *m* is the transfer function element in *Gp*(*s*). Design the compensator *Gc*(*s*) ∈ *R <sup>m</sup>*×*<sup>n</sup>* as shown in Equation (2).

$$\mathbf{G}\_{\mathcal{C}}(\mathbf{s}) = \begin{bmatrix} h\_{11}(\mathbf{s}) & \cdots & h\_{1n}(\mathbf{s}) \\ \dots & \ddots & \dots \\ h\_{m1}(\mathbf{s}) & \cdots & h\_{mn}(\mathbf{s}) \end{bmatrix} \tag{2}$$

where *hij*(*s*)(*i* = 1, 2, · · · , *m*; *j* = 1, 2, · · · , *n*) is the transfer function element in *Gc*(*s*). In order to reduce the difficulty of the designed compensator *Gc*(*s*), set *Gc*(*s*) as a constant matrix. When the compensator *G<sup>c</sup>* acts on the MIMO system *Gp*(*s*), the decoupling system *Q*(*s*) ∈ *R n*×*n* is obtained as Equation (3).

$$\begin{aligned} Q(s) &= G\_p(s)\mathbf{G}\_c \\ &= \begin{bmatrix} g\_{11}(s) & \cdots & g\_{1m}(s) \\ \cdots & \ddots & \cdots \\ g\_{n1}(s) & \cdots & g\_{nm}(s) \end{bmatrix} \begin{bmatrix} h\_{11} & \cdots & h\_{1n} \\ \cdots & \ddots & \cdots \\ h\_{m1} & \cdots & h\_{mn} \end{bmatrix} \\ &= \begin{bmatrix} f\_{11}(s) & \cdots & f\_{1n}(s) \\ \cdots & \ddots & \cdots \\ f\_{n1}(s) & \cdots & f\_{nn}(s) \end{bmatrix}. \end{aligned} \tag{3}$$

The purpose of designing the compensator is to make the decoupling system *Q*(*s*) diagonal in all frequency domains, which means non-diagonal elements *f lr*(*s*) = 0

(*l* 6= *r*, *l* = 1, 2, · · · *n*, *r* = 1, 2, · · · *n*). In this way, the interaction is minimized, and the decoupling effect is the best. However, it is not easy to find such an ideal compensator. Therefore, this paper selects a specific frequency *s* = *jω*<sup>0</sup> to design the compensator. What needs to be explained is that the selection of a specific frequency *s* = *jω*<sup>0</sup> depends on the control object and requires the designer to use design experience to verify it through repeated experiments.

Use *s* = *jω*<sup>0</sup> to denote the element in the *r*(*r* = 1, 2, · · · *n*) column of *Q*(*s*) = *Gp*(*s*)*Gc*, we can get

$$f\_{\rm lr}(j\omega\_0) = \lg(j\omega\_0)\hat{h}\_{\rm r} = (\mathbf{a}\_{\rm l} + j\beta\_{\rm l})\hat{h}\_{\rm r} \qquad l = 1, \ 2, \ \cdots \cdot \mathbf{n}\_{\rm r} \tag{4}$$

where *g<sup>l</sup>* (*jω*0) is the *Gp*(*jω*0) row vector of *l*, ˆ*hr* is the *G<sup>c</sup>* column column vector of *r*, *α<sup>l</sup>* = Re{*g<sup>l</sup>* (*jω*0)}, and *β<sup>l</sup>* = Im{*g<sup>l</sup>* (*jω*0)}.

In order to achieve *Q*(*jω*0) diagonalization, we make the absolute value square of the off-diagonal elements in the *r*(*r* = 1, 2, · · · *n*) column of *Q*(*jω*0) equal to zero, which is

$$\left| \left| f\_{lr}(j\omega\_0) \right|^2 = \hat{h}\_r^T (\mathbf{a}\_l \mathbf{a}\_l^T + \beta\_l \beta\_l^{-T}) \hat{h}\_r = 0 \quad \text{l} \neq r. \tag{5}$$

Under Equation (5), the optimal solution ˆ*h<sup>r</sup>* can be obtained, thereby obtaining the compensator *G<sup>c</sup>* and the decoupling system *Q*(*s*). However, the decoupling system *Q*(*s*) may not meet the decoupling design requirements. One reason is that under the condition of a certain frequency *s* = *jω*0, Equation (5) can only guarantee that the absolute value square of the off-diagonal elements of *Q*(*jω*0) is equal to zero, but the absolute value square of diagonal elements | *f lr*(*jω*0)| 2 (*l* = *r*) is not equal to zero or does not tend to zero. Suppose the designed compensator *G<sup>c</sup>* cannot guarantee that *Q*(*s*) at a certain frequency *s* = *jω*<sup>0</sup> achieves diagonalization. In that case, there is no guarantee that *Q*(*s*) can be decoupled in the entire frequency domain. The other reason is that the <sup>ˆ</sup>*hr*(*<sup>r</sup>* <sup>=</sup> 1, 2, · · · *<sup>n</sup>*) may be a trivial solution, so the compensator designed is meaningless. To effectively illustrate the above description, we give a concrete example next.

**Example 1.** *Consider the two-input two-output one-order inertial system, The transfer function is:*

$$\mathbf{G}\_p = \begin{bmatrix} 2 & -3.6 \\ \hline 6s+1 & \frac{-3.6}{4s+1} \\ \frac{0.4}{9s+1} & \frac{4}{42s+1} \end{bmatrix}. \tag{6}$$

*we select the frequency ω*<sup>0</sup> = 1*, and under Equation (5), use PSO to obtain the compensator as equation:*

$$\mathbf{G}\_{\mathbf{f}} = \begin{bmatrix} -1.2499 \times 10^{-16} & 3.2508 \times 10^{-22} \\ 1.7313 \times 10^{-17} & 1.0364 \times 10^{-21} \end{bmatrix} \text{.} \tag{7}$$

It can be seen from Equation (7) that without any restrictions, the calculated *G<sup>c</sup>* is meaningless. Therefore, taking into account the above deficiencies, we make the following additions based on the constraint condition of Equation (5): Firstly, we select the square | *f lr*(*jω*0)| 2 (*l* = *r*) of the absolute value of diagonal elements of *Q*(*jω*0)*r*(*r* = 1, 2, · · · *n*) column as the objective function to obtain its maximum value. Secondly, in order to prevent the trivial solution of the obtained ˆ*h<sup>r</sup>* , we add the Equation (8) as the constraint condition:

$$
\hat{h}\_r^T \hat{h}\_r = 1.\tag{8}
$$

In summary, the decoupling problem of MIMO systems is transformed into the optimization problem as follows:

$$\max \, f(\hat{h}\_{l}) = \hat{h}\_{r}^{T} (\boldsymbol{a}\_{l} \boldsymbol{\alpha}\_{l}^{T} + \beta\_{l} \boldsymbol{\beta}\_{l}^{T}) \hat{h}\_{r} \quad l = r,$$

$$\text{s.t. } \hat{h}\_{r}^{T} (\boldsymbol{a}\_{l} \boldsymbol{\alpha}\_{l}^{T} + \beta\_{l} \boldsymbol{\beta}\_{l}^{T}) \hat{h}\_{r} = 0 \quad l \neq r,\tag{9}$$

$$\hat{h}\_{r}^{T} \hat{h}\_{r} = 1.$$

#### *2.2. Interaction Measurement*

The design process of the compensator has been given in Section 2.1. Since the magnitude of the interaction between the variables of the decoupling system *Q*(*s*) does not have a specific numerical measurement, it is not clear whether the designed compensator can achieve the desired decoupling effect. Therefore, this section presents an interaction measurement for the MIMO system to evaluate the impact of the decoupling degree of the compensator. The equation is established on the basis that the diagonal elements of the diagonal matrix are equal to the reciprocal of the diagonal elements of its inverse.

Assuming that the controlled variable of the decoupling system *Q*(*s*) is *Y* = [*y*1, *y*2, · · · , *yn*] *T* , the manipulated variable is *U* = [*u*1, *u*2, · · · , *un*] *T* , and *u<sup>i</sup>* (*i* = 1 · · · *n*) controls *y<sup>i</sup>* . For the i-th channel of *Q*(*s*), the open-loop gain of the channel is obtained when all other manipulated variables are zero, that is, equality (10). The open-loop gain of the channel is obtained when all other controlled variables are zero, that is, equality (11).

$$\text{other loops are open:} \left(\frac{\partial y\_i}{\partial u\_i}\right)\_{u\_n=0, n\neq i} = f\_{ii}. \tag{10}$$

$$\text{other loops are closed:} \left(\frac{\partial y\_l}{\partial u\_l}\right)\_{y\_n = 0, n \neq i} = \tilde{f}\_{li}. \tag{11}$$

Here, measurement for *Q*(*s*) interaction in MIMO systems is given:

$$E = \sum\_{i=1}^{n} \frac{|f\_{\text{ii}} - f\_{\text{ii}}|}{|\tilde{f}\_{\text{ii}}|} \quad i = 1 \sim n. \tag{12}$$

When the decoupling system *Q*(*s*) is diagonalized, *E* = 0. Therefore, when Equation (12) is equal to zero or close to zero, it shows that other channels have no or minimal relationship with the channel, and the decoupling effect is good.

**Remark 1.** *Equations (10) and (11) are based on the steady-state of the MIMO system, but this situation is usually not maintained at other frequencies. Therefore, it can only be used as the measurement of the interaction size of the MIMO system and cannot be used as the judgment of whether the MIMO systems are decoupled.*

#### *2.3. Parameter Tuning of Compensator*

In this paper, particle swarm optimization (PSO) is used to optimize the objective function. Firstly, the fitness function is compiled. In order to facilitate programming, −| *f lr*(*jω*0)| 2 ( *l* = *r* ) is taken as the objective function to obtain its minimum value. The fitness function can be obtained as follows:

$$\left| \mathrm{Fit}[f(\hat{h}\_{l})] - -\left| f\_{lr}(j\omega\_{0}) \right|^{2} = -\hat{h}\_{r}^{T}(\mathfrak{a}\_{l}\mathfrak{a}\_{l}^{T} + \mathfrak{B}\_{l}\mathfrak{B}\_{l}^{T})\hat{h}\_{r} \quad l=r. \tag{13}$$

Secondly, through the above fitness function and constraints in the frequency domain, <sup>ˆ</sup>*h<sup>r</sup>* ( *<sup>r</sup>* <sup>=</sup> 1, 2, · · · *<sup>n</sup>* ) can be obtained through PSO debugging, and thus the compensator *G<sup>c</sup>* can be obtained. Figure 2 shows the flowchart of PSO.

**Figure 2.** The flowchart of PSO.

PSO is essentially a stochastic algorithm, which has the function of self-organization, evolution, and memory and the strong searching ability and fast optimizing speed. In order to demonstrate the superiority of PSO to other evolutionary algorithms, we execute a number of comparisons between PSO and other evolutionary algorithms, such as Genetic Algorithm (GA), Shuffled Frog Leaping Algorithm (SFLA) and Cuck Search (CS). GA originates from Darwin's idea of natural evolution and follows the natural law of competition and survival of the fittest. GA is characterized by fast search speed, strong randomness, simple process, and robust flexibility. Still, it is easy to fall into the local optimum due to the reduction of population diversity in the evolution process. CS is a new swarm intelligence algorithm based on simulating cuckoo's nesting behavior. The algorithm has been successfully applied to solve various optimization problems due to its fewer parameters and easy realization. A significant feature of the CS is that it uses Levy flight to generate new solutions. The high randomness of Levy flight is that it can make the search process throughout the whole search space so that the global search ability of the algorithm is strong. However, the Levy flight height's randomness causes the CS algorithm's poor ability to perform a refined search in the local area and the slow convergence of the algorithm. SFLA simulates the communication and cooperation behaviors of frog populations in the process of foraging in nature, which has the advantages of fewer control parameters, simple operation, and easy realization. The specific parameter settings of different evolutionary algorithms are proposed in Table 1. The population size of each algorithm is 50, and the times of iterations are 100. The crossover probability and mutation probability of GA are 0.9 and 0.1, respectively. SFLA's moving maximum distance is 0.02,

CS's maximum discovery probability is 0.05. The weight of inertia, the self-learning factor and the population-learning factor of PSO are 0.35, 1.5 and 2.5. respectively.

In order to facilitate a comparison, we randomly select *Gp*(*s*)=[0.2, 0.5; −0.3, 0.6], set *Gc*=[*h*1, *h*2; *h*3, *h*4], and only obtain the first column ˆ*h*<sup>1</sup> : *h*<sup>1</sup> and *h*<sup>3</sup> of *Gc*. Each algorithm is implemented independently 30 times. Table 2 presents the statistical results of each algorithm, including the maximum, minimum, average, standard deviation values of the objective function, and the average computational time. According to Table 2, we can see that PSO has an evident advantage of minimum, average, standard deviation values and average computational time over other algorithms. Figure 3 is the convergence graph of the optimization algorithms. It can be seen that PSO has a fast convergence speed and a good effect in finding the optimal global solution.


**Table 1.** The parameter settings of different evolutionary algorithms.

**Table 2.** Statistical results of different algorithms.


**Figure 3.** Convergence graphs of the optimization algorithms.

Furthermore, according to the research works with respect to the non-parametric statistical tests for different algorithms [22], some statistical tests have been adopted to compare the performance of GA, SFLA, CS and PSO. Table 3 proposes ranks achieved by Friedman, Friedman aligned and Quade tests for the objective function obtained by different algorithms. It is noticeable from Table 3 that PSO performs best in all statistical tests. Consequently, PSO has the superiority over other evolutionary algorithms in solving unknown parameters of compensator *Gc*.


**Table 3.** The ranks achieved by Friedman, Friedman aligned and Quade tests.

#### **3. CDM Controller Design and Measurement Noise Rejection**

In Section 2, the decoupling design can obtain the decoupling system *Q*(*s*) with minimized interaction, but its open-loop transfer function is complex, and the order is high. Therefore, when stability, response characteristics, and robustness are considered simultaneously, the designed controller will become more complicated. CDM can effectively solve such problems.

#### *3.1. CDM Controller Design*

For the SISO linear systems, the standard block diagram designed by CDM is shown in Figure 4. The CDM control system consists of two parts: the controlled object and the CDM controller.

**Figure 4.** CDM control system standard block diagram.

where *r*(*t*), *u*(*t*), *y*(*t*) and *d*(*t*) are reference signal, control quantity, output quantity and disturbance quantity, respectively. The control function of the controller *u*(*t*) may be interfered by the interference signal *d*. *N*(*s*) and *D*(*s*) are the numerator and denominator polynomials of the controlled object, respectively, defined as follows:

$$\begin{aligned} N(s) &= b\_m s^m + b\_{m-1} s^{m-1} + \dots + b\_1 s + b\_{0\prime} \\ D(s) &= d\_n s^n + d\_{n-1} s^{n-1} + \dots + d\_1 s + d\_{0\prime} \end{aligned} \tag{14}$$

where *bm*, *bm*−<sup>1</sup> · · · *b*<sup>0</sup> and *dn*, *dn*−<sup>1</sup> · · · *d*<sup>0</sup> are real coefficients and *m* ≤ *n*. *A*(*s*) and *B*(*s*) are the denominator and numerator polynomial of the controller, respectively, defined as follows:

$$A(\mathbf{s}) = \sum\_{i=0}^{p} l\_i \mathbf{s}^i, \quad B(\mathbf{s}) = \sum\_{i=0}^{q} k\_i \mathbf{s}^i. \tag{15}$$

where *l<sup>i</sup>* and *k<sup>i</sup>* are unknown coefficients of the controller and *i* ≤ *n*. There are many criteria for the selection of *A*(*s*) and *B*(*s*) polynomials. Disturbance is one of the selection

criteria. When *l*<sup>0</sup> = 0, the influence of disturbance signal can be well suppressed. *F*(*s*) is the reference numerator of the controller, which can ensure that the steady-state error in the performance of the closed-loop system is reduced to zero. The definition form is as follows:

$$F(s) = (\frac{P(s)}{N(s)})\_{|s=0^\nu} \tag{16}$$

where *P*(*s*) is the characteristic polynomial of the closed-loop system. From Figure 4, we can obtained

$$P(s) = D(s)A(s) + N(s)B(s) = \sum\_{i=0}^{n} a\_i s^i, a\_i > 0,\tag{17}$$

where *a<sup>i</sup>* is the real coefficient. The design parameters of CDM-related characteristic polynomials are equivalent to the time constant *τ* and stability index *γ<sup>i</sup>* , defined as follows:

$$\begin{cases} \tau = \frac{a\_1}{a\_0} \\ \gamma\_i = \frac{a\_i^2}{a\_{i+1}a\_{i-1}} \\ \gamma\_0 = \gamma\_n = \infty \\ \gamma\_i^\* = \frac{1}{\gamma\_{i+1}} + \frac{1}{\gamma\_{i-1}} \end{cases} \tag{18}$$

where *γ<sup>i</sup>* ∗ denotes the stability limit, which is used to constrain the value of the stability index *γ<sup>i</sup>* , and *γ<sup>i</sup>* ∗ is mainly used to ensure that it meets the Lyapunov stability conditions in the actual design process. The equivalent time constant *τ* is closely related to the setting time and bandwidth, which determines the rapid response of the system. If the setting time is represented by *t<sup>s</sup>* , according to the Manabe standard form [13], its relationship with the equivalent time constant *t<sup>s</sup>* is *τ* = *t<sup>s</sup>* (2.5 ∼ 3).

The selection of the stability index *γ<sup>i</sup>* determines the stability and time domain response characteristics of the system. Robustness is different from the system's stability, mainly considering the influence of system parameters on the speed of pole change. Control systems with other structures may have different robustness even if they have the same characteristic equation. The robustness of the system can only be determined when the open-loop system structure is determined. An essential feature of CDM in the application is that the controller structure and the characteristic polynomial can be designed simultaneously, and the robustness of the system can be guaranteed by setting the controller structure.

If Equation (17) of the corresponding system is a third-order system, according to the Routh stability criterion, the stability condition is *a*2*a*<sup>1</sup> > *a*3*a*0. According to the expression in formula (18), this is equivalent to requiring the stability index to satisfy *γ*1*γ*<sup>2</sup> > 1. Similarly, the stability conditions of the fourth-order system are *a*<sup>2</sup> > (*a*1/*a*3)*a*<sup>4</sup> + (*a*3/*a*1)*a*<sup>0</sup> and *γ*<sup>2</sup> > *γ*<sup>2</sup> ∗ . For the fifth-order and above systems, Lyapunov gives several sufficient conditions for different forms of stability and instability, among which the conditions suitable for the CDM are as follows [23]: if all the fourth-order polynomials of the system are stable and have a margin of 1.12 times, the system is stable. If some third-order polynomials in the system are unstable, the system is unstable. The stability conditions of the system can be described as :

$$\begin{cases} \ a\_i > 1.12(\frac{a\_{i-1}}{a\_{i+1}} a\_{i+2} + \frac{a\_{i+1}}{a\_{i-1}} a\_{i-2}), \\ \quad \gamma\_i > 1.12 \gamma\_i^\* \quad i = 2 \sim (n-1). \end{cases} \tag{19}$$

Manabe has proved that the system can obtain better stability and response characteristics when *γ<sup>i</sup>* > 1.12*γ<sup>i</sup>* <sup>∗</sup> and *γ<sup>i</sup>* ' values are between 1 and 4. If the stability index is selected

according to *γ<sup>i</sup>* > 1.5*γ<sup>i</sup>* ∗ , the system's robustness is improved by sacrificing stability and response characteristics [23]. With the help of some design experience, designers can consider stability, response characteristics and robustness by reasonably selecting the structure and parameters of the controller.

In this article, we use the stability index *γ<sup>i</sup>* values in the Manabe standard form. According to the Manabe standard form, the stability index *γ<sup>i</sup>* is defined as:

$$
\gamma\_1 = 2.5, \ \gamma\_0 = \gamma\_\text{\textmu} = \infty, \ \gamma\_\text{i} = 2; \ \text{ i} = 2 \sim (n-1). \tag{20}
$$

Using the equivalent time constant *τ* and the stability index *γ<sup>i</sup>* , the characteristic polynomial *P*(*s*) can be obtained as follows:

$$P(\mathbf{s}) = a\_0 \left[ \left\{ \sum\_{i=2}^{n} \left( \prod\_{j=1}^{i-1} \frac{1}{\gamma\_{i-j}^i} \right) \left( \tau \mathbf{s} \right)^i \right\} + \tau \mathbf{s} + 1 \right]. \tag{21}$$

By comparing the coefficients of the characteristic polynomial Equations (17) and (21), the CDM controller parameters can be obtained.

#### *3.2. Measurement Noise Rejection*

To meet with the design needs of real-life, we analyze the output effect of the measurement noise in the controlled variable *u* in Section 3.2. Usually the block diagram presented in Figure 4 is extended by including measuring noise. Measurement noise may have a different character, but it is typically dominated by high frequencies, and low-frequency noise would correspond to drift. High-frequency noise can be suppressed by limiting the bandwidth of the closed-loop system. CDM can restrain the influence of high-frequency noise by selecting the equivalent time constant *τ* to limit the bandwidth of the closed-loop system. The reason is that the rapid response of the closed-loop system is proportional to the bandwidth, The equivalent time constant *τ* is closely related to the setting time and bandwidth, which determines the rapid response of the system. Here, we give the analysis of low-frequency noise suppression. Those signals are represented in Figure 5.

**Figure 5.** Basic structure of a CDM controller.

The newly added signal *n*(*t*) denoting the measurement noise. We assume *n*(*t*) is bounded with |*n*(*t*)| ≤ *µ* · *h*(*t*), where *µ* and *h*(*t*) are a positive constant and a step-type signal, respectively [24]. In order to analyze the output effect of the measurement noise in the controlled variable *u*(*t*), the reference signal *r*(*t*) and disturbance signal *d*(*t*) is set to zero. This leads to a relationship between *n*(*t*) and *u*(*t*), *n*(*t*) and *y*(*t*) given by the following differential equation:

$$\begin{array}{l} -n(s) \cdot B(s) \cdot D(s) = (A(s) \cdot D(s) + N(s) \cdot B(s)) \cdot U(s), \\ -n(s) \cdot B(s) \cdot N(s) = (A(s) \cdot D(s) + N(s) \cdot B(s)) \cdot Y(s), \end{array} \tag{22}$$

where *n*(*s*) is the Laplace transforms of *n*(*t*), *U*(*s*) is the Laplace transforms of *u*(*t*), *Y*(*s*) is the Laplace transform of *y*(*t*).

Let us impose *b<sup>i</sup>* 6= 0 for *i* = 0, · · · , *m* ,*d<sup>i</sup>* 6= 0 for *i* = 0, · · · , *n* in Equation (14). The product *B*(*s*)· *N*(*s*) is a *q* + *m* order polynomial and the product *B*(*s*)· *D*(*s*) is a *q* + *n* order polynomial, respectively. The polynomials will be denoted by *C*(*s*) and E(s) defined as:

$$\mathcal{C}(\mathbf{s}) = \sum\_{i=0}^{q+m} \mathbf{g}\_i^{\prime} \cdot \mathbf{s}^i \; \; \; E(\mathbf{s}) = \sum\_{i=0}^{q+n} \mathbf{g}^{\prime \prime} \mathbf{s}^i \cdot \mathbf{s}^i. \tag{23}$$

Assuming *r*(*t*) = 0, *d*(*t*) = 0, this steady-state system behaviour will be easily handled in the Laplace domain. By applying the final value theorem, the following equality should hold:

$$\lim\_{t \to \infty} y(t) = \lim\_{s \to 0} s \cdot Y(s), \tag{24}$$

However, in order to satisfy this equality, all the *Y*(*s*) poles must have negative real parts and no more than one pole can be at the origin [25].

Assuming causality and zero initial conditions, the application of Laplace transform to (22) leads to,

$$\begin{split} \mathcal{U}(s) &= \frac{-E(s)}{A(s) \cdot D(s) + \mathcal{C}(s) \cdot \mathcal{B}(s)} \boldsymbol{\pi}(s). \\ \mathcal{Y}(s) &= \frac{-\mathcal{C}(s)}{A(s) \cdot D(s) + \mathcal{C}(s) \cdot \mathcal{B}(s)} \boldsymbol{\pi}(s). \end{split} \tag{25}$$

Applying the final value theorem to the above expression then,

$$\lim\_{s \to 0} s \cdot -\mathbb{C}(s) \cdot n(s) = 0 \tag{26}$$

Due to <sup>|</sup>*n*(*t*)<sup>|</sup> <sup>≤</sup> *<sup>µ</sup>* · *<sup>h</sup>*(*t*) and *<sup>h</sup>*(*t*), the Laplace transform is <sup>1</sup> *s* , and we use *<sup>µ</sup> s* replace *n*(*s*), thus expression (26) takes the following format:

$$\lim\_{s \to 0} s \cdot -\mathbb{C}(s) \cdot \frac{\mu}{s} = -\mu \cdot g\_0. \tag{27}$$

Since *g*<sup>0</sup> is equal to the product of *b*<sup>0</sup> and *k*<sup>0</sup> and since *b*<sup>0</sup> 6= 0 then, in order for *g*<sup>0</sup> to be zero, the controller coefficient *k*<sup>0</sup> must be equal to zero. Similarly, when the controller coefficient *k*<sup>0</sup> is equal to 0, the measurement noise has no effect on the control *u*(*t*). Therefore, when the controller coefficient *k*<sup>0</sup> is equal to zero, the measurement noise does not affect performance.

#### **4. Overall Design Ideas**

This paper designs a compensator *G<sup>c</sup>* and a centralized CDM controller for the *n* × *m* MIMO system in Figure 6. Systematic design ideas ensure the feasibility of decoupling design methods in large and small systems. At the same time, when the MIMO system interaction is minimized with high accuracy, the controller can achieve good control effects due to the robustness of the CDM. The most considerable advantages of CDM can be listed as follows:


The decoupling control and CDM controller design for the MIMO system can be summarized as the following steps.

**Figure 6.** MIMO system control block diagram.

Figure 7 shows the design steps for MIMO systems, where the design process of CDM controllers is shown as the following:


**Figure 7.** MIMO system control block diagram.

**Remark 2.** *The necessary condition for designing CDM controllers is that both denominators and molecules of the transfer function of the controlled object need to be expressed by rational polynomials. If there is a delay element in the transfer function of the controlled object, the improved Padé approximation method in reference [26] is used to deal with the delay element. According to the results of [26], the third-order improved Padé approximation is:*

$$e^{-sL} = \frac{60 - 24sL + 3(sL)^2}{60 + 36sL + 9(sL)^2 + (sL)^3}. \tag{28}$$

*where L is delay time.*

#### **5. Simulation Experiment**

This section conducts simulation experiments on four unique control targets to prove the effectiveness of this method. The experiments are evaluated with a step response of 1. The state variable is set to *xi*(*i* = 1, 2, 3, . . . , *n*), and the system output is set to *yi*(*i* = 1, 2, 3, . . . , *m*). Use the compensator for decoupling. When the interaction of the MIMO system is minimized, treat it as *n* SISO systems, and set each SISO system as *A<sup>i</sup>* = (*i* = 1, 2, 3, . . . , *m*).

**Example 2.** *Consider the two-input two-output second-order inertial system (sugar factory model) in [27]. The transfer function is:*

$$\mathbf{G}\_{p} = \begin{bmatrix} \frac{0.28}{21s^2 + 10s + 1} & \frac{-0.33}{30s^2 + 11s + 1} \\ \frac{0.4}{270s^2 + 39s + 1} & \frac{0.5}{432s^2 + 42s + 1} \end{bmatrix}. \tag{29}$$

Without the decoupling design, the step response curve is shown in Figure 8.

**Figure 8.** Step response curve of the original system (29) without decoupling design. (**a**) *x*<sup>1</sup> 6= 0; (**b**) *x*<sup>2</sup> 6= 0.

It can be seen from Figure 8a that *x*<sup>1</sup> 6= 0. As shown in Figure 8b, *x*<sup>2</sup> 6= 0. The two loops are obviously related, so the system (29) is a related system to the interaction. Therefore a decoupling design method is used to eliminate the interaction of the original system.

Select the angular frequency *ω*<sup>0</sup> = 0.13, and obtain the compensator as Equation (30).

$$\mathbf{G}\_{\mathcal{L}} = \begin{bmatrix} 0.7807 & 0.7606 \\ -0.6256 & 0.6507 \end{bmatrix} \text{.} \tag{30}$$

Apply the compensator (30) to the original system (29), then the decoupling system *Q*(*s*) is obtained. Figure 9 draws the step response curve of the original system (29) after the decoupling design. As can be seen from Figure 9, the interaction of *Q*(*s*) is effectively suppressed, especially in the static response part of the system. However, there is still a weak interaction in the dynamic response part. Overall, the decoupling effect is good.

**Figure 9.** Step response curve of the original system (29) with decoupling design. (**a**) *A*<sup>1</sup> ; (**b**) *A*2.

The two SISO systems after decoupling are set to *A*<sup>1</sup> and *A*2. For *A*<sup>1</sup> and *A*2, use the stability index *γ<sup>i</sup>* and equivalent time constant *τ* in Table 4 to calculate CDM control polynomial parameters. Table 4 shows the equivalent time constant *τ* and CDM control polynomial parameter values.

**Table 4.** The equivalent time constant *τ* and CDM control polynomial parameter values.


Using the CDM control polynomial parameters in Table 4 to control *A*<sup>1</sup> and *A*2, the results are shown in Figure 10.

**Figure 10.** CDM controls *A*<sup>1</sup> and *A*<sup>2</sup> step response.

Figure 11 shows the result of inserting the compensator (30) in front of the controlled object (29) and using the CDM parameters in Table 4 for control. Affected by the interaction of the dynamic response part, the system overshoot increases.

**Figure 11.** Use this method to control the MIMO system (29) step response.

Here, the compensator (31) designed in [27] is compared with the method in this paper. Table 5 summarizes the results of evaluating the compensators (30) and (31) using formula (12). Table 5 shows that the decoupling effect of the compensator designed by the method in this paper is better.

$$\mathbf{G}\_{\mathcal{C}} = \begin{bmatrix} 0.174 & 0.479 \\ -0.219 & 0.503 \end{bmatrix} \text{.} \tag{31}$$

**Table 5.** Comparison of evaluation.


In [27], Masaya et al. designed a PID controller according to Shunji's optimization method. We also use the design method proposed in [27] to design the controller for the decoupling system *Q*(*s*), which uses the compensator (30), and Figure 12 shows the results of the Masaya-PID controller and the CDM controller in this paper to control the decoupling system *Q*(*s*). From this figure and the performance values appearing in Table 6, it is seen that the CDM controller has a more successful time-domain performance.

**Figure 12.** Use the PID controller in [27] and CDM controller to control the decoupling system *Q*(*s*) step response. (**a**) controlled variable *x*<sup>1</sup> ; (**b**) controlled variable *x*2.


**Table 6.** Performance values of the time response curves shown in Figure 12.

In order to verify the robustness of the method in this paper, the system with disturbances and modeling errors is simulated. When there is a step disturbance in the original system, the control result is shown in Figure 13. According to Figure 13, the influence of the disturbance signal subsides in a short time. Suppose that the correct system model is represented by Equation (32), and the system model with errors is represented by Equation (29). The changes in the parameter of Equation (29) are in the interval ±15%. The compensator (30) is applied to the correct system model (32), and the CDM controller is used to control. The experimental results are shown in Figure 14. From the results in Figure 14, it can be seen that when the model has measurement errors, the control effect of the method in this paper is good. Figures 13 and 14 show that the method proposed in this paper is robust.

**Figure 13.** Use the method in this paper to control the step response of the original system (29) with the step disturbance.

**Figure 14.** Use the method of this article to control the step response of the system (32).

$$\mathbf{G}\_P = \left[ \begin{array}{cc} \frac{0.238}{21s^2 + 10s + 1} & \frac{-0.3795}{30s^2 + 11s + 1} \\ \frac{0.34}{270s^2 + 39s + 1} & \frac{0.575}{432s^2 + 42s + 1} \end{array} \right]. \tag{32}$$

**Example 3.** *The multivariable four-tank system has a tunable transmission of zero [28,29]. With appropriate "tuning", this system will exhibit nonminimum-phase characteristics. Applying the nominal operating parameters given in [28,29] yields the four-tank system model:*

$$\mathbf{G}\_p = \begin{bmatrix} 0.1987 & -0.3779 \\ \frac{65s+1}{65s+1} & \frac{-0.3779}{(65s+1)(34s+1)} \\ 0.4637 & \frac{0.16194}{54s+1} \end{bmatrix}.\tag{33}$$

For the four-tank system of the controlled object (33), the frequency *ω*<sup>0</sup> = 0.34 is selected, and the precompensator is obtained.

$$\mathbf{G}\_{\mathfrak{c}} = \begin{bmatrix} -0.3262 & 0.8884 \\ 0.9455 & -0.4597 \end{bmatrix} \text{.} \tag{34}$$

Use the evaluation formula (12) to evaluate the decoupling system *Q*(*s*) after the compensator formula (34) acts on the controlled object formula (33), and the result is 0.000013. It shows that after the decoupling design, the interaction of the controlled object (33) is weak, and the decoupling effect is well.

The two SISO systems after decoupling are set to *A*<sup>1</sup> and *A*2. In order to suppress measurement noise, we select the controller coefficient *k*<sup>0</sup> = 0. Then, use the stability index *γ<sup>i</sup>* and equivalent time constant *τ* in Table 7 to calculate CDM control polynomial parameters. Table 7 shows the equivalent time constant *τ* and CDM control polynomial parameter values.

**Table 7.** The equivalent time constant *τ* and CDM control polynomial parameter values.


**Figure 15.** CDM controller controls *A*<sup>1</sup> and *A*<sup>2</sup> step response.

Figure 16 shows the result of inserting the compensator (34) in front of the controlled object (33) and using the CDM parameters in Table 7 for control. It can be seen that the system overshoot is slightly increased due to the interaction.

**Figure 16.** Use this method to control the MIMO system (33) step response.

Figure 17 shows the result of the controlled MIMO system (33) under the measurement noise whose magnitude is limited within [−0.0016, 0.0016]. It can be seen that the system response has not changed, and the measurement noise does not affect performance.

**Figure 17.** Controlled MIMO system (33) under the measurement noise.

**Example 4.** *The controlled object (29) increases the delay link to become the accused object (35).*

$$\mathbf{G}\_P = \left[ \begin{array}{cc} \frac{0.28}{21s^2 + 10s + 1} e^{-0.71s} & \frac{-0.33}{30s^2 + 11s + 1} e^{-2.24s} \\ \frac{0.4}{270s^2 + 39s + 1} e^{-0.59s} & \frac{0.5}{432s^2 + 42s + 1} e^{-0.68s} \end{array} \right]. \tag{35}$$

Using the method in this paper, select the angular frequency *ω*<sup>0</sup> = 0.28 and obtain the compensator.

$$\mathbf{G}\_{\mathbf{c}} = \begin{bmatrix} 0.7809 & 0.7625 \\ -0.6247 & 0.6470 \end{bmatrix} \text{.} \tag{36}$$

Use Equation (12) to evaluate the effect of the compensator (Equation (36)) on the controlled object (Equation (35)) to obtain the decoupling system *Q*(*s*), and the result is 0.000000057. It shows that the system interaction effect of using compensator decoupling is minimal, and the decoupling effect is good.

The two SISO systems after decoupling are set to *A*<sup>1</sup> and *A*2. For *A*<sup>1</sup> and *A*2, the delay link is approximated by the improved Padé approximation method in [26]. Then use the stability index *γ<sup>i</sup>* and Table 8 equivalent time constant *τ* to calculate the CDM control polynomial parameters. Table 8 shows the equivalent time constant *τ* and CDM control polynomial parameter values.


**Table 8.** The equivalent time constant *τ* and CDM control polynomial parameter values.

Using the CDM control polynomial parameters in Table 8 to control *A*<sup>1</sup> and *A*2, the results are shown in Figure 18.

*Time(s)*

**Figure 18.** CDM control *A*<sup>1</sup> and *A*<sup>2</sup> step response.

Figure 19 is the result of inserting the compensator (36) before the controlled object (35) and using the CDM parameters in Table 8 for control. Figure 18 is the same as Figure 19, which proves that the decoupling effect is well.

**Figure 19.** Use this method to control the MIMO system (33) step response.

**Example 5.** *The controlled object (29) adds a line of input and a column output to become the controlled object (37).*

$$\mathbf{G}\_p = \begin{bmatrix} \frac{0.28}{21s^2 + 10s + 1} & \frac{-0.33}{30s^2 + 11s + 1} & \frac{0.38}{45s^2 + 12s + 1} \\\\ \frac{0.4}{270s^2 + 39s + 1} & \frac{0.5}{432s^2 + 42s + 1} & \frac{0.6}{543s^2 + 68s + 1} \\\\ \frac{0.9}{500s^2 + 30s + 1} & \frac{0.45}{440s^2 + 45s + 1} & \frac{1}{600s^2 + 89s + 1} \end{bmatrix}. \tag{37}$$

Using the method in this paper, select the angular frequency *ω*<sup>0</sup> = 0.13, and obtain the compensator:

$$\mathbf{G}\_{\mathcal{L}} = \begin{bmatrix} 0.6251 & -0.7607 & 0.8234 \\ 0.3317 & 0.0941 & 0.0350 \\ -0.7061 & 0.6423 & -0.5667 \end{bmatrix} \text{.} \tag{38}$$

Use the evaluation (12) to evaluate the decoupling system *Q*(*s*) after the compensator formula (38) acts on the controlled object formula (37), and the result is 0.0005411. The two SISO systems after decoupling are set to *A*1, *A*<sup>2</sup> and *A*3. For *A*1, *A*<sup>2</sup> and *A*3, use the stability index *γ<sup>i</sup>* and Table 9 equivalent time constant *τ* to calculate the CDM control polynomial parameters. Table 9 shows the equivalent time constant *τ* and CDM control polynomial parameter values.

**Table 9.** The equivalent time constant *τ* and CDM control polynomial parameter values.


Using the CDM control polynomial parameters in Table 9 to control *A*1, *A*<sup>2</sup> and *A*3, the results are shown in Figure 20.

**Figure 20.** CDM control *A*<sup>1</sup> , *A*<sup>2</sup> and *A*<sup>3</sup> step response.

Figure 21 is the result of inserting the compensator (38) before the controlled object (37) and using the CDM parameters in Table 9 for control.


*Time(s)*

**Figure 21.** Use this method to control the MIMO system (37) step response.

#### **6. Conclusions**

This paper proposes a multivariable system controller design method based on the CDM and analyzes the controller's suppression effect on measurement noise based on the CDM. The decoupling design is realized by designing the compensator in the frequency domain, and the compensator parameters are optimized through PSO. At the same time, use statistical tests to compare four evolutionary algorithms, including PSO, GA, SFLA, CS, to prove the advantages of PSO. After decoupling, the open-loop transfer function of the system is complex. Therefore, the controller structure design and parameter tuning are based on CDM. Finally, simulation experiments are carried out for four unique control targets. The results show that the decoupling effect of the MIMO system is good, and the designed system can take into account stability, response characteristics, and robustness at the same time, which confirms the effectiveness of the method.

**Author Contributions:** All authors discussed and agreed on the idea and scientific contribution. Conceptualization, K.L., F.M. and S.M.; methodology, K.L. and F.M.; software, S.M. and C.W.; validation, K.L., F.M. and S.M.; writing—original draft preparation, K.L. and S.M.; writing—review and editing, F.M., S.M. and C.W.; funding and supervision by F.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (12162007), Natural Science Foundation of Hebei Province (F2019501012).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Observer Based Multi-Level Fault Reconstruction for Interconnected Systems**

**Mei Zhang <sup>1</sup> , Boutaïeb Dahhou <sup>2</sup> , Qinmu Wu <sup>1</sup> and Zetao Li 1,\***


**Abstract:** The problem of local fault (unknown input) reconstruction for interconnected systems is addressed in this paper. This contribution consists of a geometric method which solves the fault reconstruction (FR) problem via observer based and a differential algebraic concept. The fault diagnosis (FD) problem is tackled using the concept of the differential transcendence degree of a differential field extension and the algebraic observability. The goal is to examine whether the fault occurring in the low-level subsystem can be reconstructed correctly by the output at the high-level subsystem under given initial states. By introducing the fault as an additional state of the low subsystem, an observer based approached is proposed to estimate this new state. Particularly, the output of the lower subsystem is assumed unknown, and is considered as auxiliary outputs. Then, the auxiliary outputs are estimated by a sliding mode observer which is generated by using global outputs and inverse techniques. After this, the estimated auxiliary outputs are employed as virtual sensors of the system to generate a reduced-order observer, which is caplable of estimating the fault variable asymptotically. Thus, the purpose of multi-level fault reconstruction is achieved. Numerical simulations on an intensified heat exchanger are presented to illustrate the effectiveness of the proposed approach.

**Keywords:** local unknown input; interconnected system; local reconstrucability; global reconstrucability; reduce-order uncertain observer

### **1. Introduction**

Increasing developments in modern technologies have led to a high complexity of control systems. Thus, either due to physical or analytical purpose, modern control systems are frequently tackled as interconnected systems. Potential faults in interconnected systems have also become inevitable and increasingly complex since faults of the interconnected system can be represented at either the local subsystem level, or at the global system level with the whole system in view, considering faults such as unknown external disturbance, or parameter variations. Faults at either level may not only cause the decline of the performance of both the global system or the local subsystem, but also may trigger a series of fault subsystems. Compared with residual fault diagnosis methodologies, fault reconstruction is capable of identifying the size, location, and dynamics of the fault. In addition, the fault can usually be regarded as an unknown input to the system. The problem of reconstructing the inaccessible inputs from the available measurements is therefore motivated and has attracted remarkable interest in the last decades. Particularly, reconstruction of unknown or inaccessible inputs from noise or indirect measures is very common in many real industrial situations.

In the case of fault diagnosis and unknown input reconstruction for interconnected systems, centralized structure-based fault reconstruction approaches are well investigated, e.g., in Refs. [1–19]. A significant approach of FD and FR for dynamic systems are the

**Citation:** Zhang, M.; Dahhou, B.; Wu, Q.; Li, Z. Observer Based Multi-Level Fault Reconstruction for Interconnected Systems. *Entropy* **2021**, *23*, 1102. https://doi.org/ 10.3390/e23091102

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 16 July 2021 Accepted: 18 August 2021 Published: 25 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

observer based methodologies [1–7], with differential geometry-based techniques also representing another attractive method [8–13]. Investigations aimed at solving problems of FD and FR of nonlinear dynamic systems via algebraic and differential techniques can be found in studies such as Refs. [12–17]. These approaches are normally applications of dynamic inversion to achieve the purpose of FD and FR, just as the familiar idea of dynamic inversion is used in the control problem of dynamic systems. Basic notions of this kind of analysis method include the concepts of input reconfigurability [12], left invertibility of dynamic system [9], relative degree of dynamic system and zero dynamics [16].

However, the application of individual system-based methodologies is mainly limited. First, the identification of internal dynamics at local level is incomplete; second, it lacks the dynamics information of the global system. In real applications, it is rather difficult to utilize a centralized scheme to solve the problem of fault reconfiguration in interconnected systems. Luckily, due to advances in computing and communications, it is becoming increasingly popular to directly adopt hierarchical, decentralized, and distributed schemes to deal with fault reconfiguration [20]. In fact, naturally, the architecture of the underlying subsystem is decentralized or distributed, which means that it is necessary to develop distributed FD and FR frameworks. In other words, local fault diagnosis and reconstruction should be performed [21–39]. However, since the interconnected systems are becoming increasingly complex, the problem of system fault reconstruction has also become increasingly difficult, especially problems related to fault propagation, due to the fact that faults occurring in one subsystem influence adjacent subsystems. Therefore, in order to better understand the fault propagation problems, there is research concerning both local and global systems such as in Refs. [20–37]. An important method is to propose a local observer for individual subsystems using its own input and output measurements. All local observers work together to achieve the purpose of estimation and diagnosis of the global system. In this way, the intensive traditional observer design method, based on a single dynamic system, can be employed, such as the high gain observer in Ref. [24], sliding mode observer in Ref. [32], adaptive observer in Ref. [23], etc.

However, the operation of distributed FD and FR approaches greatly relies on reliable information about the full measurement of all subsystems. Such a dependence makes theses methodologies much more challenging, since online measurements available for each subsystem are either difficult to obtain or are inaccurate and or expensive. The matching conditions may be truly too harsh to be satisfied for many physical systems, which makes these methods for unknown input reconstruction not available.

Therefore, it is of great importance to solve the above-mentioned difficulties when analyzing the interconnected systems, which has also motivated us to carry out this research. In this work, system inversion and observer design techniques are combined and extended, aimed at tackling multi-level faults (unknown input) reconstruction problems of the interconnected system. A distributed fault reconstruction scheme is developed and the propagation of the fault effects among interconnected subsystems is investigated. The initial objective is to recognize unknown inputs at the low-level subsystem by using information provided at the global level. A remarkable benefit is that it is capable of reconstructing the system state and local fault signals simultaneously, including incipient faults, for which the fault is considered as an unknown input uncertainty. By introducing the fault as an additional state of the low subsystem, an extended reduced-order observer is developed to produce an estimation of this state. In particular, the output of the lower subsystem is assumed unknown, and is considered as auxiliary output. An inversebased high order sliding mode observer is developed, aimed at estimating the auxiliary output and its derivatives via measurements of global system. By using this estimation information of auxiliary output, an extended reduced-order observer is generated, aimed at reconstructing the unknown inputs locally. The applicable system categories of this method include systems that depend on polynomial input and its time derivatives. Encouraging numerical simulation results confirm the effectiveness of the proposed multi-level fault reconstruction approach.

The rest of this article is organized as follows: in Section 2, condition of fault reconstructability both locally and globally is given, while in Section 3, a multi-level fault reconstruction scheme for an interconnected system is proposed. First, at the local level, by introducing an auxiliary output to replace its inaccessible output, an extended reduce-order observer is designed to estimate both the states and the fault signals. Second, in order to give an estimation of the auxiliary output and its derivatives, a high order sliding mode observer is introduced. Finally, by gathering all the estimates from both observers, the local fault reconstruction via global information is achieved. In Section 4, the effectiveness of the proposed approach is illustrated by numerical simulations implemented on an intensified heat exchanger. Conclusions and further works are discussed in Section 5. structability both locally and globally is given, while in Section 3, a multi-level fault reconstruction scheme for an interconnected system is proposed. First, at the local level, by introducing an auxiliary output to replace its inaccessible output, an extended reduceorder observer is designed to estimate both the states and the fault signals. Second, in order to give an estimation of the auxiliary output and its derivatives, a high order sliding mode observer is introduced. Finally, by gathering all the estimates from both observers, the local fault reconstruction via global information is achieved. In Section 4, the effectiveness of the proposed approach is illustrated by numerical simulations implemented on an intensified heat exchanger. Conclusions and further works are discussed in Section 5.

method include systems that depend on polynomial input and its time derivatives. Encouraging numerical simulation results confirm the effectiveness of the proposed multi-

The rest of this article is organized as follows: in Section 2, condition of fault recon-

*Entropy* **2021**, *23*, x FOR PEER REVIEW 3 of 18

#### **2. Model Description and Problem Formulation 2. Model Description and Problem Formulation**

level fault reconstruction approach.

Analytically, the system can be decomposed into several subsystems, and different control or supervision algorithms can then be developed from both local and global viewpoints, as shown in Figure 1. Analytically, the system can be decomposed into several subsystems, and different control or supervision algorithms can then be developed from both local and global viewpoints, as shown in Figure 1.

**Figure 1.** Interconnected system structure. **Figure 1.** Interconnected system structure.

The important aspect is to develop models of individual subsystems that can describe cause ωଶ(t) and effect y(t) relationships between the 1st and 2nd subsystems. In this The important aspect is to develop models of individual subsystems that can describe cause ω2(t) and effect y(t) relationships between the 1st and 2nd subsystems. In this case, estimation technologies on states and parameters are capable.

case, estimation technologies on states and parameters are capable. It is supposed that the 1st subsystem can be described with the following state affine form by (1): It is supposed that the 1st subsystem can be described with the following state affine form by (1):

$$\sum\_{\mathbf{1st}} \colon \begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{f}\_1(\mathbf{x}\_1) + \mathbf{g}\_1(\mathbf{x}\_1)\mathbf{u}\_1\\ \mathbf{y} = \mathbf{h}\_1(\mathbf{x}\_1, \mathbf{u}\_1) \end{cases} \tag{1}$$
 
$$\begin{bmatrix} \mathbf{x}\_1 \ \mathbf{x}\_2 \ \mathbf{x}\_3 \ \vdots \ \mathbf{x}\_n \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \end{bmatrix} \tag{2}$$

<sup>y</sup> = hଵ(xଵ, uଵ) (1) where xଵ ∈ ℜ<sup>୬</sup> ∈ Μ is the state of the 1st subsystem, uଵ ∈ ℜ୫ is the input of 1st subsystem, which represents elements such as the control input, reference signal, etc., and is also the output of the 2nd subsystem; y∈ℜ୮ is output of the 1st subsystem, as well as the overall system. fଵ, gଵ are smooth vector fields on Μ. xଵ(t) = xଵ is the initial condition. where <sup>x</sup><sup>1</sup> ∈ <<sup>n</sup> <sup>∈</sup> <sup>M</sup> is the state of the 1st subsystem, <sup>u</sup><sup>1</sup> ∈ <<sup>m</sup> is the input of 1st subsystem, which represents elements such as the control input, reference signal, etc., and is also the output of the 2nd subsystem; <sup>y</sup> ∈ <<sup>p</sup> is output of the 1st subsystem, as well as the overall system. f1, g<sup>1</sup> are smooth vector fields on M. x1(t0) = x<sup>10</sup> is the initial condition. In addition, it is assumed that u<sup>1</sup> is inaccessible and can be recovered through available measures of the global system.

In addition, it is assumed that uଵ is inaccessible and can be recovered through available measures of the global system. Consider the following nonlinear systems for the 2nd subsystem subject to either actuator or sensor faults ω<sup>2</sup> by (2):

$$\sum\_{\mathbf{2nd}} : \begin{cases} \dot{\mathbf{x}}\_2 = f\_2(\mathbf{x}\_2, \mathbf{u}\_\prime \,\mathrm{w}\_2) \\ \mathbf{u}\_1 = h\_2(\mathbf{x}\_2, \mathbf{u}\_\prime \,\mathrm{w}\_2) \end{cases} \tag{2}$$

∑ଶ୬ୢ : ൜ xሶ uଵ = hଶ(xଶ, u, ωଶ) (2) where the state is represented by xଶ ∈ ℛ୬; u ∈ ℛ୪ is the input of the 2nd subsystem, as well as the overall system; uଵ ∈ R୫ is the output; ωଶ = (ωଶଵ, ωଶଶ, …,ωଶ୩)∈ℛ<sup>୩</sup> represents the either actuator or sensor faults of the system. fଶ, hଶ are assumed to be analytical vector functions. Specifically, each fault is related to the variables of a specific device and where the state is represented by <sup>x</sup><sup>2</sup> ∈ R<sup>n</sup> ; <sup>u</sup> ∈ R<sup>l</sup> is the input of the 2nd subsystem, as well as the overall system; is the output; <sup>ω</sup><sup>2</sup> <sup>=</sup> (ω21, <sup>ω</sup>22, . . . , <sup>ω</sup>2k) ∈ R<sup>k</sup> represents the either actuator or sensor faults of the system. f2, h<sup>2</sup> are assumed to be analytical vector functions. Specifically, each fault is related to the variables of a specific device and subcomponent. Each of these faults implies an abnormal physical change, such as sticking, leakage or actuator blockage.

subcomponent. Each of these faults implies an abnormal physical change, such as sticking, leakage or actuator blockage. In this way, the studied interconnected system is composed of the two local subsystems ∑1st and ∑2nd; for the global system, the vector u and y represent its input and output, respectively.

For the interconnected systems described by (1) and (2), the main purpose of the study is to reconstruct fault vector ω<sup>2</sup> at the local level using information at the global level; meanwhile, performance supervision of the global system, as well as individual subsystems, is obliged. A significant objective is to examine whether the unknown inputs ω<sup>2</sup> at local 2nd subsystem can be reconstructed uniquely by output of the 1st subsystem at a global level, given initial states. The initial task is to propose conditions under which both the unknown input and initial state of a known model can be determined from output measurements. For that, the concept of the differential transcendence degree of a differential field extension and the algebraic observability concept of the variable are employed. An interconnected observer based scheme is then developed and analyzed to perform local fault variables reconstruction. A reduced-order uncertainty observer combined with a high order sliding mode observer is developed to achieve this purpose. Finally, the performance of a traditional distributed UIO approach and the proposed multi-level FR approach are compared in detail through numerical simulations, which are presented in an intensified heat exchanger.

#### **3. On Condition of Fault Reconstructability Locally and Globally**

In this section, the assumptions and main results on condition of fault reconstructability locally and globally are discussed. An initial task is to prove that the fault vector ω<sup>2</sup> at local level and output vector y at the global are implicitly causal. Moreover, it is also necessary to provide condition to guarantee that local fault impacts on global information are distinguishable. Basic notions are introduced first, and related concepts can be found in Refs. [16,30].

#### *3.1. Fault Reconstructability Condition*

To cope with the problem, faults are regarded as local unknown inputs of the interconnected system. Thus, local faults' reconstructability can be treated equivalent with the capability of reconstructing unknown inputs at the local level. In solving the problem of input reconstruction, the primary task is to evaluate the observability of input, so as to distinguish whether the change of input of dynamic systems can be reflected in the change of the output. In order to ensure that the local unknown input can be reconstructed from the global outputs by means of a finite number of ordinary differential equations, there are conditions involving observability and reconstructionability to be met.

From Ref. [30], if any unknown variable x in a dynamic satisfies a differential algebraic equation, the coefficients κ of the equation are greater than in the components of u and y, and the number of its derivatives is finite, then the x is algebraically observable with respect to κ(u1, ω2). Any dynamic with output y is said to be algebraically observable if, and only if, any variable has this property. In addition, a fault (unknown input) is defined as a transcendent element over κ(u), in which case a faulty system can be viewed as extension of differential transcendence with both fault (unknown input) and its time derivatives. Motivated by this, fault observability of an interconnected system can be defined from multi-level viewpoints:

**Definition 1. (Local Algebraic observability).** *For subsystem (2), a fault element ω*<sup>2</sup> ∈ *κ*(*u*, *ω*2) *is said to be locally algebraically observable if ω*<sup>2</sup> *satisfies a differential algebraic equation with coefficients over κ*(*u*, *u*1, *ω*2).

**Definition 2. (Global Algebraic observability).** *For interconnected systems depicted by (1) and (2), a fault element ω*<sup>2</sup> ∈ *κ*(*u*, *ω*2) *is said to be globally algebraically observable if ω*<sup>2</sup> *satisfies a differential algebraic equation with coefficients over κ*(*u*, *y*, *ω*2).

Typically, the problem of observability and left invertibility of dynamic system can be equivalently tackled, while the property of left over invertibility usually means a recontructability of the system input from the output. From Refs. [16,30], if invertibility of the interconnected system, denoted by (1) and (2), can be insured, then it is capable of obtaining the fault element ω2i(i = 1, . . . , k) globally from information of overall system output y. Equivalently, if subsystems depicted by (1) and (2) are invertible, respectively, then their inputs and unknown inputs vectors u<sup>1</sup> and ω<sup>2</sup> can be expressed locally by their corresponding local measured outputs y and u1. To accomplish the aims, the central issue is to provide conditions which can guarantee invertibility of both individual systems and the interconnected system. Luckily, this has been discussed in previous paper in Ref. [30]. It can be seen that a differential output rank is defined to determine invertibility of single dynamic system, while invertibility of all the subsystems are the necessary and sufficient condition for ensuring invertibility of the interconnected system.

**Definition 3. (Local reconstructability).** *For system (2), it is said to be locally reconstructable if the system is invertible. In this way, it is capable of estimating the unknown input ω*<sup>2</sup> *from local system information u and u*1.

For the concept of algebraic observability, it is required that each fault component can be written as the solution of the polynomial equation in ω2i and the finite number of time derivatives of u and u<sup>1</sup> with coefficients in k.

$$\mathcal{H}(\mathbf{w}\_{2\mathbf{i}\prime}\mathbf{u}, \dot{\mathbf{u}}, \dots, \mathbf{u}\_{1\prime}\dot{\mathbf{u}}\_{1\prime}\dots) = \begin{array}{c} \text{0} \tag{3}$$

**Definition 4. (Global reconstructability).** *For the interconnected nonlinear system described by (1) and (2), it is said to be globally reconstructable if the interconnected system is invertible, in this way, it is capable of estimating the unknown input ω*<sup>2</sup> *from global system information u and y*.

In other words, it is required that the local unknown input vector can be expressed as a solution of a polynomial equation in ω2i and the finite number of time derivatives of u and y with coefficients in k.

$$\mathcal{H}(\omega\_{2i}, \mathbf{u}, \dot{\mathbf{u}}, \dots, \mathbf{y}, \dot{\mathbf{y}}, \dots) = \mathbf{0} \tag{4}$$

As mentioned before, requirements of health measurement of all the subsystems increases the difficulty the procedure. In this work, u<sup>1</sup> is supposed to be inaccessible. Therefore, it is also critical for estimating a reliable u<sup>1</sup> and to ensure that reconstructed u<sup>1</sup> has a one-to-one relationship with fault vector ω2i. If it can prove that the reconstructed uˆ <sup>1</sup> is converged to u<sup>1</sup> with acceptable accuracy, then by substituting u<sup>1</sup> as its estimates uˆ <sup>1</sup> in (3), the fault vector (ω2*<sup>i</sup>* , i = 1, . . . , k) is capable of obtaining by a solution of a polynomial equation in ω2i and the finite number of time derivatives of u and uˆ <sup>1</sup>, with coefficients in k.

$$\mathcal{H}\left(\omega\_{2i}, \mathbf{u}, \dot{\mathbf{u}}, \dots, \dot{\mathbf{u}}\_{1}, \dot{\mathbf{u}}\_{1}, \dots\right) = \begin{array}{c} \mathbf{0} \\ \end{array} \tag{5}$$

In summary, if ω<sup>2</sup> is algebraically observable with respect to u and y, then ω<sup>2</sup> is said to be reconstructable. If, and only if, the interconnected system is invertible both locally and globally, the task of reconstruction of the local unknown input vector ω<sup>2</sup> from global measures, y can be achieved. That is, if the overall interconnected system is invertible, then the impacts of the unknown input ω<sup>2</sup> on the global system, output y is distinguished.

#### *3.2. Minimum Number of Measurements and Reconstructable Unknown Inputs*

In this work, accessible measurements are of great importance when implementing the proposed FR method. Therefore, the minimum number of measurements is an essential prerequisite for determining whether a fault in the dynamic system is reconstructable or not. This problem is also related to the problem of invertibility of the dynamic system. According to [16], in order to insure invertibility of the system, the differential output rank of the system should equal to the number of the fault candidates. The differential output rank is also defined as the maximum number of outputs associated with differential polynomial equations with coefficients over K (independent of x). It means that the available measurable outputs of the system must be greater than or equal to the possible faults.

**Remark 1.** *For a subsystem described by (1), invertibility cannot be guaranteed if the available outputs are less than the inputs. Conversely, if there are more outputs than inputs, then the redundant outputs are unneeded*.

**Remark 2.** *For subsystem in failure mode depicted in (2), invertibility of the system cannot be guaranteed if the available outputs is less than the possible faults.*

**Proposition 1.** *From remarks 1 and 2, the simultaneous reconstructable failure number (ω*2*<sup>i</sup>* , *i* = 1, . . . , *k) depends on the number of the measurable outputs.*

**Remark 3.** *For interconnected system depicted in (1) and (2), the minimum number of available measurements predetermined the reconstructable unknown inputs, thus, equal dimensions of both subsystems and the whole interconnected system is more meaningful.*

#### **4. Observer Design for Unknown Input Reconstruction**

As mentioned before, existing observers for fault reconstruction are mainly focused on individual systems. Although there is some research concerning both local and global subsystems, the associated match criteria are usually overly strict to be satisfied in real industrial applications. In order to cope with this difficulty, this work is concerned with the challenges by deriving a fault reconstruction method based on some auxiliary outputs. The architecture of the proposed multi-level fault reconstruction method is shown in Figure 2. *Entropy* **2021**, *23*, x FOR PEER REVIEW 7 of 18

**Figure 2.** Structure of the proposed local unknown input reconstructor. **Figure 2.** Structure of the proposed local unknown input reconstructor.

*4.1. Asymptotic Reduced-Order Observer Design with Auxiliary Output*  Considered system (2), the unknown fault vector ωଶ(ݐ (is assimilated as an extra state of the system with uncertain dynamics. It is expressed according to the states, the unknown inputs (faults) and the known inputs of the system. The dynamics of this new state are unknown. The original system is then converted into an extended system where the dynamics of the extra state are unknown, and it is assumed to be bounded. The original problem is then an observation problem, where the aim is to observe this extra state of the system. The new extended system is given by: ቐ xሶ <sup>ଶ</sup> = fଶ(xଶ, u, ωଶ) ωሶ <sup>ଶ</sup> ሶ = ψ(xଶ, u, ωଶ) uଵ = hଶ(xଶ, u, ωଶ) (6) The main idea is based on distributed observer design, since distribution resources of dynamic systems are said to be particularly effective for estimation of interconnected systems, due to the fact that they can update internal states using local measurement outputs. However, the significant challenge here is the inaccessible of the interconnection, which is the input of the first subsystem and the output of second subsystem. To cope with this difficulty, first, in order to reconstruct local unknown fault of the first subsystem, the interconnection is extended as an additional state of the first subsystem, an asymptotic reduced-order observer is proposed for the first subsystem, using local input and output measurement information. Then, it is considered the problem that local output is not available directly. An inverse and sliding mode observer based estimator for the second subsystem is then designed to generate an estimation of the local output, and the estimated auxiliary output is applied to the reduced-order observer to replace its measured output. A kind of multi-level fault reconstruction is achieved by gathering estimation of these two observers.

where ߰(xଶ, u, ωଶ) is a bounded uncertain function, ωଶ(ݐ (is algebraically observable

known. Therefore, in order to estimate the unknown input variable ωଶ, a proportional reduced-order uncertainty observer using differential algebraic techniques is applied to

where ωෝଶ denotes the estimate of the unknown input vector ωଶ(ݐ (and the convergence

ωଶ୧ = α୧uሶ

where ߙ is a constant vector and ߚ)u, uଵ) is a bounded function.

order unknown input observer exists, for system (6) it can be written as:

Normally, time derivatives of the output are included in the algebraic equation of the unknown input vector, which may enhance computation burden and cause significant computation error even under minor measurement noise, then it is practical and worthwhile to employ an auxiliary variable rather than the computations of the time derivatives. If the unknown fault vector is algebraically observable and can be written in the fol-

If a C1 real-valued function ߛ exists, such that a proportional asymptotic reduced-

**Theorem 1.** *Supposed that the auxiliary output vector* ݑଵ *is available for measurement, then the* 

An asymptotic reduced-order observer with a corresponding quadratic-type Lya-

ωሶ ଶ୧ = K୧(ωଶ୧ − ωෝଶ୧), 1 ≤ i ≤ ε (7)

<sup>ଵ</sup> + β୧(u, uଵ) (8)

punov function can be constructed for system (6):

of the observer is determined by K୧.

lowing form:

*system* 

the fault estimation is constructed to overcome the above problem.

#### *4.1. Asymptotic Reduced-Order Observer Design with Auxiliary Output*

Considered system (2), the unknown fault vector ω2(t) is assimilated as an extra state of the system with uncertain dynamics. It is expressed according to the states, the unknown inputs (faults) and the known inputs of the system. The dynamics of this new state are unknown. The original system is then converted into an extended system where the dynamics of the extra state are unknown, and it is assumed to be bounded. The original problem is then an observation problem, where the aim is to observe this extra state of the system.

The new extended system is given by:

$$\begin{cases}
\dot{\mathbf{x}}\_2 = \mathbf{f}\_2(\mathbf{x}\_2, \mathbf{u}, \omega\_2) \\
\dot{\omega}\_2 = \psi(\mathbf{x}\_2, \mathbf{u}, \omega\_2) \\
\mathbf{u}\_1 = \mathbf{h}\_2(\mathbf{x}\_2, \mathbf{u}, \omega\_2)
\end{cases} \tag{6}$$

where ψ(x2, u, ω2) is a bounded uncertain function, ω2(t) is algebraically observable over κ(u, u1). It should be noted that a typical structure observer, similarly to a classic Luenberger observer, is not available in the literature because the term ψ(x2, u, ω2) is unknown. Therefore, in order to estimate the unknown input variable ω2, a proportional reduced-order uncertainty observer using differential algebraic techniques is applied to the fault estimation is constructed to overcome the above problem.

An asymptotic reduced-order observer with a corresponding quadratic-type Lyapunov function can be constructed for system (6):

$$
\dot{\mathfrak{w}}\_{\text{2i}} = \mathbb{K}\_{\text{i}} (\mathfrak{w}\_{\text{2i}} - \hat{\mathfrak{w}}\_{\text{2i}}), \; 1 \le \text{i} \le \varepsilon \tag{7}
$$

where ωˆ <sup>2</sup> denotes the estimate of the unknown input vector ω2(t) and the convergence of the observer is determined by K<sup>i</sup> .

Normally, time derivatives of the output are included in the algebraic equation of the unknown input vector, which may enhance computation burden and cause significant computation error even under minor measurement noise, then it is practical and worthwhile to employ an auxiliary variable rather than the computations of the time derivatives.

If the unknown fault vector is algebraically observable and can be written in the following form: .

$$
\omega\_{2\mathbf{i}} = \alpha\_{\mathbf{i}} \dot{\mathbf{u}}\_{\mathbf{l}} + \beta\_{\mathbf{i}}(\mathbf{u}, \mathbf{u}\_{\mathbf{l}}) \tag{8}
$$

where α<sup>i</sup> is a constant vector and β*<sup>i</sup>* (u, u1) is a bounded function.

If a C<sup>1</sup> real-valued function γ exists, such that a proportional asymptotic reduced-order unknown input observer exists, for system (6) it can be written as:

**Theorem 1.** *Supposed that the auxiliary output vector u*<sup>1</sup> *is available for measurement, then the system*

$$\begin{cases}
\dot{\mathbf{v}}\_{\text{i}} = -\mathbf{K}\_{\text{i}}\mathbf{\dot{y}}\_{\text{i}} + \mathbf{K}\_{\text{i}}\boldsymbol{\beta}\_{\text{i}}(\mathbf{u}\_{\text{i}}\mathbf{u}\_{\text{1}}) - \mathbf{K}\_{\text{i}}^{2}\mathbf{a}\_{\text{i}}\mathbf{u}\_{\text{1}} \\
\dot{\mathbf{w}}\_{\text{2i}} = \boldsymbol{\gamma}\_{\text{i}} + \mathbf{K}\_{\text{i}}\mathbf{a}\_{\text{i}}\mathbf{u}\_{\text{1}}
\end{cases} \tag{9}$$

*is a proportional asymptotic reduced-order unknown input observer for system (6).*

The observer (9) can be implemented under assumption that u<sup>1</sup> is measured. However, in our design u<sup>1</sup> is assumed to be unavailable, it is therefore obliged to produce an estimate of the auxiliary output to substitute the measured one.

**Remark 4.** *By optimizing the observer gain, the optimum tradeoff between the speed of state reconstruction and the robustness to model uncertainty is realized. In this way, the designed observer is not only capable of recovering the system state but also of minimizing the impacts of the measurement noise.*

**Remark 5.** *It is worth noting that better robustness can be achieved by adding integral action to the proportional asymptotically reduced-order fault observer during the implementation of the observer.*

#### *4.2. Auxiliary Output Estimation*

The premise of implementation of Theorem 1 is that the auxiliary output u<sup>1</sup> is measurable. It is therefore first required to reconstruct a smooth function of this auxiliary output together with its derivatives from the output data records. To deal with the actual situation, a system inverse-based high order sliding mode observer is considered to accurately estimate the auxiliary output vector u<sup>1</sup> and the derivative in the subsystem (1). If this estimation can be well achieved, then the estimated u<sup>1</sup> in (9) and its derivatives can be utilized to substitute u<sup>1</sup> in (1) to complete the purpose of unknown input reconstruction. In order to achieve this purpose, we first need to construct a dynamic system, which is indeed the realization of the inverse of the original system.

Specifically, for an invertible nonlinear system with the form of (1), a finite relative degree of the output r<sup>i</sup> , i = 1, . . . , m is first defined as the smallest integer as follows:

$$\mathbf{L\_{\mathcal{G}\_1}} \mathbf{L\_{\mathfrak{f}\_1}^{\mathbf{r}\_1 - 1}} \mathbf{h\_i}(\mathbf{x}\_1) \;= \begin{bmatrix} \mathbf{L\_{\mathcal{G}\_{11}}} \mathbf{L\_{\mathfrak{f}\_1}^{\mathbf{r}\_1 - 1}} \mathbf{h\_i}(\mathbf{x}\_1) \mathbf{L\_{\mathfrak{g}\_{12}}} \mathbf{L\_{\mathfrak{f}\_1}^{\mathbf{r}\_1 - 1}} \mathbf{h\_i}(\mathbf{x}\_1) \; \dots \mathbf{L\_{\mathfrak{g}\_{1m}}} \mathbf{L\_{\mathfrak{f}\_1}^{\mathbf{r}\_1 - 1}} \mathbf{h\_i}(\mathbf{x}\_1) \end{bmatrix} \neq \begin{bmatrix} 0, \; 0, \dots, 0 \end{bmatrix}$$

Then, by calculating expressions for their derivatives, one gets:

$$
\begin{bmatrix}
\mathbf{y}\_{1}^{(\mathbf{r}\_{1})} \\
\vdots \\
\mathbf{y}\_{\mathbf{m}}^{(\mathbf{r}\_{\mathbf{m}})}
\end{bmatrix} = \begin{bmatrix}
\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{r}\_{1}}\mathbf{h}\_{11}(\mathbf{x}\_{1}) \\
\vdots \\
\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{m}}\mathbf{h}\_{1\mathbf{m}}(\mathbf{x}\_{1})
\end{bmatrix} + \begin{bmatrix}
\mathbf{L}\_{\mathbf{g}\_{11}}\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{r}\_{1}-1}\mathbf{h}\_{11}(\mathbf{x}\_{1}) & \dots & \mathbf{L}\_{\mathbf{g}\_{1m}}\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{r}\_{1}-1}\mathbf{h}\_{11}(\mathbf{x}\_{1}) \\
\vdots & \dots & \dots \\
\mathbf{L}\_{\mathbf{g}\_{11}}\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{r}\_{m}-1}\mathbf{h}\_{1m}(\mathbf{x}\_{1}) & \dots & \mathbf{L}\_{\mathbf{g}\_{1m}}\mathbf{L}\_{\mathbf{f}\_{1}}^{\mathbf{r}\_{m}-1}\mathbf{h}\_{1m}(\mathbf{x}\_{1})
\end{bmatrix} \mathbf{u}\_{1} \tag{10}
$$

Although the algebraic polynomial (10) is based on a system inversion, and has already been able to compute u1, the requirements of calculating of the successive derivative of the output may burden the reconstruction process. In practical applications especially, the measurements often subject to noise, it may result in large overshoot, even failure. An inverse-based high order sliding mode observer is then generated to tackle this problem.

Define the following change of the coordinates:

$$\boldsymbol{\xi}\_{\text{i}} = \left[\boldsymbol{\mathsf{L}}\_{\text{i}}^{1}, \boldsymbol{\mathsf{L}}\_{\text{i}}^{2}, \dots, \boldsymbol{\mathsf{L}}\_{\text{i}}^{\text{n}}\right]^{\mathrm{T}} = \left[\boldsymbol{\mathsf{o}}\_{\text{i}}^{1}(\mathbf{x}\_{\text{l}}), \, \boldsymbol{\mathsf{o}}\_{\text{i}}^{2}(\mathbf{x}\_{\text{l}}), \dots, \boldsymbol{\mathsf{o}}\_{\text{i}}^{\text{n}}(\mathbf{x}\_{\text{l}})\right]^{\mathrm{T}} = \left[\mathbf{h}\_{\text{li}}(\mathbf{x}\_{\text{l}}), \, \mathbf{L}\_{\text{l}}\mathbf{h}\_{\text{li}}(\mathbf{x}), \dots, \boldsymbol{\mathsf{L}}\_{\text{i}}^{\text{n}-1}\mathbf{h}\_{\text{li}}(\mathbf{x})\right]^{\mathrm{T}} \text{ i } = \mathbf{1}, \dots, \mathbf{m}\_{\text{i}}^{\text{n}}$$

Next, to construct:

$$\begin{array}{c} \mathbf{y\_i} = \boldsymbol{\mathsf{L}}\_{\mathbf{i}}^{1} \dot{\boldsymbol{\xi}}\_{\mathbf{i}}^{\mathbf{j}} = \boldsymbol{\mathsf{L}}\_{\mathbf{i}}^{\mathbf{j}+1}; \; 1 \le \mathbf{j} \le \mathbf{r\_i} - \mathbf{1} \\\ \dot{\boldsymbol{\xi}}\_{\mathbf{i}}^{\mathbf{r\_i}} = \boldsymbol{\mathsf{L}}\_{\mathbf{f\_1}}^{\mathbf{r\_i}} \mathbf{h\_{1i}}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\xi}, \boldsymbol{\eta}) + \boldsymbol{\sum}\_{\mathbf{j}=1}^{\mathbf{m}} \mathbf{L}\_{\mathbf{g\_{1i}}} \mathbf{L}\_{\mathbf{f\_1}}^{\mathbf{r\_i}-1} \mathbf{h\_{1i}}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\xi}, \boldsymbol{\eta})) \text{ } \mathbf{u\_{1j}}; \; \mathbf{j} = \mathbf{r\_i} \end{array}$$

The expression of input vector u<sup>1</sup> is then issued:

$$\mathbf{u}\_{\rm l} = \mathbf{A} \left( \boldsymbol{\Phi}^{-1} (\boldsymbol{\xi}, \boldsymbol{\eta}) \right)^{-1} \left( \begin{bmatrix} \boldsymbol{\xi}\_{1}^{(\rm r\_{l})} \\ \vdots \\ \boldsymbol{\xi}\_{\rm m}^{(\rm r\_{m})} \end{bmatrix} - \begin{bmatrix} \mathbf{L}\_{\rm f1}^{\rm r\_{l}} \mathbf{h}\_{11} (\boldsymbol{\Phi}^{-1} (\boldsymbol{\xi}, \boldsymbol{\eta}) \\ \vdots \\ \mathbf{L}\_{\rm f1}^{\rm r\_{m}} \mathbf{h}\_{\rm lm} (\boldsymbol{\Phi}^{-1} (\boldsymbol{\xi}, \boldsymbol{\eta}) \end{bmatrix} \right) \tag{11}$$

The inversed based sliding mode observer can then be designed as follows:

$$\begin{array}{rcl} \mathfrak{H}\_{\mathbf{i}} &=& \mathfrak{E}\_{\mathbf{i}}^{1} \dot{\mathfrak{E}}\_{\mathbf{i}}^{\dot{\mathfrak{j}}} = \mathfrak{E}\_{\mathbf{i}}^{\mathbf{j}+1} + \lambda\_{\mathbf{i}}^{\dot{\mathfrak{j}}} |\mathfrak{j}\_{\mathbf{i}} - \mathbf{y}\_{\mathbf{i}}|^{1/2} \text{sgn}(\mathfrak{y}\_{\mathbf{i}} - \mathbf{y}\_{\mathbf{i}}); \ 1 \le \mathbf{j} \le \mathbf{r}\_{\mathbf{i}} - 1\\ & & \dot{\mathfrak{E}}\_{\mathbf{i}} = \lambda\_{\mathbf{i}}^{\mathbf{r}\_{\mathbf{i}}} |\mathfrak{y}\_{\mathbf{i}} - \mathbf{y}\_{\mathbf{i}}|^{\frac{1}{2}} \text{sgn}(\mathfrak{y}\_{\mathbf{i}} - \mathbf{y}\_{\mathbf{i}}); \ \mathbf{j} = \mathbf{r}\_{\mathbf{i}} \end{array} \tag{12}$$

Finally, estimation of ξ<sup>i</sup> is achieved finitely:

$$\begin{array}{rcl} \hat{\xi}\_{\text{i}} &=& \left[ \hat{\xi}\_{\text{i}}^{1}, \hat{\xi}\_{\text{i}}^{2}, \dots, \hat{\xi}\_{\text{i}}^{\text{r}} \right]^{\text{T}} = \left[ \hat{\Phi}\_{\text{i}}^{1}(\mathbf{x}\_{\text{1}}), \hat{\Phi}\_{\text{i}}^{2}(\mathbf{x}\_{\text{1}}), \dots, \hat{\Phi}\_{\text{i}}^{\text{r}\_{\text{i}}}(\mathbf{x}\_{\text{1}}) \right]^{\text{T}} \text{i} = \text{1}, \dots, \text{m} \\\ \hat{\xi}\_{\text{i}} &=& \left[ \hat{\xi}\_{\text{1}}, \hat{\xi}\_{\text{2}}, \dots, \hat{\xi}\_{\text{m}} \right] = \left[ \hat{\Phi}\_{\text{1}}(\mathbf{x}\_{\text{1}}), \hat{\Phi}\_{\text{2}}(\mathbf{x}\_{\text{1}}), \dots, \hat{\Phi}\_{\text{m}}(\mathbf{x}\_{\text{1}}) \right] \end{array} \tag{13}$$

#### *4.3. Reconstruction of the Unknown Inputs by Asymptotic Reduced-Order Observer with Auxiliary Output*

Now that the estimation of auxiliary output vector u<sup>1</sup> and its derivatives has been achieved, then the unknown input can be reconstructed by this information. The uˆ <sup>1</sup> is the exact estimate of the auxiliary output vector u<sup>1</sup> in a finite time obtained from the high order sliding mode observer.

**Proposition 2.** *If it can be insured that reconstructed u*ˆ<sup>1</sup> *is correctly converged, the conclusion can be obtained that the fault vector ω*<sup>2</sup> *and u*ˆ<sup>1</sup> *has one-to-one correspondence.*

Since estimation of the auxiliary output vector is now possible with acceptable accuracy, observer (7) can then be extended in the following form in (14).

**Theorem 2.** *Supposed that the auxiliary output vector u*ˆ<sup>1</sup> *is obtained, then an asymptotical reduced-order observer in accordance with the original system (2) can be generated as follows:*

$$\begin{cases}
\dot{\gamma}\_{\text{i}} = -\mathbf{K}\_{\text{i}}\mathbf{\gamma}\_{\text{i}} + \mathbf{K}\_{\text{i}}\beta\_{\text{i}}(\mathbf{u}\_{\text{i}}\,\hat{\mathbf{u}}\_{\text{1}}) - \mathbf{K}\_{\text{i}}^{2}\alpha\_{\text{i}}\hat{\mathbf{u}}\_{\text{1}} \\
\dot{\omega}\_{\text{2i}} = \mathbf{\gamma}\_{\text{i}} + \mathbf{K}\_{\text{i}}\alpha\_{\text{i}}\hat{\mathbf{u}}\_{\text{1}}
\end{cases} \tag{14}$$

*system (14) is capable of asymptotically reconstructing local unknown input vector finitely.*

**Proof.** Subtracting the first equation of (14) from the first one of original system (2), error dynamic of the observer can be reached.

While it has been proven that the estimated uˆ <sup>1</sup> is the accurate estimation of the auxiliary output vector u<sup>1</sup> in a finite time, the convergence of (14) is straightforward because the error dynamic system is not corrupted.

#### **5. Numerical Simulation Implementation on a Pilot Intensified Heat Exchanger**

In this section, the effectiveness of our proposed methods is illustrated on a pilot intensified heat exchanger which can be found in Ref. [31] for physical details. Here, the heat exchanger system is regarded as an interconnected system, in which the heat exchanger itself is a subsystem, and the actuator is regarded as the other subsystem cascaded with the heat exchanger. The purpose of the simulation is to prove that the unknown local internal signals of the actuator, like unknown air pressure change, can be recovered by measuring the outlet temperature of the heat exchanger.

#### *5.1. Interconnected System Modelling*

Define measured outlet temperatures Tp, T<sup>u</sup> of both fluids as two states x11, x<sup>12</sup> of the heat exchanger subsystem, flow rates Fp, F<sup>u</sup> of the two fluids are defined as two inputs u11, u12, which are also the interconnection of the interconnected system, outputs y<sup>1</sup> , y<sup>2</sup> are specified as x11, x12,

The state space form of heat exchanger subsystem can then be written as:

$$\begin{cases}
\dot{\mathbf{x}}\_{11} = \frac{\mathbf{u}\_{11}}{\mathbf{V}\_{\mathbf{P}}} (\mathbf{T}\_{\mathbf{P}i} - \mathbf{x}\_{11}) + \frac{\mathbf{h}\_{\mathbf{P}} \mathbf{A}}{\rho\_{\mathbf{P}} \mathbf{C}\_{\mathbf{P} \mathbf{P}} \mathbf{V}\_{\mathbf{P}}} (\mathbf{x}\_{12} - \mathbf{x}\_{11}) \\
\dot{\mathbf{x}}\_{12} = \frac{\mathbf{u}\_{12}}{\mathbf{V}\_{\mathbf{u}}} (\mathbf{T}\_{\mathbf{c}i} - \mathbf{x}\_{12}) + \frac{\mathbf{h}\_{\mathbf{u}} \mathbf{A}}{\rho\_{\mathbf{u}} \mathbf{C}\_{\mathbf{p} \mathbf{u}} \mathbf{V}\_{\mathbf{u}}} (\mathbf{x}\_{11} - \mathbf{x}\_{12})
\end{cases} \tag{15}$$

The actuators in this process are two pneumatic control valves, it is to define the stem displacement X1, X<sup>2</sup> and their derivatives dX<sup>1</sup> dt , dX<sup>2</sup> dt as four states x<sup>2</sup> <sup>T</sup> = - x<sup>21</sup> x<sup>22</sup> x<sup>23</sup> x<sup>24</sup> of the actuator subsystem, two local inputs v <sup>T</sup> = - v<sup>1</sup> v<sup>2</sup> are defined as the pneumatic pressure of two valves u<sup>1</sup> <sup>T</sup> = - F<sup>1</sup> F<sup>2</sup> , two fluid flow rate F<sup>1</sup> F<sup>2</sup> are outputs of the subsystem, which correspond to inputs Fp, F<sup>u</sup> in the heat exchanger subsystem, and are assumed unmeasured in this subsystem.

The state space form of actuator subsystem can then be written as:

$$\begin{cases} \begin{aligned} \dot{\mathbf{x}}\_{2} &= \begin{bmatrix} 0 & 1 & 0 & 0 \\ -\frac{\mathbf{k}\_{1}}{\mathbf{m}} & -\frac{\mu\_{1}}{\mathbf{m}} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -\frac{\mathbf{k}\_{2}}{\mathbf{m}} & -\frac{\mu\_{2}}{\mathbf{m}} \end{bmatrix} \mathbf{x}\_{2} + \begin{bmatrix} \frac{\mathbf{A}\_{\mathrm{s}}}{\mathbf{m}} & 0 \\ 0 & 0 \\ 0 & \frac{\mathbf{A}\_{\mathrm{s}}}{\mathbf{m}} \\ 0 & 0 \end{bmatrix} \mathbf{v} + \begin{bmatrix} 0 \\ \omega\_{21} \\ 0 \\ \omega\_{22} \end{bmatrix} \end{cases} \end{cases} \end{cases} \end{cases} $$

ω<sup>21</sup> ω<sup>22</sup> are defined as two local fault variables. Each one of these faults represents a variation in the respective control valve gain, which can be originated by an electronic component malfunction, leakage, or an obstruction in the control valve.

#### *5.2. Observer Design for Unknown Input Reconstruction*

#### 5.2.1. Reduce-Order Observer Design

By calculating output differential rank, it is obvious that both subsystem and the overall system are invertible. Then, it is necessary to verify the condition provided by 3.1 and to construct an algebraic equation for each component of the unknown inputs with coefficients in Π(v, u1).

By obtaining a second time derivative of u1, it is possible to obtain a differential algebraic polynomial for the unknown inputs whose coefficients are in Π(v, u1).

$$\begin{cases} \boldsymbol{\omega}\_{21} = \dot{\mathbf{x}}\_{22} + \frac{\mu\_1}{\mathrm{m}} \mathbf{x}\_{22} + \frac{\mathbf{k}\_1}{\mathrm{m}} \mathbf{u}\_{11} - \frac{\mathbf{A}\_2}{\mathrm{m}} \mathbf{v}\_1\\ \boldsymbol{\omega}\_{22} = \dot{\mathbf{x}}\_{24} + \frac{\mu\_2}{\mathrm{m}} \mathbf{x}\_{24} + \frac{\mathbf{k}\_2}{\mathrm{m}} \mathbf{u}\_{12} - \frac{\mathbf{A}\_2}{\mathrm{m}} \mathbf{v}\_2 \end{cases} \tag{17}$$

Obviously, the time derivates of outputs and the states appear in the algebraic equation of the unknown input, then, according to (13), an auxiliary variable is used to avoid using them.

#### 5.2.2. System Inversion Based Interconnection Reconstruction

The input of the first subsystem can also be represented by means of the output and its derivatives.

Differential all two outputs in (15), and one can obtain:

$$\begin{cases}
\dot{\mathbf{y}}\_1 = \frac{\mathbf{h}\_\mathrm{p}\mathbf{A}}{\rho\_\mathrm{p}\mathbf{C}\_\mathrm{pp}\mathbf{V}\_\mathrm{p}}(\mathbf{y}\_2 - \mathbf{y}\_1) + \frac{\mathbf{u}\_{\mathrm{II}}}{\mathbf{V}\_\mathrm{P}}(\mathbf{T}\_\mathrm{pi} - \mathbf{y}\_1) \\
\dot{\mathbf{y}}\_2 = \frac{\mathbf{h}\_\mathrm{u}\mathbf{A}}{\rho\_\mathrm{u}\mathbf{C}\_\mathrm{pu}\mathbf{V}\_\mathrm{u}}(\mathbf{y}\_1 - \mathbf{y}\_2) + \frac{\mathbf{u}\_{\mathrm{II}}}{\mathbf{V}\_\mathrm{u}}(\mathbf{T}\_\mathrm{ui} - \mathbf{y}\_2)
\end{cases} \tag{18}$$

Denoted estimates of the two inputs of the heat exchanger subsystem as <sup>u</sup>e<sup>1</sup> <sup>=</sup> - <sup>u</sup>e<sup>11</sup> <sup>u</sup>e<sup>12</sup> , the following expression can be achieved by using above results:

$$\begin{cases}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\text{ $V\_{p}$ } \\
\text{ $\mathbf{T}\_{\text{pl}}$ }
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\text{ $\mathbf{y}\_{\text{p}}$ }
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\text{ $\mathbf{y}\_{\text{p}}$ }
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\text{ $\mathbf{y}\_{\text{p}}$ }
\end{array}
\end{array}
\end{array}
\end{array}
\end{bmatrix}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\text{ $\mathbf{y}\_{\text{p}}$ }
\end{array}
\end{array}
\end{array}
\end{cases}
\end{cases}
$$

Obviously, successive derivatives of outputs y<sup>1</sup> and y<sup>2</sup> are required to develop an inversed based second order sliding mode observer to produce exact estimates of them finitely.

Construct new ordinates as:

$$\mathbf{y}\_1 = \mathbf{T}\_\mathbf{p} = \boldsymbol{\xi}\_1^1 \mathbf{y}\_2 = \mathbf{T}\_\mathbf{u} = \boldsymbol{\xi}\_2^1 \tag{20}$$

The sliding observer of Formula (10) is obtained. Then, the estimated <sup>u</sup>e<sup>11</sup> andue<sup>12</sup> can be used to obtain observer of (21).

By construction:

$$\begin{cases} \begin{array}{c} \gamma\_1 = \ \updownarrow \mathfrak{o}\_{21} + \mathsf{K}\_1 \mathsf{x}\_{22} \\ \gamma\_2 = \ \updownarrow \mathfrak{o}\_{24} + \mathsf{K}\_2 \mathsf{x}\_{24} \end{array} \tag{21}$$

The following reduce-order observer are obtained:

$$\begin{cases}
\dot{\hat{\mathbf{y}}}\_{1} = -\mathbf{K}\_{1}\mathbf{\boldsymbol{\gamma}}\_{1} + \mathbf{K}\_{1} \left(\frac{\mu\_{1}}{\mathbf{m}}\mathbf{\boldsymbol{x}}\_{22} + \frac{\mathbf{k}\_{1}}{\mathbf{m}}\widetilde{\mathbf{u}}\_{11} - \frac{\mathbf{A}\_{4}}{\mathbf{m}}\mathbf{v}\_{1}\right) - \mathbf{K}\_{1}^{2}\mathbf{x}\_{22} \\
\dot{\hat{\mathbf{y}}}\_{2} = -\mathbf{K}\_{2}\mathbf{\boldsymbol{\gamma}}\_{2} + \mathbf{K}\_{2} \left(\frac{\mu\_{2}}{\mathbf{m}}\mathbf{\boldsymbol{x}}\_{24} + \frac{\mathbf{k}\_{2}}{\mathbf{m}}\widetilde{\mathbf{u}}\_{12} - \frac{\mathbf{A}\_{4}}{\mathbf{m}}\mathbf{v}\_{2}\right) - \mathbf{K}\_{2}^{2}\mathbf{x}\_{24}
\end{cases} \tag{22}$$

Then, an asymptotic observer is constituted.

#### *5.3. Simulation Results and Discussion*

Aimed at illustrating the effectiveness of the proposed multi-level fault reconstruction method, two numerical simulations are carried out in this section. Two kinds of faults are considered, containing sudden changes and incipient variations. In addition, a simulation comparison between the well-known UIO proposed in [30] and the proposed FR is also provided. Detailed values of the variables used for the simulation can be found in [30].

#### **Case 1.** *Abrupt fault situation.*

In this simulation, the fault variables are considered to be abrupt ones. The simulation is implemented with initial conditions γ<sup>1</sup> = γ<sup>2</sup> = 0, and the observer gains are given by K<sup>1</sup> = K<sup>2</sup> = 5. Two unknown inputs ω21, ω<sup>22</sup> are considered. Dynamics of ω<sup>21</sup> remains zero from the beginning, and at t = 50 s, it changes to 10 and never comes back. The value of ω<sup>22</sup> jump to 60 at 120 s and drops back down at 160 s. Simulation results are reported in Figures 3–8. *Entropy* **2021**, *23*, x FOR PEER REVIEW 12 of 18 *Entropy* **2021**, *23*, x FOR PEER REVIEW 12 of 18

**Figure 3.** Measured global output: process fluid temperature Tp. **Figure 3.** Measured global output: process fluid temperature Tp. **Figure 3.** Measured global output: process fluid temperature Tp.

**Figure 4.** Measured global output: utility fluid temperature Tu. **Figure 4.** Measured global output: utility fluid temperature Tu. **Figure 4.** Measured global output: utility fluid temperature Tu.

ous that the value of computed F<sup>u</sup>

ous that the value of computed F<sup>u</sup>

**Figure 5.** Auxiliary output: process fluid flowrate Fp.

**Figure 5.** Auxiliary output: process fluid flowrate Fp.

variables.

variables.

As shown in Figures 5 and 6, the computed fluid flowrates are denoted by the black

As shown in Figures 5 and 6, the computed fluid flowrates are denoted by the black

and its estimated value F̂

and its estimated value F̂

u

u

converged adequately af-

converged adequately af-

the tracking capacities of designed sliding mode observer. It can be seen that after a short transient time, the estimated curves converge to the computed lines with ready accuracy. From Figure 5, at t = 50 s, the process fluid flowrate F<sup>p</sup> increases suddenly and stables at a new level after a short transient time again, which is in accordance with the assumption. Figure 6 shows the computed and estimated result of utility fluid flow rate Fu. It is obvi-

transient time, the estimated curves converge to the computed lines with ready accuracy. From Figure 5, at t = 50 s, the process fluid flowrate F<sup>p</sup> increases suddenly and stables at a new level after a short transient time again, which is in accordance with the assumption. Figure 6 shows the computed and estimated result of utility fluid flow rate Fu. It is obvi-

ter a relatively short transient period. Then, at 120 s, it jumps abruptly and drops to the original value at 160 s, and the estimated dash line tracks the computed solid line again after about 2 s. These variations are influenced by variation of unknown input ω22. Since both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-order observer to recover the local fault

ter a relatively short transient period. Then, at 120 s, it jumps abruptly and drops to the original value at 160 s, and the estimated dash line tracks the computed solid line again after about 2 s. These variations are influenced by variation of unknown input ω22. Since both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-order observer to recover the local fault

**Figure 3.** Measured global output: process fluid temperature Tp.

**Figure 4.** Measured global output: utility fluid temperature Tu.

*Entropy* **2021**, *23*, x FOR PEER REVIEW 13 of 18

ous that the value of computed F<sup>u</sup> and its estimated value F̂

As shown in Figures 5 and 6, the computed fluid flowrates are denoted by the black solid lines, and the dash green lines represent the estimated values. The two figures verify the tracking capacities of designed sliding mode observer. It can be seen that after a short transient time, the estimated curves converge to the computed lines with ready accuracy. From Figure 5, at t = 50 s, the process fluid flowrate F<sup>p</sup> increases suddenly and stables at a new level after a short transient time again, which is in accordance with the assumption. Figure 6 shows the computed and estimated result of utility fluid flow rate Fu. It is obvi-

ter a relatively short transient period. Then, at 120 s, it jumps abruptly and drops to the original value at 160 s, and the estimated dash line tracks the computed solid line again after about 2 s. These variations are influenced by variation of unknown input ω22. Since both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-order observer to recover the local fault

solid line. From Figure 7, at 50 s, the estimated ω<sup>21</sup> unexpectedly increase, and finally it

<sup>u</sup> converged adequately af-

**Figure 5. Figure 5.**  Auxiliary output: process fluid flowrate F Auxiliary output: process fluid flowrate <sup>p</sup>. Fp. lines and dash-dot lines give accurate estimation values to the simulated real values in

variables.

**Figure 6.** Auxiliary output: utility fluid flowrate Fu. **Figure 6.** Auxiliary output: utility fluid flowrate Fu. measures.

timated ω<sup>22</sup> of unknown input in Figure 8. At time 120 s, as expected, both simulated and reconstructed curves of the unknown inputs ω<sup>22</sup> jump with corresponding to the **Figure 7.** Simulated and reconstructed unknown input ω21. **Figure 7.** Simulated and reconstructed unknown input ω21.

**Figure 8.** Simulated and reconstructed unknown input ω22. **Figure 8.** Simulated and reconstructed unknown input ω22.

The simulation curves indicate that the proposed observer is proper for reconstructing the dynamics of the local unknown inputs with acceptable accuracy, using global

The safe and reliable operation of dynamic systems through the early detection of a small fault before it becomes a serious failure is a crucial component of the overall system's performance and sustainability. In this case, an incipient variation is considered on individual unknown inputs. The simulation is implemented with initial conditions γ<sup>1</sup> = γ<sup>2</sup> = 0 , and the observer gains are given by <sup>1</sup> = 10,<sup>2</sup> = 5 . Two unknown inputs

The measured global outputs, temperature of both fluid T<sup>p</sup> and Tu, are shown in Figures 9 and 10. It can be seen that both temperature curves change irregularly and incipiently, with these changes coinciding with the changes of two unknown inputs. This measured information are fed to the sliding mode observer to estimate the interconnection of

−0.1

−0.05 )].

)]. Simulation results are reported

ω21, ω<sup>22</sup> are considered. The dynamics of ω<sup>21</sup> is generated by 10[1 + sin(0.2

the two subsystems, which are also the auxiliary outputs of the low subsystem.

**Figure 7.** Simulated and reconstructed unknown input ω21.

Dynamics of ω<sup>22</sup> is generated by 3[1 + sin(0.5

**Figure 9.** Measured global output: process fluid temperature Tp.

**Figure 10.** Measured global output: utility fluid temperature Tu.

measurements.

in Figures 9–14.

**Case 2**. *Incipient fault situation.*

The measured global outputs, temperature of both fluid T<sup>p</sup> and Tu, are shown in Figures 3 and 4. It can be seen that both temperature curves change abruptly at 50 s, 120 s and 160 s. Interestingly, these changes coincide with the changes of two unknown inputs. The measured information is fed to the inverse-based sliding mode observer to correctly estimate the interconnection of the two subsystems, which are also the auxiliary outputs of the low subsystem.

As shown in Figures 5 and 6, the computed fluid flowrates are denoted by the black solid lines, and the dash green lines represent the estimated values. The two figures verify the tracking capacities of designed sliding mode observer. It can be seen that after a short transient time, the estimated curves converge to the computed lines with ready accuracy. From Figure 5, at t = 50 s, the process fluid flowrate F<sup>p</sup> increases suddenly and stables at a new level after a short transient time again, which is in accordance with the assumption. Figure 6 shows the computed and estimated result of utility fluid flow rate Fu. It is obvious that the value of computed F<sup>u</sup> and its estimated value Fˆ <sup>u</sup> converged adequately after a relatively short transient period. Then, at 120 s, it jumps abruptly and drops to the original value at 160 s, and the estimated dash line tracks the computed solid line again after about 2 s. These variations are influenced by variation of unknown input ω22. Since both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-order observer to recover the local fault variables.

Dynamics of the fault (unknown inputs) are shown in Figures 7 and 8. The real simulated values are denoted by the black solid lines, and the dash lines and dash-dot lines represent the reconstructed values by a traditional unknown UIO and the proposed FR, respectively, where local measures are available for UIO. From Figures 7 and 8, it is clear that both reconstructed unknown inputs follow closely their corresponding true values. After a short transient time, the reconstructed unknown inputs ω<sup>21</sup> and ω<sup>22</sup> in both dash lines and dash-dot lines give accurate estimation values to the simulated real values in solid line. From Figure 7, at 50 s, the estimated ω<sup>21</sup> unexpectedly increase, and finally it stabilizes at a new level, and an increase of 10 is observed. These changes satisfy the assumption of the unknown inputs ω<sup>21</sup> correctly. It is also obvious that the traditional UIO method converges quickly than the proposed FR. The similar result is obtained in the estimated ω<sup>22</sup> of unknown input in Figure 8. At time 120 s, as expected, both simulated and reconstructed curves of the unknown inputs ω<sup>22</sup> jump with corresponding to the assumption, an increase of 60 is observed, then another drop happens at t = 160 s and it returns to zero with a −60 reduction. It also proves that the reconstructed value in dash line and dash-dot line track well the real simulated value in the solid line. Again, they demonstrate that traditional UIO has better rapidity for fault reconstruction than the proposed FR, and they have the same accuracy as fault reconstruction. However, the proposed FR is more suitable for real engineering world since it does not need local output measures.

The simulation curves indicate that the proposed observer is proper for reconstructing the dynamics of the local unknown inputs with acceptable accuracy, using global measurements.

#### **Case 2.** *Incipient fault situation.*

The safe and reliable operation of dynamic systems through the early detection of a small fault before it becomes a serious failure is a crucial component of the overall system's performance and sustainability. In this case, an incipient variation is considered on individual unknown inputs. The simulation is implemented with initial conditions γ<sup>1</sup> = γ<sup>2</sup> = 0, and the observer gains are given by K<sup>1</sup> = 10, K<sup>2</sup> = 5. Two unknown inputs ω21, ω<sup>22</sup> are considered. The dynamics of ω<sup>21</sup> is generated by 10- 1 + sin 0.2te−0.05t. Dynamics of ω<sup>22</sup> is generated by 3 - 1 + sin 0.5te−0.1t. Simulation results are reported in Figures 9–14. measurements.

measurements.

in Figures 9–14.

in Figures 9–14.

**Case 2**. *Incipient fault situation.*

**Case 2**. *Incipient fault situation.*

the two subsystems, which are also the auxiliary outputs of the low subsystem.

the two subsystems, which are also the auxiliary outputs of the low subsystem.

ures 9 and 10. It can be seen that both temperature curves change irregularly and incipi-

**Figure 8.** Simulated and reconstructed unknown input ω22.

**Figure 8.** Simulated and reconstructed unknown input ω22.

*Entropy* **2021**, *23*, x FOR PEER REVIEW 14 of 18

Dynamics of ω<sup>22</sup> is generated by 3[1 + sin(0.5

Dynamics of ω<sup>22</sup> is generated by 3[1 + sin(0.5

The simulation curves indicate that the proposed observer is proper for reconstruct-

The safe and reliable operation of dynamic systems through the early detection of a

−0.1

−0.1

−0.05

)].

)]. Simulation results are reported

and Tu, are shown in Fig-

)]. Simulation results are reported

and Tu, are shown in Fig-

−0.05

)].

small fault before it becomes a serious failure is a crucial component of the overall system's performance and sustainability. In this case, an incipient variation is considered on individual unknown inputs. The simulation is implemented with initial conditions γ<sup>1</sup> = γ<sup>2</sup> = 0 , and the observer gains are given by <sup>1</sup> = 10,<sup>2</sup> = 5 . Two unknown inputs

The safe and reliable operation of dynamic systems through the early detection of a

ures 9 and 10. It can be seen that both temperature curves change irregularly and incipiently, with these changes coinciding with the changes of two unknown inputs. This meas-

ω21, ω<sup>22</sup> are considered. The dynamics of ω<sup>21</sup> is generated by 10[1 + sin(0.2

ω21, ω<sup>22</sup> are considered. The dynamics of ω<sup>21</sup> is generated by 10[1 + sin(0.2

small fault before it becomes a serious failure is a crucial component of the overall system's performance and sustainability. In this case, an incipient variation is considered on individual unknown inputs. The simulation is implemented with initial conditions γ<sup>1</sup> = γ<sup>2</sup> = 0 , and the observer gains are given by <sup>1</sup> = 10,<sup>2</sup> = 5 . Two unknown inputs

The measured global outputs, temperature of both fluid T<sup>p</sup>

The measured global outputs, temperature of both fluid T<sup>p</sup>

ing the dynamics of the local unknown inputs with acceptable accuracy, using global

The simulation curves indicate that the proposed observer is proper for reconstruct-

ing the dynamics of the local unknown inputs with acceptable accuracy, using global

**Figure 9.** Measured global output: process fluid temperature Tp. **Figure 9.** Measured global output: process fluid temperature Tp. **Figure 9.** Measured global output: process fluid temperature Tp.

**Figure 10.** Measured global output: utility fluid temperature **Figure 10.** Measured global output: utility fluid temperature T Tu. <sup>u</sup>. order observer to recover the unknown inputs. accurate estimation values to the computed values, they can be used as auxiliary outputs to reduceorder observer to recover the unknown inputs.

**Figure 11.** Auxiliary output: process fluid flowrate Fp. **Figure 11.** Auxiliary output: process fluid flowrate Fp. **Figure 11.** Auxiliary output: process fluid flowrate Fp.

Dynamics of the unknown inputs are shown in Figures 13 and 14. The real simulated values are denoted by the black solid lines, and the dash lines represent the reconstructed values. From Figures 13 and 14, it is clear that the reconstructed unknown input follows

Dynamics of the unknown inputs are shown in Figures 13 and 14. The real simulated values are denoted by the black solid lines, and the dash lines represent the reconstructed values. From Figures 13 and 14, it is clear that the reconstructed unknown input follows closely their corresponding true values. After a short transient period, the reconstructed

simulated real values indicated by the solid line. It can also illustrate that the reconstructed value in dash line tracks well the real simulated value as shown by the solid line.

simulated real values indicated by the solid line. It can also illustrate that the reconstructed value in dash line tracks well the real simulated value as shown by the solid line.

**Figure 12.** Auxiliary output: utility fluid flowrate Fu. **Figure 12.** Auxiliary output: utility fluid flowrate Fu. **Figure 12.** Auxiliary output: utility fluid flowrate Fu.

**Figure 13.** Simulated and reconstructed unknown input ω21.

**Figure 13.** Simulated and reconstructed unknown input ω21.

order observer to recover the unknown inputs.

**Figure 11.** Auxiliary output: process fluid flowrate Fp.

**Figure 12.** Auxiliary output: utility fluid flowrate Fu.

As shown in Figures 11 and 12, the computed fluid flowrates are denoted by the black solid

Dynamics of the unknown inputs are shown in Figures 13 and 14. The real simulated

values are denoted by the black solid lines, and the dash lines represent the reconstructed values. From Figures 13 and 14, it is clear that the reconstructed unknown input follows closely their corresponding true values. After a short transient period, the reconstructed unknown inputs ω<sup>21</sup> and ω<sup>22</sup> in dash lines produce an accurate estimation value to the simulated real values indicated by the solid line. It can also illustrate that the reconstructed value in dash line tracks well the real simulated value as shown by the solid line.

lines, and the dash lines represent the estimated values. The two figures verify the tracking capacities of designed sliding mode observer. It can be seen that after a short transient time, the estimated curves converge to the computed lines with ready accuracy. Both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-

**Figure 13.** Simulated and reconstructed unknown input ω21. **Figure 13.** Simulated and reconstructed unknown input ω21.

**Figure 14.** Simulated and reconstructed unknown input ω22. **Figure 14.** Simulated and reconstructed unknown input ω22.

The obtained results clearly put forward the following features. The results demonstrate that traditional UIO has a faster speed of fault reconstruction than the proposed FR, and both methods can obtain high accuracy in incipient fault reconstruction procedure. The measured global outputs, temperature of both fluid T<sup>p</sup> and Tu, are shown in Figures 9 and 10. It can be seen that both temperature curves change irregularly and incipiently, with these changes coinciding with the changes of two unknown inputs. This measured information are fed to the sliding mode observer to estimate the interconnection of the two subsystems, which are also the auxiliary outputs of the low subsystem.

Therefore, the proposed multi-level local fault (unknown input) reconstruction approach is effective for an interconnected system with unmeasured information. **6. Conclusions and Discussion** This paper addresses the multi-level local fault (unknown input reconstruction) As shown in Figures 11 and 12, the computed fluid flowrates are denoted by the black solid lines, and the dash lines represent the estimated values. The two figures verify the tracking capacities of designed sliding mode observer. It can be seen that after a short transient time, the estimated curves converge to the computed lines with ready accuracy. Both estimated fluid flowrates give accurate estimation values to the computed values, they can be used as auxiliary outputs to reduce-order observer to recover the unknown inputs.

problem of interconnected nonlinear systems. By introducing the local fault as an additional state and auxiliary outputs of the low subsystem, then the extended states, the auxiliary outputs and their derivatives are then accurately estimated by combining functions of an asymptotical reduce-order observer and an inverse-based second order sliding mode observer. Effectiveness of the proposed schemes is verified by using simulations on an Dynamics of the unknown inputs are shown in Figures 13 and 14. The real simulated values are denoted by the black solid lines, and the dash lines represent the reconstructed values. From Figures 13 and 14, it is clear that the reconstructed unknown input follows closely their corresponding true values. After a short transient period, the reconstructed unknown inputs ω<sup>21</sup> and ω<sup>22</sup> in dash lines produce an accurate estimation value to the simulated real values indicated by the solid line. It can also illustrate that the reconstructed value in dash line tracks well the real simulated value as shown by the solid line.

intensified heat exchanger system, and the satisfactory performances are validated by good simulation results. However, large bias and computation errors are observed when significant measured output noise is involved. The applicable system categories of this method include systems that depend on polynomial input and its time derivatives. In ad-The obtained results clearly put forward the following features. The results demonstrate that traditional UIO has a faster speed of fault reconstruction than the proposed FR, and both methods can obtain high accuracy in incipient fault reconstruction procedure. Therefore, the proposed multi-level local fault (unknown input) reconstruction approach is effective for an interconnected system with unmeasured information.

#### dition, the results of this work can easily explore the application scenarios, such as fault **6. Conclusions and Discussion**

5103.

**References**

2008.

*Frankl. Inst.* **2018**, *355*, 9351–9373.

detection and fault reconstruction. In this paper, model uncertainty and external disturbances are not taken into consideration during the FR designing process. Therefore, enhancing the robustness to model This paper addresses the multi-level local fault (unknown input reconstruction) problem of interconnected nonlinear systems. By introducing the local fault as an additional state and auxiliary outputs of the low subsystem, then the extended states, the auxiliary

uncertainty and external disturbance is a meaningful direction for further research, and

by the proposed FR could be used in active fault tolerant control of dynamic system for

**Author Contributions:** M.Z. and Z.L. conceived and designed the study. M.Z. and Q.W. carried out simulations, and wrote the original draft. B.D. reviewed and edited the manuscript. All authors

**Funding:** This research was funded by the National Natural Science Foundation of China, Grant No. 62003106 and No. 51867006, Provincial Natural Science Foundation of Guizhou, Grant No. ZK (2021) 321. Talent Project of GZU (2018) 02. Key Lab construction project of Guizhou Province (2016)

better achieving its effectiveness, and could be another focus of further research.

have read and agreed to the published version of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

1. Ding, S.X. *Model-Based Fault Diagnosis Techniques-Design Schemes, Algorithms, and Tools*; Springer: Berlin/Heidelberg, Germany,

2. Gao, Z.; Cecati, C.; Ding, S.X. A Survey of Fault Diagnosis and Fault-Tolerant Techniques Part II: Fault Diagnosis with Knowledge-Based and Hybrid/Active Approaches. *IEEE Trans. Ind. Electron.* **2015**, *62*, 1, doi:10.1109/tie.2015.2419013. 3. Mironova, A.; Mercorelli, P.; Zedler, A. A multi input sliding mode control for Peltier Cells using a cold–hot sliding surface. *J.*  outputs and their derivatives are then accurately estimated by combining functions of an asymptotical reduce-order observer and an inverse-based second order sliding mode observer. Effectiveness of the proposed schemes is verified by using simulations on an intensified heat exchanger system, and the satisfactory performances are validated by good simulation results. However, large bias and computation errors are observed when significant measured output noise is involved. The applicable system categories of this method include systems that depend on polynomial input and its time derivatives. In addition, the results of this work can easily explore the application scenarios, such as fault detection and fault reconstruction.

In this paper, model uncertainty and external disturbances are not taken into consideration during the FR designing process. Therefore, enhancing the robustness to model uncertainty and external disturbance is a meaningful direction for further research, and relevant investigation has already been started. Moreover, the reconstructed information by the proposed FR could be used in active fault tolerant control of dynamic system for better achieving its effectiveness, and could be another focus of further research.

**Author Contributions:** M.Z. and Z.L. conceived and designed the study. M.Z. and Q.W. carried out simulations, and wrote the original draft. B.D. reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, Grant No. 62003106 and No. 51867006, Provincial Natural Science Foundation of Guizhou, Grant No. ZK (2021) 321. Talent Project of GZU (2018) 02. Key Lab construction project of Guizhou Province (2016) 5103.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows**

**Huan Luo 1,3 , Yinhe Wang <sup>1</sup> , Ruidian Zhan 1,2,4,\*, Xuexi Zhang <sup>1</sup> , Haoxiang Wen <sup>3</sup> and Senquan Yang <sup>3</sup>**


**Abstract:** This paper investigates the cluster-delay mean square consensus problem of a class of first-order nonlinear stochastic multi-agent systems with impulse time windows. Specifically, on the one hand, we have applied a discrete control mechanism (i.e., impulsive control) into the system instead of a continuous one, which has the advantages of low control cost, high convergence speed; on the other hand, we considered the existence of impulse time windows when modeling the system, that is, a single impulse appears randomly within a time window rather than an ideal fixed position. In addition, this paper also considers the influence of stochastic disturbances caused by fluctuations in the external environment. Then, based on algebraic graph theory and Lyapunov stability theory, some sufficiency conditions that the system must meet to reach the consensus state are given. Finally, we designed a simulation example to verify the feasibility of the obtained results.

**Keywords:** cluster-delay mean square consensus; multi-agent systems; stochastic disturbances; impulse time windows; impulsive control

### **1. Introduction**

In today's era, automation and intelligence are the mainstream directions of technological development. As a typical representative among them, multi-agent systems (MASs) [1] are widely used in epidemiology [2,3], sociology [4,5], engineering circles [6–8], and other fields with their powerful distributed integration capabilities. In [9,10], a concept called Holonic MAS was proposed, and subsequent researchers have achieved a series of meaningful results on this basis. As a key subject in the field of distributed collaborative control, the research on the consensus of MASs has also received increasingly more attention from the academic community, including group or cluster consensus [11–13], leader-following consensus [14–16], *H*<sup>∞</sup> consensus [17–19], finite-time or fixed-time consensus [20–22], etc. In practical applications, MASs are required to simultaneously tend to multiple consensus states according to different task requirements. Specifically, MASs is divided into multiple clusters (i.e., subgroups) based on the degree of association between agents, and the states of all individuals included in each cluster eventually tend to be the same.

In particular, if a virtual state is selected as the consensus state of a certain cluster, and the remaining clusters' consensus states are different delay states corresponding to the virtual state, such a case is called cluster-delay consensus, and it is also a special case of the group consensus. In [23], for a class first-order nonlinear MASs, the authors proposed the cluster-delay consensus problem for the first time and studied it through a continuity control strategy. Furthermore, in [24], a new type of pinning consensus protocol with intermittent effect was designed to ensure that the system can achieve the cluster-delay consensus. Moreover, by using the pinning leader-following approach, the cluster-delay

**Citation:** Luo, H.; Wang, Y.; Zhan, R.; Zhang, X.; Wen, H.; Yang, S. Cluster-Delay Mean Square Consensus of Stochastic Multi-Agent Systems with Impulse Time Windows. *Entropy* **2021**, *23*, 1033. https:// doi.org/10.3390/e23081033

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 29 June 2021 Accepted: 8 August 2021 Published: 11 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

consensus of first-order nonlinear MASs with aperiodic intermittent communication was studied in [25]. On the basis of these research work, the cluster-delay consensus problem with intermittent effects and layered intermittent communication was studied in [26] through tracking approach. In [27], the authors extend the research work on the first-order integrator system to more complex second-order system, and investigated the cluster-delay consensus problem of a class of second-order nonlinear MASs.

However, the above-mentioned works are all based on the continuity control protocol, which requires the agent to maintain continuous communication with its neighbors. First, it has higher requirements for communication guarantee capability. Second, it also increases the control cost. In applications, the agent may not be able to obtain the neighbor's information continuously, and the above research results will no longer be applicable. At this time, it is conservative. Different from the traditional continuous control method, impulsive control has the advantages of low control cost, high control efficiency, and strong adaptability. Consequently, it is widely used in the research on leader-following consensus or group consensus of MASs [28–30]. Therefore, it is necessary to study the cluster-delay consensus of MASs via impulsive control [31]. In addition, there are some other interesting control mechanisms, such as the fuzzy control-based on sampled data [32,33], which is widely used in the consensus or synchronization problems research of MASs. Actually, the impulsive controller may not accurately act on the system at an ideal fixed impulse instant, it may be earlier or later. Therefore, the impulse appears randomly within a time window that is defined as an impulse time window in [34], and the window must be known. In order to obtain more general results, it is undoubtedly necessary to introduce the concept of impulse time window into the study of cluster-delay consensus. In general, MASs is also affected by stochastic disturbances caused by fluctuations in the external environment. Therefore, it is also necessary to study the cluster-delay consensus of nonlinear stochastic MASs (SMASs) [35].

Inspired by the above discussion, based on impulsive control strategy, we study the cluster-delay consensus of a class of SMASs with impulse time windows. The main contributions are as follows.


The organization of the rest of this paper is shown below. Section 2 introduces the commonly used symbols and the content of algebraic graph theory. In Section 3, the research problem is described and the corresponding system model is constructed. In Section 4, the corresponding consensus criterion is derived through the analysis method. Then, numerical simulation is given in Section 5 to verify the validity of the obtained results. Section 6 summarizes the work of the full text.

#### **2. Notation and Preliminaries**

The symbols R, R*m*×*<sup>n</sup>* , and N denote the sets of real numbers, *m* × *n* matrices, and natural numbers, respectively. R*<sup>n</sup>* denotes *n*-dimensional Euclidean space. N<sup>+</sup> denotes the set of positive integers. Symbols |*x*| and k*x*k represent the absolute value and the Euclidean norm for *<sup>x</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* , respectively. The Kronecker product and the Kolmogorov operator are denoted by <sup>⊗</sup> and <sup>L</sup>, respectively. For *\$* <sup>∈</sup> <sup>R</sup>*m*×*<sup>n</sup>* , (*\$*) *<sup>T</sup>* and *λmax*(*\$*) denote the transpose and the maximal eigenvalue of the matrix *\$*, respectively. *E*(·) denotes the mathematic expectation of corresponding variable. Let *w*(*t*) be the Wiener process with *m*-dimensional, which defined on the complete probability space (Ω, F, {F*t*}*t*≥0, *P*) with filtration {F*t*}*t*≥0. diag(·) represents a diagonal matrix.

Consider a class of MASs of *N* agents, and the system's communication topology can be denoted by digraph G = (D, E, *A*) without self-circulation, where D = {D1, . . . , D*N*} is the set of nodes, E = {(D*<sup>j</sup>* , D*i*) : *i*, *j* = 1, . . . , *N*} ⊂ D × D is the set of edges, *A* = [*aij*] is the weighted adjacency matrix with order *N* × *N*. If D*<sup>i</sup>* receives the state information of D*<sup>j</sup>* , the weight of edge (D*<sup>j</sup>* , D*i*) is greater than 0, for convenience, let *aij* = 1. Otherwise, *aij* = 0. The degree matrix is denoted by *D* = diag(*d<sup>i</sup>* , *i* = 1, . . . , *N*), where *d<sup>i</sup>* = ∑ *N j*=1,*j*6=*i aij*. Then,

*L* = *D* − *A* = [*lij*] denotes the Laplacian matrix, where *lij* = ( −*aij*, *i* 6= *j* − ∑ *N j*=1,*j*6=*i lij*, *i* = *j* . If

MASs contains a leader D0, then the connection matrix is denoted by C = diag(*c*1, . . . , *cN*). When agent *i* receives the leader's information, for convenience, let the weight of edge (D0, D*i*) be *c<sup>i</sup>* = 1. Otherwise, *c<sup>i</sup>* = 0. If all agents can receive the leader's information, the leader is called a globally reachable node (i.e., C is an N-dimensional identity matrix).

Similar to the work in [23], we give the explanation and description of the following concepts in advance to facilitate the understanding of the cluster-delay consensus. If MASs is divided into multiple clusters labeled by <sup>D</sup><sup>ˆ</sup> <sup>1</sup>,. . .,D<sup>ˆ</sup> *<sup>Q</sup>*, respectively, and let the index sets of *<sup>Q</sup>* clusters be <sup>D</sup><sup>ˆ</sup> <sup>1</sup> <sup>=</sup> {1, 2, · · · , *<sup>m</sup>*1}, . . ., <sup>D</sup><sup>ˆ</sup> ¯*<sup>i</sup>* <sup>=</sup> {*m*<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> · · · <sup>+</sup> *<sup>m</sup>*¯*i*−<sup>1</sup> <sup>+</sup> 1, . . . , *<sup>m</sup>*<sup>1</sup> <sup>+</sup> · · · <sup>+</sup> *<sup>m</sup>*¯*i*−<sup>1</sup> <sup>+</sup> *<sup>m</sup>*¯*<sup>i</sup>* }, . . ., <sup>D</sup><sup>ˆ</sup> *<sup>Q</sup>* <sup>=</sup> {*m*<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> · · · <sup>+</sup> *<sup>m</sup>Q*−<sup>1</sup> <sup>+</sup> 1, . . . , *<sup>N</sup>*}, where *<sup>N</sup>* <sup>=</sup> *<sup>m</sup>*<sup>1</sup> <sup>+</sup> · · · <sup>+</sup> *<sup>m</sup>Q*, ¯*<sup>i</sup>* ∈ {1, 2, · · · , *<sup>Q</sup>*}, *<sup>Q</sup>* <sup>∈</sup> <sup>N</sup>+, *<sup>m</sup>*¯*<sup>i</sup>* <sup>∈</sup> <sup>N</sup>+. If the *<sup>i</sup>*-th agent belongs to a certain cluster, let the subscript of the index set of the cluster be <sup>ˆ</sup>*i*, that is, *<sup>i</sup>* <sup>∈</sup> <sup>D</sup><sup>ˆ</sup> ˆ*i* and ˆ*i* = 1, . . . , *Q*. As for why these concepts are introduced, we will describe them in detail in the following part.

#### **3. Problem Description and Model Construction**

We consider a first-order nonlinear SMASs composed of N agents, the *i*-th agent's dynamic is defined by

$$\begin{split} d\mathbf{x}\_{i}(t) &= [f(t, \mathbf{x}\_{i}(t)) + \mathcal{A}\mathbf{x}\_{i}(t) - \rho\_{\hat{i}}(\mathbb{S}\_{\hat{i}}(t) - \mathbb{S}\_{1}(t - \tau\_{\hat{i}})) \\ &+ u\_{i}(t)]dt + \mathfrak{F}(t, \mathbf{x}\_{i}(t))dw(t), \end{split} \tag{1}$$

where *<sup>x</sup>i*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state vector (or displacement state vector in some physical systems), <sup>A</sup> is a known constant matrix, *<sup>f</sup>* : <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* is a continuous nonlinear function, *<sup>u</sup>i*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the control input, *S*ˆ*<sup>i</sup>* (*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state vector of the virtual leader of the cluster where the *i*-th agent belongs, *S*<sup>1</sup> *t* − *τ*ˆ*<sup>i</sup>* is the delay state of the virtual leader of the first cluster, *τ*ˆ*<sup>i</sup>* is the time delay, and *τ*<sup>1</sup> = 0, *ρ*ˆ*<sup>i</sup>* is the coupling strength, *<sup>ξ</sup>* : <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*n*×*<sup>m</sup>* stands for the noisy intensity function.Besides, *w*(*t*) is an m-dimensional Wiener process defined on the complete probability space (Ω, F, {F*t*}*t*≥0, *P*) with filtration {F*t*}*t*≥<sup>0</sup> which satisfies the usual conditions (i.e., F<sup>0</sup> contains all *P*-null sets and F*<sup>t</sup>* is right continuous), and *wi*(*t*) and *wj*(*t*) are independent of each other when *i* 6= *j*.

**Assumption 1.** *Each agent has a communication connection with the virtual leader of the cluster to which it belongs, and the first cluster's virtual leader has a communication connection with the virtual leaders of all other clusters.*

Different from continuous control strategy, the following impulsive controller is designed.

$$\begin{split} u\_i(t) &= \sum\_{k=1}^{+\infty} \delta(t - t\_k) (\mathcal{K}(\mathfrak{a} \sum\_{j=1}^N a\_{ij}(\mathfrak{x}\_j(t) - \mathcal{S}\_{\hat{f}}(t)) \\ &- (\mathfrak{x}\_i(t) - \mathcal{S}\_{\hat{l}}(t))) - \beta (\mathfrak{x}\_i(t) - \mathcal{S}\_{\hat{l}}(t)))), \end{split} \tag{2}$$

where *δ*(*t*) is the Dirac function, *K* is an impulsive gain matrix, *α* ∈ (0, 1) and *β* ∈ (0, 1) are the coupling strengths, {*tk*} satisfies <sup>0</sup> ≤ *<sup>t</sup>*<sup>0</sup> < · · · < *<sup>t</sup><sup>k</sup>* and lim*k*→+<sup>∞</sup> *<sup>t</sup><sup>k</sup>* = +∞, *<sup>x</sup>i*(*t*) is right continuous at each *t<sup>k</sup>* , i.e., lim*h*→0<sup>+</sup> *<sup>x</sup>i*(*t<sup>k</sup>* <sup>+</sup> *<sup>h</sup>*) <sup>=</sup> *<sup>x</sup>i*(*tk*), *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>+.

**Remark 1.** *In [23], the continuity control protocol was designed as ui*(*t*) = −*k*ˆ*<sup>i</sup>* (*si*(*t*) − *s*1(*t* − *τ*ˆ*i* )) − *σi*(*xi*(*t*) − *s*ˆ*<sup>i</sup>* (*t*)) + <sup>∑</sup>*j*∈/*V*ˆ*<sup>i</sup> lij*(*xj*(*t*) − *xi*(*t*))*. It is easy to see that the i-th agent needs to continuously obtain the state information of its neighbors j to update the control signal so that the control cost, as well as the communication burden, are higher. In other words, once the communication between agents cannot be maintained continuously, the above-mentioned controller will lose its effectiveness. However, the impulsive controller is shown in (2) only acts on the system at a series of discrete-time points, which reduces the control cost and the communication volume effectively in the control process. Therefore, the impulsive control mechanism is suitable for some actual environments with a limited communication load, and its adaptability is stronger. In addition, when the state error between the agent and its leader is large, the agent's state will have a large instantaneous jump via the impulsive control, so the response speed is faster than that in other methods.*

For the impulsive control mechanism, we need to preset an impulsive time sequence and assume that the impulse acts on the system at these given ideal moments. However, due to the limitations of physical equipment and objective environments, in practical applications, the real instant of impulse appearance is earlier or later than the ideal moment. In [34], the authors proposed a concept called impulse time window to describe this common phenomenon, as shown in Figure 1, where *Z l k* and *Z r k* are the left and right end points of the *k*-th window, respectively, *t<sup>k</sup>* is the real impulsive control moment, *t*<sup>0</sup> ≤ *Z l* <sup>1</sup> < *t*<sup>1</sup> < *Z r* <sup>1</sup> < *Z l* <sup>2</sup> < · · · . It can be seen from Figure 1 that impulse appears randomly in the window, and each window corresponds to only one impulse.

**Figure 1.** Impulse time windows on the time axis.

We introduce the corresponding virtual leaders into each cluster of SMASs, and their dynamic equations are described by

$$dS\_{\mathcal{Y}}(t) = \left[f(t, \mathcal{S}\_{\mathcal{Y}}(t)) + \mathcal{A}\mathcal{S}\_{\mathcal{Y}}(t) - \rho\_{\mathcal{Y}}(\mathcal{S}\_{\mathcal{Y}}(t) - \mathcal{S}\_{1}(t - \tau\_{\mathcal{Y}}))\right]dt + \tilde{\mathcal{\xi}}(t, \mathcal{S}\_{\mathcal{Y}}(t))dw(t), \tag{3}$$

where *y* = 1, . . . , *Q*.

**Remark 2.** *Because our research object is a SMASs without real leaders, in order to facilitate group control, we assign corresponding virtual leaders to each cluster in the system. Note that to make it easier to construct an error system, the virtual leader and the follower agent have the same dynamics. As the follower agents in each cluster need to reach their respective consensus states, the number of virtual leaders is the same as the number of clusters in the system. At the same time, suppose there is a coupling relationship of state information between some virtual leaders, as shown in Figure 2.*

Let *ey*(*t*) = *Sy*(*t*) − *S*1(*t* − *τy*), *e*(*t*) = (*e T* 1 (*t*), · · · ,*e T Q* (*t*))*<sup>T</sup>* , *F*(*t*,*ey*(*t*)) = *f*(*t*, *Sy*(*t*)) − *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>y*, *<sup>S</sup>*1(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>y*)), *<sup>F</sup>*¯(*t*,*e*(*t*)) = (*<sup>F</sup> T* (*t*,*e*1(*t*)), · · · , *F T* (*t*,*eQ*(*t*)))*<sup>T</sup>* , *ξ*e(*t*,*ey*(*t*)) = *ξ*(*t*, *Sy*(*t*)) − *ξ*(*t* − *τy*, *S*1(*t* − *τy*)), ¯*ξ*(*t*,*e*(*t*)) = (*ξ*e*<sup>T</sup>* (*t*,*e*1(*t*)), · · · , *<sup>ξ</sup>*e*<sup>T</sup>* (*t*,*eQ*(*t*)))*<sup>T</sup>* . Then, according to (3), we can get the following error system.

$$de(t) = \left[ (I\_{\mathbb{Q}} \otimes I\_{\mathbb{H}}) \tilde{\mathbf{F}}(t, e(t)) + (I\_{\mathbb{Q}} \otimes \mathcal{A})e(t) - (\Lambda \otimes I\_{\mathbb{H}})e(t) \right] dt + \tilde{\xi}(t, e(t)) dw(t), \tag{4}$$

where Λ = diag *ρ*1, *ρ*2, · · · , *ρ<sup>Q</sup>* .

Next, based on (1) and (2), we have the system model with impulse time windows as follows.

 *dxi*(*t*) =[ *f*(*t*, *xi*(*t*)) + A*xi*(*t*) − *ρ*ˆ*<sup>i</sup>* (*S*ˆ*i* (*t*) − *S*1(*t* − *τ*ˆ*<sup>i</sup>* ))]*dt* + *ξ*(*t*, *xi*(*t*))*dw*(*t*), *t* ∈ [*t*0, *Z l* 1 ] ∪ [*Z l k* , *tk* ) ∪ (*t<sup>k</sup>* , *Z r k* ], ∆*xi*(*t*) =*xi*(*t*) − *xi*(*t* −) =*K*(*α N* ∑ *j*=1 *aij*(*xj*(*t* <sup>−</sup>) − *S*ˆ*<sup>j</sup>* (*t* <sup>−</sup>) − (*xi*(*t* <sup>−</sup>) − *S*ˆ*<sup>i</sup>* (*t* <sup>−</sup>))) − *β*(*xi*(*t* <sup>−</sup>) − *S*ˆ*<sup>i</sup>* (*t* <sup>−</sup>))), *t* = *t<sup>k</sup>* . (5) Let *x*ˆ*i*(*t*) = *xi*(*t*) − *S*ˆ*<sup>i</sup>* (*t*), ˘ *f*(*t*, *x*ˆ*i*(*t*)) = *f*(*t*, *xi*(*t*)) − *f t*, *S*ˆ*<sup>i</sup>* (*t*) , ˘*ξ*(*t*, *x*ˆ*i*(*t*)) = *ξ*(*t*, *xi*(*t*)) − *ξ t*, *S*ˆ*<sup>i</sup>* (*t*) . Then, error system (6) can be obtained as *dx*ˆ*i*(*t*) = [ ˘ *<sup>f</sup>*(*t*, *<sup>x</sup>*ˆ*i*(*t*)) + <sup>A</sup>*x*ˆ*i*(*t*)]*dt* <sup>+</sup> ˘*ξ*(*t*, *<sup>x</sup>*ˆ*i*(*t*))*dw*(*t*), *<sup>t</sup>* <sup>∈</sup> [*t*0, *<sup>Z</sup> l* 1 ] ∪ [*Z l k* , *tk* ) ∪ (*t<sup>k</sup>* , *Z r k* ], ∆*x*ˆ*i*(*t*) =*x*ˆ*i*(*t*) − *x*ˆ*i*(*t* −) =*K*(*α N* ∑ *j*=1 *aij*(*x*ˆ*j*(*t* <sup>−</sup>) − *x*ˆ*i*(*t* <sup>−</sup>)) − *βx*ˆ*i*(*t* <sup>−</sup>)), *t* = *t<sup>k</sup>* . (6) Let *x*ˆ(*t*) = (*x*ˆ *T* 1 (*t*), . . . , *x*ˆ *T N* (*t*))*<sup>T</sup>* , *F*˘(*t*, *x*ˆ(*t*)) = ( ˘ *f T* (*t*, *x*ˆ1(*t*)), · · · , ˘ *f T* (*t*, *x*ˆ*N*(*t*)))*<sup>T</sup>* , ˆ*ξ*(*t*, *x*ˆ(*t*)) = ( ˘*ξ T* (*t*, *x*ˆ1(*t*)), · · · , ˘*ξ T* (*t*, *x*ˆ*N*(*t*)))*<sup>T</sup>* . Therefore, system (6) can be rewritten as *l l*

$$\begin{cases} d\mathfrak{x}(t) = [(I\_N \otimes \mathcal{A})\mathfrak{x}(t) + (I\_N \otimes I\_\mathbf{n})\mathfrak{f}(t, \mathfrak{x}(t))]dt + \mathfrak{f}(t, \mathfrak{x}(t))dw(t), t \in [t\_\mathcal{V}, \mathcal{Z}\_1^l] \cup [\mathcal{Z}\_k^l, t\_k) \\ \qquad \cup (t\_k, \mathcal{Z}\_k^r], \\ \mathfrak{x}(t) = \Omega \mathfrak{x}(t^-), t = t\_k. \end{cases} \tag{7}$$

where Ω = *INn* − (*βI<sup>N</sup>* + *αL*) ⊗ *K*, *I<sup>N</sup>* and *INn* are the identity matrices with *N*-order and *Nn*-order, respectively.

**Remark 3.** *In [23–26], the authors have adopted continuity control strategies to study the clusterdelay consensus problem of deterministic MASs. Obviously, this control method will greatly increase control costs and risks [36]. In contrast, this paper is characterized in that the influence of stochastic disturbances is considered, and what is more, it adopts a more advantageous impulsive control strategy. Therefore, the results obtained in this paper are suitable for actual scenarios in the presence of stochastic disturbances and limited communication load. Compared with the work in [31], the system model researched in this paper is more complicated, that is, the concepts of stochastic disturbances and impulse time window are introduced in the construction of the model and the controller, respectively. When the impulse signal appears jitter or drift, the obtained results effectively solve the problem of how to preset the impulse time sequence. In addition, compared with the research work related to the impulse time window, this paper studies the cluster-delay consensus problem of a class of nonlinear SMASs for the first time, and our work is mainly to explore the feasibility of combining these two different research fields. Although the authors considered the influence of random noises in [35,37], the continuity control strategy they applied may bring a great communication burden to the actual control. In this regard, by applying impulsive control mechanism, our paper avoids this problem well.*

For the subsequent consensus analysis, we give the following necessary lemma, assumption, and definitions.

**Lemma 1** ([38])**.** *For vectors <sup>x</sup>*, *<sup>y</sup>*<sup>ˆ</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and constant σ* > 0*, we can get x <sup>T</sup>y*ˆ + *y*ˆ *<sup>T</sup><sup>x</sup>* <sup>≤</sup> *<sup>σ</sup><sup>x</sup> <sup>T</sup>x* + *σ* <sup>−</sup>1*y*ˆ *<sup>T</sup>y.*ˆ

**Assumption 2.** ∀*x<sup>i</sup>* , *<sup>x</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> , there exist Lipschtiz constants <sup>φ</sup> and <sup>φ</sup>*<sup>ˆ</sup> *such that* <sup>k</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>i*) <sup>−</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>j*)k ≤ *<sup>φ</sup>*k*x<sup>i</sup>* <sup>−</sup> *<sup>x</sup>j*<sup>k</sup> *and* <sup>k</sup>*ξ*(*t*, *<sup>x</sup>i*) <sup>−</sup> *<sup>ξ</sup>*(*t*, *<sup>x</sup>j*)k ≤ *<sup>φ</sup>*ˆk*x<sup>i</sup>* <sup>−</sup> *<sup>x</sup>j*k*.*

**Definition 1** ([23])**.** *The SMASs with (3) and (5) are said to reach cluster mean square consensus, if there exist the solutions of (3) and (5) such that* lim*t*→+<sup>∞</sup> E(k*x*ˆ*i*(*t*)k 2 ) = 0*, where x*ˆ*i*(*t*) = *xi*(*t*) − *S*ˆ*<sup>i</sup>* (*t*)*.*

**Definition 2** ([23])**.** *The SMASs with (3) are said to reach delay mean square consensus, if there exist the solutions of (3) such that* lim*t*→+<sup>∞</sup> E(k*ey*(*t*)k 2 ) = 0*, where ey*(*t*) = *Sy*(*t*) − *S*1(*t* − *τy*)*.*

**Definition 3.** *The SMASs with (3) and (5) are said to reach cluster-delay mean square consensus, if there exist the solutions of (3) and (5) such that* lim*t*→+<sup>∞</sup> E(k*x*ˆ*i*(*t*)k 2 ) = 0*, and* lim*t*→+<sup>∞</sup> E(k*ey*(*t*)k 2 ) = 0*, where x*ˆ*i*(*t*) = *xi*(*t*) − *S*ˆ*<sup>i</sup>* (*t*) *and ey*(*t*) = *Sy*(*t*) − *S*1(*t* − *τy*)*.*

**Remark 4.** *As mentioned above, for ease of understanding, we have provided three different definitions of consensus. Obviously, only when the given conditions in Definitions 1 and 2 are met at the same time, Definition 3 related to cluster-delay consensus needed in this paper can be established. In other words, Definition 3 includes Definitions 1 and 2, and Definitions 1 and 2 are independent of each other. This also facilitates the step-by-step proof of the following consensus analysis part.*

#### **4. Consensus Analysis**

In this section, based on the Lyapunov stability theory and combined with the It*o*ˆ formula, we conduct a theoretical analysis of the cluster-delay consensus problem of the uncertain MASs and give the corresponding consensus criterion. The core idea of the proof is to transform the consensus problem of the original system into the stability analysis problem of the error system. According to Definition 3, the work of this part needs to be divided into two parts, namely, the proof of cluster mean square consensus and delay mean square consensus.

**Theorem 1.** *Under Assumptions 1–2, for the involved scalars σ* > 0*, φ* > 0*, and φ*ˆ > 0 *satisfying the following conditions (1)–(2), if there exist the solutions of (3) and (5) such that* lim*t*→+<sup>∞</sup> E(k*x*ˆ*i*(*t*)k 2 ) = 0*, and* lim*t*→+<sup>∞</sup> E(k*ey*(*t*)k 2 ) = 0*, then the SMASs with (3) and (5) will achieve cluster-delay mean square consensus.*

*(1) There exists a constant ϑ* > 1 *such that* ln(*ϑλ*∗ ) + *ρ*ˆ *Z l <sup>k</sup>*+<sup>1</sup> − *Z l k* ≤ 0*, where λ* <sup>∗</sup> = *λ*max Ω*T*Ω *,* Ω = *INn* − (*βI<sup>N</sup>* + *αL*) ⊗ *K, ρ*ˆ = *γ* + *σ* + *σ* <sup>−</sup>1*φ* <sup>2</sup> + *φ*ˆ<sup>2</sup> *, and γ* = *λ*max *I<sup>N</sup>* ⊗ <sup>A</sup> <sup>+</sup> <sup>A</sup>*<sup>T</sup> .*

*(2) There exists a negative definite matrix I<sup>Q</sup>* ⊗ A − Λ ⊗ *I<sup>n</sup> such that ρ*˜ = *σ* + *σ* <sup>−</sup>1*φ* <sup>2</sup> + *φ*ˆ<sup>2</sup> + 2*λ*max *I<sup>Q</sup>* ⊗ A − Λ ⊗ *I<sup>n</sup>* < 0*, where* Λ = diag *ρ*1, *ρ*2, · · · , *ρ<sup>Q</sup> .*

**Proof.** *(a): cluster mean square consensus*

Construct the following Lyapunov function:

$$V(t, \mathfrak{X}(t)) = \mathfrak{X}^T(t)\mathfrak{X}(t). \tag{8}$$

The stochastic derivative of (8) is derived by the *Ito*ˆ formula along the trajectory of system (7) as follows.

$$dV(t, \hat{\mathfrak{x}}(t)) = \mathcal{L}V(t, \hat{\mathfrak{x}}(t)) + 2\hat{\mathfrak{x}}^T(t)\hat{\xi}(t, \hat{\mathfrak{x}}(t))dw(t),\tag{9}$$

$$\mathcal{L}V(t,\mathfrak{x}(t)) = \mathfrak{X}^{\mathsf{T}}(t)[(I\_N \otimes \mathcal{A})\mathfrak{X}(t) + (I\_N \otimes I\_\mathbb{R})\mathfrak{F}(t,\mathfrak{x}(t))] + \text{trace}[\xi^{\mathsf{T}}(t,\mathfrak{x}(t))\xi(t,\mathfrak{x}(t))].\tag{10}$$

According to Assumption 2 and Lemma 1, from (10), we have

$$\mathcal{L}\hat{\mathfrak{x}}^T(t)(I\_N \otimes \mathcal{A})\hat{\mathfrak{x}}(t) \le \gamma V(t, \hat{\mathfrak{x}}(t)),\tag{11}$$

$$\begin{array}{l} \mathsf{2}\mathfrak{x}^{T}(t)(I\_{N}\otimes I\_{\mathfrak{n}})\mathfrak{F}(t,\mathfrak{x}(t)) \\ \leq \sigma\mathfrak{x}^{T}(t)\mathfrak{x}(t) + \sigma^{-1}\mathfrak{F}^{T}(t,\mathfrak{x}(t))\mathfrak{F}(t,\mathfrak{x}(t)) \\ \leq \left(\sigma + \sigma^{-1}\mathfrak{\phi}^{2}\right)V(t,\mathfrak{x}(t)) \end{array} \tag{12}$$

and

$$\text{trace}[\xi^T(t, \mathfrak{x}(t))\xi(t, \mathfrak{x}(t))] \le \delta^2 V(t, \mathfrak{x}(t)). \tag{13}$$

For *<sup>t</sup>* ∈ [*tk*−<sup>1</sup> , *tk* ), assume that ∆*t* is a small enough positive constant such that *t* + ∆*t* ∈ (*tk*−<sup>1</sup> , *tk* ), then one has

$$EV(t + \Delta t, \hat{\mathbf{x}}(t + \Delta t)) - EV(t, \hat{\mathbf{x}}(t)) = \int\_{t}^{t + \Delta t} E\mathcal{L}V(s, \hat{\mathbf{x}}(s))ds.\tag{14}$$

By (11)–(14), we can obtain

$$D^{+}EV(t, \pounds(t)) = E\mathcal{L}V(t, \pounds(t)) \le \rho EV(t, \pounds(t)).\tag{15}$$

When *t* ∈ h *t*0, *Z l* 1 and *t* ∈ h *Z r k* , *Z l k*+1 , from (15), we have

$$EV\left(Z\_{1}^{l}, \pounds\left(Z\_{1}^{l}\right)\right) \le EV(t\_{0}, \pounds(t\_{0})) \exp\left(\bigwedge^{l} \left(Z\_{1}^{l} - t\_{0}\right)\right),\tag{16}$$

$$EV(Z\_{k+1}^{l}, \pounds(Z\_{k+1}^{l})) \le EV(Z\_{k'}^{l}, \pounds(Z\_{k}^{r})) \exp(\oint (Z\_{k+1}^{l} - Z\_{k}^{r})).\tag{17}$$

Let *k* = 1. For *t* ∈ h *Z l* 1 , *t*<sup>1</sup> , it holds that

$$EV(t\_1^-, \hat{x}(t\_1^-)) \le EV(t\_0, \hat{x}(t\_0)) \exp(\hat{\rho}(t\_1 - t\_0)).\tag{18}$$

When *t* = *t<sup>k</sup>* , one has

$$EV(t\_k) = E\left(\hat{\mathfrak{x}}^T(t\_k^-) \Omega^T \Omega \hat{\mathfrak{x}}(t\_k^-)\right) \le \lambda^\* EV(t\_k^-, \hat{\mathfrak{x}}(t\_k^-)).\tag{19}$$

Thus, from (19), we have

$$EV(t\_1, \pounds(t\_1)) \le \lambda^\* EV(t\_1^-, \pounds(t\_1^-)). \tag{20}$$

For *t* ∈ *t*1, *Z r* 1 , we can get

$$\begin{split} &EV(Z\_1^r, \mathfrak{k}(Z\_1^r)) \\ &\leq EV(t\_1, \mathfrak{k}(t\_1)) \exp(\mathfrak{\rho}(Z\_1^r - t\_1)) \\ &\leq \lambda^\* EV(t\_1^-, \mathfrak{k}(t\_1^-)) \exp(\mathfrak{\rho}(Z\_1^r - t\_1)) \\ &\leq \lambda^\* EV(t\_0, \mathfrak{k}(t\_0)) \exp(\mathfrak{\rho}(Z\_1^r - t\_0)). \end{split} \tag{21}$$

When *t* ∈ h *Z r* 1 , *Z l* 2 , by (17) and (21), it follows that

$$\begin{split} EV\left(\mathbf{Z}\_{2}^{l}, \mathfrak{k}\left(\mathbf{Z}\_{2}^{l}\right)\right) &\leq EV(\mathbf{Z}\_{1}^{r}, \mathfrak{k}\left(\mathbf{Z}\_{1}^{r}\right)) \exp\left(\mathfrak{\delta}\left(\mathbf{Z}\_{2}^{l} - \mathbf{Z}\_{1}^{r}\right)\right) \\ &\leq \lambda^{\*} EV(t\_{0}, \mathfrak{k}\left(t\_{0}\right)) \exp\left(\mathfrak{\delta}\left(\mathbf{Z}\_{2}^{l} - t\_{0}\right)\right). \end{split} \tag{22}$$

Let *k* = 2. When *t* ∈ h *Z r* 2 , *Z l* 3 , it yields

$$EV(Z\_3^l, \mathfrak{X}(Z\_3^l)) \le (\lambda^\*)^2 EV(t\_0, \mathfrak{X}(t\_0)) \exp(\not p(Z\_3^l - t\_0)).$$

By analogy, for *t* ∈ h *Z l k* , *Z l k*+1 , if there exists a constant *ϑ* > 1 such that ln(*ϑλ*∗ ) + *ρ*ˆ *Z l <sup>k</sup>*+<sup>1</sup> − *Z l k* ≤ 0, then we have

$$\begin{split} EV(t, \mathfrak{f}(t)) &\leq \left(\lambda^\*\right)^k EV(t\_0, \mathfrak{f}(t\_0)) \exp(\mathfrak{f}(t - t\_0)) \\ &\leq EV(t\_0, \mathfrak{f}(t\_0)) \exp\left(\mathfrak{f}\left(t - Z\_k^l\right)\right) \lambda^\* \exp(\mathfrak{f}(Z\_k^l - Z\_{k-1}^l)) \cdot \dots \lambda^\* \exp(\mathfrak{f}(Z\_1^l - t\_0)) \\ &\leq \frac{1}{\theta^k} EV(t\_0, \mathfrak{f}(t\_0)) \exp\left(\mathfrak{f}\left(t - Z\_k^l\right)\right). \end{split} \tag{23}$$

From (23), it can be seen that *EV*(*t*, *x*ˆ(*t*)) → 0 when *t* → ∞. That is, lim*t*→<sup>∞</sup> *E*(k*xi*(*t*) − *S*ˆ*i* (*t*)k 2 ) = 0. Therefore, the SMASs with (3) and (5) can achieve the cluster mean square consensus.

*(b): delay mean square consensus*

Construct the following Lyapunov function:

$$V(t,e(t)) = e^T(t)e(t). \tag{24}$$

The stochastic derivative of (24) is derived by the *Ito*ˆ formula along the trajectory of system (4) as follows.

$$dV(t, e(t)) = \mathcal{L}V(t, e(t)) + 2e^T(t)\tilde{\xi}(t, e(t))dw(t),\tag{25}$$

$$\begin{split} \mathcal{L}V(t, e(t)) &= 2e^{T}(t)[(I\_{Q} \otimes \mathcal{A} - \Lambda \otimes I\_{\mathfrak{n}})e(t) + (I\_{Q} \otimes I\_{\mathfrak{n}})\tilde{F}(t, e(t))] \\ &+ \text{trace}[\tilde{\xi}^{T}(t, e(t))\tilde{\xi}(t, e(t))]. \end{split} \tag{26}$$

Similar to (12) and (13), we have

$$\begin{array}{l} \mathsf{2}e^{T}(t) \left( I\_{\mathbb{Q}} \otimes I\_{\mathbb{N}} \right) \mathsf{F}(t, e(t)) \\ \leq \sigma e^{T}(t)e(t) + \sigma^{-1} \mathsf{F}^{T}(t, e(t)) \mathsf{F}(t, e(t)) \\ \leq \left( \sigma + \sigma^{-1} \mathfrak{\phi}^{2} \right) \mathsf{V}(t, e(t)). \end{array} \tag{27}$$

and

$$\text{trace}\left[\xi^T(t,e(t))\xi(t,e(t))\right] \le \hat{\phi}^2 V(t,e(t)).\tag{28}$$

Furthermore, one has

$$2e^T(t)\left(I\_Q \otimes \mathcal{A} - \Lambda \otimes I\_\hbar\right)e(t) \le 2\lambda\_{\text{max}}\left(I\_Q \otimes \mathcal{A} - \Lambda \otimes I\_\hbar\right)V(t, e(t)).\tag{29}$$

In the same way, we can get the following inequality similar to (14).

$$EV(t + \Delta t, e(t + \Delta t)) - EV(t, e(t)) = \int\_{t}^{t + \Delta t} E\mathcal{L}V(s, e(s))ds.\tag{30}$$

According to (27)–(30), we can obtain

$$
\Delta D^{+}EV(t, e(t)) = E\mathcal{L}V(t, e(t)) \le \nexists \mathcal{PE}V(t, e(t)).\tag{31}
$$

From (31), one has

$$EV(t, e(t)) \le EV(t\_0, e(t\_0)) \exp(\tilde{\rho}(t - t\_0)).\tag{32}$$

At this time, if matrix *I<sup>Q</sup>* ⊗ A − Λ ⊗ *I<sup>n</sup>* is negative definite and its maximum eigenvalue satisfying *ρ*˜ < 0, then it can be known from (32) that *EV*(*t*,*e*(*t*)) → 0 when *t* → ∞. That is, lim*t*→<sup>∞</sup> *E*(k*Sy*(*t*) − *S*1(*t* − *τy*)k 2 ) = 0. Consequently, the SMASs with (3) can reach the delay mean square consensus.

According to parts (a) and (b), we can say that the SMASs with (3) and (5) can reach the cluster-delay mean square consensus. This completes the proof.

**Remark 5.** *By condition (1), we have Z l <sup>k</sup>*+<sup>1</sup> − *Z l k* ≤ − ln(*ϑλ*<sup>∗</sup> ) *ρ*ˆ *, where parameters λ* ∗ < 1 *and ρ*ˆ *can be obtained by simple calculations. Without loss of generality, ϑ can be equivalently regarded as an adjustable variable that satisfies ϑλ*∗ < 1*. Obviously, the artificial preset of the impulse time* *windows and the selection of the value of ϑ influence each other. When the interval between adjacent impulse time windows is designed to be larger, this means that ϑ needs to be larger to ensure that ϑλ*∗ < 1 *holds. At this time, it can be seen from (23) that the convergence speed of the error system will decrease. Reflected in the actual control, the impulsive interval may become larger due to the above-mentioned design changes, and the number of impulses within a certain period of time will be reduced, resulting in a slower system convergence speed, and vice versa.*

**Remark 6.** *We know that* A *is a known real matrix, and its value depends on the inherent dynamic behavior of SMASs. In other words, for a particular system, the value of* A *cannot be adjusted. Therefore, to satisfy condition (2), we can only adjust the diagonal matrix* Λ *composed of virtual coupling strengths ρ*1*,* · · · *, and ρQ. According to condition (2), we can see that the stronger the coupling strengths are, the easier the inequality ρ*˜ < 0 *is satisfied. At the same time, the delay mean-square consensus of SMASs may be realized faster.*

**Remark 7.** *Different from the general literature, the proof method in this paper combines the characteristics of multiple current methods and has been successfully applied to the study of the SMASs' cluster-delay mean square consensus problem. In a sense, this is an extension of current research methods. Moreover, how to construct the dynamic equation of the virtual leaders, how to design an impulsive controller and adjust its parameters, how to design a reasonable impulse time sequence, and how to design a reasonable simulation program to verify the effectiveness of the research method are challenging jobs. In addition, through the above research, we can reasonably preset the impulse time sequence to avoid the possible adverse effects of the digital signal's jitter or drift on the system when the MASs are facing stochastic disturbances. In practical applications, the target MASs studied in this paper can be cluster drones flying in formation, numerous unmanned vehicles on the road, or a network of multiple power stations.*

#### **5. Numerical Simulation**

Next, we design a simulation example to verify the validity of the obtained results.

**Example 1.** *Consider a first-order nonlinear SMASs composed of 9 agents, and its topology graph is shown in Figure 2. In order to easily identify the state trajectory of each agent in the simulation diagram, we choose a class of one-dimensional variable as the agent's state, namely, n* = 1*.*

**Figure 2.** Multi-agent systems with virtual leaders.

Let the initial states *x*1(*t*0) = 1, *x*2(*t*0) = 10, *x*3(*t*0) = −12, *x*4(*t*0) = −3, *x*5(*t*0) = −16, *x*6(*t*0) = 5, *x*7(*t*0) = 15, *x*8(*t*0) = −7, *x*9(*t*0) = 9, *S*1(*t*0) = −1, *S*2(*t*0) = −6, and *S*3(*t*0) = 3. Let functions *f*(*t*, *xi*(*t*)) = *xi*(*t*) sin(tan *t*), *f Sy*(*t*), *t* = *Sy*(*t*) sin(tan *t*), and *ξ*(*t*, *xi*(*t*)) = 0.16| cos(*t*)|*xi*(*t*). Obviously, we can choose Lipschtiz constants *φ* = 1 and *<sup>φ</sup>*<sup>ˆ</sup> <sup>=</sup> 0.16. Furthermore, let <sup>A</sup> <sup>=</sup> 1, *<sup>α</sup>* <sup>=</sup> 0.2, *<sup>β</sup>* <sup>=</sup> 0.8, *<sup>K</sup>* <sup>=</sup> diag(0.3, · · · , 0.3), *<sup>ρ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ρ</sup>*<sup>3</sup> <sup>=</sup> 20,

*τ*<sup>2</sup> = 0.2, and *τ*<sup>3</sup> = 0.3. Based on the above parameters, we have *λ* ∗ = 0.6111, *γ* = 2. We can choose parameters *ϑ* = 1.2 and *σ* = 1. Then, it can be calculated according to condition 1) in Theorem 1 that *Z l <sup>k</sup>*+<sup>1</sup> − *Z l <sup>k</sup>* ≤ 0.103. In addition, it is clear that the matrix *I<sup>Q</sup>* ⊗ A − Λ ⊗ *I<sup>n</sup>* is negative definite and satisfies the condition *ρ*˜ < 0.

Finally, for convenience, we designed a class of layout of the impulse time windows as shown in Figure 3. Specifically, we stipulate that the width of each window in the figure is 0.05 and that the impulse appears in the center point of each window. In other words, <sup>∀</sup>*<sup>k</sup>* <sup>∈</sup> <sup>N</sup>+, we have *<sup>Z</sup> l <sup>k</sup>*+<sup>1</sup> − *Z l <sup>k</sup>* = 0.05 and *tk*+<sup>1</sup> − *t<sup>k</sup>* = 0.05.

**Figure 3.** Design layout of impulse time window (ITW).

Based on the above work, Figures 4–6 are obtained by Matlab platform as follows.

**Figure 4.** State trajectory of each agent under impulsive control.

**Figure 5.** The state error trajectory for each agent in clusters.

**Figure 6.** The error trajectories between *S*2(*t*), *S*3(*t*) and their virtual leaders' states when *ρ*<sup>2</sup> = *ρ*<sup>3</sup> = 20.

According to Figure 4, we can find that the MASs are divided into three clusters, and the states of the three agents in each cluster gradually tend to a common state (i.e., the virtual leader's state). Correspondingly, the state error between each agent and its virtual leader also gradually tends to 0, as shown in Figure 5. Thus, the cluster mean square consensus of system (5) can be achieved. Obviously, it can be seen from Figure 6 that system (3) has achieved the delay mean square consensus. In summary, based on the impulse time windows, the cluster-delay mean square consensus of the SMASs with (3) and (5) can be realized.

We know that by adjusting the size of the impulsive interval during the simulation process and observing the impact of this operation on the speed of the multi-agent systems to achieve cluster mean square consensus, it can verify the dynamic relationship between the selection of parameter *ϑ* and the preset layout of the impulse time windows. We assume that there exists a parameter 1 < *ϑ* < 1.2 such that *Z l <sup>k</sup>*+<sup>1</sup> − *Z l <sup>k</sup>* = 0.12 and *tk*+<sup>1</sup> − *t<sup>k</sup>* = 0.12, and Figure 7 is obtained. By Figure 7, as described in Remark 5, it takes longer for SMASs to achieve cluster mean square consensus. Thus, the discussions in Remark 5 are reasonable.

**Figure 7.** The state error trajectory for each agent in clusters.

To verify the correctness of the theoretical analysis in Remark 6, we increase the value of the coupling strengths. That is, let *ρ*<sup>2</sup> = *ρ*<sup>3</sup> = 30, and Figure 8 is obtained. According to Figure 8, it can be found that the two error trajectories in the figure can approximately converge to 0 at about 0.15. This convergence speed is obviously faster than that in Figure 6. Therefore, the obtained results in Remark 6 are correct.

**Figure 8.** The error trajectories between *S*2(*t*), *S*3(*t*) and their virtual leaders' states when *ρ*<sup>2</sup> = *ρ*<sup>3</sup> = 30.

#### **6. Conclusions**

Based on the discrete impulsive control strategy, this paper studies the cluster-delay mean square consensus problem of a class of SMASs with impulse time windows. According to the algebraic graph theory and Lyapunov stability theory, sufficient consensus criteria are given, and the obtained results are more general than the existing work. Moreover, according to the obtained conditions, the upper bound of the interval between the left endpoints of the two adjacent windows can be derived, which is conducive to the reasonable setting of the windows, so as to ensure that the cluster-delay consensus of SMASs in the mean square sense can be realized. Finally, a simulation example is designed to analyze and verify the feasibility of the relevant results. However, the research work in this paper still has some shortcomings. For instance, the dynamic model of each agent is homogeneous, and there are fewer objective factors considered in the system. Due to the wide application of heterogeneous MASs in practical applications, it is necessary to extend existing research work to heterogeneous MASs. In addition, considering the influence of factors such as time delay and switching topology in this paper is also a meaningful direction for work in the future.

**Author Contributions:** Conceptualization, H.L. and Y.W.; methodology, H.L. and Y.W.; software, H.W. and S.Y.; validation, Y.W., R.Z. and X.Z.; formal analysis, H.L.; investigation, H.L.; resources, R.Z.; data curation, X.Z.; writing—original draft preparation, H.L.; writing—review and editing, Y.W. and H.L.; visualization, H.W.; supervision, Y.W. and X.Z.; project administration, Y.W.; funding acquisition, R.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research work was funded by the Key-Area Research and Development Program of Guangdong Province (2019B010142001 and 2019B010140002), and the National Natural Science Foundation of China (61803104 and 61673120).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


## *Article* **Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems**

**Yang Li <sup>1</sup> , Jianhua Zhang 1,\* , Xinli Xu <sup>1</sup> and Cheng Siong Chin <sup>2</sup>**


**Abstract:** In this article, a novel adaptive fixed-time neural network tracking control scheme for nonlinear interconnected systems is proposed. An adaptive backstepping technique is used to address unknown system uncertainties in the fixed-time settings. Neural networks are used to identify the unknown uncertainties. The study shows that, under the proposed control scheme, each state in the system can converge into small regions near zero with fixed-time convergence time via Lyapunov stability analysis. Finally, the simulation example is presented to demonstrate the effectiveness of the proposed approach. A step-by-step procedure for engineers in industry process applications is proposed.

**Keywords:** adaptive fixed-time; neural network; nonlinear interconnected systems

**Citation:** Li, Y.; Zhang, J.; Xu, X.; Chin, C.S. Adaptive Fixed-Time Neural Network Tracking Control of Nonlinear Interconnected Systems. *Entropy* **2021**, *23*, 1152. https:// doi.org/10.3390/e23091152

Academic Editor: José A. Tenreiro Machado

Received: 31 July 2021 Accepted: 29 August 2021 Published: 1 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

### **1. Introduction**

In actual industrial processes, after decades of development, several control strategies based on classical control theory and modern control theory have been developed. However, most of these control methods are based on single-input single-output linear systems. There are many nonlinear, uncertain, unmodeled dynamic problems in actual industrial processes that pose great challenges to the design of control systems. With the development of engineering automation requirements, the research of control strategies based on multi-input multi-output nonlinear systems has attracted growing attention. In recent decades, many control schemes have been proposed for stability analysis and the control for nonlinear systems, such as the adaptive technique [1–3], backstepping technique [4–6], U model control [7–9], sliding mode control [10–12], super twisting algorithm [13,14], neural network technique [6,15,16], etc. In particular, neural network technology has attracted many researchers' attention because of the following aspects: (1) a neural network has the strong ability to learn any function and can approximate any nonlinear system, and (2) because of the self-learning ability of neural networks, the controller does not need much system model and parameter information, so neural network control can be widely used to solve the control problems caused by uncertain models [17]. In [18], the control problem of time-varying output constraints was investigated using the neural network technique. The use of adaptive neural network control for an uncertain nonlinear system with external disturbance was presented in [19]. In [20], neural network controller designs were presented for several classes of nonlinear systems, including single-input single-output nonlinear systems, strict feedback nonlinear systems, nonaffine nonlinear systems, and multi-input multi-output triangular nonlinear systems.

Most of the above research was proven based on Lyapunov stability theory. However, actual systems often have various disturbances that cannot strictly meet the definition of Lyapunov stability. To solve this problem, scholars have mainly introduced new concepts based on two aspects. On the one hand, the concept of input-to-state stability has been

introduced; on the other hand, the concept of practical stability has been introduced with the aim of making systems stable in finite time. If a closed-loop system reaches a stable state in a limited time, it is called finite-time stability. Further, if a system meets the convergence time and does not depend on the initial parameters, it is called fixed-time stability. Compared with the traditional finite-time control method, the convergence time of the fixed-time control method is independent of the initial conditions. Exponential stability, finite-time stability, and fixed-time stability are all concepts related to the convergence rate of a system. They are very important for many control applications, such as the explosion of missiles. In [21], taking the stability analysis of a sliding mode control system as an example, input–output stability, finite-time stability, and fixed-time stability are introduced in detail. In [14], the convergence time of the super-twisting algorithm for a nonaffine nonlinear system was calculated. In [22], the finite time input–output stability of nonlinear systems is studied. In [23], a finite-time adaptive controller is designed for interconnected systems with time-varying output constraints to make the system stable in finite time, but the fixed-time convergence problem is not considered in this paper. The use of adaptive fixed-time tracking control for a strict feedback nonlinear system was studied in [24]. However, there are still many problems to be solved in these existing control strategies, such as state constraints, the adaptive backstepping "explosion" problem, and so on.

In the actual production process, many physical models, such as power systems, process control systems, and manipulator models, can be modeled as nonlinear interconnected systems. Because the interconnected terms between nonlinear interconnected subsystems are unknown, it is physically difficult to obtain this information through sensors, and it requires significant computer resources. Therefore, designing a reliable control scheme for nonlinear interconnected systems is a challenging task. The adaptive decentralized control scheme for interconnected nonlinear systems was discussed in [25]. In [26], the robust adaptive tracking control scheme was proposed for uncertain interconnected nonlinear systems. In [27], the use of adaptive neural control for high-order interconnected systems was examined.

Based on the above analysis, the main goal of this article is to design an adaptive fixed-time neural network tracking controller for nonlinear interconnected systems. The main contributions of this paper are as follows:


The article is organized into the following sections. A nonlinear interconnected mathematical description of the problem is presented in Section 2, the adaptive fixed-time neural network control scheme for a class of nonlinear interconnected systems is proposed in Section 3, two simulation examples are provided to show the reliability of the presented control scheme in Section 4, and finally, some conclusions are given in Section 5.

#### **2. Problem Formation and Preliminaries**

Consider the interconnected nonlinear system:

$$\begin{cases}
\dot{\mathbf{x}}\_{i,m} = \mathbf{x}\_{i,m+1} + f\_{i,m}(\overline{\mathbf{x}}\_{i,m}) \\
\dot{\mathbf{x}}\_{i,n} = u\_i + f\_{i,n}(\overline{\mathbf{x}}\_{i,n}) + h\_{i,n}(\overline{\mathbf{x}}\_{1,n'} \overline{\mathbf{x}}\_{2,n'} \dots \mathbf{x}\_{N,n}) \\
y\_i = \mathbf{x}\_{i,1}
\end{cases} \tag{1}$$

where *xi*,*<sup>m</sup>* ∈ *R* is the state of the interconnected nonlinear system; *xi*,*<sup>m</sup>* = [*xi*,1, . . . , *xi*,*m*] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup> m* is the state vector of the system; *fi*,*m*(*xi*,*m*) : *R <sup>m</sup>* <sup>→</sup> *<sup>R</sup>* is the known smooth function; *hi*,*n*(*x*1,*n*, *x*2,*n*, · · · , *xN*,*n*) : *R <sup>n</sup>*×*<sup>N</sup>* <sup>→</sup> *<sup>R</sup>* is the unknown smooth function; *<sup>y</sup><sup>i</sup>* <sup>∈</sup> *<sup>R</sup>* is the output

of the system; *u<sup>i</sup>* ∈ *R* is the corresponding control input of the system; the desired trajectory *yi*,*<sup>d</sup>* and its derivative are continuous and bounded.

**Remark 1:** *In the next section, we introduce a neural network adaptive control method based on the fixed-time stability theory. The objective of the method is for the nonlinear interconnected system output to be able to track the desired signal and maintain fixed-time stability based on the adaptive fixed-time neural network controller. The designed setting time does not rely on the initial parameters and can be realized only by adjusting the controller parameters.*

#### **3. Adaptive Fixed-Time Tracking Control System Design**

#### *3.1. Control System Design*

In this section, the design of a fixed-time adaptive law for the error systems of neural networks will be presented. The tracking control system's objective is to drive the error system to fixed-time stability. To solve the tracking control problem, the neural network adaptive controller, based on fixed-time Lyapunov stability theory for nonlinear interconnected systems, is presented. The adaptive fixed-time laws were designed to update the weights of the neural networks for the error systems. Neural networks were used to approximate unknown functions. The parameters of the neural networks were iteratively based on the Lyapunov fixed-time stability theorem. The convergence time can be designed by choosing controller parameters without the initial condition. Based on the controller, the error-closed loop system achieves Lyapunov fixed-time bounded stability, which means the output trajectory can track to the desired trajectory in fixed time.

**Remark 2:** *The control structure's design for the closed loop system is shown in Figure 1. The states of the error system can be determined from the minus between the setting reference function and the actual output function of the nonlinear interconnected system.*

**Figure 1.** Control structure of the closed system.

*3.2. Control System Analysis*

Step 1: First, for the system *i*, the following variables are selected:

$$z\_{i,1} = x\_{i,1} - y\_{i,d} \tag{2}$$

The dynamics of *zi*,1 can be obtained as

$$
\dot{z}\_{i,1} = x\_{i,2} + f\_{i,1}(x\_{i,1}) - \dot{y}\_{i,d} \tag{3}
$$

Design the ideal virtual control as

$$
\overline{\mathfrak{a}}\_{i,1} = -p\_{i,1}z\_{i,1}^p - q\_{i,1}z\_{i,1}^q - f\_{i,1}(\mathbf{x}\_{i,1}) + \dot{y}\_{i,d} \tag{4}
$$

where *pi*,1 > 0, *qi*,1 > 0 Select the virtual control *αi*,1 and design the adaptive law as

$$\dot{\alpha}\_{i,1} = -l\_{i,p,1}y\_{i,1}^p - l\_{i,q,1}y\_{i,1}^q - k\_{i,a,1}y\_{i,1} - z\_{i,1}, \alpha\_{i,1}(0) = 0 \tag{5}$$

where, *li*,*p*,1 > 0, *li*,*q*,1 > 0 and the error virtual control is

$$y\_{i,1} = \mathfrak{a}\_{i,1} - \overline{\mathfrak{a}}\_{i,1} \tag{6}$$

Therefore, based on system (3) and virtual control

$$\begin{array}{lcl} \dot{z}\_{i,1} &= \mathbf{x}\_{i,2} + f\_{i,1}(\mathbf{x}\_{i,1}) - \dot{y}\_{i,d} - p\_{i,1}z\_{i,1}^p - q\_{i,1}z\_{i,1}^q - f\_{i,1}(\mathbf{x}\_{i,1}) + \dot{y}\_{i,d} - \overline{\mathbf{x}}\_{i,1} \\ &= \mathbf{x}\_{i,2} - p\_{i,1}z\_{i,1}^p - q\_{i,1}z\_{i,1}^q - a\_{i,1} + a\_{i,1} - \overline{\mathbf{a}}\_{i,1} \\ &= -p\_{i,1}z\_{i,1}^p - q\_{i,1}z\_{i,1}^q + z\_{i,2} + y\_{i,1} \end{array} \tag{7}$$

where

.

$$z\_{i,2} = \mathfrak{x}\_{i,2} - \mathfrak{a}\_{i,1} \tag{8}$$

and the dynamic error virtual control is

$$\begin{array}{rcl} \dot{y}\_{i,1} &= \dot{\mathfrak{a}}\_{i,1} - \dot{\overline{\mathfrak{a}}}\_{i,1} \\ &= -l\_{i,p,1}y\_{i,1}^p - l\_{i,q,1}y\_{i,1}^q - k\_{i,a,1}y\_{i,1} - z\_{i,1} - \dot{\overline{\mathfrak{a}}}\_{i,1} \end{array} \tag{9}$$

The Lyapunov candidate functional is chosen as

$$V\_1 = \frac{1}{2}z\_{i,1}^2 + \frac{1}{2}y\_{i,1}^2\tag{10}$$

Differentiating *V*<sup>1</sup> with respect to time *t* yields

$$\begin{array}{rcl} \dot{V}\_{1} &= z\_{i,1}\dot{z}\_{i,1} + y\_{i,1}\dot{y}\_{i,1} \\ &= -p\_{i,1}z\_{i,1}^{p+1} - q\_{i,1}z\_{i,1}^{q+1} + z\_{i,1}z\_{i,2} \\ &- l\_{i,p,1}y\_{i,1}^{p+1} - l\_{i,q,1}y\_{i,1}^{q+1} - k\_{i,q,1}y\_{i,1}^2 - y\_{i,1}\dot{\overline{\alpha}}\_{i,1} \end{array} \tag{11}$$

Step *m*: the form of the tracking error is

.

$$z\_{i,m} = \mathfrak{x}\_{i,m} - \mathfrak{a}\_{i,m-1} \tag{12}$$

The dynamics of *zi*,*<sup>m</sup>* can be obtained as

$$
\dot{z}\_{i,m} = \chi\_{i,m+1} + f\_{i,m}(\chi\_{i,m}) - \dot{\alpha}\_{i,m-1} \tag{13}
$$

Design the ideal virtual control as

$$\overline{\mathfrak{a}}\_{i,m} = -p\_{i,m}z\_{i,m}^p - q\_{i,m}z\_{i,m}^q - f\_{i,m}(\mathfrak{x}\_{i,m}) + \dot{\mathfrak{a}}\_{i,m-1} \tag{14}$$

where *pi*,*<sup>m</sup>* > 0, *qi*,*<sup>m</sup>* > 0 Select the virtual control *αi*,*m*, and the adaptive law can be obtained as

$$\dot{\mathfrak{a}}\_{i,m} = -l\_{i,p,m} y\_{i,m}^p - l\_{i,q,m} y\_{i,m}^q - k\_{i,a,m} y\_{i,m} - z\_{i,m} \mathfrak{a}\_{i,m}(0) = 0 \tag{15}$$

where, *li*,*p*,*<sup>m</sup>* > 0, *li*,*q*,*<sup>m</sup>* > 0 and the error virtual control is

$$y\_{i,m} = \mathfrak{a}\_{i,m} - \overline{\mathfrak{a}}\_{i,m} \tag{16}$$

Therefore, based on system (13) and virtual control

$$\begin{array}{lcl} \dot{z}\_{i,m} &= x\_{i,m+1} + f\_{i,m}(x\_{i,m}) - \dot{a}\_{i,m-1} - p\_{i,m}z\_{i,m}^p - q\_{i,m}z\_{i,m}^q - f\_{i,m}(x\_{i,m}) + \dot{y}\_{i,m-1} - \dot{a}\_{i,m-1} \\ &= x\_{i,m+1} - p\_{i,m}z\_{i,m}^p - q\_{i,m}z\_{i,m}^q - a\_{i,m} + a\_{i,m} - \overline{a}\_{i,m} \\ &= -p\_{i,m}z\_{i,m}^p - q\_{i,m}z\_{i,m}^q + z\_{i,m+1} + y\_{i,m} \end{array} \tag{17}$$

where

$$z\_{i,m+1} = \chi\_{i,m+1} - \alpha\_{i,m} \tag{18}$$

and the dynamic error virtual control is

$$\begin{array}{rcl} \dot{y}\_{i,m} &= \dot{\alpha}\_{i,m} - \dot{\overline{\alpha}}\_{i,m} \\ &= -l\_{i,p,m} y\_{i,m}^p - l\_{i,q,m} y\_{i,m}^q - k\_{i,a,m} y\_{i,m} - z\_{i,m} - \dot{\overline{\alpha}}\_{i,m} \end{array} \tag{19}$$

The Lyapunov candidate functional is chosen as

$$V\_m = \frac{1}{2}z\_{\mathrm{i},m}^2 + \frac{1}{2}y\_{\mathrm{i},m}^2\tag{20}$$

Differentiating *V<sup>m</sup>* with respect to time *t* yields

$$\begin{array}{ll} \dot{V}\_{m} &= z\_{i,m}\dot{z}\_{i,m} + y\_{i,m}\dot{y}\_{i,m} \\ &= -z\_{i,m-1}z\_{i,m} - p\_{i,m}z\_{i,m}^{p+1} - q\_{i,m}z\_{i,m}^{q+1} + z\_{i,m}z\_{i,m+1} \\ &- l\_{i,p,m}y\_{i,m}^{p+1} - l\_{i,q,m}y\_{i,m}^{q+1} - k\_{i,a,m}y\_{i,m}^{2} - y\_{i,m}\dot{\overline{\alpha}}\_{i,m} \end{array} \tag{21}$$

Step *n*: the time derivative of *zi*,*<sup>n</sup>* can be described as

$$z\_{i,n} = \mathfrak{x}\_{i,n} - \mathfrak{a}\_{i,n-1} \tag{22}$$

Based on dynamics and tracking errors, the dynamics of *zi*,*<sup>n</sup>* can be obtained as

$$\dot{z}\_{i,\mathfrak{n}} = u\_{\mathfrak{i}} + f\_{\mathfrak{i},\mathfrak{n}}(\overline{\mathfrak{x}}\_{i,\mathfrak{n}}) + h\_{\mathfrak{i},\mathfrak{n}}(\overline{\mathfrak{x}}\_{1,\mathfrak{n}}, \overline{\mathfrak{x}}\_{2,\mathfrak{n}}, \dots, \overline{\mathfrak{x}}\_{N,\mathfrak{n}}) - \dot{a}\_{i,\mathfrak{n}-1} \tag{23}$$

Design the NNs' approximate nonlinear systems as

$$\mathcal{W}\_{\rm i,n}(\mathbb{Z}\_{1,\mathbb{N}}, \mathbb{Z}\_{2,\mathbb{N}}, \dots, \mathbb{Z}\_{N,\mathbb{n}}) = \mathcal{W}\_{\rm i}^{T}\Psi(Z) + \varepsilon\_{\rm i}(Z) \tag{24}$$

where *Z* = [*x*1,*n*, *x*2,*n*, · · · , *xN*,*n*] *T* , *<sup>θ</sup><sup>i</sup>* <sup>=</sup> <sup>k</sup>*Wi*k, and is estimated by <sup>ˆ</sup>*θ<sup>i</sup>* . The controller is designed as

$$u\_i = -z\_{i,n-1} - p\_{i,n}z\_{i,n}^p - q\_{i,n}z\_{i,n}^q - k\_{i,n}z\_{i,n} - f\_{i,n}(\overline{x}\_{i,n}) - \text{sign}(z\_{i,n})\hat{\theta}\_i \|\Psi\| + \dot{a}\_{i,n-1} \tag{25}$$

where *ki*,*<sup>n</sup>* > <sup>1</sup> 2 . Design the adaptive law as

$$\dot{\theta}\_i = \mu\_i \Big( |z\_{i,n}| |\Psi\_i| - \varsigma\_i \theta\_i^p - \chi\_i \theta\_i^q \Big) \tag{26}$$

where *µ<sup>i</sup>* > 0, *ς<sup>i</sup>* > 0, *χ<sup>i</sup>* > 0 Therefore, based on system (23) and controller

$$\dot{z}\_{i,\mathfrak{n}} = W\_i^T \Psi(Z) + \varepsilon\_i(Z) - z\_{i,\mathfrak{n}-1} - p\_{i,\mathfrak{n}} z\_{i,\mathfrak{n}}^p - q\_{i,\mathfrak{n}} z\_{i,\mathfrak{n}}^q - k\_{i,\mathfrak{n}} z\_{i,\mathfrak{n}} - \text{sign}(z\_{i,\mathfrak{n}}) \hat{\theta}\_i ||\mathfrak{Y}|| \tag{27}$$

the Lyapunov candidate functional is chosen as

$$V\_n = \frac{1}{2}z\_{i,n}^2 + \frac{1}{2\mu\_i}\widehat{\theta}\_i^2\tag{28}$$

Differentiating *V<sup>n</sup>* with respect to time *t* yields

$$\begin{split} \dot{V}\_{n} &= z\_{i,n} \dot{z}\_{i,n} + \frac{1}{\mu\_{i}} \widetilde{\theta}\_{i} \dot{\hat{\theta}}\_{i} \\ &= z\_{i,n} \mathsf{W}\_{i}^{T} \mathsf{Y}(\boldsymbol{Z}) + z\_{i,n} \varepsilon\_{i}(\boldsymbol{Z}) - z\_{i,n-1} z\_{i,n} - p\_{i,n} z\_{i,n}^{p+1} - q\_{i,n} z\_{i,n}^{q+1} \\ &- k\_{i,n} z\_{i,n}^{2} - \dot{\theta}\_{i} |z\_{i,n}| ||\,\boldsymbol{\Psi}\,\,\,|| + \widetilde{\theta}\_{i} \left( |z\_{i,n}| |\Psi\_{i}| - \varsigma\_{i} \theta\_{i}^{p} - \chi\_{i} \theta\_{i}^{q} \right) \\ &\leq \theta\_{i} |z\_{i,n}| ||\,\boldsymbol{\Psi}\,\,|| - \left( k\_{i,n} - \frac{1}{2} \right) z\_{i,n}^{2} + \frac{1}{2} \widetilde{\varepsilon}\_{i} - z\_{i,n-1} z\_{i,n} - p\_{i,n} z\_{i,n}^{p+1} - q\_{i,n} z\_{i,n}^{q+1} \\ &- \dot{\theta}\_{i} |z\_{i,n}| ||\,\boldsymbol{\Psi}\,\,|| + \widetilde{\theta}\_{i} |z\_{i,n}| ||\boldsymbol{\Psi}\_{i}| - \varsigma\_{i} \widetilde{\theta}\_{i} \theta\_{i}^{p} - \chi\_{i} \widetilde{\theta}\_{i} \theta\_{i}^{q} \end{split} \tag{29}$$

where *ε* 2 *i* (*Z*) ≤ *ε<sup>i</sup>* . Based on Lemma 4, we have

$$\begin{array}{l} -\varsigma\_{i}\widetilde{\theta}\_{i}\widehat{\theta}\_{i}^{p} \leq -\gamma\_{i}\widetilde{\theta}\_{i}^{p+1} + \lambda\_{i}\theta\_{i}^{p+1} \\ -\chi\_{i}\widetilde{\theta}\_{i}\widehat{\theta}\_{i}^{q} \leq -\nu\_{i}\widetilde{\theta}\_{i}^{q+1} + \upsilon\_{i}\theta\_{i}^{q+1} \end{array} \tag{30}$$

where *ς<sup>i</sup>* , *γ<sup>i</sup>* , *λ<sup>i</sup>* , *χ<sup>i</sup>* , *νi* , *υ<sup>i</sup>* are real numbers, *γ<sup>i</sup>* , *λ<sup>i</sup>* is determined by *ς<sup>i</sup>* , *p*, and *ν<sup>i</sup>* , *υ<sup>i</sup>* is determined by *χ<sup>i</sup>* , *q*. Therefore, we have

$$\begin{array}{ll} \dot{V}\_{n} \leq & -\left(k\_{i,n} - \frac{1}{2}\right)z\_{i,n}^{2} + \frac{1}{2}\overline{\varepsilon}\_{i} - z\_{i,n-1}z\_{i,n} - p\_{i,n}z\_{i,n}^{p+1} - q\_{i,n}z\_{i,n}^{q+1} \\ & -\gamma\_{l}\widehat{\theta}\_{l}^{p+1} + \lambda\_{l}\theta\_{l}^{p+1} - \nu\_{l}\widehat{\theta}\_{l}^{q+1} + \upsilon\_{l}\theta\_{l}^{q+1} \end{array} \tag{31}$$

**Theorem 1.** *For the interconnected nonlinear system (1), based on the feasible virtual control signal (5), (15) actual controller (25), and adaptive law (26), the error state between the system output and the desired function is fixed-time Lyapunov stability, and the setting time does not rely on the initial parameters.*

**Proof.** Based on the Lyapunov candidate functionals (10), (20), and (28), differentiating the Lyapunov functional with respect to time *t* yields (11), (21), and (31). Choosing the Lyapunov candidate functional as

$$V = \frac{1}{2} \sum\_{j=1}^{n} z\_{i,j}^2 + \frac{1}{2} \sum\_{j=1}^{n-1} y\_{i,j}^2 + \frac{1}{2\mu\_i} \hat{\theta}\_i^2 \tag{32}$$

and differentiating the Lyapunov functional with respect to time *t* yields

$$\begin{array}{rl} \dot{V} \leq & -\sum\_{j=1}^{n} p\_{i,j} z\_{i,j}^{p+1} - \sum\_{j=1}^{n} q\_{i,j} z\_{i,j}^{q+1} - \sum\_{j=1}^{n} l\_{i,p,j} y\_{i,j}^{p+1} - \sum\_{j=1}^{n} l\_{i,q,j} y\_{i,j}^{q+1} \\ & -\sum\_{j=1}^{n-1} k\_{i,\alpha,j} y\_{i,j}^{2} - \sum\_{j=1}^{n-1} y\_{i,j} \bar{\varpi}\_{i,j} - \left( k\_{i,n} - \frac{1}{2} \right) z\_{i,n}^{2} + \frac{1}{2} \bar{\varepsilon}\_{i} \\ & -\gamma\_{i} \widehat{\theta}\_{i}^{p+1} + \lambda\_{i} \theta\_{i}^{p+1} - \nu\_{i} \widehat{\theta}\_{i}^{q+1} + \nu\_{i} \theta\_{i}^{q+1} \end{array} \tag{33}$$

where . *αi*,1 2 ≤ *p<sup>i</sup>* , and

$$\begin{split} \dot{V} &\leq \quad -\sum\_{j=1}^{n} p\_{i,j} z\_{i,j}^{p+1} - \sum\_{j=1}^{n} q\_{i,j} z\_{i,j}^{q+1} - \sum\_{j=1}^{n} l\_{i,p,j} y\_{i,j}^{p+1} - \sum\_{j=1}^{n} l\_{i,q,j} y\_{i,j}^{q+1} \\ &\quad - \gamma\_i \hat{\theta}\_i^{p+1} - \nu\_i \hat{\theta}\_i^{q+1} - \sum\_{j=1}^{n-1} \left( k\_{i,a,j} - \frac{1}{2} \right) y\_{i,j}^2 - \left( k\_{i,n} - \frac{1}{2} \right) z\_{i,n}^2 \\ &\quad + \sum\_{j=1}^{n-1} m\_j + \frac{1}{2} \bar{\varepsilon}\_i + \lambda\_l \theta\_i^{p+1} + \upsilon\_l \theta\_i^{q+1} \end{split} \tag{34}$$

By choosing the control parameters, *ki*,*α*,*<sup>j</sup>* > <sup>1</sup> 2 , *ki*,*<sup>n</sup>* > <sup>1</sup> <sup>2</sup> we have

$$\begin{array}{lcl}\dot{V} & \leq & -\sum\_{j=1}^{n} p\_{i,j} z\_{i,j}^{p+1} - \sum\_{j=1}^{n} q\_{i,j} z\_{i,j}^{q+1} - \sum\_{j=1}^{n} l\_{i,p,j} y\_{i,j}^{p+1} \\ & -\sum\_{j=1}^{n} l\_{i,q,j} y\_{i,j}^{q+1} - \gamma\_{i} \widehat{\vartheta}\_{i}^{p+1} - \nu\_{i} \widehat{\vartheta}\_{i}^{q+1} \\ & +\sum\_{j=1}^{n-1} m\_{j} + \frac{1}{2} \overline{\varepsilon}\_{i} + \lambda\_{i} \theta\_{i}^{p+1} + \upsilon\_{i} \theta\_{i}^{q+1} \\ & \leq -aV^{\frac{p+1}{2}} - bV^{\frac{q+1}{2}} + c \end{array} \tag{35}$$

where

$$\begin{array}{c} a = \frac{\min\left\{p\_{i,j\in N}l\_{i,p,j\in N}\gamma\_{i}\right\}}{\left(\max\left\{\frac{1}{2},\frac{1}{2\mu\_{i}}\right\}\right)^{\frac{p+1}{2}}}, b = \frac{(2n)^{\frac{1-q}{2}}\min\left\{q\_{i,j\in N}l\_{i,q,j\in N}\nu\_{i}\right\}}{\left(\max\left\{\frac{1}{2},\frac{1}{2\mu\_{i}}\right\}\right)^{\frac{q+1}{2}}} \\ c = \sum\_{j=1}^{n-1} m\_{j} + \frac{1}{2}\overline{\varepsilon}\_{i} + \lambda\_{i}\theta\_{i}^{p+1} + \upsilon\_{i}\theta\_{i}^{q+1} \end{array} \tag{36}$$

Therefore, the error system is fixed-time, practical, and stable.

**Remark 3:** *From the definition of Lyapunov stability, Lyapunov stability, asymptotic stability, and finite-time stability are the three most basic concepts, and their definitions are progressive.*

*At present, the concepts of stability related to the convergence rate mainly include exponential stability, finite-time stability, and fixed-time stability. Exponential stability is the realization of asymptotic stability, while fixed-time stability is the generalization of finite-time stability. From the point of view of standard definitions, they are strictly distinguished, and from the point of view of controller design, they also have different forms.*

*From the perspective of system convergence time, Lyapunov stability can be divided into infinite-time stability and finite-time stability. In the theoretical analysis of stability, the most common exponential stability is infinite-time stability. Infinite-time stability means that when time tends to infinity, the system state can converge exponentially by designing the parameters of the controller. Finite-time stability means that the system can be stable at a certain time and continue to maintain a stable state by designing the parameters of the controller. For practical engineering control, finite-time stability is obviously more practical than infinite-time stability, but there are some limitations to finite-time stability; for instance, the convergence time of the designed system depends on the initial state. Therefore, in this section, another scheme was introduced: the tracking control method for nonlinear interconnected systems based on fixed-time stability theory. The setting time does not depend on the initial parameters and can be realized only by adjusting the controller parameters. In other words, under the condition that the controlled system is stable in fixed time, even if the initial parameters are changed, the controlled system can still be stable within the originally designed fixed time without redesigning the controller. This greatly improves the potential of the control method to be used in practical applications.*

**Remark 4:** *The design details are summarized in Figure 2. The above-listed step procedure is detailed below.*

*Step 1: Design of the ideal virtual control laws (4) and (14), based on backstepping control technology.*

*Step 2: Design of the virtual control laws (5) and (15), based on the fixed-time low pass filter.*

*Step 3: The actual controller (25) is obtained recursively through the virtual control signal and the adaptive parameter (26).*

$$\begin{aligned} \text{\textbf{\underline{\underline{\text{\underline{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\$}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$$

**Figure 2.** Design procedure.

#### **4. Numerical Examples**

The main purposes of the simulation studies include: (1) validating the effectiveness of the adaptive fixed-time neural network tracking controller for nonlinear interconnected systems; (2) showing a step-by-step procedure for building the adaptive fixed-time tracking control system.

A. Numerical Example

Consider the following nonlinear interconnected system:

$$\begin{cases} \dot{\mathbf{x}}\_{1,1} = \mathbf{x}\_{1,2} \\ \dot{\mathbf{x}}\_{1,2} = \sin(\mathbf{x}\_{1,1}) + u\_1 + h\_{1,2} \\ y\_1 = \mathbf{x}\_{1,1} \\ \dot{\mathbf{x}}\_{2,1} = \mathbf{x}\_{2,2} \\ \dot{\mathbf{x}}\_{2,2} = \sin(\mathbf{x}\_{2,1}) + u\_2 + h\_{2,2} \\ y\_2 = \mathbf{x}\_{2,1} \end{cases} \tag{37}$$

where the function, *h*1,2 = sin(*x*2,1) − sin(*x*1,1). *h*2,2 = sin(*x*1,1) − sin(*x*2,1).

Step 1: Design of the fixed-time ideal virtual control law and fixed-time adaptive law of neural networks:

$$
\overline{\mathfrak{a}}\_{i,1} = -2z\_{i,1}^{\frac{1}{3}} - 2z\_{i,1}^3 \tag{38}
$$

$$\dot{\theta}\_i = 0.01 \left( |z\_i| |\Psi\_i| - 0.1 \dot{\theta}\_i^{\frac{5}{3}} - 0.1 \dot{\theta}\_i^{\frac{1}{3}} \right) \tag{39}$$

Step 2: Design of the fixed-time low pass filter:

$$\dot{\alpha}\_{i,1} = -2y\_{i,1}^{\frac{1}{3}} - 2y\_{i,1}^3 - 2y\_{i,1} - z\_{i,1}, \alpha\_{i,1}(0) = 0 \tag{40}$$

Step 3: The actual controller is obtained recursively through the virtual control signal and the adaptive parameter, and the control input is designed as

$$u\_i = -z\_{i,1} - 2z\_{i,2}^{\frac{1}{3}} - 2z\_{i,2}^3 - 2z\_{i,2} - \operatorname{sign}(z\_{i,2})\hat{\theta}\_i || \Psi\_i || + \dot{a}\_{i,1} \tag{41}$$

The initial condition is selected as *<sup>x</sup>*11(0) <sup>=</sup> <sup>−</sup>2, *<sup>x</sup>*21(0) <sup>=</sup> 2, <sup>ˆ</sup>*θ*1(0) <sup>=</sup> 0, <sup>ˆ</sup>*θ*2(0) <sup>=</sup> <sup>0</sup> The neural network consists of seven nodes, the centers *c* = [−3, −2, −1, 0, 1, 2, 3], and the widths *b* = 1, respectively.

Figure 1 shows the control structure of the closed-loop system. Figure 2 shows the stepby-step design procedure. Figures 3–5 are the simulation results. Figure 3 shows the output of the interconnected system and the output of the interconnected system's convergence to the origin point in finite time, which indicates the control performance of the fixed-time neural network adaptive control. Figure 4 shows the controller of the interconnected system, which is bound and realizable. Figure 5 shows the trajectories of error of the low pass filter, and the error between the virtual control and ideal virtual control indicates that the virtual control is close to the ideal virtual control and that the differential coefficient exists. The simulation results show that the controlled system can become stable in 14 s, and the convergence time can be designed. Even if the states' initial parameters change, the controlled system can still become stable in 14 s. The system includes unknown nonlinear functions, which verify that the neural network control scheme has strong adaptive ability and approximation ability.

**Figure 3.** Trajectories of *x*1,1 and *x*2,1 of the interconnected system.

**Figure 4.** Trajectories of the controller of the interconnected system.

**Figure 5.** Trajectories of error of the low pass filter.

#### B. Application Example

Consider a pendulum system [25], with two degrees, as an interconnected system (Figure 6), which has been used as an example of decentralized neural control.

$$\begin{cases} \dot{\mathbf{x}}\_{1,1} = \mathbf{x}\_{1,2} \\ \dot{\mathbf{x}}\_{1,2} = \frac{1}{f\_1}\mathbf{u}\_1 + \frac{kr}{2f\_1}(l-b) + \left(\frac{m\_{1}gr}{f\_1} - \frac{kr^2}{4f\_1}\right)\sin(\mathbf{x}\_{1,1}) + \frac{kr^2}{4f\_1}\sin(\mathbf{x}\_{2,1}) \\ y\_1 = \mathbf{x}\_{1,1} \\ \dot{\mathbf{x}}\_{2,1} = \mathbf{x}\_{2,2} \\ \dot{\mathbf{x}}\_{2,2} = \frac{1}{f\_2}\mathbf{u}\_2 + \frac{kr}{2f\_2}(l-b) + \left(\frac{m\_{2}gr}{f\_2} - \frac{kr^2}{4f\_2}\right)\sin(\mathbf{x}\_{2,1}) + \frac{kr^2}{4f\_2}\sin(\mathbf{x}\_{2,1}) \\ y\_2 = \mathbf{x}\_{2,1} \end{cases} \tag{42}$$

**Figure 6.** Two inverted pendulums connected by an unknown device.

Step 1: Design of the fixed-time ideal virtual control law and fixed-time adaptive law of neural networks:

$$
\overline{\mathfrak{a}}\_{i,1} = -2z\_{i,1}^{\frac{1}{3}} - 2z\_{i,1}^3 \tag{43}
$$

$$\dot{\theta}\_i = 0.01 \left( |z\_i| |\Psi\_i| - 0.1 \theta\_i^{\frac{5}{3}} - 0.1 \theta\_i^{\frac{1}{3}} \right) \tag{44}$$

Step 2: Design of the fixed-time low pass filter:

$$\dot{\mathfrak{a}}\_{i,1} = -2y\_{i,1}^{\frac{1}{3}} - 2y\_{i,1}^3 - 2y\_{i,1} - z\_{i,1} \mathfrak{a}\_{i,1}(0) = 0 \tag{45}$$

Step 3: The actual controller is obtained recursively through the virtual control signal and the adaptive parameter, and the control input is designed as

$$u\_i = -z\_{i,1} - 2z\_{i,2}^{\frac{1}{3}} - 2z\_{i,2}^3 - 2z\_{i,2} - \operatorname{sign}(z\_{i,2})\theta\_i \|\Psi\_i\| + \dot{a}\_{i,1} \tag{46}$$

The initial condition is selected, and *m*<sup>1</sup> = 2 kg , *m*<sup>2</sup> = 2.5 kg, *J*<sup>1</sup> = 0.5 kg, *J*<sup>2</sup> = 0.625 kg, *k* = 100 N/M, *r* = 0.5 m, *l* = 0.5 m, *g* = 9.81 m/s<sup>2</sup> , *b* = 0.4 m and the function, *y*1,*<sup>d</sup>* = 0.5 sin(*t*). *y*2,*<sup>d</sup>* = 0.5 cos(*t*) The neural network consists of seven nodes, the centers, *c* = [−3, −2, −1, 0, 1, 2, 3] and the widths, respectively.

The simulation results are shown in Figures 7–10. From the simulation results, it can be seen that the system output can track the desired signal in 2.5 and 5 s, respectively, and the setting time can be designed. Even if the states' initial parameter change, the controlled system can still become stable in the same time. The pendulum system includes unknown nonlinear functions, which verify that the neural network control scheme has strong adaptive ability and approximation ability. The pendulum system also includes interconnected items, which verify that the control scheme can be applied to interconnected nonlinear control systems.

**Figure 7.** Trajectories of *x*1,1 and *y*1,*<sup>d</sup>* of the interconnected system.

**Figure 8.** Trajectories of *x*2,1 and *y*2,*<sup>d</sup>* of the interconnected system.

**Figure 9.** Trajectories of the controller of the interconnected system.

**Figure 10.** Trajectories of error of the low pass filter.

#### **5. Conclusions**

In this paper, through the design of a timing adaptive law for a neural network error system based on the Lyapunov fixed-time stability theorem, the unknown parameters of neural networks are iterated in fixed time. The convergence time can be designed only by modifying the parameters of the system controller and the adaptive rate, and it does not depend on the initial conditions. The fixed-time Lyapunov stability theorem is proposed, and the strict mathematical proof is completed, which will have more practical significance than the ideal stability analysis based on Lyapunov stability theory. Furthermore, the algorithm flow chart is given, which can be used by engineers to realize the proposed tracking control method by a computer for practical engineering. However, the control scheme based on neural networks also has some limitations. Firstly, because it cannot guarantee the asymptotic stability of the system, it is suitable for the controlled system, whose control objective is bounded stability. Secondly, the pure feedback structure is more

general than strict feedback, and pure feedback has more application value; therefore, pure feedback interconnected system fixed-time neural network control results should be presented in future works.

**Author Contributions:** Software, X.X. and C.S.C.; Writing—original draft, Y.L.; Writing—review & editing, J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Kezhen Han \* , Changzhi Chen, Mengdi Chen and Zipeng Wang**

School of Electrical Engineering, University of Jinan, Jinan 250022, China; 202021100375@mail.ujn.edu.cn (C.C.); 202021200742@mail.ujn.edu.cn (M.C.); cse\_wangzp@ujn.edu.cn (Z.W.)

**\*** Correspondence: cse\_hankz@ujn.edu.cn

**Abstract:** A new active fault tolerant control scheme based on active fault diagnosis is proposed to address the component/actuator faults for systems with state and input constraints. Firstly, the active fault diagnosis is composed of diagnostic observers, constant auxiliary signals, and separation hyperplanes, all of which are designed offline. In online applications, only a single diagnostic observer is activated to achieve fault detection and isolation. Compared with the traditional multi-observer parallel diagnosis methods, such a design is beneficial to improve the diagnostic efficiency. Secondly, the active fault tolerant control is composed of outer fault tolerant control, inner fault tolerant control and a linear-programming-based interpolation control algorithm. The inner fault tolerant control is determined offline and satisfies the prescribed optimal control performance requirement. The outer fault tolerant control is used to enlarge the feasible region, and it needs to be determined online together with the interpolation optimization. In online applications, the updated state estimates trigger the adjustment of the interpolation algorithm, which in turn enables control reconfiguration by implicitly optimizing the dynamic convex combination of outer fault tolerant control and inner fault tolerant control. This control scheme contributes to further reducing the computational effort of traditional constrained predictive fault tolerant control methods. In addition, each pair of inner fault tolerant control and diagnostic observer is designed integratedly to suppress the robust interaction influences between estimation error and control error. The soft constraint method is further integrated to handle some cases that lead to constraint violations. The effectiveness of these designs is finally validated by a case study of a wastewater treatment plant model.

**Keywords:** active diagnosis; active reconfiguration; constrained systems; fault tolerance; interpolation control; linear programming

#### **1. Introduction**

Fault tolerance is already a common design property to be considered for most control systems. In terms of the system structure, faults can be classified as sensor faults, actuator faults, and component/parameter faults [1,2]. In general, the first two do not directly affect the intrinsic stability of the system, while the component faults tend to directly change the dynamic characteristics of the system. In the literature, the methods to handle these types of faults can be divided into active fault tolerant control (AFTC) and passive fault tolerant control (PFTC) [3]. PFTC draws on robust control theory to suppress the effects of faults, while AFTC uses fault information to adjust or reconfigure control actions to match the dynamics of the faulty system. Due to such matching adjustments, AFTC typically provides better reliability than PFTC. Many representative results can be found in the survey papers [4–7].

Recently, the design of optimal AFTC for systems with state/input constraints has been received a lot of attention. Unlike the design of unconstrained FTC, the design of constrained FTC has to take into account more requirements, including robust stability, feasibility, optimization efficiency, etc. Particularly, the faults occurring in constrained

**Citation:** Han, K.; Chen, C.; Chen, M.; Wang, Z. Constrained Active Fault Tolerant Control Based on Active Fault Diagnosis and Interpolation Optimization. *Entropy* **2021**, *23*, 924. https://doi.org/ 10.3390/e23080924

Academic Editors: Quanmin Zhu, Giuseppe Fusco, Jing Na, Weicun Zhang and Ahmad Taher Azar

Received: 24 June 2021 Accepted: 15 July 2021 Published: 21 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

systems often cause constraint violations, and the unconstrained FTC designed without considering feasibility may result in an empty set of feasible solutions for a given control objective. This often further leads to the eventual loss of closed-loop stability. In the literature, some typical design methods for constrained FTC have been reported, such as Barrier Lyapunov function method [8], command governor [9,10], saturation control [11], model predictive control (MPC) [12], etc. Among these methods, the MPC-based FTC method is widely considered, since MPC has the inherent and flexible capacity to address constrained optimization problems. The representative studies include FTC based on min-max MPC [13], FTC based on explicit MPC [14], multi-actuator/sensor FTC based on set theoretic MPC [15–18], FTC based on dual model MPC [19,20], etc.

Most of the above mentioned MPC-based FTC designs are developed for actuator and sensor faults, whereas relatively few results are reported for component/parameter faults. Since the component faults often change the structural parameters of the system, determining the real-time operating mode of the system is a prerequisite for achieving fault tolerance. A common approach to this problem is to use multiple observers to first discriminate the fault modes, and then activate the corresponding control law of the isolated mode to achieve switching control reconfiguration [21]. Such an approach can be viewed as a passive fault diagnosis (PFD)-based AFTC scheme. Actually, due to the potential lack of diagnostically relevant information in the input–output data, the PFD method may fail to isolate a fault or may isolate a fault incorrectly. Moreover, for high-dimension systems, the multiple observers for parallel applications usually occupy a large amount of memory, and the involved modal discriminant optimization problem is generally computationally demanding. One promising way is to integrate the active fault diagnosis (AFD) methods into FTC, i.e., the AFD-based AFTC. The central idea in AFD is to design a small harmful test/auxiliary input signal that can ensure maximal or full separation among the model predictions corresponding to the different modes of operation [22]. According to different design methods of AFD, some representative results have been presented, such as AFTC based on Youla–Kucera parametrization [23], AFTC based on set detection and isolation [24], AFTC based on performance transformation [25], AFTC based on distributed fault isolation [26], etc.

The above studies have provided different ideas for the construction of AFD and AFTC. Inspired by these results, we find two more problems whose handling can be further improved:


The main idea is to optimize an interpolation coefficient in real time based on the updated system states and use this coefficient to make a smooth convex combination of a outer controller and a inner controller. The outer controller is used to enlarge the controllable feasible domain, while the inner controller is used to satisfy the given control performance requirements. In general, the inner controller is optimally designed offline, while the outer controller is determined online simultaneously when the interpolation coefficient is optimized. This method of offline designing some parameters of the controller in advance helps to reduce the online calculation burden. Moreover, the optimized interpolation coefficient enables a smooth transition between the inner–outer controllers and ensures a fast convergence of the states to the set point under the constraints. In particular, the associated optimization problem belongs to standard linear programming (LP), which can be readily solved in the practical implementation. Given these characteristics, the IC-based optimization can provide a good compromise among computational load, feasible region size, performance, etc. Therefore, the development of the IC strategy to solve the constrained AFTC problem would be very promising. To the authors' knowledge, no relevant results have been reported.

Motivated by the above observation, we seek to further push the development of the field of constrained FTC for component/actuator faults by proposing a new AFD-based interpolating FTC synthesis scheme. The central ideas of the technical route are: (1) the passive fault detection (FD) is firstly designed by using a diagnostic observer in the current mode; (2) after a fault is detected, the active fault isolation (FI) and mode identification are then achieved by using a constant test signal and a separated hyperplane; and (3) after the actual mode is isolated, the constrained AFTC is finally determined by virtue of optimizing the interpolation coefficient to combine the inner FTC and outer FTC. How to comprehensively solve the problems involved in this technical route is the main research content of this paper.

Compared with the recent results on constrained AFTC studies (e.g., [13,16,20]), our main contributions can be reflected in the follows aspects: (i) A new and efficient AFDbased AFTC approach for component/actuator faults is proposed. In this work, only one observer is applied in real time to achieve FD and FI, while most of the existing studies use multiple observers for online parallel diagnosis; the fault mode separation is achieved by using auxiliary signals and separating hyperplanes designed offline, rather than by solving receding horizon optimizations and set membership discriminations online; the real-time control reconfiguration-based AFTC is achieved by solving simple LP problems instead of solving quadratic or semi-definite positive programming problems. (ii) When designing diagnostic observers and FTCs, the interaction influences between estimation error and control error is further handled based on integrated design and constraint tightening so as to improve the robust feasibility of AFTC optimization algorithm. (iii) The soft constraints IC-based AFTC strategy is also designed to address some infeasible scenarios, such as, the deviation of states from the maximum controllable invariant set after fault isolation, or the constraints violation caused by some unanticipated factors.

The remainder of this paper is structured as follows. Section 2 provides the problem formulation. In Section 3, the proposed AFD-based interpolating AFTC scheme is explained in detail and an integrated algorithm is also given to summarize the involved offline design and online application steps. In Section 4, the algorithm verification is given. Some conclusion and future work are discussed in Section 5.

**Notation 1.** *diag*{*X*1, *X*2, *X*3} *is a diagonal matrix with diagonal elements X*1*, X*2*, and X*3*. A <sup>T</sup>P*(∗) = *<sup>A</sup> <sup>T</sup>PA. 1<sup>m</sup> is a m-dimensional column vector with all elements of* 1*, while I<sup>m</sup> is a m-dimensional unitary matrix. Let <sup>p</sup>* <sup>∈</sup> *<sup>P</sup> and <sup>q</sup>* <sup>∈</sup> *<sup>Q</sup> be two sets of* <sup>R</sup>*<sup>n</sup> . Then, P* ⊕ *Q* = {*p* + *q*|*p* ∈ *P*, *q* ∈ *Q*} *is the Minkowsi sum of two sets. For two sets satisfying Q* ⊂ *P, x* ∈ *P* ∼ *Q represents x* ∈ *P, but x* ∈/ *Q. A polyhedron is the intersection of a finite number of open and/or closed half-spaces, and a polytope is a closed and bounded polyhedron.*

#### **2. System Description and Problem Formulation**

Consider the following uncertain discrete-time systems affected by unknown component faults, actuator faults and disturbances:

$$\mathbf{x}\_{k+1} = A^l \mathbf{x}\_k + B^l \boldsymbol{u}\_k + d\_{k'} \ y\_k = \mathbb{C} \mathbf{x}\_k + v\_k \tag{1}$$

where *<sup>x</sup><sup>k</sup>* ∈ X ⊂ <sup>R</sup>*<sup>n</sup>* is the state vector; *<sup>u</sup><sup>k</sup>* ∈ U ⊂ <sup>R</sup>*n<sup>u</sup>* is the actuator input vector; *<sup>d</sup><sup>k</sup>* ∈ D ⊂ <sup>R</sup>*n<sup>d</sup>* is the unknown process disturbance vector; *<sup>v</sup><sup>k</sup>* ∈ V ⊂ <sup>R</sup>*n<sup>v</sup>* is the unknown measurement disturbance vector; *y<sup>k</sup>* ∈ R *ny* is the measurement output vector. The matrices *A l* , *B <sup>l</sup>* and *C* are constant and have appropriate dimensions. The index *l* is associated with the configuration in which the system is actually operating, i.e., (*A l* , *B l* ) ∈ {(*A* 0 , *B* 0 ), (*A* 1 , *B* 1 ), · · · ,(*A nf* , *B <sup>n</sup>f*)}, *<sup>l</sup>* <sup>∈</sup> [0, *<sup>n</sup><sup>f</sup>* ]. Without loss of generality, we assume that *l* = 0 corresponds to the healthy condition (*A* 0 , *B* 0 ) while any other *l* ≥ 1 corresponds to a faulty condition. In addition, X , U, D, V are defined as the bounded polyhedral constraint sets [30,31]: <sup>X</sup> <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : *<sup>H</sup><sup>x</sup> <sup>x</sup>* <sup>≤</sup> *<sup>b</sup>x*}, <sup>U</sup> <sup>=</sup> {*<sup>u</sup>* <sup>∈</sup> <sup>R</sup>*n<sup>u</sup>* : *<sup>H</sup>u<sup>u</sup>* <sup>≤</sup> *<sup>b</sup>u*}, <sup>D</sup> <sup>=</sup> {*<sup>d</sup>* <sup>∈</sup> <sup>R</sup>*n<sup>d</sup>* : *<sup>H</sup>d<sup>d</sup>* <sup>≤</sup> *<sup>b</sup>d*}, <sup>V</sup> <sup>=</sup> {*<sup>v</sup>* <sup>∈</sup> <sup>R</sup>*b<sup>v</sup>* : *Hvv* ≤ *bv*}, where *Hx*, *Hu*, *H<sup>d</sup>* , *Hv*, *bx*, *bu*, *b<sup>d</sup>* , *b<sup>v</sup>* are predetermined.

**Remark 1.** *The model* (1) *can represent some uncertainties. Firstly, the changes in the configuration of the system (i.e., l* ∈ [0, *n<sup>f</sup>* ]*) due to the appearance or disappearance of faults are essentially a description of the uncertainty of the system [1]. Secondly, the disturbance terms* (*d<sup>k</sup>* , *v<sup>k</sup>* ) *included in the model can directly reflect the multiple uncertainties in the system. For instance, let A <sup>l</sup>* = *A* 0 *, d<sup>k</sup>* = ∆*Ax<sup>k</sup>* + *δ<sup>k</sup> with unknown but bounded term* ∆*A,* (1) *can represent a class of additive parametric uncertainty models; let A <sup>l</sup>* = *A* 0 *, d<sup>k</sup>* = *A* 0 (*I* − ∆*A*)*x<sup>k</sup>* + *δ<sup>k</sup> ,* (1) *can represent a class of multiplicative parametric uncertainty models; let d<sup>k</sup> be a time-varying/time-invariant uncertainty term only,* (1) *can represent the uncertainty case for a class of mechanistic models with bounded offsets of modeling error, etc. All of these scenarios can be used to reflect a mismatch between the model and the reality.*

**Remark 2.** *The model* (1) *can represent both component and actuator faults [1]. For example, A <sup>l</sup>* = *A* <sup>0</sup> + ∑ *n <sup>i</sup>*=<sup>1</sup> *A i θ i <sup>k</sup> with unknown faulty factor θ i k can represent some component/parameter faults; B <sup>l</sup>* = *B* <sup>0</sup>*diag*{*<sup>θ</sup>* 1 *k* , *θ* 2 *k* , · · · , *θ nu k* } *with θ i k* ∈ [0, 1] *can describe some actuator effectiveness loss faults.*

For the sake of simplicity, the dynamics of the *l*-th system configuration can be rewritten as

$$\begin{aligned} \mathbf{x}\_{k+1}^{l} &= A^{l}\mathbf{x}\_{k}^{l} + B^{l}u\_{k}^{l} + d\_{k\prime} \ y\_{k}^{l} = \mathbb{C}\mathbf{x}\_{k}^{l} + v\_{k} \\ \text{s.t. } \mathbf{x}\_{k} &\in \mathcal{X}, \ u\_{k} \in \mathcal{U}, \ d\_{k} \in \mathcal{D}, \ v\_{k} \in \mathcal{V} \end{aligned} \tag{2}$$

The following assumptions are given for systems (1) and (2).

**Assumption 1.** *The typical system configurations of concern can be modeled in advance, and these system configurations are controllable.*

**Remark 3.** *We recognize that not all systems and faults can be tolerant by only one FTC method. Therefore, we make the above assumptions to explain the situations in which the proposed method can be applied.*

**Definition 1.** *Let* S *be a neighborhood of the origin. The closed-loop trajectory of* (1) *is said to be Uniformly Ultimately Bounded (UUB) in* S*, if* ∀ *x*0*,* ∃*T*(*x*0) > 0 *such that x<sup>k</sup>* ∈ S *for k* ≥ *T*(*x*0)*.*

The control objective is to construct an AFD-based robust and feasible AFTC strategy such that the states of the controlled system (1) can be steered inside a neighborhood of origin (i.e., UUB) in a way of minimizing the following optimization problem

$$\begin{aligned} \min\_{\boldsymbol{u}\_{k}} \mathcal{J}\_{\mathbf{x}\_{k}, \boldsymbol{u}\_{k}} &= \sum\_{t=0}^{\infty} \mathcal{U}(\mathbf{x}\_{k+t}, \boldsymbol{u}\_{k+t})\\ \text{s.t. } \mathbf{x}\_{k+1}^{l} &= A^{l} \mathbf{x}\_{k}^{l} + B^{l} \boldsymbol{u}\_{k}^{l} + d\_{k}, \ y\_{k}^{l} = \mathbf{C} \mathbf{x}\_{k}^{l} + v\_{k} \\ \quad & \boldsymbol{x}\_{k} \in \mathcal{X}, \ \boldsymbol{u}\_{k} \in \mathcal{U}, \ d\_{k} \in \mathcal{D}\_{\ \ \ } v\_{k} \in \mathcal{V} \\ \quad & l \in [0, \ n\_{f}] \end{aligned} \tag{3}$$

where *U*(*x<sup>k</sup>* , *u<sup>k</sup>* ) = *x T k* Ξ*x<sup>k</sup>* + *u T <sup>k</sup>* Θ*u<sup>k</sup>* , Ξ > 0, Θ > 0 is a utility function.

#### **3. Main Results**

#### *3.1. The Overall Scheme of the Proposed AFD-Based Interpolation AFTC Method*

The overall scheme of the proposed AFD-based interpolation AFTC method is shown in Figure 1. In the subsequent analysis, we let that the index *l* ∈ [0, *n<sup>f</sup>* ] denotes the unknown actual system operating condition and the index *i* ∈ [0, *n<sup>f</sup>* ] denotes the recently identified system operating condition. Then, according to the flowchart in Figure 1, the AFTC method works as explained below. First, the I/O data of the practical system (i.e., the *l*th model) is collected by the *i*th estimator to give the state estimates *x*ˆ *i k* and generate the residuals *r i k* . Second, the fault detection unit performs change detection based on the estimator outputs. When there is no change (i.e., *l* = *i*), the interpolation control algorithm currently in use continues to regulate the system. When a change/fault is detected (i.e., *l* 6= *i*), the fault isolation unit is activated, and in this case the pre-designed auxiliary test signal *u i FI* is injected into the system and the estimator to perform modal discrimination. Next, after the practical system condition is isolated (i.e., *i* = *l*), the decision results of the fault isolation unit will update the operating condition index of the estimator and the reconfiguration controller. Next, the suitable interpolation optimization should be selected according to the location of states in relation to the feasible set of controller (i.e., robust control invariant set). Namely, if the states belong to the feasible set of the isolated controller, the general interpolation control is applied; otherwise, the relaxed interpolation control should be activated. Finally, these control actions will adjust the system states to the desired operating region. The design of each unit in this flowchart is given in detail below.

**Figure 1.** Scheme of AFD-based interpolation AFTC. The fault detection and active fault isolation constitute AFD, which will be designed in Section 3.2; AFTC consists of outer FTC, inner FTC and interpolation optimization, where outer FTC and interpolation optimization are designed in Section 3.4 and inner FTC with observer is designed in Section 3.3.

#### *3.2. AFD: Fault/Mode Change Detection and Isolation*

Without loss of generality, the following *i*-th observer is adopted to estimate states and generate residuals

$$\begin{aligned} \hat{\mathfrak{x}}\_{k+1}^{i} &= (A^{i} + L^{i}\mathbb{C})\hat{\mathfrak{x}}\_{k}^{i} + \mathcal{B}^{i}u\_{k}^{i} - L^{i}y\_{k}^{i} \\ \hat{y}\_{k}^{i} &= \mathbb{C}\hat{\mathfrak{x}}\_{k'}^{i}r\_{k}^{i} = y\_{k}^{i} - \hat{y}\_{k}^{i} \end{aligned} \tag{4}$$

where *x*ˆ *i k* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* denotes the estimated state vector; *<sup>y</sup>*<sup>ˆ</sup> *i k* ∈ R *ny* is the estimated output vector; *r i k* ∈ R *ny* is the generated residual signal that is used to provide key information of abnormal condition for achieving AFD. *L i* is the observer gain.

**Assumption 2.** *For the sake of discussion, we assume that the observer* (4) *for each i* ∈ [0, *n<sup>f</sup>* ] *has been designed in advance, and* (*A <sup>i</sup>* + *L <sup>i</sup>C*) *is Schur stable. The detailed design conditions of L i are given in Theorem 1.*

**Remark 4.** *In a cycle of AFD, the FI is always triggered by the FD [22]. Moreover, when a fault is detected at time k<sup>d</sup> , the closed-loop FTC controller that is currently being used should preferably be put on standby to avoid that the feedback function hides the effect of the fault. In this setting, only the auxiliary input is used to stimulate the faulty system. In principle, the design of such an auxiliary input should (1) minimize the harmful influence to the currently matched system operation and (2) accurately identify and isolate the real system operating condition l.*

In Figure 1, there are two cases about the generated residual signal *r i k* . One is that the *i*th observer currently in use is matched to the real system mode *l*, and the other is the opposite. In the sequel, we will discuss the characteristics of the corresponding residuals for each of these two cases.

*(1) Case I (design of FD logic for i* = *l)*: First, based on Remark 4 and (1)–(4), the following estimation error system can be established:

$$e\_{\mathbf{x}^i,k+1} = (A^i + L^i \mathbb{C})e\_{\mathbf{x}^i,k} + L^i v\_k + d\_{k\prime} \ r^i\_k = \mathbb{C}e\_{\mathbf{x}^i,k} + v\_k \tag{5}$$

where *e x i* ,*<sup>k</sup>* = *x i <sup>k</sup>* − *x*ˆ *i k* . Given *i* ∈ [0, *n<sup>f</sup>* ], the relevant disturbance term *L <sup>i</sup>v<sup>k</sup>* + *d<sup>k</sup>* is bounded by a deterministic set ∆ *i*,*i <sup>e</sup>* = (*L <sup>i</sup>*V) ⊕ D. Then, based on a series of finite set iterations along (5) using ∆ *i*,*i e* , an approximate maximal RPI set Ω *i*,*i e* (see Definition 2) can be computed and the limit set of residual *r i* can be directly obtained as R *i*,*i FD* = *C*Ω *i*,*i <sup>e</sup>* ⊕ V. According to Figure 1 and Remark 4, the detection of mode changes and the triggered action can be formulated as

$$\begin{cases} r\_k^i \notin \mathcal{R}\_{FD}^{i,i} \Rightarrow \text{Mode change} \Rightarrow \text{Active FI} \\ r\_k^i \in \mathcal{R}\_{FD}^{i,i} \Rightarrow \text{No change} \quad \Rightarrow \text{Continuous detection} \end{cases} \tag{6}$$

**Remark 5.** *Considering the possibility of fault occurrence, transformation, or recovery, we uniformly use mode change in* (6) *to indicate any phenomenon that causes a change in the system behavior.*

*(2) Case II (design of FI logic for i* 6= *l)*: The case *i* 6= *l* implies that the real status of system has changed and it generally leads to *r i k* ∈ R/ *i*,*i FD*. In this case, the fault/mode isolation should be activated. According to the analysis method in [27] and Remark 4, an auxiliary input *u i FI* will be used to replace the AFTC input *u i C*,*k* . A relevant augmentation representation is firstly constructed as

$$\mathbf{x}\_{k+1}^{l,i} = \mathbf{A}\_{\chi}^{l,i}\mathbf{x}\_{k}^{l,i} + \mathbf{B}\_{\chi}^{l,i}\mathbf{u}\_{\text{FI}}^{i} + \mathbf{E}\_{\chi}^{i}\sigma\_{\mathbf{k}\prime}\mathbf{r}\_{k}^{l,i} = \mathbf{C}\_{\chi}\chi\_{\mathbf{k}}^{l,i} + \mathbf{D}\_{\chi}\sigma\_{\mathbf{k}}\tag{7}$$

$$\begin{aligned} \text{where } \chi\_k^{l,i} = \begin{bmatrix} (\mathfrak{x}\_k^l)^T & (\mathfrak{x}\_k^i)^T \end{bmatrix}^T, A\_\chi^{l,i} = \begin{bmatrix} A^l & 0 \\ -L^i \mathbb{C} & A^i + L^i \mathbb{C} \end{bmatrix}^\prime, \sigma\_k \in \mathcal{E} = \{ \begin{bmatrix} d\_k^T & v\_k^T \end{bmatrix}^T : d \in \mathcal{D} \} \\ \mathcal{D}. v \in \mathcal{V} \}, B\_\chi^{l,i} = \begin{bmatrix} B^l \\ -\frac{1}{2} \end{bmatrix}, E\_\chi^i = \begin{bmatrix} I & 0 \\ -\frac{1}{2} \end{bmatrix}, \mathcal{C}\_\chi = \begin{bmatrix} \mathbb{C} & -\mathbb{C} \end{bmatrix}, D\_\chi = \begin{bmatrix} 0 & I \end{bmatrix}. \text{ Clearly, the} \end{aligned}$$

D, *v* ∈ V }, *B <sup>χ</sup>* = *B i* , *E <sup>χ</sup>* = 0 −*L i* , *C<sup>χ</sup>* = *C* −*C* , *D<sup>χ</sup>* = 0 *I* . Clearly, the term *E i <sup>χ</sup>σ<sup>k</sup>* lies in the set ∆ *i <sup>χ</sup>* = *E i <sup>χ</sup>*E. Then, given *u i FI* and based on Assumption 1, an approximate maximal RPI set Ω *l*,*i <sup>χ</sup>* for each pair (*l*, *i*), *i* 6= *l*, can be determined by finite set iterations along (7). Accordingly, the limit set that is used to achieve modal isolation can be obtained as R *l*,*i FI* = *Cχ*Ω *l*,*i <sup>χ</sup>* ⊕ *Dχ*E. The approximated calculation method of R *l*,*i FI* is given in Appendix A.

A crucial condition for the existence of *u i FI* that discriminates between configurations *ζ* and *η* in finite time is R *ζ*,*i FI* ∩ R*η*,*<sup>i</sup> FI* = ∅, *ζ* 6= *η*. According to [27], such discrimination can be achieved by checking whether the distance between the two sets is positive. Without loss of generality, the following distance metric is defined as

$$dis\_{\zeta,\eta}^{i} = \inf\_{(q\_{\zeta} \in \mathcal{R}\_{FI}^{\zeta,i}, p\_{\eta} \in \mathcal{R}\_{FI}^{\zeta,i})} \parallel q\_{\zeta} - p\_{\eta} \parallel\_{2} \tag{8}$$

Clearly, for each pair (*ζ*, *η*) ∈ {[0, *n<sup>f</sup>* ] ∼ *i*}, *ζ* 6= *η*, we need to solve (8) to determine a suitable auxiliary input *u i FI* such that the distance metric *dis<sup>i</sup> ζ*,*η* is positive. The distance metric (8) has the following properties.

**Lemma 1.** *[27] The distance metric function dis<sup>i</sup> ζ*,*η is convex and hence its maximum is reached on certain vertices of the input constraint set.*

Based on Remark 4 and Lemma 1, the optimization design problem of auxiliary input signal *u i FI*, ∀*i* ∈ [0, *n<sup>f</sup>* ] can then be formulated as

$$\begin{cases} \min \quad \gamma\\ \text{s.t.} \, dis\_{\zeta,\eta}^{i} > 0; \, u\_{FI}^{i} \in \text{vert} (\gamma \mathcal{U}); \, \sigma\_{k} \in \mathcal{E};\\ \quad \zeta\_{\prime}, \eta \in \{ [0, n\_{f}] \sim i \}; \, \zeta \neq \eta. \end{cases} \tag{9}$$

Once the problem in (9) is solved for each *i*, the corresponding separation hyperplane (denoted as Π*<sup>i</sup> ζ*,*η* ) that is used to isolate the new mode can be further calculated through

$$\begin{split} \Pi^{i}\_{\vec{\mathsf{T}},\eta} &= \{ r : (r - \mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\prime$$

where *r*˘ *<sup>ζ</sup>* ∈ R*ζ*,*<sup>i</sup> FI* and *r*˘ *<sup>η</sup>* ∈ R*η*,*<sup>i</sup> FI* are two points at minimum distance from <sup>Π</sup>*<sup>i</sup> ζ*,*η* , and they can be determined when solving (9). Then, these off-line designed separation hyperplanes will be used for real-time isolation. For simplicity, the isolation function is constructed as

$$Iso^i\_{\tilde{\mathbb{L}}, \eta} = \operatorname{sign}[(\mathbb{H}^\zeta - \mathbb{H}^\eta)^T r\_k - \frac{(\mathbb{H}^\zeta - \mathbb{H}^\eta)^T (\mathbb{H}^\zeta + \mathbb{H}^\eta)}{2}] \tag{11}$$

Then, for the residual signals generated in real time, the online FI logic can be designed as

$$\begin{cases} Iso^i\_{\zeta,\eta} > 0 \Rightarrow \text{Mode } \zeta \text{ is effective} \\ Iso^i\_{\zeta,\eta} < 0 \Rightarrow \text{Mode } \eta \text{ is effective} \end{cases} \tag{12}$$

The current system mode can thus be discerned by making no more than *n<sup>f</sup>* comparisons using (11) and (12).

#### *3.3. Integrated Design of Observer and Unconstrained Controller*

When the practical system mode index *l* ∈ [0, *n<sup>f</sup>* ] is isolated, the control reconfiguration should be activated immediately, i.e., the control action *u l C*,*k* is reconfigured with the new isolated mode index *l*. Now we will design the control policy *u l C*,*k* . Here, we consider for now the case where the constraints (*x* ∈ X , *u* ∈ U) are not triggered and *u l C*,*k* can then be designed only as an estimator-based robust feedback control policy *u l <sup>C</sup>*,*<sup>k</sup>* = *K lx*ˆ *l k* , ∀*l* ∈ [0, *n<sup>f</sup>* ]. Under such settings, the closed-loop system dynamics can be obtained as

$$\begin{split} \mathbf{x}\_{k+1}^{l} &= A^{l} \mathbf{x}\_{k}^{l} + B^{l} K^{l} \mathbf{x}\_{k}^{l} + d\_{k} \\ &= (A^{l} + B^{l} K^{l}) \mathbf{x}\_{k}^{l} - B^{l} K^{l} e\_{\mathbf{x}^{l},k} + d\_{k} \\ &= \bar{A}^{l} \mathbf{x}\_{k}^{l} + \bar{B}^{l} e\_{\mathbf{x}^{l},k} + d\_{k} \end{split} \tag{13}$$

where *A*¯*<sup>l</sup>* = *A <sup>l</sup>* + *B lK l* , and *<sup>B</sup>*¯*<sup>l</sup>* <sup>=</sup> <sup>−</sup>*<sup>B</sup> lK l* .

On the other hand, by defining a virtual output variable vector *z l <sup>k</sup>* = Ξ 1/2 0 *x l <sup>k</sup>* + 0 Θ1/2 *u l k* , the unity function of cost function (3) can be represented by *U*(*x l k* , *u l k* ) = (*z l k* ) *T z l k* . Then, the closed-loop virtual output by *u l C*,*k* can be deduced as

$$z\_k^l = \mathbb{C}^l x\_k^l + \mathbf{D}^l e\_{x^l,k} \tag{14}$$

where *C*¯*<sup>l</sup>* = Ξ 1/2 0 + 0 Θ1/2 *K <sup>l</sup>* and *<sup>D</sup>*¯ *<sup>l</sup>* <sup>=</sup> <sup>−</sup> 0 Θ1/2 *K l* .

According to [20,32], there may exist robustness interaction influences between estimation accuracy and unconstrained control performance, since the estimation error *e x l* ,*k* disturbs the closed-loop system (13) and (14) whilst the unmodeled dynamics *d<sup>k</sup>* usually containing states can affect the estimation system (5). Hence, an integrated design of composite closed-loop system (5), (13) and (14) must be adopted to obtain the satisfactory observer gain *L <sup>l</sup>* and control gain *K l* , ∀*l* ∈ [0, *n<sup>f</sup>* ]. The following composite closed-loop system is firstly established:

$$\begin{aligned} \psi\_{k+1}^{l} &= \tilde{A}^{l} \psi\_{k}^{l} + \tilde{B}^{l} \varrho\_{k} \\ z\_{k}^{l} &= \tilde{C}^{l} \psi\_{k}^{l} \end{aligned} \tag{15}$$

where *ψ l <sup>k</sup>* = h *e T x l* ,*k* (*x l k* ) *T* i*T* , *\$<sup>k</sup>* = - *v T k d T k T* , *A*˜*<sup>l</sup>* = *A*˜*l* <sup>11</sup> 0 *A*˜*l* <sup>21</sup> *<sup>A</sup>*˜*<sup>l</sup>* 22 , *B*˜*<sup>l</sup>* = *B*˜*l* 1 *B*˜*l* 2 , *A*˜*<sup>l</sup>* <sup>11</sup> = *A <sup>l</sup>* + *L <sup>l</sup>C*, *A*˜*<sup>l</sup>* <sup>21</sup> = −*B lK l* , *A*˜*<sup>l</sup>* <sup>22</sup> = *A <sup>l</sup>* + *B lK l* , *B*˜*<sup>l</sup>* <sup>1</sup> = - *L l I* , *B*˜ <sup>2</sup> = - 0 *I* , and *C*˜*<sup>l</sup>* = - *D*¯ *<sup>l</sup> C*¯*<sup>l</sup>* .

The following theorem presents the integrated design conditions of observer gain and unconstrained feedback gain.

**Theorem 1.** *For each l* ∈ [0, *n<sup>f</sup>* ]*, a robust observer* (5) *and associated robust feedback control policy uC*,*<sup>k</sup>* = *K lx*ˆ *l k can be integratedly determined, if some decision variables α* > 0*, β* > 0*,* *P l* <sup>1</sup> = (*P l* 1 ) *<sup>T</sup>* > 0*, P l* <sup>2</sup> = (*P l* 2 ) *<sup>T</sup>* > 0*, Y l* 1 *, Y l* 2 *, K*¯*<sup>l</sup> , L*¯ *<sup>l</sup> exist as the solutions to the following optimization problem:*

$$\begin{array}{ccccc}\min\limits\_{P\_1', P\_2', Y\_1 Y\_2', I, Y\_1 I} \xi a + (1 - \xi)\beta\\ \text{s.t.} & \begin{bmatrix} I - P\_1^l & \diamond & \diamond\\ 0 & -a^2 I & \diamond\\ \Gamma\_{31l} & \Gamma\_{32l} & \diamond\\ \Gamma\_{31l} & \Gamma\_{32l} & P\_1^l - Y\_1^l - (Y\_1^l)^T \end{bmatrix} < 0\\ & \begin{bmatrix} P\_2' - Sym(Y\_2^l) & \diamond & \diamond & \diamond & \diamond\\ 0 & P\_2 - Sym(Y\_2^l) & \diamond & \diamond & \diamond\\ 0 & 0 & -\beta^2 I & \diamond & \diamond\\ \Upsilon\_{41l} & \Upsilon\_{42l} & \tilde{B}\_2 & -P\_2^l & \diamond\\ \Upsilon\_{51l} & \Upsilon\_{52l} & 0 & 0 & -I \end{bmatrix} < 0\\ \end{array} \tag{16b}$$

*where Sym*(*Y l* 2 ) = *Y l* <sup>2</sup> + (*Y l* 2 ) *T ,* Γ31*<sup>l</sup>* = *Y l* 1*A <sup>l</sup>* + *L*¯ *<sup>l</sup>C,* Γ32*<sup>l</sup>* = - *L*¯ *<sup>l</sup> Y l* 1 *,* Υ41*<sup>l</sup>* = −*B lK*¯*l ,* Υ42*<sup>l</sup>* = *A l* (*Y l* 2 ) *<sup>T</sup>* + *B lK*¯*l , B*˜*<sup>l</sup>* <sup>2</sup> = - 0 *I ,* Υ51*<sup>l</sup>* = − 0 Θ1/2 *K*¯*l ,* Υ52*<sup>l</sup>* = Ξ 1/2 0 (*Y l* 2 ) *<sup>T</sup>* + 0 Θ1/2 *K*¯*l , ς* ∈ (0, 1)*. Once the above optimization is solved, the parameters of observer and feedback gain can be calculated by K<sup>l</sup>* = *K*¯*<sup>l</sup>* ((*Y l* 2 ) *T* ) −1 *and L<sup>l</sup>* = (*Y l* 1 ) <sup>−</sup>1*L*¯ *<sup>l</sup> , respectively.*

**Proof.** The proof of Theorem 1 is given in Appendix B.

*3.4. Constrained AFTC: Reconfigured Interpolating Control*

Based on the set-theoretic concepts in [28,31], several invariant sets are defined.

**Definition 2.** *Given the controller u l <sup>C</sup>*,*<sup>k</sup>* = *K lx*ˆ *l k , the set* Ω*<sup>l</sup> RPI* ⊆ X *is a robust positive invariant set (RPI-set) for closed-loop system* (13) *subject to constraint x l k* ∈ X *if for any x l* <sup>0</sup> <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> RPI we have x l k* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> RPI for all <sup>B</sup>*¯*<sup>l</sup> e x l* ,*<sup>k</sup>* + *d<sup>k</sup> , k* > 0*. Moreover,* Ω*<sup>l</sup> MRPI is the maximal RPI-set if* <sup>Ω</sup>*<sup>l</sup> MRPI contains all the RPI-sets of constrained closed-loop system* (13) *in* <sup>X</sup> *. For simplicity,* <sup>Ω</sup>*<sup>l</sup> MRPI is represented in the polyhedral form of* Ω*<sup>l</sup> MRPI* = {*x l* : *F l I x <sup>l</sup>* <sup>≤</sup> *<sup>g</sup> l I* }*.*

The following enlarged invariant set is further defined for some constrained allowable control inputs.

**Definition 3.** *Given the lth model of* (2) *and the constraints* (<sup>X</sup> , <sup>U</sup>)*, the set* <sup>Ω</sup>*<sup>l</sup> RCI* ⊆ X *is a robust control invariant set (RCI-set), if for any x l* <sup>0</sup> ∈ Ω*RCI there exists an admissible control input u l k* ∈ U *such that all the state updates satisfy x l k* ∈ Ω*RCI for all d<sup>k</sup> and e x l* ,*k , k* > 0*. Similarly, the maximal RCI-set* Ω*MRCI contains all robust RCI-sets.*

Generally, the determination of Ω*MRCI* is computationally demanding, in particular for high-dimension systems. As an alternative, the *M*-step robust control invariant set can be used.

**Definition 4.** *The set P l <sup>M</sup>* ⊆ X *is defined as a M-step robust control invariant set for the lth model of* (2) *with respect to the constraints* (X , U)*, if there exists an admissible control sequence such that all states x l k* ∈ *P l <sup>M</sup> can be steered into* <sup>Ω</sup>*<sup>l</sup> MRPI in no more than M steps. For simplicity, P l <sup>M</sup> is described as P<sup>l</sup> <sup>M</sup>* = {*x l* : *F l <sup>M</sup>x <sup>l</sup>* <sup>≤</sup> *<sup>g</sup> l <sup>M</sup>*}*.*

In general, two cases exist for the location of the states of system after the active FI is completed, namely *x l k* ∈ *P l <sup>M</sup>* and *x l k* ∈/ *P l <sup>M</sup>*. In the sequel, we will construct an interpolating FTC strategy for each of these two cases.

*(1) Case I (x l k* ∈ *P l <sup>M</sup> after FI)*: Firstly, in order to get <sup>Ω</sup>*<sup>l</sup> MRPI* of (13), the bounded set of *B*¯*<sup>l</sup> e x l* ,*<sup>k</sup>* + *d<sup>k</sup>* should be determined. By a series of finite set iterations along (5), the

disturbance invariant set of *e x <sup>l</sup>* subject to *v<sup>k</sup>* ∈ V and *d<sup>k</sup>* ∈ D has been computed as Ω *l*,*l e* . Then, we have *B*¯*<sup>l</sup> e x l* ,*<sup>k</sup>* <sup>+</sup> *<sup>d</sup><sup>k</sup>* <sup>∈</sup> (*B*¯*l*<sup>Ω</sup> *l*,*l <sup>e</sup>* ⊕ D). Further, the *Procedure 2.1* in [28] can be referred to calculate Ω*<sup>l</sup> MRPI* of (13).

In order to describe the control actions that can regulate the state *x l k* from *P l <sup>M</sup>* back to Ω*l MRPI* in no more than *M* steps, an augmented control sequence *Ul <sup>M</sup>*,*<sup>k</sup>* = h (*u l IC*,*k* ) *T* (*u l IC*,*k*+1 ) *T* · · · (*u l IC*,*k*+*M*−1 ) *T* i*T* is defined. In fact, these actions are expected to regulate the dynamic behavior of the system in the following manner:

$$\begin{aligned} \mathbf{x}\_{k+1}^{l} &= A^{l} \mathbf{x}\_{k}^{l} + B^{l} u\_{\text{IC},k}^{l} + d\_{k} \in \mathbf{P}\_{M}^{l} \\ &\vdots\\ \mathbf{x}\_{k+M-1}^{l} &= A^{l} \mathbf{x}\_{k+M-2}^{l} + B^{l} u\_{\text{IC},k+M-2}^{l} + d\_{k+M-2} \in \mathbf{P}\_{M}^{l} \\ \mathbf{x}\_{k+M}^{l} &= A^{l} \mathbf{x}\_{k+M-1}^{l} + B^{l} u\_{\text{IC},k+M-1}^{l} + d\_{k+M-1} \in \mathbf{O}\_{M\text{MR}}^{l} \end{aligned} \tag{17}$$

Obviously, in (17) we can observe that the migration process of states can be approximately deduced by the current initial state *x l k* and a sequence of inputs *U<sup>l</sup> M*,*k* . Considering the constraints with Definition 4, we can further describe the maximal admissible control domain of the system (1) with respect to the corresponding control inputs in terms of the following half-space representation for the augmented state space *Q<sup>l</sup> <sup>M</sup>* = {*x l* , *U<sup>l</sup> <sup>M</sup>*}:

$$Q\_M^l = \{ \mathbf{x}^l \, \mathcal{U}\_M^l : \mathbb{F}\_M^l \begin{bmatrix} \mathbf{x}^l \\ \mathbf{U}\_M^l \end{bmatrix} \le \mathbf{g}\_M^l \} \tag{18}$$

**Remark 6.** *Given the previously obtained* Ω*<sup>l</sup> MRPI and certain M, the augmented set <sup>Q</sup><sup>l</sup> <sup>M</sup> can be calculated by following the algorithm in [28]. In addition, by comparing the definition in* (18) *and Definition 4, it can be seen that P<sup>l</sup> <sup>M</sup> is a projection of Q<sup>l</sup> <sup>M</sup> onto the state space.*

Without loss of generality, any state vector *x l k* ∈ *P l <sup>M</sup>* can be decomposed as a convex combination form

$$\mathbf{x}\_k^l = \mathbf{s}\_k^l \mathbf{x}\_{O,k}^l + (\mathbf{1} - \mathbf{s}\_k^l) \mathbf{x}\_{I,k}^l \tag{19}$$

where *x l I*,*k* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> MRPI* denotes an inner state vector while *x l O*,*k* ∈ *P l <sup>M</sup>* <sup>∼</sup> <sup>Ω</sup>*<sup>l</sup> MRPI* denotes an outer state vector. *s l k* ∈[0, 1] is the so-called interpolation coefficient. Since *xI*,*<sup>k</sup>* has already inside Ω*<sup>l</sup> MRPI*, the previously designed unconstrained optimal control law by *K lx l I*,*k* can be directly adopted to achieve UUB regulation of *x l I*,*k* robustly. Thus, for *x l k* <sup>∈</sup>/ <sup>Ω</sup>*<sup>l</sup> MRPI*, (19) means that the problem of finding *U<sup>l</sup> M*,*k* to regulate state *x l k* back to Ω*<sup>l</sup> MRPI* can be transformed into the problem of solving *U<sup>l</sup> M*,*k* to regulate state *x l O*,*k* back into Ω*<sup>l</sup> MRPI*.

In line with the above state decomposition (19), the following interpolated FTC strategy for the *l*th model is constructed

$$\boldsymbol{u}^{l}\_{\mathsf{C},k} = \boldsymbol{s}^{l}\_{k}\boldsymbol{u}^{l}\_{\mathsf{IC},k} + (1 - \boldsymbol{s}^{l}\_{k})\boldsymbol{u}^{l}\_{I,k} \tag{20}$$

where *u l <sup>I</sup>*,*<sup>k</sup>* = *K lx l I*,*k* is the inner FTC law while *u l IC*,*k* is the outer FTC law to be determined. It should be noted that *u l I*,*k* is the optimal unconstrained terminal control law, and it generally presents high control performance. However, for *x l O*,*k* ∈ *P l <sup>M</sup>* <sup>∼</sup> <sup>Ω</sup>*<sup>l</sup> MPRI*, the constraints will be activated and the performance might be poor. Thus, in order to make the high-performance inner controller as dominant as possible and minimize the constraint

activation influence simultaneously, it is desirable to set *s l k* as small as possible. This can be achieved by solving the following optimization problem:

$$\begin{cases} \mathfrak{s}\_{k}^{l} = \min\_{s\_{k}^{l}, \mathbf{x}\_{I,k}^{l}, \mathbf{x}\_{O,k}^{l}, \mathbf{M}\_{M,k}^{l}} & \mathbf{s}\_{k}^{l} \\\\ \text{s.t. } \boldsymbol{\mathcal{F}}\_{I}^{l} \mathbf{x}\_{I,k}^{l} \leq \mathbf{g}\_{I}^{l}; \ \boldsymbol{\mathcal{F}}\_{M}^{l} \begin{bmatrix} \boldsymbol{\mathcal{X}}\_{O,k}^{l} \\ \boldsymbol{\mathcal{U}}\_{M,k}^{l} \end{bmatrix} \leq \tilde{\mathbf{g}}\_{M}^{l} \mathbf{y} \\\ \mathbf{x}\_{k}^{l} = \mathbf{s}\_{k}^{l} \mathbf{x}\_{O,k}^{l} + (1 - \mathbf{s}\_{k}^{l}) \boldsymbol{\mathcal{x}}\_{I,k}^{l} \mathbf{y} \ 0 \leq \mathbf{s}\_{k}^{l} \leq 1. \end{cases} \tag{21}$$

The first constraint in (21) is used to ensure *x l I*,*k* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> MRPI*; the second inequality is used to ensure that there exists *U<sup>l</sup> M*,*k* such that *x l O*,*k* <sup>∈</sup> *<sup>P</sup>*˜*<sup>l</sup> <sup>M</sup>* ⊆ *P l <sup>M</sup>* and *x l <sup>O</sup>*,*k*+*<sup>M</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> MRPI*; the third equation guarantees a smooth convex interpolation between *x l I*,*k* and *x l O*,*k* and also achieves a smooth interpolation between the associated two control laws.

**Remark 7.** *In view of the influence of estimation error on the feasibility of optimization, we have contracted the constraint condition in* (18)*, and obtained the second constraint condition in* (21)*. Specifically, by setting u l IC*,*<sup>k</sup>* = *K lx l <sup>O</sup>*,*<sup>k</sup>* + *c l k , there is u l <sup>C</sup>*,*<sup>k</sup>* = *s l k u l IC*,*<sup>k</sup>* + (1 − *s l k* )*u l <sup>I</sup>*,*<sup>k</sup>* = *s l k* (*K lx l <sup>O</sup>*,*<sup>k</sup>* + *c l k* ) + (1 − *s l k* )*u l <sup>I</sup>*,*<sup>k</sup>* = *s l k K lx l <sup>O</sup>*,*<sup>k</sup>* + (1 − *s l k* )*K lx l <sup>I</sup>*,*<sup>k</sup>* + *s l k c l <sup>k</sup>* = *K l* (*s l k x l <sup>O</sup>*,*<sup>k</sup>* + (1 − *s l k* )*x l I*,*k* ) + *s l k c l <sup>k</sup>* = *K lx l <sup>k</sup>* + *c*¯ *l k , where c*¯ *l <sup>k</sup>* = *s l k c l k . Then, following the augmentation analysis technique in dual-mode predictive control [20], we can calculate a disturbance invariant set of* [*x<sup>k</sup>* ; *U<sup>l</sup> M*,*k* ] *that is driven by e x l* ,*k and d<sup>k</sup> . Further, based on the constraint tightening, a conservative constraint set Q*˜*<sup>l</sup> <sup>M</sup>* = {*x l* , *U<sup>l</sup> <sup>M</sup>* : *<sup>F</sup>*˜*<sup>l</sup> M x l Ul M* ≤ *g*˜ *l <sup>M</sup>*} *can be determined, where <sup>P</sup>*˜*<sup>l</sup> <sup>M</sup> is a projection of <sup>Q</sup>*˜*<sup>l</sup> <sup>M</sup> onto the state space.*

Since *s l k* , *x l I*,*k* , *x l O*,*k* are unknown, the optimization (21) is nonlinear. Let *b l <sup>O</sup>*,*<sup>k</sup>* = *s l k x l O*,*k* , *b l <sup>I</sup>*,*<sup>k</sup>* = (1 − *s l k* )*x l I*,*k* , and *T l <sup>M</sup>*,*<sup>k</sup>* = *s l kUl M*,*k* , the above optimization problem (21) can be then simplified as a linear programming problem:

$$\begin{cases} \mathfrak{s}\_k^l = \min\_{s\_k^l, b\_{O,k}^l, T\_{M,k}^l} \quad & s\_k^l \\ \text{s.t. } F\_I^l(\mathbf{x}\_k^l - b\_{O,k}^l) \le (1 - s\_k^l) g\_I^l; \\ \qquad \qquad \qquad f\_M^l \begin{bmatrix} b\_{O,k}^l \\ T\_{M,k}^l \end{bmatrix} \le s\_k^l \mathfrak{s}\_{M}^l; \ 0 \le s\_k^l \le 1. \end{cases} \tag{22}$$

When the optimal solution of (22) is obtained, the reconfigured interpolation FTC can then be constructed as *u l <sup>C</sup>*,*<sup>k</sup>* = *T l <sup>M</sup>*1,*<sup>k</sup>* + *K l* (*x l <sup>k</sup>* − *b l O*,*k* ), where *T l M*1,*k* is the first control input in *T l M*,*k* .

*(2) Case II (x l k* ∈/ *P l <sup>M</sup> after FI)*: The soft constraint methods are employed to ensure that states outside *P l <sup>M</sup>* can also be steered into <sup>Ω</sup>*<sup>l</sup> MRPI* after the fault is isolated. Depending on the requirements of the actual system for state constraints and input constraints, there exist two general ways to design soft constraints [33,34]. The first is that the input constraints must not be violated while the boundaries of the state constraints can be relaxed appropriately. The other is that the boundaries of both constraints can be adjusted. In either case, the relaxation variable introduced by the soft constraints is non-zero only when the original constraints are violated. Once the original constraints are restored, the relaxation variable must be zero. For the sake of simplicity, the second strategy is adopted and we design the following soft constrained interpolating control algorithm. First of all, we suppose that the maximal admissible control domain (18) can be relaxed to contain states *x l k* ∈/ *P l M* as follows:

$$\mathbf{Q}^{l}\_{\mathfrak{S}\_{k},M} = \{ \mathbf{x}^{l}, \mathbf{U}^{l}\_{M} : \tilde{\mathbf{f}}^{l}\_{M} \begin{bmatrix} \mathbf{x}^{l} \\ \mathbf{U}^{l}\_{M} \end{bmatrix} \le \tilde{\mathbf{g}}^{l}\_{M} + \mathbf{g}^{l}\_{k} \boldsymbol{\Lambda} \} \tag{23}$$

where *ς l <sup>k</sup>* ≥ 0 is the relaxation variable and *Λ* can be a column vector of ones or an arithmetic progression vector with the first term 1 and common difference −*κ* ∈ [−1, 0]. Note that the soft constraints by (23) implicitly define an enlarged *M*−step robust control invariant set *P*˜*<sup>l</sup> ςk* ,*<sup>M</sup>* for systems (1) with relaxed constraints of states and inputs.

Then, in a similar way to formulate (19) and (20), we can also update the interpolations of states and inputs for *x l I*,*k* <sup>∈</sup> <sup>Ω</sup>*<sup>l</sup> MRPI* and *x l O*,*k* <sup>∈</sup> *<sup>P</sup>*˜*<sup>l</sup> ςk* ,*<sup>M</sup>* <sup>∼</sup> <sup>Ω</sup>*<sup>l</sup> MRPI*. Slightly different from the optimization objective of (21), here the slack variable *ς<sup>k</sup>* also needs to be minimized, i.e., the degree of constraint violation of *P*˜*<sup>l</sup> <sup>M</sup>* should be minimized. To this point, we can further establish the following optimization problem through the same design of variables as (22):

$$\begin{cases} \boldsymbol{\uphat{\boldsymbol{\mu}}}\_{k}^{l} = \min\_{\boldsymbol{s}\_{k}^{l}, \boldsymbol{b}\_{O,k}^{l}, \boldsymbol{T}\_{M,k}^{l}, \boldsymbol{\xi}\_{k}^{l}} \boldsymbol{e}\_{1} \boldsymbol{\uphat{\boldsymbol{\varepsilon}}}\_{k}^{l} + \boldsymbol{e}\_{2} \boldsymbol{s}\_{k}^{l} \\ \quad \text{s.t.} \ \boldsymbol{F}\_{I}^{l}(\boldsymbol{x}\_{k}^{l} - \boldsymbol{b}\_{O,k}^{l}) \leq (1 - \boldsymbol{s}\_{k}^{l}) \boldsymbol{g}\_{I}^{l}; \ 0 \leq \boldsymbol{s}\_{k}^{l} \leq 1; \\ \quad \quad \quad \quad \quad \quad \boldsymbol{f}\_{M}^{l} \left[ \boldsymbol{b}\_{O,k}^{l} \right] \leq \boldsymbol{s}\_{k}^{l} \boldsymbol{g}\_{M}^{l} + \boldsymbol{\uphat{\boldsymbol{\varepsilon}}}\_{k}^{l} \Lambda; \ \boldsymbol{\xi}\_{k}^{l} = \boldsymbol{s}\_{k}^{l} \boldsymbol{g}\_{k}^{l}. \end{cases} \tag{24}$$

where *ε*<sup>1</sup> + *ε*<sup>2</sup> = 1. In order to highlight the function of soft constraint FTC, *ε*<sup>1</sup> is generally set to be larger than *ε*2.

#### *3.5. The AFD-Based Reconfigured Interpolation FTC Algorithm*

A binary parameter *ε*<sup>3</sup> is introduced to unify the optimization problems of (22) and (24):

$$\begin{cases} \mathfrak{f}\_{k}^{l} = \min\_{\substack{\boldsymbol{s}\_{k}^{l}, \boldsymbol{b}\_{O,k}^{l}, \boldsymbol{T}\_{M,k}^{l} \boldsymbol{s}\_{k}^{l}}} \boldsymbol{e}\_{3} (\boldsymbol{e}\_{1} \boldsymbol{\xi}\_{k}^{l} + \boldsymbol{e}\_{2} \boldsymbol{s}\_{k}^{l}) + (1 - \boldsymbol{e}\_{3}) \boldsymbol{s}\_{k}^{l} \\ \quad \text{s.t. } \boldsymbol{F}\_{I}^{l} (\boldsymbol{x}\_{k}^{l} - \boldsymbol{b}\_{O,k}^{l}) \le (1 - \boldsymbol{s}\_{k}^{l}) \boldsymbol{g}\_{I}^{l}; \; 0 \le \boldsymbol{s}\_{k}^{l} \le 1; \\ \quad \quad \quad \boldsymbol{f}\_{M}^{l} \left[ \begin{matrix} \boldsymbol{b}\_{O,k}^{l} \\ \boldsymbol{T}\_{M,k}^{l} \end{matrix} \right] \le \boldsymbol{s}\_{k}^{l} \boldsymbol{g}\_{M}^{l} + \boldsymbol{e}\_{3} \boldsymbol{\xi}\_{k}^{l} \boldsymbol{\Lambda}; \; 0 \le \boldsymbol{\xi}\_{k}^{l}. \end{cases} \tag{25}$$

By setting *ε*<sup>3</sup> = 1, (25) reduces to (24), which is used to achieve soft constrained interpolating control for the case *x l k* <sup>∈</sup>/ *<sup>P</sup>*˜*<sup>l</sup> <sup>M</sup>*. By setting *ε*<sup>3</sup> = 0, (25) reduces to (22) and the standard interpolating control based FTC can then be achieved. All the above developments allow us to write down Algorithm 1.

#### **Algorithm 1** AFD-based interpolation AFTC.

	- 1: (*x*ˆ*<sup>k</sup>* ,*r<sup>k</sup>* ) ← (4); C Using (4) to estimate (*x*ˆ*<sup>k</sup>* ,*r<sup>k</sup>* )

$$\begin{array}{ll} \text{4:} & \mathsf{else} \\ \text{5:} & \mathsf{Cast}^{i} \end{array} \begin{array}{ll} & \text{<0 A fault is detected} \\ \\ \text{1:} & \text{2:} \end{array}$$

*<sup>C</sup>*,*<sup>k</sup>* ← 0, *u<sup>k</sup>* ← *u FI*; 6: **end if** 7: **do**

*i*

8: (*x*ˆ*k*+*<sup>τ</sup>* ,*rk*+*<sup>τ</sup>* ) ← (4); C Generate *τ* residuals 9: Find *ζ* such that *Iso<sup>i</sup>*


5: Set *u*


#### **4. Algorithm Verification by a Wastewater Treatment Plant Model**

#### *4.1. System Model and Parameters*

The purpose of a wastewater treatment plant is to purify the sewage and return clean water to the river. Activated sludge process (ASP) is a very important part of the cleaning procedure [35]. Generally, ASP systems usually consist of a bioreactor and a settler. Bioreactors mainly rely on suspended microorganisms for biodegradation of dissolved substrate. After that, the suspended micro-organisms are completely separated in the settler. Some of the degraded biomass will be recycled to the bioreactor for further purification, while the remaining biomass will be discharged to maintain the balance of limited organisms in the ASP system. The energy needed for the reaction is provided by the dissolved oxygen, and the resulting carbon dioxide is in turn released. In [36], a simplified state-space error model describing the mass balances in ASP systems is built around the equilibrium point (*XP*, *UP*) = ([122.7342 49.4714 196.3750 6.8300] *T* , [0.06 1.35]). Here, to achieve the fault tolerant mass balance of ASP systems, some uncertain parameters along the model in [36] are additionally considered as follows:

$$\begin{aligned} A &= \begin{bmatrix} 0.7685 - \Delta\_A & 0.1551 & 0.0576 & 0.1273 \\ -0.1438 & 0.4137 + \Gamma\_A & -0.0859 & -0.013 \\ -0.0109 & -0.0175 & 0.0026 + \Delta\_A & -0.0018 \\ 0.3396 & 0.0377 & 0.0253 & 0.8335 - \Gamma\_A \end{bmatrix} \\ B &= \begin{bmatrix} -250.0774 & 0.9268 \\ 398.0189 & -1.4129 \\ -13.8515 & 2.0454 \\ 102.6287 & 0.18 \end{bmatrix} \begin{bmatrix} \Delta\_B & 0 \\ 0 & \Gamma\_B \end{bmatrix}, \mathcal{C} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}^T \end{aligned} \tag{26}$$

We assume that two types of faults can appear: (∆*<sup>A</sup>* =0.2, ∆*<sup>B</sup>* =0.7, Γ*<sup>A</sup>* =Γ*<sup>B</sup>* =0) and (∆*<sup>A</sup>* = ∆*<sup>B</sup>* =0, Γ*<sup>A</sup>* =0.3, Γ*<sup>B</sup>* =0.6). The former is identified as fault *l* = 1 (faulty mode 1) and the latter is identified as fault *l* = 2 (faulty mode 2). Clearly, the health condition *l* = 0 (healthy mode 0) is indicated when (∆*<sup>A</sup>* = Γ*<sup>A</sup>* = ∆*<sup>B</sup>* = Γ*<sup>B</sup>* =0). The other parameters are: Ξ=0.1*I*4, Θ=0.02*I*2, *n<sup>f</sup>* =2, *d<sup>k</sup>* ∈[−0.5**1**<sup>4</sup> 0.5**1**4], *v<sup>k</sup>* ∈[−0.5*I*<sup>2</sup> 0.5*I*2], and

$$
\begin{bmatrix} -90.0342 \\ -37.4714 \\ -143.8750 \\ -5.9300 \end{bmatrix} \le u\_k \le \begin{bmatrix} 137.2658 \\ 15.5286 \\ 13.6250 \\ 73.1700 \end{bmatrix}, \begin{bmatrix} -0.0600 \\ -1.3500 \end{bmatrix} \le u\_k \le \begin{bmatrix} 2.3400 \\ 13.6500 \end{bmatrix} \tag{27}
$$

#### *4.2. Offline Design of AFD and AFTC According to Algorithm 1 and Relevant Validation*

According to Algorithm 1, the following parameters of AFTC policy are designed. Firstly, by solving Theorem 1, the integrated parameters of observer *L <sup>l</sup>* and inner FTC gain matrix *K <sup>l</sup>* are obtained as

$$\begin{aligned} \;^L L^0 &= \begin{bmatrix} 4.7216 & 0.2884 \\ -1.2215 & -0.0229 \\ -0.0518 & -0.0051 \\ 9.4863 & 0.6130 \end{bmatrix}, \;^K = \begin{bmatrix} -0.0016 & -0.0010 & 0.0002 & -0.0021 \\ -0.8410 & -0.2855 & -0.0381 & -0.7295 \end{bmatrix} \\\ \;^L L^1 &= \begin{bmatrix} 0.1650 & -0.0290 \\ -0.4078 & 0.0669 \\ 0.0995 & -0.1675 \\ 0.7874 & 0.0267 \end{bmatrix}, \;^K = \begin{bmatrix} -0.0031 & -0.0017 & -0.0002 & -0.0048 \\ -0.6755 & -0.2408 & -0.0910 & -0.8303 \end{bmatrix} \\\ &\;^L L^2 = \begin{bmatrix} 0.6487 & -0.0128 \\ -0.7432 & 0.0605 \\ 0.0036 & -0.0024 \\ 0.6966 & 0.0089 \end{bmatrix}, \;^K = \begin{bmatrix} -0.0014 & -0.0021 & 0.0001 & -0.0008 \\ -0.5591 & -0.2930 & 0.0015 & -0.2738 \end{bmatrix} \end{aligned} \tag{28}$$

Secondly, by using the disturbance set ∆ *i*,*i e* for 3-step set iteration along (5), the limit sets of residual for each *i* = 0, 1, 2 are approximately calculated, where the H-representations of R 0,0 *FD*, R 1,1 *FD*, and R 2,2 *FD* have 23, 38, and 47 inequalities, respectively. Due to the page limit, they are not listed here.

Thirdly, by solving optimization problem (9), some suitable choices of test input signals are determined as *u* 0 *FI* = 1.5 × U.*V*(1), *u* 1 *FI* = 1.3 × U.*V*(1), *u* 2 *FI* = 1.1 × U.*V*(1), respectively. Here, U.*V*(1) is used to denote the first vertex of the V-representation of set U. In order to clearly describe the relationship between the FD limit set and the FI separation line, we simulated the residual responses by injecting the above test input signal excitation in different modes of the system. As shown in Figure 2, the AFD can be successfully achieved as long as the residual value exceeds the relevant separation line. Here, the isolation can be accomplished in a maximum of six steps.

Next, the robust invariant sets Ω*<sup>l</sup> MRPI* and *P l <sup>M</sup>* are calculated for *l* = 0, 1, 2, respectively. In order to describe the relationship among the interpolating AFTC, the controlled states and the corresponding invariant set for each mode, the evolution of an arbitrary initial state *x*<sup>0</sup> = [−20 10 − 10 − 1.83] *T* is simulated. The results of the first three states are shown in Figure 3. It can be seen from Figure 3a,b that *x*<sup>0</sup> belongs to *P l <sup>M</sup>* <sup>∼</sup> <sup>Ω</sup>*<sup>l</sup> MRPI*, *l* = 0, 1. Therefore, as shown in sub-Figure 3d, the corresponding interpolation coefficients are not zero and *x*<sup>0</sup> is adjusted back to Ω*<sup>l</sup> MRPI* in 2-3 steps. Figure 3c illustrates that *x*<sup>0</sup> belongs to Ω<sup>2</sup> *MRPI*. Hence, the associated interpolation coefficient in Figure 3d is zero.

**Figure 2.** Test of the isolation effect of the constructed active fault isolation method in three scenarios

**Figure 3.** Test of the control effect of the developed interpolating AFTC. The yellow area represents the set Ω*<sup>l</sup> MRPI* and the green area represents the set *P l <sup>M</sup>*. (**a**) Invariant sets for health mode *l* = 0; (**b**) invariant sets for fault mode *l* = 1; (**c**) invariant sets for fault mode *l* = 2; (**d**) interpolation coefficient *s*˜*<sup>k</sup>* .

#### *4.3. Simulation Results and Analysis of the above Designed AFD-Based AFTC Method*

Based on the parameters obtained above, we next perform performance tests on the proposed AFD-based AFTC method. First, the following fault scenarios are considered:

Fault scenarios: The system initially works in a healthy condition; when *k* ∈ [160 550), the first kind of fault occurs in the system. For *k* ≥ 550, the previous fault disappears and the second type of fault appears.

Then, the online AFTC strategy described in Algorithm 1 is implemented to deal with the above fault situations. The simulation results are collected and depicted in Figures 4–6, where the occurrence and duration of different faults have been marked using different color areas, i.e., green area for healthy condition (*l* = 0), yellow area for type I faults (*l* = 1) and gray area for type II faults (*l* = 2). As depicted in Figure 4, it takes some time after a fault occurs to achieve the state regulation to track the equilibrium point *XP*. The reason is that the fault detection, isolation, and control reconfiguration need to be completed during this time. Taking the fault-tolerant process for the first type of fault as an example, Figure 4 firstly depicts that the estimated values of the states can quickly deviate from their actual values in the moments after the fault occurs. Their estimation errors caused by the presence of the fault further generate large residual values, thus facilitating the timely triggering of FI. In fact, the interpolation coefficient in Figure 6 appears to increase rapidly at *k* > 160, which also indicates the occurrence of abnormal system conditions. The inputs of the corresponding constant value auxiliary test signals are further shown in Figure 5. It should be noted that both variables in Figure 6 are zero at this time. After a few steps, it can be seen in Figure 4 that the first three states have been accurately estimated, which indicates that the FI is completed. However, the estimation of the fourth state still deviates from the actual value. The reason is that the auxiliary signal injected during FI drives it to a large deviation (as shown in Figure 2). Hence, additional time is required to achieve its unbiased tracking.

After FI, the corresponding control reconfiguration is further activated. As shown in Figure 6, the soft constraint FTC (24) is triggered first, which also leads to a sharp increase of the control input in Figure 5. When the states are adjusted into *P* 1 *<sup>M</sup>* by the soft constraint FTC, the interpolation FTC (22) is activated timely. At the same time, as illustrated in Figure 5, the control inputs subsequently become smaller. The decreasing interpolation coefficient in Figure 6 also indicates that the system states are gradually tuned into Ω<sup>1</sup> *MRPI*. After that, the states are gradually regulated to track the equilibrium point.

**Remark 8.** *The above process constitutes a complete cycle of AFD and AFTC. Clearly, the decreasing interpolation coefficients and relaxation variables in Figure 6 fully illustrate the convergence of the proposed Algorithm 1. Correspondingly, the state and control variables in Figures 4 and 5 are also adjusted to the equilibrium point* (*XP*, *UP*)*, which further illustrates that the control system under the influence of the fault is stabilized and the tracking target is achieved.*

**Figure 4.** Simulation results of state evolution and estimation under the control of Algorithm 1.

**Figure 5.** Simulation results of interpolation-based AFTC input obtained from Algorithm 1.

**Figure 6.** Simulation results of interpolation coefficient and relaxation variable obtained from Algorithm 1.

*4.4. Multi-Performance Comparison and Discussion of Active Fault-Tolerant Control Methods*

Some qualitative comparisons with the recently reported AFTC methods are given in Table 1.



Note: interpolating control (IC), model predictive control (MPC), linear matrix inequality (LMI), linear programming (LP), quadratic programming (QP), semi-positive definite programming (SDP), *M*-step robust control invariant set (*PM*).

The involved comparisons in Table 1 are explained from the following aspects. Firstly, as shown in the second row of Table 1, both component faults and actuator faults are considered in this paper, while only actuator faults are considered in [13,16,20]. In general, the component faults can significantly affect the system dynamics. In this paper, an AFD method is embedded to identify the system operating mode in real time in order to achieve fault tolerance for component faults. Secondly, unlike the multiple-observers-based realtime diagnosis approach in [16], here only one observer needs to be employed at each moment to achieve fault mode identification. Theoretically, this facilitates the diagnosis efficiency and it is also another implicit advantage of using AFD.

In terms of the design and implementation of fault-tolerant methods (i.e., rows 5–7 in Table 1), the MPC optimization problems in [13] are constructed by relying on ellipsoidal constraint sets and LMI, which belongs to SDP and whose solution tends to be more time-consuming. In addition, approximating the feasible domain with ellipsoidal sets is generally more conservative than polyhedral sets. In [20], the dual-mode prediction mechanism is adopted to construct a predictive FTC, whose optimization problem belongs to QP and can be solved relatively efficiently. However, this FTC method is only used to handle actuator additive offset faults and is not suitable for addressing fault tolerance problems of multiplicative faults and component faults. Relatively, the receding horizon set theoretic FTC method in [16] is appealing. This method provides a way to perform the state figure using switching *M*-step controllable ellipsoidal sets under different fault conditions. However, it may be computationally demanding and takes up a large storage space because of the need to solve real-time QP when the states do not belong to the corresponding maximum allowable invariant set. In this paper, the interpolation methods are employed to combine *M*-step controllable polyhedral sets and inner feedback control laws to achieve the state figure, and the corresponding fault-tolerant optimization is formed as LP. Compared to the sets that need to be stored by the FTC method in [16], Algorithm 1 only needs to store the maximum *M*-step controllable polyhedral set for each operating condition, which helps to reduce the storage burden.

The penultimate row of Table 1 illustrates that the soft-constrained FTC method is further integrated into Algorithm 1 and used to deal with some unanticipated situations, such as uncertain fault amplitudes, system parameter drifts, disturbance overruns, etc. The last row of Table 1 implies that the design of the FTC method in [13] is more intuitive and better scalable than the FTC methods in Algorithm 1, [16,20]. It should be noted that the above comparisons are discussed mainly for the characteristics of the involved faulttolerant methods and not for the contents of the overall studies in [13,16,20]. Clearly, they have different system models and control objectives, and therefore different innovations.

**Remark 9.** *According to Remark 7, the FTC law based on dual-mode predictive control constructed in [20] can be considered as a special form of the interpolation AFTC developed in this paper. Hence, the interpolation-based AFTC theoretically has a higher degree of design freedom as well as a more efficient optimization capability. To verify this, a further numerical comparison was made. Let the system operate sequentially in two scenarios: scenario I (health l* = 0*) for* 1 ≤ *k* < 160 *and scenario II (fault l* = 2*) for* 160 ≤ *k* ≤ 500*. To be fair, the same active fault diagnosis and integration design were used. Table 2 gives the comparisons of these two methods in terms of interval cost function* (3) *and running time. It can be seen that the interpolation-based AFTC method runs faster and provides better tracking accuracy for scenario I. In scenario II, the developed interpolation-based AFTC remains feasible and continues to optimize the cost function, however the FTC method of [20] will no longer be feasible after k* = 170*. Based on the above numerical comparisons, the effectiveness of the method constructed in this paper can be further verified.*


**Table 2.** Comparisons of interval cost function (3) and running time.

#### **5. Conclusions**

In this paper, a novel activate fault tolerant control scheme is proposed to address the component/actuator faults for the uncertain systems with state/input constraints. Its significant merits are that (1) it relies on only one diagnostic observer for online fault detection and isolation, which helps to reduce the internal memory consumption of the hardware controller; (2) the auxiliary inputs and separation hyperplanes for fault isolation are designed offline in advance, which helps to reduce the online computational burden and increase the freedom of fault isolation decisions; (3) the overall fault tolerant control is reconfigured by optimizing the interpolation coefficient to dynamically regulate the convex combination of inner and outer fault tolerant control laws, which can further reduce the online optimization effort; (4) the inner fault tolerant control and the diagnostic observer are designed offline in advance, and by such design the robust interaction influence on the feasibility of the reconfigured fault tolerant control algorithm can be reduced; (5) the soft constraint method is embedded to achieve a relaxed fault tolerance, which can handle some cases that lead to infeasible constrained optimization in an emergency. The simulation with detailed discussions is given to demonstrate the above benefits of the proposed method.

Some issues need to be further addressed in the future work. For instance, the application of semi-active fault diagnosis to enhance the design flexibility of auxiliary signals; the combination of soft constraint theory and period invariant sets to construct an outer fault tolerant control with flexible and adjustable feasible domains; the construction of parametrization method of interpolated coefficient to avoid solving linear programming problems, etc.

**Author Contributions:** Conceptualization, K.H.; methodology, K.H.; software, C.C.; validation, C.C. and M.C.; formal analysis, K.H.; investigation, K.H.; resources, K.H.; data curation, K.H.; writing original draft preparation, K.H.; writing—review and editing, K.H.; visualization, C.C.; supervision, Z.W.; project administration, K.H.; funding acquisition, K.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China under Grant number 61803178, and Shandong Provincial Natural Science Foundation under Project number ZR2019BF036.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are thankful to the reviewers for their comments and suggestions to improve the quality of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

The approximated calculation method of R *l*,*i FI* is given below.

By [31], R *l*,*i FI* can be explicitly represented as R *l*,*i FI* = {*Cχ*(*I* − *A l*,*i χ* ) <sup>−</sup>1*B l*,*i <sup>χ</sup> u i FI*} ⊕*CχO l*,*i χ*,∞ ⊕ *Dχ*E, where *O l*,*i <sup>χ</sup>*,<sup>∞</sup> = {*χ* : ∑ ∞ *j*=0 (*A l*,*i χ* ) *jE i <sup>χ</sup>σ<sup>k</sup>* , *σ<sup>k</sup>* ∈ E }. Generally, *O l*,*i <sup>χ</sup>*,<sup>∞</sup> is difficult to determine, especially for high-dimensional systems. In [27], an external approximation method

is proposed to enable *O l*,*i <sup>χ</sup>*,<sup>∞</sup> ⊆ (1 + *µT*)*O l*,*i χ*,*T* , where *O l*,*i <sup>χ</sup>*,*<sup>T</sup>* = {*χ* : ∑ *T j*=0 (*A l*,*i χ* ) *jE i <sup>χ</sup>σ<sup>k</sup>* , *σ<sup>k</sup>* ∈ E } can be calculated in a finite time. Then, for given *u i FI*, the internal point of residual limit set R *l*,*i FI* can be parameterized as *Cχ*(*I* − *A l*,*i χ* ) <sup>−</sup>1*B l*,*i <sup>χ</sup> u i FI* + (1 + *µT*) ∑ *T <sup>j</sup>*=<sup>0</sup> *Cχ*(*A l*,*i χ* ) *jE i <sup>χ</sup>σ*1,*<sup>k</sup>* + *Dχσ*2,*<sup>k</sup>* , ∀*σ*1,*<sup>k</sup>* , *σ*2,*<sup>k</sup>* ∈ E.

#### **Appendix B**

The proof of **Theorem 1** is given below.

**Proof.** Let *V l* 1,*<sup>k</sup>* = *e T x l* ,*k P l* 1 *e x l* ,*k* and *V l* 2,*<sup>k</sup>* = (*x l k* ) *T* (*P l* 2 ) <sup>−</sup>1*x l k* be the Lyapunov functions of (5), (13) and (14), respectively. Equivalently, *V l* 1,*<sup>k</sup>* + *V l* 2,*k* is a Lyapunov function of (15). Define *K l* (*Y l* 2 ) *<sup>T</sup>* = *K*¯*<sup>l</sup>* and *Y l* 1 *L <sup>l</sup>* = *L*¯ *<sup>l</sup>* . Then, by using −*Y l* 1 (*P l* 1 ) −1 (*Y l* 1 ) *<sup>T</sup>* <sup>≤</sup> *<sup>P</sup> l* <sup>1</sup> − *Y l* <sup>1</sup> − (*Y l* 1 ) *<sup>T</sup>* and congruence transformation *diag*{*I*, *I*,(*Y l* 1 ) <sup>−</sup>1} to (16a), the inequality <sup>−</sup>*diag*{*<sup>P</sup> l* <sup>1</sup> − *I*, *α* 2 *I*} + - *A*˜*l* <sup>11</sup> *<sup>B</sup>*˜*<sup>l</sup>* 1 *T P l* 1 (∗) < 0 can be deduced. It further implies that the relation *V l* 1,*k*+<sup>1</sup> − *V l* 1,*<sup>k</sup>* + *e T x l* ,*k e x l* ,*<sup>k</sup>* − *α* 2*\$ T k \$<sup>k</sup>* < 0 holds.

Similarly, by using the inequality −*Y l* 2 (*P l* 2 ) −1 (*Y l* 2 ) *<sup>T</sup>* <sup>≤</sup> *<sup>P</sup> l* <sup>2</sup> − *Y l* <sup>2</sup> − (*Y l* 2 ) *<sup>T</sup>* and congruence transformation factor *diag*{(*Y l* 2 ) −1 ,(*Y l* 2 ) −1 , *I*, *I*, *I*} to (16b), we get the inequality

−*diag*{(*P l* 2 ) −1 ,(*P l* 2 ) −1 , *β* 2 *I*} + - *A*˜*l* <sup>21</sup> *<sup>A</sup>*˜*<sup>l</sup>* <sup>22</sup> *<sup>B</sup>*˜*<sup>l</sup>* 2 *T* (*P l* 2 ) −1 (∗)+- *D*¯ *<sup>l</sup> C*¯*<sup>l</sup>* 0 *T* (∗) < 0. Based on the Lyapunov function *V l* 2,*k* and the system model (13) and (14), we can further derive *V l* 2,*k*+<sup>1</sup> − *V l* 2,*<sup>k</sup>* + (*z l k* ) *T z l <sup>k</sup>* − *e T x l* ,*k* (*P l* 2 ) −1 *e x l* ,*<sup>k</sup>* − *β* 2*\$ T k \$<sup>k</sup>* < 0.

Under zero initial conditions, the summation of *V l* 2,*k*+<sup>1</sup> − *V l* 2,*<sup>k</sup>* + (*z l k* ) *T z l <sup>k</sup>* − *e T x l* ,*k* (*P l* 2 ) −1 *e x l* ,*<sup>k</sup>* − *β* 2*\$ T k \$<sup>k</sup>* < 0 over *k* = 0 to *k* = ∞ can be bounded in the form of k *z l k* k 2 <sup>2</sup>≤ *e* 2 k *e x l* ,*k* k 2 <sup>2</sup> +*β* <sup>2</sup> <sup>k</sup> *\$<sup>k</sup>* <sup>k</sup> 2 2 , where *e T x l* ,*k* (*P l* 2 ) −1 *e x l* ,*k* is relaxed by *e T x l* ,*k* (*P l* 2 ) −1 *e x l* ,*<sup>k</sup>* ≤ *eigmax*((*P l* 2 ) −1 ) *e T x l* ,*k e x l* ,*<sup>k</sup>* = *e* 2 *e T x l* ,*k e x l* ,*k* . Similarly, ∑ ∞ *k*=0 {*V l* 1,*k*+<sup>1</sup> − *V l* 1,*<sup>k</sup>* + *e T x l* ,*k e x l* ,*<sup>k</sup>* − *α* 2*\$ T k \$k*} < 0 leads to k *e x l* ,*k* k 2 <sup>2</sup>≤ *α* <sup>2</sup> <sup>k</sup> *\$<sup>k</sup>* <sup>k</sup> 2 2 under zero initial conditions. Then, we further have k *z l k* k 2 2≤ (*e* 2*α* <sup>2</sup> + *β* 2 ) k *\$<sup>k</sup>* k 2 2 . Clearly, the integrated optimization of *α* and *β* contributes to improving the synthesized *H*<sup>∞</sup> performances of observer (5) and unconstrained robust control policy *u l <sup>C</sup>*,*<sup>k</sup>* = *K lx*ˆ *l k* . The proof is completed.

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