Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems

Abstract

This paper studies the asymptotic control problem of a class of large-scale high-order nonlinear systems (LSHONSs), and an asymptotic fuzzy adaptive dynamic surface controller is developed. Unknown nonlinear terms are learned online by fuzzy logic systems (FLSs) such that the accurate nonlinear model is released in the controller design procedure, where the parameters of FLSs are updated by developing adaptive laws. To compensate for the “boundary error” caused by the dynamic surface control method where a linear filter is added in the backstepping procedure to handle the “explosion of complexity” problem, a nonlinear filter is proposed to eliminate the boundary layer error. Some simulations are given to demonstrate the effectiveness of the proposed algorithm.

1. Introduction

The nonlinear control problem has been widely studied because many practice systems are usually described as nonlinear models [1,2,3], such as permanent magnet synchronous motors [4], induction motor systems [5,6], and four-tank systems [7]. To achieve a better control effect, a series of nonlinear control techniques, such as adaptive control and the backstepping technique, are developed [8,9,10,11,12]. An adaptive backstepping method, which comes from the combination of adaptive control and the backstepping technique, is an important technique to handle the parameters uncertainty of high-order nonlinear systems [13,14,15,16]. However, the above methods were proposed for the system with the accurate nonlinear model or the nonlinear system with unknown parameters. In addition, at each step of backstepping, a virtual control law should be differentiated. The repeated differentiation can lead to an “explosion of complexity” problem such that the traditional backstepping method cannot be used in practice. Therefore, this paper focuses on a class of high-order nonlinear systems control problems with unmodeled nonlinear term, low complexity, and asymptotic tracking performance.

A large-scale system is modeled to describe a group of interrelated subsystems, thereby reducing control complexity and improving the application feasible for multiple nonlinear systems [17,18]. Therefore, a decentralized control strategy has been applied to many large-scale nonlinear systems in the existing research, and it has received constant attention from different perspectives [19,20,21]. To obtain the prescribed tracking performance, a nonlinear feedback function was constructed and used instead of the linear feedback function to improve the dynamic performance for uncertain MIMO nonlinear systems in [22]. To ensure tracking performance, a modified DSC method was proposed by introducing a differentiator in [23]. An adaptive DSC algorithm is developed to optimize the convergence speed of the tracking error in [24]. For large-scale systems, a decentralized stabilizer is designed, and the transient performance is also analyzed and evaluated by the $L_2$ bound oft he tracking error in [25]. However, the above research is based on the nonlinear system that can be modeled.

In order to realize the necessary accuracy for a system model, neural networks or fuzzy logic are used to parameterize nonlinear terms or approximate the control input directly in recent research. In [26], a neural network is directly used to design a controller such that the unknown nonlinear term was not used in the controller design procedure. For an LSHONS with unknown interconnections and non-affine nonlinear subsystems, RBF NNs are used to develop a controller whose stability is analyzed by the Lyapunov stability theory in [27]. For a large-scale non-affine nonlinear system, an adaptive fuzzy method, where only the system state is used for feedback, is developed to achieve the stable control in [28]. For strong interconnections in large-scale nonlinear systems, a nonlinearly parameterized NNs is used to design a controller such that the a priori knowledge for the nonlinear term is released in [29]. The above researchers handle the nonlinear control problem without an accurate system model, but all of these only achieve the bounded tracking performance. The main reasons are (1) the approximation error caused by neural networks and fuzzy logic and (2) the boundary layer error caused by the DSC technique for LSHONS control. How to design an asymptotic tracking controller without an accurate system model for LSHONS plays an important role in theory and practice.

This paper develops a fuzzy adaptive DSC method for LSHONS. The main contributions are as follows: (1) an unknown nonlinear term is approximated by FLSs without an accurate nonlinear model of the system; (2) DSC is designed with a nonlinear filter to handle the “explosion of complexity” problem; (3) asymptotic tracking performance is guaranteed since the designed nonlinear filter can eliminate the boundary layer error in traditional DSC.

In Section 2, the problem formulation is presented. The detail steps of the fuzzy adaptive dynamic surface controller are shown in Section 3. Section 4 shows comparative simulations. Section 5 concludes this paper.

2. Preliminaries and Problem Formulation

The LSHONS is modeled as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{x}}_{k,r_k}\hfill & =x_{k,(k_r+1)}+f_{k,r_k}\left({\overline{x}}_{k,r_k}\right),\hfill \\ {\dot{x}}_{k,n_k}\hfill & =u_k+f_{k,n_k}\left({\overline{x}}_{k,n_k}\right),k=1,\dots ,N,r_k=1,\dots ,n_k-1\hfill \end{array}\right.\end{array}$$

(1)

where ${\overline{x}}_{k,r_k}={[x_{k,1},\dots ,x_{k,r_k}]}^T,{\overline{x}}_{k,n_k}={[x_{k,1},\dots ,x_{k,n_k}]}^T\in {\Re }^n,n=n_1+\dots +n_N$ denote the system states; $f_{k,r_k}(·):{\Re }^{r_k}\to {\Re }^{n_k}$ denotes the unknown smooth functions; $n_k$ describes the order of the kth subsystem; $u_k\in {\Re }^N$ and $y_k\in {\Re }^N$ describe the control input and output; N represents the number of subsystems.

Due to the nonlinear terms $f_{k,r_k}(·):{\Re }^{r_k}\to {\Re }^{n_k}$ are unknown functions, which cannot be used to design a controller directly. In this paper, $f_{k,r_k}(·)$ is approximated and parameterized by FLSs, and its detailed definition is introduced in Lemma 1.

Lemma 1

([30]). If function $g\left(z\right)$ defined in a compact set ${\Omega }_z$ is continuous, there exists an FLS $Q^{\mathrm{T}}\Psi \left(z\right)$ that causes

$$\begin{array}{c}\hfill \underset{z\in {\Omega }_z}{sup}\left|g\left(z\right)-Q^{\mathrm{T}}\Psi \left(z\right)\right|\le {\overline{\epsilon }}^{*}\end{array}$$

(2)

where $z={[z_1\left(t\right),z_2\left(t\right),\cdots ,z_n\left(t\right)]}^{\mathrm{T}}\in {\Re }^n$ is an input vector; and ${\overline{\epsilon }}^{*}$ denotes the bounds of the approximation error. $Q={[q_1,\dots ,q_m]}^{\mathrm{T}}$ describes the fuzzy basic function, and m describes the number of fuzzy rules. $q_p(p=1,\dots ,m)$ can be expressed as $q_p=\frac{\mathit{exp}\left(\frac{-{(z-{\mu }_p)}^{\mathrm{T}}(z-{\mu }_p)}{{\chi }_p^2}\right)}{{\sum }_{p=1}^m\mathit{exp}\left(\frac{-{(z-{\mu }_p)}^{\mathrm{T}}(z-{\mu }_p)}{{\chi }_p^2}\right)},1\le p\le m$ with center vector ${\mu }_p={[{\mu }_{p1},{\mu }_{p2},\dots ,{\mu }_{pn}]}^{\mathrm{T}}$ if a Gaussian function is chosen as the fuzzy membership function.

If an optimal vector $Q^{*}$ exists, which means $Q^{*}:=arg\underset{Q\in {\Omega }_Q}{min}\left\{\underset{z\in {\Omega }_z}{sup}\left|g\left(z\right)-Q^{\mathrm{T}}\Psi \left(z\right)\right|\right\}$, $g\left(z\right)$ is described as

$$\begin{array}{c}\hfill g\left(z\right)=Q^{*}\Psi \left(z\right)+{\epsilon }^{*}\left(z\right)\end{array}$$

(3)

where $Q^{*}\in {\Omega }_{Q^{*}}$, $z\in {\Omega }_z$; ${\Omega }_{Q^{*}}$ and ${\Omega }_z$ are compact sets; ${\epsilon }^{*}\left(z\right)$ satisfies $|{\epsilon }^{*}\left(z\right)|\le {\overline{\epsilon }}^{*}$.

Leveraging Lemma 1, nonlinear terms are approximated by FLSs, and (1) is written as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{x}}_{k,r_k}\hfill & =x_{k,r_k+1}+{Q_{k,r_k}^{*}}^{\mathrm{T}}{\Psi }_{k,r_k}\left({\overline{x}}_{k,r_k}\right)+{\epsilon }_{k,r_k}\left({\overline{x}}_{k,r_k}\right),\hfill \\ {\dot{x}}_{k,n_k}\hfill & =u_k+{Q_{k,n_k}^{*}}^{\mathrm{T}}{\Psi }_{k,n_k}\left({\overline{x}}_{k,n_k}\right)+{\epsilon }_{k,n_k}\left({\overline{x}}_{k,n_k}\right)\hfill \end{array}\right.\end{array}$$

(4)

where $\left|{\epsilon }_{k,r_k}\left({\overline{x}}_{k,r_k}\right)\right|\le {\overline{\epsilon }}_{k,r_k}\le \infty$ with ${\overline{\epsilon }}_{k,r_k}$ is an unknown small positive constant.

This paper aims to design $u_k$ for each subsystem without an accurate nonlinear model such that the following conditions are met: $u_k$ only contains system states; $u_k$ can avoid the “explosion of complexity” problem of the backstepping technique for a high-order system using the DSC technique; and the approximation error caused by FLSs and the boundary layer error caused by DSC can be compensated such that $y_k$ can track the reference trajectory $y_{k,d}$, asymptotically. Thus, the boundedness of all states in the closed-loop system can be guaranteed.

To design the controller and achieve the above objectives, the following Assumption and Lemmas are given.

Assumption 1.

We assume $y_{k,d}$ and its derivatives $y_{k,d}^1,y_{k,d}^2,\dots ,y_{k,d}^n$ are bounded and continuous.

Lemma 2

([31]). For any $\eta \in \Re$, inequality $0\le \left|\eta \right|-\eta tanh\left(\frac{\eta }{{\sigma }_0}\right)\le \alpha {\sigma }_0$ is satisfied, where ${\sigma }_0>0$, $\alpha =e^{-(\alpha +1)},\alpha =0.2785$.

Lemma 3

([32]). For any $w\in \Re$, inequality $0\le \left|w\right|-\frac{w^2}{\sqrt{w^2+{ϵ}^2}}\le ϵ$ is satisfied, and $ϵ>0$ is a constant.

Lemma 4

([16]). For the uniform continuity function $f(·)$, if ${lim}_{t\to \infty }{\int }_0^{t}f\left(\tau \right)d\tau$ is bounded, we have $li{m}_{t\to \infty }f\left(t\right)=0$.

3. Main Results

3.1. Fuzzy Adaptive DSC Design

Define the notions $Q_k=max\{\parallel Q_{k,1}^{*}{\parallel }^2,\dots ,\parallel Q_{k,n_k}^{*}{\parallel }^2\},{\overline{\epsilon }}_k=max\{{\overline{\epsilon }}_{k,1},\dots ,{\overline{\epsilon }}_{k,n_k}\}$. Instead of $Q_{k,r_k}^{*}$ and ${\overline{\epsilon }}_{k,r_k}$, $Q_k$ and ${\overline{\epsilon }}_k$ will be used to update the parameters of FLSs such that the tedious analytic computation is avoided. Define the notions ${\widehat{Q}}_k,{\tilde{Q}}_k,{\widehat{\overline{\epsilon }}}_k,{\tilde{\overline{\epsilon }}}_k$. ${\widehat{Q}}_k$ and ${\widehat{\overline{\epsilon }}}_k$ denote the estimated values of $Q_k,{\overline{\epsilon }}_k$, respectively. ${\tilde{Q}}_k=Q_k-{\widehat{Q}}_k$ and ${\tilde{\overline{\epsilon }}}_k={\overline{\epsilon }}_k-{\widehat{\overline{\epsilon }}}_k$ represent the estimated errors.

Step ($k,1$): For the kth subsystem, the first surface error is defined:

$$\begin{array}{c}\hfill z_{k,1}=x_{k,1}-y_{k,d}\end{array}$$

(5)

Differentiating $z_{k,1}$

$$\begin{array}{c}\hfill {\dot{z}}_{k,1}=x_{k,1+1}+{Q_{k,1}^{*}}^{\mathrm{T}}{\Psi }_{k,1}\left({\overline{x}}_{k,1}\right)+{\epsilon }_{k,1}\left({\overline{x}}_{k,1}\right)-{\dot{y}}_{k,d}\end{array}$$

(6)

For the kth subsystem, defining $V_{k,1}$

$$\begin{array}{c}\hfill V_{k,1}=\frac{1}{2}{z}_{k,1}^2+\frac{1}{2{\gamma }_{k,1}}{\tilde{Q}}_k^2+\frac{1}{2{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_{k,1}^2\end{array}$$

(7)

where $V_{k,1}$ is a Lyapunov energy function; ${\gamma }_{k,1}$ and ${\gamma }_{k,2}$ are positive constants.

Differentiating $V_{k,1}$

$$\begin{array}{c}\hfill {\dot{V}}_{k,1}=z_{k,1}(x_{k,2}+{Q_{k,1}^{*}}^{\mathrm{T}}{\Psi }_{k,1}+{\epsilon }_{k,1}-{\dot{y}}_{k,d})-\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k{\dot{\widehat{Q}}}_k-\frac{1}{{\gamma }_{k,1}}{\tilde{\overline{\epsilon }}}_{k,1}{\dot{\widehat{\overline{\epsilon }}}}_{k,1}\end{array}$$

(8)

Leveraging Lemma 3, we have

$$\begin{array}{ccc}\hfill z_{k,1}{Q_{k,1}^{*}}^{\mathrm{T}}{\Psi }_{k,1}& \le & |z_{k,1}|\frac{1}{2{\eta }_{k,1}}\parallel {\Psi }_{k,1}{\parallel }^2{Q}_k+|z_{k,1}|{\eta }_{k,1}\hfill \\ & \le & z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}\parallel {\Psi }_{k,1}{\parallel }^2{Q}_k+\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k\kappa \omega (t)+\hfill \\ & & z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}+{\eta }_{k,1}\kappa \omega (t)\hfill \\ & =& z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}+\hfill \\ & & (\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+{\eta }_{k,1})\kappa \omega (t)\hfill \\ \hfill z_{k,1}{\epsilon }_{k,1}& \le & |z_{k,1}|{\overline{\epsilon }}_k\le z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\overline{\epsilon }}_k+{\overline{\epsilon }}_k\kappa \omega (t)\hfill \end{array}$$

(9)

where ${\eta }_{k,1}$ denotes a constant, and $\omega (t)$ satisfies ${lim}_{t\to \infty }{\int }_0^t\omega (\tau )d\tau \le {\omega }_1<+\infty$, where ${\omega }_1$ is a positive constant.

Combining (8) and (9) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,1}& \le & z_{k,1}{x}_{k,2}-z_{k,1}{\dot{y}}_{k,d}+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}+\hfill \\ & & z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\overline{\epsilon }}_k+(\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+{\eta }_{k,1}+{\overline{\epsilon }}_k)\kappa \omega (t)-\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k{\dot{\widehat{Q}}}_k-\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k{\dot{\widehat{\overline{\epsilon }}}}_k\hfill \\ & =& z_{k,1}(x_{k,2}-{\alpha }_{k,1})+z_{k,1}{\alpha }_{k,1}-z_{k,1}{\dot{y}}_{k,d}+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+\hfill \\ & & z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\overline{\epsilon }}_k+(\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+{\eta }_{k,1}+{\overline{\epsilon }}_k)\kappa \omega (t)-\hfill \\ & & \frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k{\dot{\widehat{Q}}}_k-\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k{\dot{\widehat{\overline{\epsilon }}}}_k\hfill \end{array}$$

(10)

Design virtual control law ${\alpha }_{k,1}$:

$$\begin{array}{ccc}\hfill {\alpha }_{k,1}& =& -K_{k,1}{z}_{k,1}-tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{\widehat{Q}}_k-tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}-\hfill \\ & & tanh(\frac{z_{k,1}}{\omega (t)}){\widehat{\overline{\epsilon }}}_k+{\dot{y}}_{k,d}\hfill \end{array}$$

(11)

Define the following notations:

$$\begin{array}{ccc}\hfill {\Xi }_1^{Q_k}& =& {\gamma }_{k,1}{z}_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2\hfill \\ \hfill {\Xi }_1^{{\overline{\epsilon }}_k}& =& {\gamma }_{k,2}{z}_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\hfill \end{array}$$

(12)

Combining (10)–(12), we have

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,1}& \le & z_{k,1}(-K_{k,1}{z}_{k,1}-tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{\widehat{Q}}_k-tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}-\hfill \\ & & tanh(\frac{z_{k,1}}{\omega (t)}){\widehat{\overline{\epsilon }}}_k+{\dot{y}}_{k,d})-z_{k,1}{\dot{y}}_{k,d}+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)})\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+\hfill \\ & & z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\eta }_{k,1}+z_{k,1}tanh(\frac{z_{k,1}}{\omega (t)}){\overline{\epsilon }}_k+(\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+{\eta }_{k,1}+{\overline{\epsilon }}_k)\kappa \omega (t)-\hfill \\ & & \frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k{\dot{\widehat{Q}}}_k-\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k{\dot{\widehat{\overline{\epsilon }}}}_k+z_{k,1}(x_{k,2}-{\alpha }_{k,1})\hfill \\ & =& -K_{k,1}{z}_{k,1}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_1^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_1^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)+z_{k,1}(x_{k,2}-{\alpha }_{k,1})+\hfill \\ & & (\frac{1}{2{\eta }_{k,1}}{\parallel {\Psi }_{k,1}\parallel }^2{Q}_k+{\eta }_{k,1}+{\overline{\epsilon }}_k)\kappa \omega (t)\hfill \end{array}$$

(13)

Design the novel nonlinear filter:

$$\begin{array}{ccc}\hfill {\tau }_{k,1}{\dot{s}}_{k,1}& =& -e_{k,1}-\frac{{\tau }_{k,1}{\widehat{M}}_{k,1}^2{e}_{k,1}}{\sqrt{{\widehat{M}}_{k,1}^2{e}_{k,1}^2+{\sigma }^2(t)}}-{\tau }_{k,1}{z}_{k,1}\hfill \\ \hfill s_{k,1}(0)& =& {\alpha }_{k,1}(0)\hfill \end{array}$$

(14)

where ${\tau }_{k,1}$ and $e_{k,1}:=s_{k,1}-{\alpha }_{k,1}$ denote the filter time constant and the first boundary layer error, respectively; ${\widehat{M}}_{k,1}$ is illustrated later; $\sigma \left(t\right)$ satisfies ${lim}_{t\to \infty }{\int }_0^t\sigma \left(q\right)dq\le {\sigma }_{k,1}<+\infty$ and $|\dot{\sigma }\left(t\right)|\le {\sigma }_{k,2}<+\infty$ with ${\sigma }_{k,1}$ and ${\sigma }_{k,2}$ as constants. A nonlinear filter is designed to avoid the repeated differentiation of virtual control law such that the noise rejection capability is improved. The filter is characterized by that it is a hyperplane, which can make the system state slide along the hyperplane to the desired state point, and it can realize a mandatory system state switching on this plane, so that the system has the characteristics of robustness for noise.

Remark 1.

Nonlinear filter (14) releases the necessaryconditions for directly using $x_{k,2}-{\alpha }_{k,1}$, thus avoiding the “complexity explosion” problem and improving the robustness of the system. In addition, compared with a traditional linear filter, $-\frac{{\tau }_{k,1}{\widehat{M}}_{k,1}^2{e}_{k,1}}{\sqrt{{\widehat{M}}_{k,1}^2{e}_{k,1}^2+{\sigma }^2\left(t\right)}}$ with an adaptive law for ${\widehat{M}}_{k,1}$ is added such that $e_{k,1}$ is compensated, thus improving the robustness of the system in case of parameters mismatch.

Step (k, $r_k$)($2\le r_k\le n_k-1$): Let $z_{k,r_k}=x_{k,r_k}-s_{k,r_k-1}$, and the time differential of $z_{k,r_k}$ is

$$\begin{array}{c}\hfill {\dot{z}}_{k,r_k}={Q_{k,r_k}^{*}}^{\mathrm{T}}{\Psi }_{k,r_k}+{\epsilon }_{k,r_k}+x_{k,r_k+1}+\frac{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}}{\sqrt{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}^2+{\sigma }^2\left(t\right)}}+z_{k,r_k-1}+\frac{e_{k,r_k-1}}{{\tau }_{k,r_k-1}}\end{array}$$

(15)

Define the Lyapunov function:

$$\begin{array}{c}\hfill V_{k,r_k}=V_{k,r_k-1}+\frac{1}{2}{z}_{k,r_k}^2\end{array}$$

(16)

Differentiating $V_{k,r_k}$ yields

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,r_k}& =& {\dot{V}}_{r_k-1}+z_{k,r_k}{\dot{z}}_{k,r_k}\hfill \\ & \le & -\sum _{q=1}^{r_k-1}{K}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{r_k-1}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{r_k-1}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)+\hfill \\ & & \sum _{q=1}^{r_k-1}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{r_k-2}{z}_{k,q}{e}_{k,q}+z_{k,r_k-1}(x_{k,r_k}-{\alpha }_{k,r_k-1})+\hfill \\ & & z_{k,r_k}({Q_{k,r_k}^{*}}^{\mathrm{T}}{\Psi }_{k,r_k}+{\epsilon }_{k,r_k}+x_{k,r_k+1}+\frac{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}}{\sqrt{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}^2+{\sigma }^2(t)}}+z_{k,r_k-1}+\frac{e_{k,r_k-1}}{{\tau }_{k,r_k-1}})\hfill \end{array}$$

(17)

Leveraging Lemma 3, we have

$$\begin{array}{ccc}\hfill z_{k,r_k}{Q_{k,r_k}^{*}}^{\mathrm{T}}{\Psi }_{k,r_k}& \le & |z_{k,r_k}|\frac{1}{2{\eta }_{k,r_k}}\parallel {\Psi }_{k,r_k}{\parallel }^2{Q}_k+|z_{k,r_k}|{\eta }_{k,r_k}\hfill \\ & \le & z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)})\frac{1}{2{\eta }_{k,r_k}}{\parallel {\Psi }_{k,r_k}\parallel }^2{Q}_k+z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)}){\eta }_{k,r_k}+\hfill \\ & & (\frac{1}{2{\eta }_{k,r_k}}{\parallel {\Psi }_{k,r_k}\parallel }^2{Q}_k+{\eta }_{k,r_k})\kappa \omega (t)\hfill \\ \hfill z_{k,r_k}{\epsilon }_{k,r_k}& \le & |z_{k,r_k}|{\overline{\epsilon }}_k\le z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)}){\overline{\epsilon }}_k+{\overline{\epsilon }}_k\kappa \omega (t)\hfill \end{array}$$

(18)

Combining (18) and (17) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,r_k}& \le & -\sum _{q=1}^{r_k-1}{K}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{r_k-1}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{r_k-1}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)+\hfill \\ & & \sum _{q=1}^{r_k-1}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{r_k-2}{z}_{k,q}{e}_{k,q}+z_{k,r_k-1}(x_{k,r_k}-{\alpha }_{k,r_k-1})+\hfill \\ & & z_{k,r_k}{\alpha }_{k,r_k}+z_{k,r_k}\frac{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}}{\sqrt{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}^2+{\sigma }^2(t)}}+z_{k,r_k}{z}_{k,r_k-1}+z_{k,r_k}\frac{e_{k,r_k-1}}{{\tau }_{k,r_k-1}}+\hfill \\ & & z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)})\frac{1}{2{\eta }_{k,r_k}}{\parallel {\Psi }_{k,r_k}\parallel }^2{Q}_k+z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)}){\eta }_{k,r_k}+z_{k,r_k}tanh(\frac{z_{k,r_k}}{\omega (t)}){\overline{\epsilon }}_k+\hfill \\ & & (\frac{1}{2{\eta }_{k,r_k}}{\parallel {\Psi }_{k,r_k}\parallel }^2{Q}_k+{\eta }_{k,r_k}+{\overline{\epsilon }}_k)\kappa \omega (t)+z_{k,r_k}(x_{k,r_k+1}-{\alpha }_{k,r_k})\hfill \end{array}$$

(19)

Design virtual control law ${\alpha }_{k,r_k}$:

$$\begin{array}{ccc}\hfill {\alpha }_{k,r_k}& =& -k_{k,r_k}{z}_{k,r_k}-tanh(\frac{z_{k,r_k}}{\omega (t)})\frac{1}{2{\eta }_{k,r_k}}{\parallel {\Psi }_{k,r_k}\parallel }^2{\widehat{Q}}_k-tanh(\frac{z_{k,r_k}}{\omega (t)}){\eta }_{k,r_k}\hfill \\ & & -tanh(\frac{z_{k,r_k}}{\omega (t)}){\widehat{\overline{\epsilon }}}_k-\frac{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}}{\sqrt{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}^2+{\sigma }^2(t)}}-2z_{k,r_k-1}-\frac{e_{k,r_k-1}}{{\tau }_{k,r_k-1}}\hfill \end{array}$$

(20)

Define the notations:

$$\begin{array}{ccc}\hfill {\Xi }_{r_k}^{Q_k}& =& {\gamma }_{k,1}\sum _{q=1}^{r_k}{z}_{k,q}tanh(\frac{z_{k,q}}{\omega (t)})\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2\hfill \\ \hfill {\Xi }_{r_k}^{{\overline{\epsilon }}_k}& =& {\gamma }_{k,2}\sum _{q=1}^{r_k}{z}_{k,q}tanh(\frac{z_{k,q}}{\omega (t)})\hfill \end{array}$$

(21)

Combining (19)–(21) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,r_k}& \le & -\sum _{q=1}^{r_k}{K}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{r_k}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{r_k}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)+\hfill \\ & & \sum _{q=1}^{r_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\hfill \\ & & \sum _{q=1}^{r_k-1}{z}_{k,q}{e}_{k,q}+z_{k,r_k}(x_{k,r_k+1}-{\alpha }_{k,r_k})\hfill \end{array}$$

(22)

Design the nonlinear filter:

$$\begin{array}{ccc}\hfill {\tau }_{k,r_k}{\dot{s}}_{k,r_k}& =& -e_{k,r_k}-\frac{{\tau }_{k,r_k}{\widehat{M}}_{k,r_k}^2{e}_{k,r_k}}{\sqrt{{\widehat{M}}_{k,r_k}^2{e}_{k,r_k}^2+{\sigma }^2\left(t\right)}}-{\tau }_{k,r_k}{z}_{k,r_k}\hfill \\ \hfill s_{k,r_k}\left(0\right)& =& {\alpha }_{k,r_k}\left(0\right)\hfill \end{array}$$

(23)

where ${\tau }_{k,r_k}$ denotes the filter time constant; $e_{k,r_k}:=s_{k,r_k}-{\alpha }_{k,r_k}$ is the $r_k$th boundary layer error; ${\widehat{M}}_{k,r_k}$ is illustrated later.

Step (k, $n_k$): Let $z_{k,n_k}=x_{k,n_k}-s_{k,n_k-1}$. The time derivative of $z_{k,n_k}$ is

$$\begin{array}{c}\hfill {\dot{z}}_{k,n_k}={Q_{k,n_k}^{*}}^{\mathrm{T}}{\Psi }_{k,n_k}+{\epsilon }_{k,n_k}+u_k+\frac{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}}{\sqrt{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}^2+{\sigma }^2\left(t\right)}}+z_{k,n_k-1}+\frac{e_{k,n_k-1}}{{\tau }_{k,n_k-1}}\end{array}$$

(24)

Define the Lyapunov function:

$$\begin{array}{c}\hfill V_{k,n_k}=V_{k,n_k-1}+\frac{1}{2}{z}_{k,n_k}^2\end{array}$$

(25)

The time derivative of $V_{k,n_k}$ is

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,n_k}& =& {\dot{V}}_{k,n_k-1}+z_{k,n_k}{\dot{z}}_{k,n_k}\hfill \\ & =& -\sum _{q=1}^{n_k-1}{K}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{n_k-1}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{n_k-1}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)+\hfill \\ & & \sum _{q=1}^{n_k-1}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{n_k-2}{z}_{k,q}{e}_{k,q}+\hfill \\ & & z_{k,n_k-1}(x_{k,n_k}-{\alpha }_{k,n_k-1})+\hfill \\ & & z_{k,n_k}({Q_{k,n_k}^{*}}^{\mathrm{T}}{\Psi }_{k,n_k}+{\epsilon }_{k,n_k}+u_k+\frac{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}}{\sqrt{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}^2+{\sigma }^2(t)}}+z_{k,n_k-1}+\frac{e_{k,n_k-1}}{{\tau }_{k,n_k-1}})\hfill \end{array}$$

(26)

Leveraging Lemma 3, we obtain

$$\begin{array}{ccc}\hfill z_{k,n_k}{Q_{k,n_k}^{*}}^{\mathrm{T}}{\Psi }_{k,n_k}& \le & |z_{k,n_k}|\frac{1}{2{\eta }_{k,n_k}}\parallel {\Psi }_{k,n_k}{\parallel }^2{Q}_k+|z_{k,n_k}|{\eta }_{k,n_k}\hfill \\ & \le & z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)})\frac{1}{2{\eta }_{k,n_k}}{\parallel {\Psi }_{k,n_k}\parallel }^2{Q}_k+z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)}){\eta }_{k,n_k}\hfill \\ & & +(\frac{1}{2{\eta }_{k,n_k}}{\parallel {\Psi }_{k,n_k}\parallel }^2{Q}_k+{\eta }_{k,n_k})\kappa \omega (t)\hfill \\ \hfill z_{k,n_k}{\epsilon }_{k,n_k}& \le & |z_{k,n_k}|{\overline{\epsilon }}_k\le z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)}){\overline{\epsilon }}_k+{\overline{\epsilon }}_k\kappa \omega (t)\hfill \end{array}$$

(27)

Combining (27) and (26), we have

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,n_k}& \le & -\sum _{q=1}^{n_k-1}{k}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{n_k-1}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{n_k-1}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)\hfill \\ & & +\sum _{q=1}^{n_k-1}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{n_k-2}{z}_{k,q}{e}_{k,q}\hfill \\ & & +z_{k,n_k-1}(x_{k,n_k}-{\alpha }_{k,n_k-1})+z_{k,n_k}{u}_k+z_{k,n_k}\frac{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}}{\sqrt{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}^2+{\sigma }^2(t)}}\hfill \\ & & +z_{k,n_k}{z}_{k,n_k-1}+z_{k,n_k}\frac{e_{k,n_k-1}}{{\tau }_{k,n_k-1}}+z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)})\frac{1}{2{\eta }_{k,n_k}}{\parallel {\Psi }_{k,n_k}\parallel }^2{Q}_k\hfill \\ & & +z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)}){\eta }_{k,n_k}+z_{k,n_k}tanh(\frac{z_{k,n_k}}{\omega (t)}){\overline{\epsilon }}_k\hfill \\ & & +(\frac{1}{2{\eta }_{k,n_k}}{\parallel {\Psi }_{k,n_k}\parallel }^2{Q}_k+{\eta }_{k,n_k}+\overline{\epsilon })\kappa \omega (t)\hfill \end{array}$$

(28)

Design actual control law $u_k$:

$$\begin{array}{ccc}\hfill u_k& =& -K_{k,n_k}{z}_{k,n_k}-tanh(\frac{z_{k,n_k}}{\omega (t)})\frac{1}{2{\eta }_{k,n_k}}{\parallel {\Psi }_{k,n_k}\parallel }^2{\widehat{Q}}_k-tanh(\frac{z_{k,n_k}}{\omega (t)}){\eta }_{k,n_k}-\hfill \\ & & tanh(\frac{z_{k,n_k}}{\omega (t)}){\widehat{\overline{\epsilon }}}_k-\frac{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}}{\sqrt{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}^2+{\sigma }^2(t)}}-2z_{k,n_k-1}-\frac{e_{k,n_k-1}}{{\tau }_{k,n_k-1}}\hfill \end{array}$$

(29)

Define the notations:

$$\begin{array}{ccc}\hfill {\Xi }_{n_k}^{Q_k}& =& {\gamma }_{k,1}\sum _{q=1}^{n_k}{z}_{k,q}tanh(\frac{z_{k,q}}{\omega (t)})\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2\hfill \\ \hfill {\Xi }_{n_k}^{{\overline{\epsilon }}_k}& =& {\gamma }_{k,2}\sum _{q=1}^{n_k}{z}_{k,q}tanh(\frac{z_{k,q}}{\omega (t)})\hfill \end{array}$$

(30)

Combining (28)–(30) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,n_k}& \le & -\sum _{q=1}^{n_k}{K}_{k,q}{z}_{k,q}^2+\frac{1}{{\gamma }_{k,1}}{\tilde{Q}}_k({\Xi }_{n_k}^{Q_k}-{\dot{\widehat{Q}}}_k)+\frac{1}{{\gamma }_{k,2}}{\tilde{\overline{\epsilon }}}_k({\Xi }_{n_k}^{{\overline{\epsilon }}_k}-{\dot{\widehat{\overline{\epsilon }}}}_k)\hfill \\ & & +\sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{n_k-1}{z}_{k,q}{e}_{k,q}\hfill \end{array}$$

(31)

Design the update laws of $\widehat{Q}$ and $\widehat{\overline{\epsilon }}$:

$$\begin{array}{ccc}\hfill {\dot{\widehat{Q}}}_k& =& {\Xi }_{n_k}^{Q_k}-{\gamma }_{k,1}\omega (t){\widehat{Q}}_k\hfill \\ \hfill {\dot{\widehat{\overline{\epsilon }}}}_k& =& {\Xi }_{n_k}^{{\overline{\epsilon }}_k}-{\gamma }_{k,2}\omega (t){\widehat{\overline{\epsilon }}}_k\hfill \end{array}$$

(32)

Thus, (31) is written as:

$$\begin{array}{ccc}\hfill {\dot{V}}_{k,n_k}& \le & -\sum _{q=1}^{n_k}{k}_{k,q}{z}_{k,q}^2+{\tilde{Q}}_k\omega (t){\widehat{Q}}_k+{\tilde{\overline{\epsilon }}}_k\omega (t){\widehat{\overline{\epsilon }}}_k+\hfill \\ & & \sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega (t)+\sum _{q=1}^{n_k-1}{z}_{k,q}{e}_{k,q}\hfill \end{array}$$

(33)

3.2. Stability Analysis

Differentiating the surface errors:

$$\begin{array}{ccc}\hfill {\dot{z}}_{k,1}& =& {Q_{k,1}^{*}}^{\mathrm{T}}{\Psi }_{k,1}\left({\overline{x}}_{k,1}\right)+{\epsilon }_{k,1}\left({\overline{x}}_{k,1}\right)+x_{k,1+1}-{\dot{y}}_{k,d}\hfill \\ \hfill {\dot{z}}_{k,r_k}& =& {Q_{k,r_k}^{*}}^{\mathrm{T}}{\Psi }_{k,r_k}+{\epsilon }_{k,r_k}+x_{k,r_k+1}+\frac{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}}{\sqrt{{\widehat{M}}_{k,r_k-1}^2{e}_{k,r_k-1}^2+{\sigma }^2\left(t\right)}}+z_{k,r_k-1}+\frac{e_{k,r_k-1}}{{\tau }_{k,r_k-1}}\hfill \\ \hfill {\dot{z}}_{k,n_k}& =& {Q_{k,n_k}^{*}}^{\mathrm{T}}{\Psi }_{k,n_k}+{\epsilon }_{k,n_k}+u_k+\frac{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}}{\sqrt{{\widehat{M}}_{k,n_k-1}^2{e}_{k,n_k-1}^2+{\sigma }^2\left(t\right)}}+z_{k,n_k-1}+\frac{e_{k,n_k-1}}{{\tau }_{k,n_k-1}}\hfill \end{array}$$

(34)

Differentiating the boundary layer errors $e_{k,r_k}=s_{k,r_k}-{\alpha }_{k,r_k}$:

$$\begin{array}{ccc}\hfill {\dot{e}}_{k,r_k}& =& -\frac{e_{k,r_k}}{{\tau }_{k,r_k}}-\frac{{\widehat{M}}_{k,r_k}^2{e}_{k,r_k}}{\sqrt{{\widehat{M}}_{k,r_k}^2{e}_{k,r_k}^2+{\sigma }^2\left(t\right)}}-z_{k,r_k}+B_{k,r_k}(z_{k,1},\dots ,z_{k,(r_k+1)},e_{k,1},\dots ,e_{k,r_k},{\widehat{Q}}_1,\hfill \\ & & \dots ,{\widehat{Q}}_k,{\widehat{\overline{\epsilon }}}_1,\dots ,{\widehat{\overline{\epsilon }}}_k,y_{k,d},{\dot{y}}_{k,d},{\ddot{y}}_{k,d},\sigma \left(t\right),\dot{\sigma }\left(t\right),{\widehat{M}}_1,\dots ,{\widehat{M}}_i),r_k=1,\dots ,n_k-1\hfill \end{array}$$

(35)

where

$$\begin{array}{ccc}\hfill B_{k,1}(·)& =& -\frac{\partial {\alpha }_{k,1}}{\partial x_{k,1}}{\dot{x}}_{k,1}-\frac{\partial {\alpha }_{k,1}}{\partial y_{k,d}}{\dot{y}}_{k,d}-\frac{\partial {\alpha }_{k,1}}{\partial {\dot{y}}_{k,d}}{\ddot{y}}_{k,d}-\frac{\partial {\alpha }_{k,1}}{\partial {\widehat{W}}_k}{\dot{\widehat{W}}}_k-\frac{\partial {\alpha }_{k,1}}{\partial {\widehat{\overline{\epsilon }}}_k}{\dot{\widehat{\overline{\epsilon }}}}_k\hfill \\ \hfill B_{k,r_k}(·)& =& -\sum _{q=1}^{r_k}\frac{\partial {\alpha }_{k,r_k}}{\partial x_{k,q}}{\dot{x}}_{k,q}-\frac{\partial {\alpha }_{k,r_k}}{\partial {\widehat{Q}}_k}{\dot{\widehat{Q}}}_k-\frac{\partial {\alpha }_{k,r_k}}{\partial {\widehat{\overline{\epsilon }}}_k}{\dot{\widehat{\overline{\epsilon }}}}_k-\frac{\partial {\alpha }_{k,1}}{\partial y_{k,d}}{\dot{y}}_{k,d}-\frac{\partial {\alpha }_{k,r_k}}{\partial e_{k,(r_k-1)}}{\dot{e}}_{k,(r_k-1)}\hfill \\ & & -\frac{\partial {\alpha }_{k,1}}{\partial {\dot{y}}_{k,d}}{\ddot{y}}_{k,d}-\frac{\partial {\alpha }_{k,1}}{\partial \sigma \left(t\right)}\dot{\sigma }\left(t\right)-\frac{\partial {\alpha }_{k,r_k}}{\partial {\widehat{M}}_{k,(r_k-1)}}{\dot{\widehat{M}}}_{k,(r_k-1)},r_k=2,\dots ,n_k-1\hfill \end{array}$$

(36)

Define Lyapunov function $V_k$:

$$\begin{array}{c}\hfill V_k=V_{k,n_k}+\sum _{q=1}^{n_k-1}\frac{1}{2}{e}_{k,q}^2+\sum _{q=1}^{n_k-1}\frac{1}{2{\beta }_{k,q}}{\tilde{M}}_{k,q}^2\end{array}$$

(37)

where ${\beta }_{k,q},q=1,2,\dots ,n_k-1$ are constants that need to be designed.

According to the above design steps of the controller, we have the following theorem.

Theorem 1.

Consider system (1), Assumption 1, and $V_k\left(0\right)\le p$ for any initial with p being a positive constant, virtual control laws (11), (20), actual control law (29), nonlinear filters (14), (23), and the adaptive laws (32) guarantee the following:

• An output signal tracks the desired trajectory asymptotically, i.e., ${lim}_{t\to \infty }{z}_{k,1}\to 0$;
• The boundedness of all signals in the closed-loop system.

Proof. 

According to Assumption 1 and the bounded initial states, ${\Omega }_{k,0}=\{{[y_{k,d},{\dot{y}}_{k,d},{\ddot{y}}_{k,d}]}^T:y_{k,d}^2+{\dot{y}}_{k,d}^2+{\ddot{y}}_{k,d}^2\le B_{k,0}\}$ is compact in ${\Re }^3$, where $B_{k,0}>0$, the set ${\Omega }_{k,1}=\{V_k\left(t\right)\le p\}$ is compact in ${\Re }^{3n}$. Thus, ${\Omega }_{k,0}\times {\Omega }_{k,1}$ are compact in ${\Re }^{3n+3}$. In addition, $\sigma \left(t\right)$ satisfies ${lim}_{t\to \infty }{\int }_0^t\sigma \left(q\right)dq\le {\sigma }_{k,1}<+\infty$ and $|\dot{\sigma }\left(t\right)|\le {\sigma }_{k,2}<+\infty$ with ${\sigma }_{k,1}$ and ${\sigma }_{k,2}$ as constants. Thus, $|B_{k,q}(·)|\le M_{k,q}$ on ${\Omega }_{k,0}\times {\Omega }_{k,1}$, where $M_{k,q}$ denotes an unknown positive constant that can be estimated by ${\widehat{M}}_{k,q}$.

Differentiating $V_k$ yields

$$\begin{array}{ccc}\hfill \dot{V_k}& =& {\dot{V}}_{k,n_k}+\sum _{q=1}^{n_k-1}{\dot{e}}_{k,q}{e}_{k,q}-\sum _{q=1}^{n_k-1}\frac{1}{{\beta }_{k,q}}{\tilde{M}}_{k,q}{\dot{\widehat{M}}}_{k,q}\hfill \\ & =& -\sum _{q=1}^{n_k}{k}_{k,q}{z}_{k,q}^2+{\tilde{Q}}_k\omega \left(t\right){\widehat{Q}}_k+{\tilde{\overline{\epsilon }}}_k\omega \left(t\right){\widehat{\overline{\epsilon }}}_k+\sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \omega \left(t\right)-\hfill \\ & & \sum _{q=1}^{n_k-1}(\frac{e_{k,q}^2}{{\tau }_{k,q}}+z_{k,q}{e}_{k,q}-\frac{{\widehat{M}}_{k,q}^2{e}_{k,q}^2}{\sqrt{{\widehat{M}}_{k,q}^2{e}_{k,q}^2+{\sigma }^2\left(t\right)}}-e_{k,q}{z}_{k,q}+e_{k,q}{\widehat{M}}_{k,q}-\frac{1}{{\beta }_{k,q}}{\tilde{M}}_{k,q}{\dot{\widehat{M}}}_{k,q})\hfill \end{array}$$

(38)

Leveraging the inequalities:

$$\begin{array}{ccc}\hfill \omega \left(t\right){\tilde{Q}}_k{\widehat{Q}}_k& =& \omega \left(t\right)\left({\tilde{Q}}_k(-{\tilde{Q}}_k+Q_k)\right)\hfill \\ & =& -\omega \left(t\right){\tilde{Q}}_k^2+\omega \left(t\right){\tilde{Q}}_k{Q}_k\le \frac{1}{2}\omega \left(t\right)Q_k^2\hfill \\ \hfill \omega \left(t\right){\tilde{\overline{\epsilon }}}_k{\widehat{\overline{\epsilon }}}_k& \le & \frac{1}{2}\omega \left(t\right){\overline{\epsilon }}_k^2\hfill \end{array}$$

(39)

Then, considering $\parallel {\Psi }_{k,q}\parallel \le \sqrt{{\lambda }_{k,q}}$ with ${\lambda }_{k,q}>0$ yields

$$\begin{array}{ccc}\hfill \dot{V_k}& \le & -\sum _{q=1}^{n_k}{K}_{k,q}{z}_{k,q}^2+\left(\frac{1}{2}{Q}_k^2+\frac{1}{2}{\overline{\epsilon }}_k^2+\sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \right)\omega \left(t\right)-\hfill \\ & & \sum _{q=1}^{n_k-1}\left(\frac{e_{k,q}^2}{{\tau }_{k,q}}-\frac{{\widehat{M}}_{k,q}^2{e}_{k,q}^2}{\sqrt{{\widehat{M}}_{k,q}^2{e}_{k,q}^2+{\sigma }^2\left(t\right)}}+\left|e_{k,q}\right|{\widehat{M}}_{k,q}-\frac{1}{{\beta }_{k,q}}{\tilde{M}}_{k,q}{\dot{\widehat{M}}}_{k,q}\right)\hfill \end{array}$$

(40)

According to Lemma 1, we obtain

$$\begin{array}{ccc}\hfill M_{k,q}|e_{k,q}|& =& {\widehat{M}}_{k,q}|e_{k,q}|+{\tilde{M}}_{k,q}|e_{k,q}|\hfill \\ & \le & \frac{{\widehat{M}}_{k,q}^2{e}_{k,q}^2}{\sqrt{{\widehat{M}}_{k,q}^2{e}_{k,q}^2+{\sigma }^2\left(t\right)}}+\sigma \left(t\right)+{\tilde{M}}_{k,q}|e_{k,q}|\hfill \end{array}$$

(41)

Then, (40) is written as

$$\begin{array}{ccc}\hfill \dot{V_k}& \le & -\sum _{q=1}^{n_k}{K}_{k,q}{z}_{k,q}^2+\left(\frac{1}{2}{Q}_k^2+\frac{1}{2}{\overline{\epsilon }}_k^2+\sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \right)\omega \left(t\right)-\hfill \\ & & \sum _{q=1}^{n_k-1}\frac{e_{k,q}^2}{{\tau }_{k,q}}+(n_k-1)\sigma \left(t\right)-\sum _{q=1}^{n_k-1}\frac{1}{{\beta }_{k,q}}{\tilde{M}}_{k,q}({\dot{\widehat{M}}}_{k,q}-{\beta }_{k,q}|e_{k,q}|)\hfill \end{array}$$

(42)

Design update laws for ${\widehat{M}}_{k,q}$:

$$\begin{array}{c}\hfill {\dot{\widehat{M}}}_{k,q}={\beta }_{k,q}\left|e_{k,q}\right|,q=1,\dots ,n_k-1\end{array}$$

(43)

Define notation:

$$\begin{array}{c}\hfill {\phi }_k=\frac{1}{2}{Q}_k^2+\frac{1}{2}{\overline{\epsilon }}_k^2+\sum _{q=1}^{n_k}(\frac{1}{2{\eta }_{k,q}}{\parallel {\Psi }_{k,q}\parallel }^2{Q}_k+{\eta }_{k,q}+{\overline{\epsilon }}_k)\kappa \end{array}$$

(44)

Thus, (42) is written as

$$\begin{array}{ccc}\hfill \dot{V_k}& \le & -\sum _{q=1}^{n_k}{k}_{k,q}{z}_{k,q}^2-\sum _{q=1}^{n_k-1}\frac{e_{k,q}^2}{{\tau }_{k,q}}+{\phi }_k\omega \left(t\right)+(n_k-1)\sigma \left(t\right)\hfill \end{array}$$

(45)

Integrating (45), we have

$$\begin{array}{ccc}\hfill V_k\left(t\right)& \le & V_k\left(0\right)+{\phi }_k{\int }_0^t\omega \left(w\right)dw+(n_k-1){\int }_0^t\sigma \left(w\right)dw-\hfill \\ & & {\int }_0^t\left\{\sum _{q=1}^{n_k}{k}_{k,q}{z}_{k,q}^2+\sum _{q=1}^{n_k-1}\frac{e_{k,q}^2\left(w\right)}{{\tau }_{k,q}}\right\}dw\hfill \\ & \le & V_k\left(0\right)+{\phi }_k{\omega }_1+(n_k-1){\sigma }_1\hfill \end{array}$$

(46)

From (46), we can conclude that $z_{k,q},z_{k,n_k},{\widehat{\theta }}_{k,q},{\widehat{\theta }}_{k,n_k},e_{k,q}$, and ${\widehat{M}}_{k,q}$, $q=1,\dots ,n_k-1$, are bounded. Then, $x_{k,q}$, $x_{k,n_k},s_{k,q},{\alpha }_{k,q},q=1,\dots ,n_k-1$, and $u_k$ are bounded. In addition, from (46), we have

$$\begin{array}{c}\hfill {\int }_0^t\sum _{q=1}^{n_k}{k}_{k,q}{z}_{k,q}^2\left(w\right)dw\le V_k\left(0\right)+{\phi }_k{\omega }_1+(n_k-1){\sigma }_1\end{array}$$

(47)

Leveraging Lemma 4, we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{z}_{k,1}\to 0\end{array}$$

(48)

Define Lyapunov function V for LSHONS:

$$\begin{array}{ccc}\hfill V& =& \sum _{k=1}^N{V}_k\le \sum _{k=1}^N(V_k\left(0\right)+{\phi }_k{\omega }_1+(n_k-1){\sigma }_1)\hfill \end{array}$$

(49)

Using Lemma 4, the stability of the total LSHONS is guaranteed. □

4. Simulations and Discussion

To demonstrate the effectiveness of the proposed algorithm, the modified DSC (MDSC) and traditional DSC (TDSC) are used in theory and practice nonlinear system, respectively.

4.1. Nonlinear System

Consider a nonlinear system [33], which contains two subsystems

$$\begin{array}{ccc}\hfill {\dot{x}}_{1,1}& =& {\theta }_{1,1}{x}_{1,1}^3{sin}^2\left(x_{1,1}\right)+x_{1,2}\hfill \\ \hfill {\dot{x}}_{1,2}& =& {\theta }_{1,2}{x}_{1,1}^3{sin}^2\left(x_{1,1}{x}_{1,2}{x}_{2,1}{x}_{2,2}\right)tanh\left(x_{1,1}{x}_{2,1}\right)+u_1\hfill \\ \hfill y_1& =& x_{1,1}\hfill \\ \hfill {\dot{x}}_{2,1}& =& {\theta }_{2,1}{x}_{2,1}^3{sin}^2\left(x_{2,1}\right)+x_{2,2}\hfill \\ \hfill {\dot{x}}_{2,2}& =& {\theta }_{2,2}{x}_{2,1}^3{sin}^2\left(x_{1,1}{x}_{1,2}{x}_{2,1}{x}_{2,2}\right)+u_2\hfill \\ \hfill y_2& =& x_{2,1}\hfill \end{array}$$

(50)

where ${\theta }_{1,1}=2,{\theta }_{1,2}=1,{\theta }_{2,1}=2,{\theta }_{2,2}=1$. $f_{1,1}\left({\overline{x}}_{1,1}\right)$ are nonlinear terms of the system only used to update the system states; they are not used in the designed controller.

The initial conditions are chosen as: $x_{1,1}\left(0\right)=0,x_{2,1}\left(0\right)=0,x_{1,2}\left(0\right)=1,x_{2,2}\left(0\right)=1$. The referenced trajectory $y_{k,d}=sin\left(t\right)$. FLSs inputs are chosen as: $[z_{k,1},\dots ,z_{k,r_k},x_{k,1},\dots ,x_{k,r_k}]$ for the kth subsystem at the $r_k$th step; ${\mu }_p=[-1,-0.5,0,0.5,1]$ and ${\chi }_p=0.5$. The control parameters are chosen as: $k_{1,1}=10,k_{1,2}=10$, $k_{2,1}=25,k_{2,2}=15$, ${\gamma }_{1,1}=15,{\gamma }_{1,2}=15$, ${\gamma }_{2,1}=10,{\gamma }_{2,2}=10$, ${\tau }_{1,1}=0.1,{\tau }_{2,1}=0.1$, $\sigma \left(t\right)=\nu e^{-\nu t}$ with $\nu =0.001$, ${\beta }_{1,1}=5,{\beta }_{2,1}=5$.

Figure 1a,b show the output tracking performance of each subsystem using the MDSC; systems $y_1$ and $y_2$ can track the referenced trajectory asymptotically. Figure 1c,d show the smooth and bounded control input of each subsystem. Figure 1e shows the curves of the system parameters; it is clear that they are bounded. Figure 1f shows the system state, which is guaranteed to be bounded.

4.2. Comparative Simulation

To illustrate the advantages of the developed MDSC, the TDSC with a linear filter is applied to (50). The detailed steps of the TDSC are shown as follows:

$$\begin{array}{ccc}\hfill {\alpha }_{k,1}& =& \frac{1}{g_{k,1}}(-k_{k,1}{z}_{k,1}-{\widehat{\theta }}_{k,1}{f}_{k,1}\left({\overline{x}}_{k,1}\right)+{\dot{y}}_{k,d})\hfill \\ \hfill {\alpha }_{k,1}& =& {\phi }_{k,1}+{\tau }_{k,1}{\dot{\phi }}_{k,1},{\phi }_{k,1}\left(0\right)={\alpha }_{k,1}\left(0\right)\hfill \\ \hfill {\alpha }_{k,r_k}& =& \frac{1}{g_{k,r_k}}\left(-k_{k,r_k}{z}_{k,r_k}-{\widehat{\theta }}_{k,r_k}{f}_{k,r_k}\left({\overline{x}}_{k,r_k}\right)+\frac{({\alpha }_{k,(r_k-1)}-s_{1,(r_k-1)})}{{\tau }_{k,(r_k-1)}}\right)\hfill \\ \hfill {\alpha }_{k,r_k}& =& {\phi }_{k,r_k}+{\tau }_{k,r_k}{\dot{\phi }}_{k,r_k},{\phi }_{k,r_k}\left(0\right)={\alpha }_{k,r_k}\left(0\right)\hfill \\ \hfill u_k& =& \frac{1}{g_{k,n_k}}\left(-k_{k,n_k}{z}_{k,n_k}-{\widehat{\theta }}_{k,n_k}{f}_{k,n_k}\left({\overline{x}}_{k,n_k}\right)+\frac{({\alpha }_{k,(n_k-1)}-s_{1,(n_k-1)})}{{\tau }_{k,(n_k-1)}}\right)\hfill \\ \hfill {\dot{\widehat{\theta }}}_{k,1}& =& {\gamma }_{k,1}{z}_{k,1}{f}_{k,1}\left({\overline{x}}_{k,1}\right)\hfill \\ \hfill {\dot{\widehat{\theta }}}_{k,r_k}& =& {\gamma }_{k,r_k}{f}_{k,r_k}\left({\overline{x}}_{k,r_k}\right)\hfill \\ \hfill {\dot{\widehat{\theta }}}_{k,n_k}& =& {\gamma }_{k,n_k}{f}_{k,n_k}\left({\overline{x}}_{k,n_k}\right)\hfill \end{array}$$

(51)

The TDSCs were proposed in many studies in the existing literature; more details can be found in the original text in [13,34]. In addition, the meaning of notations in (51) is the same as in the definition in the MDSC. The initial state of the system and the values of control parameters are selected according to the above example. It should be pointed out that the nonlinear term should be accurately modeled in (51). In addition, (51) adopts the tradition linear filter.

The simulation results of algorithm in (51) for system (50) are shown in Figure 2. In detail, Figure 2a,b perform the output tracking curves of subsystem 1 and subsystem 2; Figure 2c,d show the control input and virtual control law curves of each subsystem; Figure 2e depicts the adaptive parameters in the LSHONS; Figure 2f gives the curves of $x_{1,2}$ and $x_{2,2}$.

4.3. Case Study: Two Inverted Pendulums on Connected Cars

The MDSC is applied to control two inverted pendulums on connected cars, whose dynamics are given as:

$$\begin{array}{ccc}\hfill {\dot{x}}_{1,1}& =& x_{1,2}\hfill \\ \hfill {\dot{x}}_{1,2}& =& \left[\frac{g}{cl}-z\left(t\right)\right]x_{1,1}+z\left(t\right)x_{2,1}+\frac{m\sin \left(x_{1,1}\right)}{M}{x}_{1,2}^2+\frac{z\left(t\right)}{a\left(t\right)}(\sin \left(2t\right)-\sin \left(3t\right)-L)+\hfill \\ & & \frac{1}{cm{l}^2}{u}_1\hfill \\ \hfill {\dot{x}}_{2,1}& =& x_{2,2}\hfill \\ \hfill {\dot{x}}_{2,2}& =& \left[\frac{g}{cl}-z\left(t\right)\right]x_{2,1}+z\left(t\right)x_{2,1}+\frac{m\sin \left(x_{1,1}\right)}{M}{x}_{2,2}^2+\frac{z\left(t\right)}{a\left(t\right)}(L+\sin \left(3t\right)-\sin \left(2t\right))+\hfill \\ & & \frac{1}{cm{l}^2}{u}_2\hfill \\ \hfill y_1& =& x_{1,1}\hfill \\ \hfill y_2& =& x_{2,1}\hfill \end{array}$$

(52)

where $z\left(t\right)=[\left(ka\left(t\right)(a\left(t\right)-cl)\right]/\left(cm{l}^2\right)$, and the coefficients of the system are the same as in [35,36]: $M=m=10$ kg, $L=2$ m, $l=1$ m, $g=9.81$ m/s${}^2$, $k=1$ N/m, $c=m/(m+M)=0.5$. The system initial states are $x_{1,1}\left(0\right)=x_{1,2}\left(0\right)=\pi /3.6,x_{2,1}\left(0\right)=x_{2,2}\left(0\right)=\pi /4$. The control parameters are $K_{k,1}=25,K_{k,2}=28,k=1,2$. The output reference trajectory is set as $y_{1,d}=sin\left(t\right)+0.5sin\left(2t\right),y_{2,d}=0.5sin\left(2t\right)$.

Figure 3a,b perform the output tracking curves of subsystem 1 and subsystem 2; Figure 2c,d show the control input and virtual control law curves of each subsystem. The developed MDSC achieves asymptotic output tracking while ensuring the boundedness of all signals. The effectiveness of the developed algorithm has been well demonstrated.

5. Conclusions

An asymptotic fuzzy adaptive dynamic surface controller was developed for the LSHONS. Uncertain nonlinear terms were well approximated by FLSs whose parameters are updated by an adaptive technique such that the necessary factors for an accurate nonlinear model of the system were released. An MDSC with a nonlinear filter approach was proposed to avoid the “explosion of complexity" problem and eliminate the boundary layer error in TDSC with a linear filter and the approximation error of FLSs.

Compared with the traditional fuzzy adaptive control method for the LSHONS, which needs to design update laws in each step, the proposed method only updates two parameters in the final step for the each subsystem. Compared with the TDSC, the proposed MDSC can achieve the asymptotic tracking for mathematical and practical LSHONSs. This advantage depends on the designed nonlinear filter, which can eliminate the approximation error of fuzzy logic and the boundary layer error of the TDSC. However, to achieve asymptotic tracking, an additional adaptive law should be designed. In the future work, we are preparing to design an adaptive law for the overall LSHONS, rather than for each subsystem, in order to further simplify the complexity of the control algorithm. In addition, we would like to apply the designed controller to a physical system to prove the effectiveness of the designed algorithm in our future work.