Disturbance Compensator Design Based on Dilated LMI for Linear Parameter-Varying Systems

Abstract

This paper presents a new dilated linear matrix inequality (LMI) representation to design a state feedback controller and a dynamic feedforward disturbance compensator for linear parameter-varying (LPV) systems. The improved LMIs are convex and finite-dimensional without any iterative approach. The designs are based on a new proposed equivalent bounded real lemma (BRL) by means of matrix dilation for LPV systems and uncertain linear systems under time-varying parametric uncertainties (TVPUs). This dilated BRL provides lower conservative results than existing methods in terms of robust stability. Accordingly, a dynamic disturbance compensator is designed in addition to a state feedback controller. This paper mainly focuses on the design of compensators against disturbances in addition to the design of state feedback controllers. The dynamic matrices of the compensator change with the time-varying parameters of the LPV or uncertain system during operation, assuming that the disturbances and the parameters are measurable or observable. The compensator can be designed to attenuate the disturbances/noises or to improve reference tracking. Finally, numerical and simulation outcomes are presented to prove both the effectiveness and lower conservativeness of the proposed LMIs.

1. Introduction

Robust performance analyses of LPV systems, which are a special class of nonlinear system, and linear systems with time-varying parametric uncertainties (TVPUs) have drawn considerable attention. Over the past two decades, many LMIs based on Lyapunov’s stability criteria have been proposed for these systems since the efficiency of LMI-based methods is well known, such as [1,2,3,4]. Most of them are based on the standard quadratic Lyapunov function known as quadratic stability where a single quadratic Lyapunov function is used as in [5]. For example, a numerical comparison is made in terms of the conservativeness of quadratic stability conditions in [6]. However, many studies have shown that a parameter-dependent Lyapunov matrix (PDLM) overcomes the conservativeness derived from a single quadratic Lyapunov function for the systems such as those in [6,7,8,9,10,11,12,13]. However, it is not appropriate for LPV systems or the continuous-time system (CTS) under TVPUs without an extra method since the PDLM or parameterized LMI (PLMI) causes a nonconvex optimization problem. However, the PDLM decreases the conservatism arising from the quadratic stability conditions. To address this problem, some approaches have been proposed such as the grid method [14], the s-procedure [15], the multi-convexity method [15,16], and the methods in [17,18]. Nevertheless, the grid method is valid only for local operation conditions contrary to all-admissible parameter trajectories. In addition, the other approaches involve the extra LMI conditions, some restrictions or complexity apart from the existing LMIs to obtain a convex solution when the PDLM is used. Furthermore, these approaches do not enable the design of multi-purpose control in [12]. Moreover, a state feedback controller designed by using the methods depends on a Lyapunov function that is restricted by the symmetric property.

Recently, to provide lower conservativeness for uncertain and LPV systems, Lyapunov matrices depending on the parameters or PLMIs have been researched without extra LMI conditions. Accordingly, “dilated” or “extended” LMIs have been suggested to address nonconvex problems and have been very popular over the past two decades. S-variables and slack-variable LMIs (SV-LMIs) are also known since slack variables are generally used for LMIs [19]. The benefit of dilated LMIs is that lower conservative results are obtained for very large parameter sets since they do not contain the multiplication of the Lyapunov matrix with the state-space matrices of the system in contrast to the common LMIs as in [5]. This case also allows us to check the robust performance or stability, and a convex LMI formulation is obtained for the systems owing to the separation. In addition, dilated LMIs have been valuable tools for designing multi-objective controllers with lower conservativeness, as in [20]. In brief, the dilated LMI approach presents a convex solution and lower conservative results for the LPV or uncertain linear systems without any extra conditions despite the PDLM, which leads to nonconvexity or infinite LMIs in common designs. For example, a stability condition is proposed in [8,21], but an H_∞ control design is not addressed. In [20], an obvious benefit of dilated LMIs over classical LMIs is presented for designing multi-objective H₂/D-stability controllers, but this formulation cannot be extended to H_∞ control synthesis. In [22], H₂ control via a slack variable is proposed, but H_∞ control is not addressed. In [23], lower conservative dilated LMI conditions are proposed under rationally varying parameters, but they are not appropriate for a CTS with TVPU and LPV systems. Apart from these studies, the improvements for the state feedback control in [24,25,26,27,28,29,30,31] by using dilated LMIs are more remarkable in the literature in terms of conservativeness and/or simplicity. These studies are less conservative than the standard state feedback controller design in [1]. However, none of them is fully appropriate for LPV systems and CTS under the TVPU whereas they are improved for CTS under time-invariant parametric uncertainties (TIPUs). Additionally, the proposed dilated LMI outperforms TIPU in terms of conservativeness. Accordingly, this specific topic is still open in the literature especially for the CTS with TVPU and LPV systems. Therefore, the first contribution of this paper is that a new dilated BRL is proposed for LPV systems and CTS with TVPU. This is subsequently extended to solve the state feedback H_∞ control problem. Therefore, less conservative results are obtained for LPV systems or the CTS with TVPU according to existing results in the literature.

Feedforward compensators or controllers are used to track the reference signal or to attenuate the external or internal disturbances, which are mostly predictable or measurable as in [32,33]. The performance of the system is improved owing to an extra generated control signal in accordance with the measured disturbance or reference values. In particular, disturbance compensators have been realized for many systems, such as [34]. A disturbance compensator is used together with a feedback controller as in [35,36], because the feedback design guarantees plant stability. However, the compensator does not provide stability to the unstable system. There are two main methods. The first is that they are concurrently designed as in [37,38]. For the second one, a certain performance is previously achieved with the feedback controller, and then the feedforward part is designed as in [39]. This paper is based on the second approach. Many feedforward compensators or controllers have been presented for LTI systems such as those in the above studies. There are a few works via the H_∞ method, especially for LPV systems, as in [40,41,42,43,44,45]. In [40], an H_∞ controller, which is reduced-order, has been designed to address the disturbances of the vibration system. In [41], a dynamic H_∞ feedforward controller design has been addressed by obtaining the system inverse. In [42], switched robust feedforward control has been presented for a class of uncertain linear systems defined in a classic linear fractional transformation form. However, none of them is appropriate for LPV systems. In [43,44], static LPV feedforward designs are used, but they are not dynamic controllers. A dynamic feedforward controller has been designed for LPV systems in [45], but many LMIs exist because some extra LMI conditions are needed because of nonconvexity from using the PDLM. In addition, the LMIs are based on a common BRL, so LMIs are somewhat conservative because of the multiplication of the Lyapunov matrix. Therefore, the feedforward controller or compensator based on L₂ gain has been rarely discussed, especially for LPV systems. Moreover, a dynamic feedforward controller or compensator design has not been used in the above studies, achieving the advantages of the dilated LMI. Therefore, the second contribution of this paper is the use of a dynamic LPV feed-forward compensator based on an L₂-gain controller design to achieve the advantages of the extended LMI approach, which according to the literature survey has not been reported in the literature so far. In [45], which is closer to this paper, the study is quite complex because of the occurrence of infinite LMIs. In addition, the design is based on standard BRL, which produces more conservative results.

In brief, in this study, solutions based on noniterative LMIs are proposed to design an LPV state feedback controller and a dynamic LPV feedforward compensator. They are convenient for both LPV and CTS with TVPU (linear systems with parametric uncertainty). The main contributions of the paper are as follows:

(1)

A dilated BRL based on a noniterative LMI is proposed for the CTS with TVPUs and LPV system by providing the advantages of a slack variable and dilated LMI. Then, a new state feedback controller is designed on the basis of a new equivalent BRL. With respect to its superiority, it has the best result among recent studies in [5,24,25,26,27,28,29,30] in view of its being less conservative as shown in Table 1, and an approximately 10% improvement has been achieved from the best result among them.

(2)

All proposed solutions do not need any additional LMI conditions as in [45] owing to the proposed dilated BRL even if the PDLM is used, which leads to nonconvexity or infinite LMIs. They are simpler and less conservative than the common approaches in the literature and less conservative than similar studies in the literature as proven via numerical tests. Therefore, the LMIs provide both simplicity and lower conservativeness.

(3)

A dynamic LPV feedforward compensator based on the dilated BRL is proposed for LPV systems with measurable or observable disturbance or noise. Therefore, the compensator works online with the time-varying parametric uncertainties of uncertain systems or the time-varying parameters for LPV systems.

Table 1. Comparisons of the achieved γ_min.

	[5]	[24]	[25]	[26]	[27]	[28]	[29]	[30]	Proposed Theorem 1
γmin	1.557	1.498	1.478	1.476	1.251	1.241	1.240	1.241	1.118

Table 1. Comparisons of the achieved γ_min.

	[5]	[24]	[25]	[26]	[27]	[28]	[29]	[30]	Proposed Theorem 1
γmin	1.557	1.498	1.478	1.476	1.251	1.241	1.240	1.241	1.118

2. Preliminaries

In this section, some definitions and lemmas are given to shed light on the proposed lemma and theorems.

Definition 1. 

The system described in (1) is named an LPV system. Here,  $x\left(t\right)\in {\Re }^{n_w}$ represents the states,  $u\left(t\right)\in {\Re }^{n_u}$ represents the control input signals,  $z\left(t\right)\in {\Re }^{n_z}$ represents the controlled outputs,   $y\left(t\right)\in {\Re }^{n_y}$ represents the measured outputs,  $\omega \left(t\right)\in {\Re }^{n_w}$ represents the disturbances and the whole matrices are of appropriate size and dependent on  $\alpha$ parameters changing with time. The parameter sets are defined by (2) where  ${\mathsf{\Delta }}_R$ and  ${\mathsf{\Delta }}_D$ denote the parameter sets for  $\alpha$ and its derivative, respectively. Additionally, the matrix  $A\left(\alpha \right)$ is the state matrix that defines the relationship between the current state and its rate of change,  $B_1\left(\alpha \right)$ is the disturbance input matrix,  $B_2\left(\alpha \right)$ is the control input matrix,  $C_1\left(\alpha \right)$ is the matrix that relates the state vector  $x\left(t\right)$ to the controlled outputs  $z\left(t\right)$,  $D_{11}\left(\alpha \right)$ is the feedforward matrix for the disturbance input  $\omega \left(t\right)$ directly linking the disturbance input  $\omega \left(t\right)$ to the controlled output  $z\left(t\right)$,  $D_{12}\left(\alpha \right)$ is the feedforward matrix for the control input  $u\left(t\right)$ directly linking the control input to the controlled output z(t), $C_2\left(\alpha \right)$ is the measured output matrix defining the relationship between the state and the measured output,  $D_{21}\left(\alpha \right)$ is the feedforward matrix for the disturbance input directly linking the disturbance input  $\omega \left(t\right)$ to the measured output  $y\left(t\right)$ and  $D_{22}\left(\alpha \right)$ is the feedforward matrix for the control input  $u\left(t\right)$ directly linking the control input to the measured output  $y\left(t\right)$.

$$\begin{array}{l}\dot{x}\left(t\right)=A\left(\alpha \left(t\right)\right)x\left(t\right)+B_1\left(\alpha \left(t\right)\right)\omega \left(t\right)+B_2\left(\alpha \left(t\right)\right)u\left(t\right)\\ z\left(t\right)=C_1\left(\alpha \left(t\right)\right)x\left(t\right)+D_{11}\left(\alpha \left(t\right)\right)\omega \left(t\right)+D_{12}\left(\alpha \left(t\right)\right)u\left(t\right)\\ y\left(t\right)=C_2\left(\alpha \left(t\right)\right)x\left(t\right)+D_{21}\left(\alpha \left(t\right)\right)\omega \left(t\right)+D_{22}\left(\alpha \left(t\right)\right)u\left(t\right)\end{array}$$

(1)

$$\begin{array}{l}{\Delta }_R=\left\{\alpha \in {\Re }^n:\overline{{\alpha }_i} \le {\alpha }_i\le \underset{¯}{{\alpha }_i}, {\forall }_i=1,\dots ,n\right\}\\ {\Delta }_D=\left\{\dot{\alpha }\in {\Re }^n:\overline{{\dot{\alpha }}_i} \le {\dot{\alpha }}_i\le \underset{¯}{{\dot{\alpha }}_i}, {\forall }_i=1,\dots ,n\right\}\end{array}$$

(2)

In addition, the closed-loop form of the system can be written as in (3) where Z indicates the outputs z and y. Additionally, the matrices A, B, C and D are considered as the closed-loop matrices.

$$\begin{array}{l}\dot{x}\left(t\right)=A\left(\alpha \left(t\right)\right)x\left(t\right)+B\left(\alpha \left(t\right)\right)\omega \left(t\right)\\ Z\left(t\right)=C\left(\alpha \left(t\right)\right)x\left(t\right)+D\left(\alpha \left(t\right)\right)\omega \left(t\right)\end{array}$$

(3)

It is assumed that the parameters are available online (measurable or computable). Additionally, the parameters are derived from the states of the nonlinear system although it normally looks like an uncertain system. If the parameters are not derived from states, unlike those of the LPV system, (1) is the linear system under time-varying uncertainties as in Example 1.

The H_∞ norm for the system in (3) can be calculated via the common LMI formulation given by Lemma 1, which is well known as the BRL in the literature. Accordingly, for a positive scalar $\gamma$, we define the performance index by (4), where $\omega \left(t\right)\in L_2^q\left.\begin{array}{cc}(0& \infty \end{array}\right)$ is an exogenous disturbance.

$$J\left(\omega ,z\right)=\left({\displaystyle \int z^{T}z-{\gamma }^2{\omega }^T\omega }\right)d\tau$$

(4)

Lemma 1. 

Let $G\left(\alpha \right)$ denote the stable transfer matrix of the LPV system in (3) for all admissible parameter trajectories. If and only if ${\gamma }^{2}I-D^T\left(\alpha \right)D\left(\alpha \right)\succ 0$, $G{\left(s\right)}_{\infty }\prec \gamma$ and the following statements are equivalent for all $\forall \left(\alpha \right)\in {\Delta }_R and \forall \left(\dot{\alpha }\right)\in {\Delta }_D$.

(i) 

If there exists a positive definite differentiable symmetric matrix  $P\left(\alpha \right)\in {\Re }^{n\times n}$, then

$$\begin{array}{l}{A}^T\left(\alpha \right)P\left(\alpha \right)+P\left(\alpha \right)A\left(\alpha \right)+\dot{P}\left(\alpha \right)+C^T\left(\alpha \right)C\left(\alpha \right)\\ +{\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)}^T{\left({\gamma }^{2}I-D^T\left(\alpha \right)D\left(\alpha \right)\right)}^{-1}\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)\prec 0 \end{array}$$

(5)

(ii) 

If there exists a positive definite differentiable symmetric matrix  $P\left(\alpha \right)\in {\Re }^{n\times n}$, then

$$\left(\begin{array}{ccc}{A}^T\left(\alpha \right)P\left(\alpha \right)+P\left(\alpha \right)A\left(\alpha \right)+\dot{P}\left(\alpha \right)& P\left(\alpha \right)B\left(\alpha \right)& C^T\left(\alpha \right)\\ B^T\left(\alpha \right)P\left(\alpha \right)& -{\gamma }^{2}I& D^T\left(\alpha \right)\\ C\left(\alpha \right)& D\left(\alpha \right)& -I\end{array}\right)\prec 0$$

(6)

Remark 1. 

LMIs such as those in (6) are not convenient to design controllers for linear systems with polytopic or affine-type uncertain parameters or LPV systems since the inequality involves the multiplication of the Lyapunov matrix by the matrices of the plant. Therefore, this affair results in a nonconvex optimization and the infinite LMIs. Nevertheless, some methods have been suggested to eliminate this difficulty such as the grid method [14], the s-procedure [15], the multi-convexity method [11] and methods in [17,18]. However, the multi-convexity method and the s-procedure result in many LMIs, and the grid method is only valid for local operation conditions in contrast to all-admissible parameter trajectories. In addition, such techniques do not enable the design of a multi-purpose controller in [12]. The use of the parameter-independent (fixed) Lyapunov function in (7) as in [5,41,46] is common, but this causes highly conservative results. Conversely, when the Lyapunov function is isolated from the plant matrices by using a slack variable, which is generally called the dilated or extended LMI approach due to the enlarging dimension of the LMI, this enables one to diminish the conservatism for analysis and design, as in [23,24,27,31]. Therefore, attaining a dilated condition equivalent to the BRL is an effective approach to overcome this complexity in robust control designs such as H₂ and H_∞. Accordingly, a new notation for H_∞ computation including the TVPU has been proposed in Lemma 3 for controller design.

The standard LMI in (7) derived from Lemma 1 is generally employed for the L₂-gain or H_∞ design ($K\left(\alpha \right)=R\left(\alpha \right)P^{-1}$) for LPV systems in the literature [5].

$$\left(\begin{array}{ccc}{A}^T\left(\alpha \right)P+B_{2}R\left(\alpha \right)+{\left(*\right)}^T& B_1\left(\alpha \right)& P{C}_1^T\left(\alpha \right)+R^T\left(\alpha \right)D_{12}^T\\ *& -{\gamma }^{2}I& D_{11}^T\left(\alpha \right)\\ *& *& -I\end{array}\right)\prec 0$$

(7)

The following Lemma 2, which is based on the projection lemma, is useful for deriving the dilated LMIs.

Lemma 2. 

[21]: If there exist a symmetric matrix $\Omega$ and compatible dimensional matrices  ${\Theta }_1, {\Theta }_2$, then, the following conditions are equivalent:

i. 

$\Omega <0$ and  $\Omega +{\Theta }_1{\Theta }_2^T+{\Theta }_2{\Theta }_1^T<0$

ii. 

The following LMI problem is feasible in accordance with  $Q$.

$$\left(\begin{array}{cc}\Omega & {\Theta }_2+{\Theta }_{1}Q\\ {\Theta }_2^T+Q^T{\Theta }_1^T& -Q-Q^T\end{array}\right)\prec 0$$

3. A New Dilated Bounded Real Lemma

In this section, a dilated BRL is proposed for the LPV and uncertain systems. Accordingly, an LMI-based solution is improved to design the H_∞ state feedback controller. Lemma 3 shows the dilated BRL.

Lemma 3. 

Let G($\alpha$) denote the transfer matrix of the LPV system in (3), then the following two statements are equivalent.

i.

If and only if the matrix A($\alpha$) is stable for all admissible parameter trajectories and $D^T\left(\alpha \right)D\left(\alpha \right)-{\gamma }^{2}I\prec 0$, the H_∞ performance for the system in (3) is bounded such that the index γ > 0. Thus,  ${\Vert G_{z\omega }\Vert }_{\infty }\prec \gamma$.

ii.

If there exist a positive definite parameter-dependent differentiable symmetric matrix $P\left(\alpha \right)\in {\Re }^{n\times n}$, a real positive constant μ satisfying  $\dot{P}\left(\alpha \right)-2\mathsf{\mu }P\left(\alpha \right)\prec 0$ and a compatible dimensional matrix $Q$, then the LMI problem in (8) holds for all $\forall \alpha \in {\Delta }_R \mathrm{and} \forall \dot{\alpha }\in {\Delta }_D$.

$$\left(\begin{array}{ccc}\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& 0& P\left(\alpha \right)+A^T\left(\alpha \right)Q+\mu Q\\ *& -{\gamma }^{2}I& \begin{array}{c}{B}^T\left(\alpha \right)Q\end{array}\\ *& *& -Q-Q^T\\ *& *& *\end{array}\right. \left.\begin{array}{c}{C}^T\left(\alpha \right)\\ D^T\left(\alpha \right)\\ 0\\ -I\end{array}\right)\prec 0$$

(8)

Proof of Lemma 3. 

The LPV form of the inequality in statement i of Lemma 1 is as follows.

$$\begin{array}{l}{A}^T\left(\alpha \right)P\left(\alpha \right)+P\left(\alpha \right)A\left(\alpha \right)+\dot{P}\left(\alpha \right)+C^T\left(\alpha \right)C\left(\alpha \right)\\ +{\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)}^T{\left({\gamma }^{2}I-D^T\left(\alpha \right)D\left(\alpha \right)\right)}^{-1}\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)\prec 0 \end{array}$$

(9)

The inequality in (9) is modified as follows, when a real constant variable μ is included in the inequality.

$$\begin{array}{l}\left(A^T\left(\alpha \right)+\mu I\right)P\left(\alpha \right)+P\left(\alpha \right)\left(A\left(\alpha \right)+\mu I\right)+C^T\left(\alpha \right)C\left(\alpha \right)+\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)\\ +{\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)}^T{\left({\gamma }^{2}I-D^T\left(\alpha \right)D\left(\alpha \right)\right)}^{-1}\left(B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\right)\prec 0 \end{array}$$

By using the Schur lemma, a new matrix inequality is obtained as follows.

$$\left(\begin{array}{cc}{\mathsf{\Gamma }}_{11}\left(\alpha \right)& {\mathsf{\Gamma }}_{12}\left(\alpha \right)\\ {\mathsf{\Gamma }}_{13}\left(\alpha \right)& {\mathsf{\Gamma }}_{14}\left(\alpha \right)\end{array}\right)\prec 0$$

where

$$\begin{array}{l}{\mathsf{\Gamma }}_{11}\left(\alpha \right)=\left(A^T\left(\alpha \right)+\mu I\right)P\left(\alpha \right)+P\left(\alpha \right)\left(A\left(\alpha \right)+\mu I\right)+C^T\left(\alpha \right)C\left(\alpha \right)+\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)\\ {\mathsf{\Gamma }}_{12}\left(\alpha \right)=P\left(\alpha \right)B\left(\alpha \right)+C^T\left(\alpha \right)D\left(\alpha \right),\\ {\mathsf{\Gamma }}_{13}\left(\alpha \right)=B^T\left(\alpha \right)P\left(\alpha \right)+D^T\left(\alpha \right)C\left(\alpha \right)\\ {\mathsf{\Gamma }}_{14}\left(\alpha \right)=D^T\left(\alpha \right)D\left(\alpha \right)-{\gamma }^{2}I\end{array}$$

We can rewrite the inequality as follows.

$$\left(\begin{array}{cc}\left(A^T\left(\alpha \right)+\mu I\right)P\left(\alpha \right)+{\left(*\right)}^T& P\left(\alpha \right)B\left(\alpha \right)\\ B^T\left(\alpha \right)P\left(\alpha \right)& 0\end{array}\right)+\left(\begin{array}{cc}{C}^T\left(\alpha \right)C\left(\alpha \right)+\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& C^T\left(\alpha \right)D\left(\alpha \right)\\ D^T\left(\alpha \right)C\left(\alpha \right)& D^T\left(\alpha \right)D\left(\alpha \right)-{\gamma }^{2}I\end{array}\right)\prec 0$$

(10)

This is rewritten as follows.

$$\left(\begin{array}{cc}{C}^T\left(\alpha \right)C\left(\alpha \right)+\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& C{\left(\alpha \right)}^{T}D\left(\alpha \right)\\ D^T\left(\alpha \right)C\left(\alpha \right)& D^T\left(\alpha \right)D\left(\alpha \right)-{\gamma }^{2}I\end{array}\right)+\left(\begin{array}{c}{A}^T\left(\alpha \right)+\mu I\\ B^T\left(\alpha \right)\end{array}\right)\left(\begin{array}{cc}P\left(\alpha \right)& 0\end{array}\right)+\left(\begin{array}{c}P\left(\alpha \right)\\ 0\end{array}\right)\left(\begin{array}{cc}\begin{array}{l}A\left(\alpha \right)\\ +\mu I\end{array}& B\left(\alpha \right)\end{array}\right)\prec 0$$

(11)

Afterwards, we can obtain the inequality (12) if we apply Lemma 2 to (11).

$$\left(\begin{array}{ccc}{\phi }_{11}\left(\alpha \right)& {\phi }_{12}\left(\alpha \right)& {\phi }_{13}\left(\alpha \right)\\ *& {\phi }_{22}\left(\alpha \right)& {\phi }_{23}\left(\alpha \right)\\ *& *& {\phi }_{33}\end{array}\right)\prec 0$$

(12)

where

$$\begin{array}{l}{\phi }_{11}\left(\alpha \right)=C^T\left(\alpha \right)C\left(\alpha \right)+\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right), \\ {\phi }_{12}\left(\alpha \right)=C^T\left(\alpha \right)D\left(\alpha \right)\\ {\phi }_{13}\left(\alpha \right)=P\left(\alpha \right)+\left(A^T\left(\alpha \right)+\mu I\right)Q, \\ {\phi }_{22}\left(\alpha \right)=D^T\left(\alpha \right)D\left(\alpha \right)-{\gamma }^{2}I\\ {\phi }_{23}\left(\alpha \right)=B^T\left(\alpha \right)Q, {\phi }_{33}=-Q-Q^T\end{array}$$

However, the expression on the left side of (11) must be negative definite according to Lemma 2. Accordingly, we rewrite the expression including $\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)\prec 0$ as follows.

$$\left(\begin{array}{cc}\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& 0\\ *& -{\gamma }^{2}I\end{array}\right)+\left(\begin{array}{c}{C}^T\left(\alpha \right)\\ D^T\left(\alpha \right)\end{array}\right)I\left(\begin{array}{cc}C\left(\alpha \right)& D\left(\alpha \right)\end{array}\right)\prec 0$$

Therefore, we obtain the inequality (13) via the Schur lemma.

$$\left(\begin{array}{ccc}\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& 0& C^T\left(\alpha \right)\\ *& -{\gamma }^{2}I& D^T\left(\alpha \right)\\ *& *& -I\end{array}\right)\prec 0$$

(13)

Finally, we obtain (8) if we apply the Schur lemma after rewriting the inequality (12) as the following form. In addition, the condition (8) also covers the condition (13).

$$\left(\begin{array}{ccc}\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& 0& P\left(\alpha \right)+A^T\left(\alpha \right)Q+\mu Q\\ *& -{\gamma }^{2}I& \begin{array}{c}{B}^T\left(\alpha \right)Q\end{array}\\ *& *& -Q-Q^T\end{array}\right)+\left(\begin{array}{c}{C}^T\left(\alpha \right)\\ D^T\left(\alpha \right)\\ 0\end{array}\right)I\left(\begin{array}{ccc}C\left(\alpha \right)& D\left(\alpha \right)& 0\end{array}\right)\prec 0$$

Hence, the proof is complete. □

All the designs can be used for polytopic-type or affine-type parameters by making some regulations. However, affine-type varying parameters are used for the examples throughout this paper, and thus, all the time-varying matrices affinely change such as the Lyapunov matrix in (14). Therefore, affine quadratic stability is obtained throughout the paper as in [11].

$$P\left(\alpha \right)=P_0+{\displaystyle \sum _{i=1}^n{\alpha }_i{P}_i},\hspace{1em}\dot{P}\left(\alpha \right)={\displaystyle \sum _{i=1}^n{\dot{\alpha }}_i{P}_i}$$

(14)

Theorem 1. 

Let the LPV system be in (1) with the time-varying parameter set in (2). If there exist a positive real scalar μ satisfying $\dot{L}\left(\alpha \right)-2\mu L\left(\alpha \right)\prec 0$, the symmetric positive definite differentiable matrix $L\left(\alpha \right)$ with the compatible dimension matrices $R, F\left(\alpha \right)$, then the LMI in (15) holds for $\forall \left(\alpha \right)\in {\Delta }_R and \forall \dot{\left(\alpha \right)}\in {\Delta }_D$. Thus, there exists a state feedback H_∞ controller $K\left(\alpha \right)=F\left(\alpha \right)R^{-1}$.

$$\left(\begin{array}{cccc}\dot{L}\left(\alpha \right)-2\mu L\left(\alpha \right)& *& *& *\\ 0& -{\gamma }^{2}I& \begin{array}{c}*\end{array}& *\\ L\left(\alpha \right)+A\left(\alpha \right)R+B_{2}F\left(\alpha \right)+\mu R& B_1\left(\alpha \right)& -R-R^T& *\\ C_1\left(\alpha \right)R+D_{12}F\left(\alpha \right)& D_{11}\left(\alpha \right)& 0& -I\end{array}\right)\prec 0$$

(15)

Proof of Theorem 1. 

We obtained the closed-loop matrices in (16) as in (17) when the controller $u=K\left(\alpha \right)x$ is substituted in the system (1) where the output y(t) is eliminated since it is considered state feedback.

$$\begin{array}{l}{\dot{x}}_{cl}\left(t\right)=A_{cl}\left(\alpha \left(t\right)\right)x_{cl}\left(t\right)+B_{cl}\left(\alpha \left(t\right)\right)\omega \left(t\right),\\ z_{cl}\left(t\right)=C_{cl}\left(\alpha \left(t\right)\right)x_{cl}\left(t\right)+D_{cl}\left(\alpha \left(t\right)\right)\omega \left(t\right)\end{array}$$

(16)

$$\begin{array}{l}{A}_{cl}\left(\alpha \right)=A\left(\alpha \right)+B_{2}K\left(\alpha \right), B_{cl}\left(\alpha \right)=B_1\left(\alpha \right), \\ C_{cl}\left(\alpha \right)=C\left(\alpha \right)+D_{12}K\left(\alpha \right), D_{cl}\left(\alpha \right)=D_{11}\left(\alpha \right)\end{array}$$

(17)

If the state space matrices in (17) are replaced in (8), we obtain the following inequality.

$$\left(\begin{array}{ccc}\dot{P}\left(\alpha \right)-2\mu P\left(\alpha \right)& *& *\\ 0& -{\gamma }^{2}I& \begin{array}{c}*\end{array}\\ P\left(\alpha \right)+Q^T\left(A\left(\alpha \right)+B_{2}K\left(\alpha \right)\right)+\mu Q^T& Q^T{B}_1\left(\alpha \right)& -Q-Q^T\\ C\left(\alpha \right)+D_{12}K\left(\alpha \right)& D_{11}\left(\alpha \right)& 0\end{array}\right.\left. \begin{array}{c}*\\ *\\ *\\ -I\end{array}\right)\prec 0$$

However, there are multiplications of unknown matrices. To obtain the LMI, we apply congruence transformation to the inequality by multiplying the transpose of (18) and (18), respectively. Thus, (15) is obtained, where $L\left(\alpha \right)=Q^{-T}P\left(\alpha \right)Q^{-1}, \dot{L}\left(\alpha \right)=Q^{-T}\dot{P}\left(\alpha \right)Q^{-1}, R=Q^{-1}, F\left(\alpha \right)=K\left(\alpha \right)R$. Hence, the proof is complete. □

$$diag\left(Q^{-1},I,Q^{-1},I\right)$$

(18)

Remark 2. 

The proposed solution provides three significant advantages. First, contrary to (7), the Lyapunov matrix can be in the form of a parameter-dependent matrix without extra approaches such as the s-procedure, grid and multi-convexity, which reduces conservativeness. Second, contrary to the common design as in (7) in [5], the controller gain is independent of the Lyapunov matrix, which is restricted by the symmetric property. Instead, it changes with a nonsymmetric matrix R for the proposed LMI in Theorem 1. Third, the real slack variable μ yields feasible outcomes even for large ranges of parameters because a large increase in the range affects the γ performance value less than similar values in terms of feasibility and performance, as proven in the numerical results.

If we consider CTS with TVPU, we can easily change the LMI for parameter-independent (static or fixed) controller design as in Theorem 1 assuming that the uncertain parameters are both unknown at the design time and not known by online measurement but known to be bounded. Thus, some matrices are converted to parameter-dependent matrices whereas some matrices are converted to parameter-independent matrices to obtain the form of the LMI. In addition, the theorem can be used for the affine-type or polytopic-type time-varying parameters or uncertainties by making some regulations. However, affine-type time-varying parameters are considered for the designs throughout this paper. Two numerical examples have been realized for the proposed state feedback controller. All the numerical computations were performed via the Yalmip parser [47] and Sedumi solver [48], which were used on a PC (2 cores 2.8 GHz CPU with 4GB RAM). Example 1 presents a controller design for the CTS with the TIPU, whereas Example 2 presents a gain-scheduled control for the CTS with TVPU.

Example 1. 

The state feedback controller for the satellite in [25,49] is addressed. The system arises from two rigid bodies joined by a flexible link. The system is as in (19). The uncertain parameters are modeled as damping f and spring k. The range of k is [0.09 0.4] whereas the range of f is [0.0038 0.04]. The performances of the methods in [5,24,25,26,27,28,29,30] and the proposed solution in Theorem 1 are compared in terms of level γ which is directly related to conservativeness. The derivatives of the parameters are zero; that is, the uncertainties are not differentiable.

$$\begin{array}{l}\left(\begin{array}{c}{\dot{\theta }}_1\\ {\dot{\theta }}_2\\ {\dot{\theta }}_3\\ {\dot{\theta }}_4\end{array}\right)=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -k& k& -f& f\\ k& -k& f& -f\end{array}\right)\left(\begin{array}{c}{\theta }_1\\ {\theta }_2\\ {\dot{\theta }}_1\\ {\dot{\theta }}_2\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right)\omega +\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right)u,\\ z=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{\theta }_1\\ {\theta }_2\\ {\dot{\theta }}_1\\ {\dot{\theta }}_2\end{array}\right)+\left(\begin{array}{c}0\\ 0\end{array}\right)\omega +\left(\begin{array}{c}0\\ 0.01\end{array}\right)u\end{array}$$

(19)

Table 1 shows the minimal γ values for the results in [5,24,25,26,27,28,29,30] and the proposed solution in Theorem 1 for this example. The results indicate that the proposed solution has much better results than those of the other methods. The feedback gain of the common solution is K = −10⁵[6.1116 44.042 1.1054 81.096], whereas that of the proposed solution is K = −[205.82 2224.6 58.393 2850.9]. The superiority of the proposed solution is that it is less conservative than [5,24,25,26,27,28,29,30], as in Table 1. In addition, Figure 1a shows the relationship between μ and γ. The γ level is almost never affected by changes in μ after a certain value for this system. Moreover, the proposed Theorem 1 has the best result even if μ is between 2.5 and 35, as shown in Figure 1a.

Remark 3. 

The proposed solution for H_∞ robust control can be used for the CTS with TVPU by obtaining advantages of dilated LMI/slack variables, but the other solutions in [5,24,25,26,27,28,29,30] can be only used for the CTS with TIPU. Additionally, the LMI is noniterative. Therefore, the noniterative dilated LMI solution includes both TIPU and TVPU by providing the advantages of the dilated LMI. Its existence has not been reported in the literature. Accordingly, Example 1 presents a controller design for the CTS with the TIPU whereas Example 2 presents a controller design for the CTS with the TVPU. Finally, Example 2 presents only a comparison between this paper and the standard method in [5] since the studies in Table 1 except for the standard design in [1] are not fully valid for the TVPU. The performances are tested by enlarging the uncertain set for the TVPU in Example 2.

Example 2. 

In this example, the gain scheduling LPV control is designed by using Theorem 1 for the randomly generated LPV system in (20) where the ranges of the time-varying parameters ${\alpha }_1$ and ${\alpha }_2$ are [0.01, 2] and [0.5, 1.5], respectively. Additionally, the rates of variation for the parameters ${\dot{\alpha }}_1$ and ${\dot{\alpha }}_2$ are [0, 3] and [0, 1], respectively.

$$\begin{array}{l}\left(\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right)=\left(\begin{array}{ccc}-{\alpha }_1& {\alpha }_1& 0.8\\ 0& 0.2& -{\alpha }_2-1\\ 0.5& {\alpha }_2& 0.1\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left(\begin{array}{c}0.1\\ 0.2\\ 0\end{array}\right)\omega +\left(\begin{array}{c}1\\ 0\\ 0.1\end{array}\right)u, \\ z=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left(\begin{array}{c}0\\ 0\end{array}\right)\omega +\left(\begin{array}{c}0\\ 0.8\end{array}\right)u\end{array}$$

(20)

As in

Table 2. The γ_min performance values for TVPU.

γ Values
	${\alpha }_1$= [0.01 2], ${\alpha }_2$= [0.5 1.5]			${\alpha }_1$= [0.01 5], ${\alpha }_2$= [0.1 19]			${\alpha }_1$= [0.01 50], ${\alpha }_2$= [0.1 190]
[5]	1.0280			6.3151			22.745
Theorem 1	0.890 (μ = 3) 14.42% decreased			2.2466 (μ = 2.5) 64.42% decreased			4.869 (μ = 3.5) 78.59% decreased
Slack variable sets for feasibility
Theorem 1	μ = (0, 9 × 105]			μ = (0, 8 × 105]			μ = (0, 8 × 105]
	Number of iterations—Number of variables—Solver time
[5]	15	18	0.0174 s	28	18	0.126 s	49	18	0.215 s
Theorem 1	39	41	9.556 s	41	41	9.904 s	56	41	13.503 s

, the improvement substantially increases as the uncertain set enlarges compared with the common set. For example, the method decreases the γ level at a rate of 78.59% for ${\alpha }_1$ = [0.01 50], ${\alpha }_2$ = [0.1 190].

In addition, the relationship between μ and γ is shown in Figure 1b. Therefore, the γ level is almost never changed by changing μ after a certain value, so its determination is easy such that the proposed solution has better performance than the common solution even if μ is between 1 and 85.

With reference to this, the results are obtained for the proposed solution in Theorem 1 and the common design in [5], where the Lyapunov matrix is fixed. For ${\alpha }_1$ = [0.01 2], ${\alpha }_2$ = [0.5 1.5], the common design yields a γ level of 1.028, whereas the proposed method yields 0.890 for μ = 3. Accordingly, the feedback gains of the standard design and the design in Theorem 1 are K₀ = 10³[−23.383 170.568 −269.482], K₁ = [ 0.30251 −2.2045 3.4833], K₂ = [ 0.11503 −0.84184 1.33] and K₀ = [−12.17 74.915 −102.05], K₁ = [0.9135 −3.0938 3.5387], K₂ = [5.306 −40.69 47.744], respectively. The relationship between μ and γ is shown in Figure 1b.

From the results, the proposed dilated LMI solution is less conservative than the common LMI in [5] for the gain scheduling design although it has disadvantages which are a greater number of iterations, number of variables and solver time. The feasibility is obtained for the large uncertain set in Table 2, and the slack variable μ provides feasible results for a large set (0, 8 × 10⁵]. Although the number of LMI variables increased from 18 to 41 according to the standard one in [5], an improvement of up to 78.59% has been achieved. Finally, good improvement and less conservative results are achieved for the TVPU even if the dilated LMIs are more costly than the standard LMI is. Notably, the comparisons in the example are made for the parameter-independent Lyapunov matrix in [5]. As emphasized in Remark 1, if PLDF is used, methods such as the multi-convexity method [11] and methods in [17,18] are used. Therefore, the standard LMI variables and the complexity of [5] increase substantially. For example, another LMI condition of the same size needs to be added. Hence, the proposed LMI condition is considered to be less costly since it has a single LMI condition.

Two examples have shown that the proposed solution presents less conservative results. Furthermore, it can be fully used for the LPV system and the uncertain system under the TVPU, but the others cannot be used except for the common LMI in [5]. On the other hand, increasing bounds for time-varying uncertainties or parameters increase the conservativeness of LMIs. However, the design based on the proposed LMI achieves less conservative and more successful results in addition to a simpler structure.

4. Dynamic Disturbance Compensator Design for LPV Systems

The block diagram of the disturbance compensator design is shown in Figure 2 where NCF is the normalization coefficient or filter. Here, the dynamic feedforward compensator is as in (21).

$$\begin{array}{l}{\dot{x}}_c\left(t\right)=A_c\left(\alpha \right)x_c\left(t\right)+B_c\left(\alpha \right)\omega \left(t\right)\\ u_c\left(t\right)=C_c\left(\alpha \right)x_c\left(t\right)+D_c\left(\alpha \right)\omega \left(t\right)\end{array}$$

(21)

Accordingly, Theorem 2 presents a dynamic feedforward disturbance compensator design according to the dilated LMI in Lemma 3.

Theorem 2. 

Let a stable LPV system be in (1) for all admissible trajectories of parameters whose set is as in (2). If there exist a real scalar μ > 0, the positive definite differentiable symmetric matrix $Y\left(\alpha \right)$ and the compatible matrices $M, N\left(\alpha \right)$, then the LMIs in (22) and (23) with the definitions in (24) hold for $\forall \left(\alpha \right)\in {\Delta }_R and \forall \dot{\left(\alpha \right)}\in {\Delta }_D$. In such a case, a dynamic H_∞ feedforward compensator in (21) is obtained from (25).

$$\left(\begin{array}{cccc}{\Psi }_{11}\left(\alpha \right)& {\Psi }_{12}\left(\alpha \right)& {\Psi }_{13}\left(\alpha \right)& {\Psi }_{14}\left(\alpha \right)\\ *& {\Psi }_{22}\left(\alpha \right)& {\Psi }_{23}\left(\alpha \right)& {\Psi }_{24}\left(\alpha \right)\\ *& *& {\Psi }_{33}\left(\alpha \right)& {\Psi }_{34}\\ *& *& *& {\Psi }_{44}\end{array}\right)\prec 0$$

(22)

$$Y\left(\alpha \right)=\left(\begin{array}{cc}{Y}_1\left(\alpha \right)& Y_2\left(\alpha \right)\\ *& Y_3\left(\alpha \right)\end{array}\right)\succ 0$$

(23)

$$\begin{array}{l}{\Psi }_{11}\left(\alpha \right)=\left(\begin{array}{cc}{\dot{Y}}_1\left(\alpha \right)-2\mu Y_1\left(\alpha \right)& {\dot{Y}}_2\left(\alpha \right)-2\mu Y_2\left(\alpha \right)\\ *& {\dot{Y}}_3\left(\alpha \right)-2\mu Y_3\left(\alpha \right)\end{array}\right), \\ {\Psi }_{12}\left(\alpha \right)=\left(\begin{array}{cc}0& Y_1\left(\alpha \right)+R_1^T{A}^T\left(\alpha \right)+N^T\left(\alpha \right)B_2^T+\mu R_1^T\\ 0& Y_1\left(\alpha \right)+R_1^T{A}^T\left(\alpha \right)+N^T\left(\alpha \right)B_2^T+\mu R_2^T\left(\alpha \right)\end{array}\right),\\ {\Psi }_{13}\left(\alpha \right)=\left(\begin{array}{cc}{Y}_2\left(\alpha \right)+M^T\left(\alpha \right)& R_1^T{C}_1^T\left(\alpha \right)+N^T\left(\alpha \right)D_{12}^T\\ Y_3\left(\alpha \right)+M^T\left(\alpha \right)& R_1^T{C}_1^T\left(\alpha \right)+N^T\left(\alpha \right)D_{12}^T\end{array}\right), \\ {\Psi }_{14}\left(\alpha \right)=\left(\begin{array}{c}{R}_1^T{C}_2^T\left(\alpha \right)+N^T\left(\alpha \right)D_{22}^T\\ R_1^T{C}_2^T\left(\alpha \right)+N^T\left(\alpha \right)D_{22}^T\end{array}\right), {\Psi }_{22}\left(\alpha \right)=\left(\begin{array}{cc}-{\gamma }^{2}I& B_1\left(\alpha \right)+D_F^T\left(\alpha \right)B_2^T\\ *& -R_1-R_1^T\end{array}\right), \\ {\Psi }_{23}\left(\alpha \right)=\left(\begin{array}{cc}{B}_F^T\left(\alpha \right)& D_{11}^T\left(\alpha \right)+D_F^T\left(\alpha \right)D_{12}^T\\ -R_1-R_2^T\left(\alpha \right)& 0\end{array}\right), {\Psi }_{24}\left(\alpha \right)=\left(\begin{array}{c}{D}_{21}^T\left(\alpha \right)+D_F^T\left(\alpha \right)D_{22}^T\\ 0\end{array}\right), \\ {\Psi }_{33}\left(\alpha \right)=\left(\begin{array}{cc}-R_2\left(\alpha \right)-R_2^T\left(\alpha \right),& 0\\ *& -I\end{array}\right), {\Psi }_{34}=\left(\begin{array}{c}0\\ 0\end{array}\right), {\Psi }_{44}=-I\end{array}$$

(24)

$$\begin{array}{l}{A}_c=M\left(\alpha \right)R_2^{-1}\left(\alpha \right), B_c\left(\alpha \right)=B_F\left(\alpha \right),\\ C_c=N\left(\alpha \right)R_2^{-1}\left(\alpha \right), D_c=D_F\left(\alpha \right)\end{array}$$

(25)

Proof of Theorem 2. 

The LMI in (8) is multiplied by the following matrix and the transpose of the matrix from the left–right side, respectively when we apply the congruence transformation.

$$\left(\begin{array}{cccc}{Q}^{-1}& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& Q^{-1}& 0\\ 0& 0& 0& I\end{array}\right)$$

Therefore, the following LMI is acquired under the definitions in (27).

$$\left(\begin{array}{ccc}\dot{Y}\left(\alpha \right)-2\mu Y\left(\alpha \right)& *& *\\ 0& -{\gamma }^{2}I& \begin{array}{c}*\end{array}\\ Y\left(\alpha \right)+A\left(\alpha \right)R+\mu R& B\left(\alpha \right)& -R-R^T\\ C\left(\alpha \right)R& D\left(\alpha \right)& 0\end{array}\right. \left.\begin{array}{c}*\\ *\\ *\\ -I\end{array}\right)\prec 0$$

(26)

$$Y\left(\alpha \right)=Q^{-T}P\left(\alpha \right)Q^{-1},\dot{Y}\left(\alpha \right)=Q^{-T}\dot{P}\left(\alpha \right)Q^{-1},R=Q^{-1}$$

(27)

The state space form from $\omega \left(t\right)$ to $Z\left(t\right)$ is as in (28) when the compensator in (21) is realized to (1). Here, the vector Z represents the outputs z and y, the vector X represents the states x in (1) and x_c in (21) and ω is the disturbance signal.

$$\left(\begin{array}{c}\dot{X}\left(t\right)\\ Z\left(t\right)\end{array}\right)=\left(\begin{array}{cc}{A}_{cl}\left(\alpha \right)& B_{cl}\left(\alpha \right)\\ C_{cl}\left(\alpha \right)& D_{cl}\left(\alpha \right)\end{array}\right)\left(\begin{array}{c}x\left(t\right)\\ \omega \left(t\right)\end{array}\right)$$

(28)

Accordingly, the state space matrices in (28) are obtained as in (29) with the realization. Therefore, the optimal controller is achieved by minimizing the L₂ gain from ω to Z.

$$\begin{array}{l}{A}_{cl}\left(\alpha \right)=\left(\begin{array}{cc}A\left(\alpha \right)& B_2{C}_c\left(\alpha \right)\\ 0& A_c\end{array}\right),B_{cl}\left(\alpha \right)=\left(\begin{array}{c}{B}_1\left(\alpha \right)+B_2{D}_c\left(\alpha \right)\\ B_c\end{array}\right),\\ C_{cl}\left(\alpha \right)=\left(\begin{array}{cc}{C}_1\left(\alpha \right)& D_{12}{C}_c\left(\alpha \right)\\ C_2\left(\alpha \right)& D_{22}{C}_c\left(\alpha \right)\end{array}\right), D_{cl}\left(\alpha \right)=\left(\begin{array}{c}{D}_{11}\left(\alpha \right)+D_{12}{D}_c\left(\alpha \right)\\ D_{21}+D_{22}{D}_c\left(\alpha \right)\end{array}\right)\end{array}$$

(29)

Let us define the Lyapunov matrix as in (23), and the $R\left(\xi \right)$ matrix is defined as follows.

$$R\left(\alpha \right)=\left(\begin{array}{cc}{R}_1& R_1\\ R_2\left(\alpha \right)& R_2\left(\alpha \right)\end{array}\right)$$

Accordingly, if Lyapunov matrix Y$\left(\alpha \right)$ in (23), its derivative, the $R\left(\alpha \right)$ matrix and the state space matrices in (29) are placed in (26) in place of A$\left(\alpha \right)$, B$\left(\alpha \right)$, C$\left(\alpha \right)$ and D$\left(\alpha \right)$, we obtain a matrix inequality. If some expressions are defined as in (30), we obtain the LMI in (22) with the definitions in (24) and the proof is complete. □

$$\begin{array}{l}M\left(\alpha \right)=A_c\left(\alpha \right)R_2\left(\alpha \right), N=C_c\left(\alpha \right)R_2\left(\alpha \right), \\ B_F\left(\alpha \right)=B_c\left(\alpha \right), D_F\left(\alpha \right)=D_c\left(\alpha \right)\end{array}$$

(30)

Remark 4. 

The proposed LMIs provide four significant advantages. Some are mentioned in Remark 2. Primarily, the Lyapunov matrix can be designated in the LPV form, which reduces conservatism, without additional approaches such as grid, s-procedure, multi-convexity and the conditions in [17,18]. For example, many LMIs exist in [45] since multi-convexity conditions are used because of these products. In addition, the LMIs are based on the common BRL in (6), so LMIs are somewhat conservative because of the multiplication of the Lyapunov matrix. However, the proposed extended BRL is the best among those available, as shown in Table 1. Second, because of the products of system matrices with a Lyapunov matrix as in (6), infinite LMIs and nonconvex optimization occur, if the Lyapunov matrix can be in the LPV form. However, such a case does not occur for the proposed LMI formulation in (8). Thus, it is clear that the proposed solution is simpler. Third, the LPV controllers designed by PDLF as in [45,50] are generally dependent on the derivative of the Lyapunov matrix (matrix A_c$\left(\alpha \right)$ of the controller). However, the derivative operation causes noise problem in real-time practical applications. The proposed designs are not dependent on the derivative of the Lyapunov matrix. Finally, the controller matrices do not depend on the Lyapunov matrix, which is restricted by the symmetric condition in common designs. Instead, the matrices depend on a full matrix in the proposed designs. Moreover, it has a simple structure.

Remark 5. 

The proposed compensator does not include a feedback controller. It is well known that a feedforward method is applied to improve the reference tracking performance or to drop the disturbance or noise, and it does not affect stability, thus a feedback controller is needed. However, a feedback design is not needed for some plants, such as many mechanical and electrical systems, since they are stable systems. Therefore, the compensator can also be used for the stable systems with disturbances in addition to stabilized systems with a feedback controller. In addition, it can be used to improve the reference tracking by exchanging reference signals instead of disturbance inputs for the stabilized systems with a feedback controller. Accordingly, it is assumed that the systems are stable or stabilized with a feedback controller for all the examples throughout the paper. However, the internal stability of the LPV feedforward controller/compensator should be provided for all time-varying parameters, as in the above theorems. Otherwise, the stable system becomes unstable. This stability is achieved by decomposing the Lyapunov matrix without the need for any extra stability method for all the proposed theorems.

5. Simulation Studies

In this section, a numerical example has been realized regarding compensator design. All numerical optimization computations have been performed by Yalmip version R20210331 [47] and Sedumi version 1.30 [48], which have been used on a PC (2 cores 2.8 GHz CPU with 4 GB Ram).

Example 3. 

This example shows the LPV compensator design for the randomly generated LPV system in (31). The time-varying parameters ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$ are bounded by (32). Although it normally appears as an indeterminate linear system, it is considered an LPV system by assuming that the measurable or computable parameters are states of the nonlinear system. An LPV system is derived from a nonlinear system, unlike an uncertain linear system.

$$\begin{array}{l}\dot{x}\left(t\right)=\left(\begin{array}{cc}1+{\alpha }_1\left(t\right)& -3-{\alpha }_1\left(t\right)\\ 6& -4-{\alpha }_2\left(t\right)\end{array}\right)x\left(t\right)+\left(\begin{array}{c}0.5+{\alpha }_1\left(t\right)\\ 0.8\end{array}\right)\omega \left(t\right)+\left(\begin{array}{c}0.2\\ 0.1\end{array}\right)u\left(t\right)\\ z\left(t\right)=\left(\begin{array}{cc}1& 0.1\end{array}\right)x\left(t\right)+\left(0.01\right)\omega \left(t\right)\\ y\left(t\right)=\left(\begin{array}{cc}1& 0\end{array}\right)x\left(t\right)\end{array}$$

(31)

$$\begin{array}{l}0.1\le {\alpha }_1\left(t\right)\le 0.7 \mathrm{with} -0.445\le {\dot{\alpha }}_1\left(t\right)\le 1.82\\ 1\le {\alpha }_2\left(t\right)\le 5 \mathrm{with} -11.7\le {\dot{\alpha }}_2\left(t\right)\le 2.295\end{array}$$

(32)

When the system is analyzed in view of time-varying parameters, the results in Figure 3 and Figure 4 are obtained. The different time-varying parameters are tested for analysis in Figure 3. However, one of them is used for simulation which is in Figure 3b. Figure 3a is only used for analysis (Bode diagram and pole mapping) because the parameter change is assumed to be linear. Typically, both parameters are unlikely to vary linearly. The parameter change in Figure 3b occurs during the simulation. The aim is to observe the reactions to different parametric changes. For the analysis, the Bode diagram of the system is shown in Figure 4 with a sampling time of 50 ms and a simulation sampling time of 100 $\mathsf{\mu }s$. Figure 4a shows the Bode diagram of the system assuming the time-varying parameters in Figure 3a, whereas Figure 4b shows the Bode diagram assuming the time-varying parameters in Figure 3b. The figure shows that the time-varying parameters strongly affect the nonlinearity of the system according to Bode diagrams of the outputs from the inputs. From the Bode diagrams, the disturbance is more effective at approximately 3–6 rad/s. Therefore, the disturbance test signals have a low frequency.

Similarly, Figure 5 presents the pole map of the system for the various time-varying parameters in Figure 3b where the sampling time is 50 ms. From the pole map, two cases can be deduced. The poles of the whole transfer matrices of the system are in the left half of the plane. This case confirms the stability of the system in accordance with all the time-varying parameters mentioned in Remark 5. Otherwise, the compensator cannot be applied since it does not affect the stability. The other is that the parameters strongly affect the system.

For the design, Figure 6 shows a schematic block diagram for the LPV compensator design where W_u and W_z represent the input and output weighting filters, respectively. However, an appropriate compensator is designed without these filters for this example. However, the filters can be used for reasons such as structural limitations as mentioned in Remark 6.

Remark 6. 

The inputs and outputs are bounded because of the mechanical/electrical structure. Therefore, the constraints should be considered according to the problem of the related field. For example, the suspension deflection should not exceed the upper limit, whereas the tire deflection should be minimized as much as possible for active vehicle suspension systems. There are some LMI-based studies on the input and output constraints such as [51]. These are regulated with the NCF, as shown in Figure 2. NCF can be a coefficient or a filter such as a low-pass filter that is determined according to system behavior by considering the frequency and gain limits of inputs and outputs. The other way is the input and output weighting functions for the H_∞ controller design as shown in Figure 6 and the block diagram in [45]. Therefore, it can be decided for NCF considering the system structure. In particular, a filter for NCF can be needed at a high frequency instead of a fixed NC. This is because the compensator needs it to regulate the actuator limit at a high frequency.

The matrices are obtained in (33) when the LMIs in Theorem 2 are solved to minimize the γ value according to the parameter-dependent variables $Y_i, B_{Fi}, D_{Fi}, M_i, N_i$ for i = 0, 1, 2 and the parameter-independent variables R₁ and R₂. Therefore, the state space matrices of LPV compensator $A_{ci}, B_{ci}, C_{ci}$ and $D_{ci}$ in (21) for i = 0, 1 and 2 are affinely obtained from the matrices in (33) according to the time-varying parameters ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$ in (32) for all t during the online operation. Hence, (34) shows the state space LPV controller matrices.

$$\begin{array}{l}{M}_0=\left[\begin{array}{cc}-163.81& 12.724\\ -1.7999& -202.82\end{array}\right],M_1=\left[\begin{array}{cc}5.1074& -22.914\\ -11.194& 36.474\end{array}\right],\\ M_2=\left[\begin{array}{cc}-10.704 & 16.627\\ 15.009& -84.506\end{array}\right],R_1=\left[\begin{array}{cc}7.1371& -3.0487\\ -20.361& 92.818\end{array}\right],\\ R_2=\left[\begin{array}{cc}7.139& -3.0494\\ -20.361& 92.826\end{array}\right],N_0=\left[\begin{array}{cc}-1175.1& 1477.5\end{array}\right],\\ N_1=\left[\begin{array}{cc}-111.95& 364.76\end{array}\right],N_2=\left[\begin{array}{cc}-53.522& 83.135\end{array}\right],\\ B_{C0}=\left[\begin{array}{c}-0.07727\\ 0.51139\end{array}\right],B_{C1}=\left[\begin{array}{c}0.070224\\ -0.46491\end{array}\right],B_{C2}=\left[\begin{array}{c}0\\ 0\end{array}\right],\\ D_{C0}=-3.3213, D_{C1}=-4.6489, D_{C2}=-0.0003355\end{array}$$

(33)

$$\begin{array}{l}{A}_c\left({\alpha }_1,{\alpha }_2\right)=\left[\begin{array}{cc}-24.886+0.012541{\alpha }_1-1.0907{\alpha }_2& -0.68047-0.24644{\alpha }_1+0.14329{\alpha }_2\\ -7.1544-0.49361{\alpha }_1-0.54525{\alpha }_2& -2.42+0.37671{\alpha }_1-0.92828{\alpha }_2\end{array}\right]\\ B_c\left({\alpha }_1\right)=\left[\begin{array}{c}-0.07727+0.070224{\alpha }_1\\ 0.51139-0.46491{\alpha }_1\end{array}\right]\\ C_c\left({\alpha }_1,{\alpha }_2\right)=\left[\begin{array}{cc}-131.53-4.9366{\alpha }_1-5.4537{\alpha }_2& 11.596+3.7674{\alpha }_1+0.71644{\alpha }_2 \end{array}\right]\\ D_c\left({\alpha }_1,{\alpha }_2\right)=-2.8863-4.6489{\alpha }_1-0.0003355{\alpha }_2\end{array}$$

(34)

Accordingly, Figure 7a shows the Bode diagram of the compensator for the time-varying parameters in Figure 3a, whereas Figure 7b shows it for the time-varying parameters in Figure 3b for a sampling time of 50 ms. According to the pole maps and Bode diagrams, the LPV compensator significantly changes with the time-varying parameters.

The value of μ is determined to be 0.42 by line searching for a good response. Accordingly, the performance index γ value is 0.037268. This value clearly changes according to the system and input-output filters. It provides feasibility for very large time-varying parameters, as proven in state feedback controller design examples. Figure 8a also shows the Bode diagram of the system combined with the compensator from the disturbance input to the z and y outputs assuming the time-varying parameters in Figure 3a, whereas Figure 8b shows the Bode diagram assuming the time-varying parameters in Figure 3b. Figure 9 shows the applied disturbance test signals for the simulations. These are a bump signal as shown in Figure 9a; 5 rad/s sine signal shown in Figure 9b; and white noise signal shown in Figure 9c.

As for the results, Figure 10 shows the outputs z and y for the external disturbance bump signal in Figure 9a. As seen from the outputs, the designed LPV compensator effectively attenuates the disturbance at a rate of approximately 100% compared with the case without the compensator.

Similarly, Figure 11 and Figure 12 show the outputs z and y for the disturbance test signals in Figure 9b and Figure 9c, respectively. The signals generated by the parameter-dependent compensator against three disturbances are shown in Figure 13.

The system with an LPV feedforward compensator is compared with the system without a compensator. It should be noted that the system must be a stable system or stabilized by a feedback controller. As seen from the outputs in Figure 11 and Figure 12, the parameter-dependent compensator is quite successful, similar to the results in Figure 10. Thus, the effects of the external disturbances are reduced almost 100%. Therefore, the designed LPV feedforward compensator has significant performance against the disturbances. Finally, the compensator design is quite successful for the LPV system. As a result, the proposed compensator is quite successful for the rejection of measurable or observable disturbances.

In the design of the proposed LPV compensator based on the dilated LMI, the Lyapunov matrix can be designated in the LPV form, which reduces conservatism, without additional approaches such as grid, s-procedure, multi-convexity in [11] and the conditions in [18,19]. For example, many LMI conditions exist for the LPV controller design in [45] since multi-convexity conditions are used because of the products A(α) × P(α) in (6). Therefore, the proposed solution is simpler and less conservative. Otherwise, either inefficient bilinear matrix inequality (BMI) solvers or iterative LMI algorithms are needed. The solutions do not require any iterative LMI or BMI algorithms.

6. Conclusions and Future Works

This paper focuses on a feedforward compensator against disturbances in addition to a state feedback controller for the LPV system. The Lyapunov matrix is divided from the matrices of system by injecting a slack real variable in the LMI-based designs; thus, the PDLM is designated. Therefore, the proposed LMI formulations for both state feedback controller and feedforward compensator are finite-dimensional and do not need extra conditions such as the s-procedure, grid and multi-convexity even if the PDLM is used. Thus, they have a simpler structure and lower conservativeness than similar state feedback and feedforward designs do. The numerical results of the state feedback controller show that the solution is less conservative and simpler even for very large parameter ranges. Especially for the compensator, the simulation results show that both the parameter-dependent and parameter-independent feedforward compensators have significant performance against disturbances because the effects of the external disturbances are almost completely eliminated. Therefore, the validity and advantages of the proposed LMIs are effectively obtained with numerical and simulation examples. As a result, the proposed LMI solutions are better for the systems in terms of simplicity and conservativeness.

For further studies, the first of these can be considered as obtaining the optimum real scalar μ via metaheuristic optimization methods. The second can design both feedback control and a feedforward compensator at the same time for unstable systems. The third is the design of switching the LPV feedforward compensator via the dilated LMI. Furthermore, this proposed feedforward compensator with a switched LPV feedforward compensator can be compared with respect to criteria such as performance and complexity.