Resilient asynchronous H ∞ control designed for discrete-time Markov jump systems: Static output feedback case

Abstract

This article is concerned with the problem of resilient asynchronous H_∞ static output feedback control for discrete-time Markov jump linear systems. By Finsler’s Lemma, and with the help of two sets of slack variables, the product terms of system matrices and Lyapunov matrices are decoupled. Resilient asynchronous controller is designed to improve the robustness of the controller and overcome the drawback that the controller cannot get the information of the system’s mode. The controller that makes sure the closed-loop system is stochastically stable and with prescribed H_∞ performance is designed. The bilinear matrix inequalities are given as the sufficient conditions for the controller design, which can be solved using linear matrix inequalities along with line search. This control strategy can be used in many practical application fields, such as robot control, aircraft, and traffic control.

Introduction

In practice, because of the impact of external environment, internal failure, the communication delay such as considered in Lu et al., ¹ the noise such as considered in Lu et al., ² and other reasons, most of the subjects to be described are dynamic and time-variant. Markov jump linear systems (MJLSs) ³ are hybrid systems to describe the dynamic subjects. Markov chain determines the different modes in the MJLSs. The values of the Markov chain constitute a finite set. Due to the above-mentioned characteristics, a lot of application^4,5 and research^6–10 about MJLSs have been done in the past few decades. In Zhang et al., ⁴ MJLSs apply to model the neural network systems. Economic systems are described by MJLSs in Zhao and Zhang. ⁶ The controller in form of observer based is designed for MJLSs in Zhao and Zhang. ⁶ The problem of stability for MJLSs is solved in Costa and Fragoso. ⁷ The observability and detectability of MJSs is studied in Tan and Zhang. ⁸ In Wu et al., ⁹ the filtering problem for MJSs is considered. The character of Markov not only used in the MJLSs but also used in other process such as in Ma and Jia. ¹⁰

The operating principle of H_∞ controller is that designing a controller for the given system and making sure the interconnection between system and controller is internally stable. So, the H_∞ norm can be used in the condition when the worst disturbance put on the controlled output. ¹¹ The H_∞ control theory has been applied to many control systems, and it plays an important role in this area. This theory can be used to analyze and solve the problems in the worst situation. Correspondingly, the theory can not only be devoted to constitute-time system but also be used to solve the problem of discrete-time linear system. ¹² Based on the mentioned reasons, the H_∞ control has been used in many areas.^13–15 The Lyapunov functions are necessary part in the H_∞ control; however, it is not only used in H_∞ control it also has lot of applications such as in literatures,^16,17 and it is used in adaptive control for a kind of nonlinear systems.

Lot of the advanced control strategies have been put into application.^18–21 However, the feedback control is still one of the most important control strategies. State feedback control such as Zhang et al. ²² and output feedback control are important parts of the problem of feedback control. In Gabriel et al., ¹² the problem of state feedback control for MJLSs is studied. The problem of dynamic output feedback control is researched in Zhu et al. ²³ Distributed output feedback control is considered in Wang and Zhang. ²⁴ The state and output feedback control are both considered in Yan et al. ²⁵ In Liu et al., ²⁶ adaptive control is designed for feedback systems. Compared with other output feedback control forms, static output feedback control can be put into practical use with low cost, so static output feedback is very useful and more realistic. Static output feedback control problem has been extensively investigated in the past decades.^27–31 However, for many existing works of static output feedback control, they assume that the controller can get access to the mode of the system. ²⁹ The influence of environment that can affect the controller is not considered by Che et al. ³¹ So, to the limit of the author’s knowledge, there still are a lot of work to do in the research of resilient asynchronous control in form of static output feedback for MJLSs.

Motivated by the above observations, the resilient asynchronous H_∞ static output feedback control problem for discrete-time MJLSs is studied in this article. Resilient asynchronous controller is designed to improve the robustness of the controller and make sure that the controller can work in the different mode from the system. By Finsler’s Lemma, and with the help of two sets of slack variables, the interconnection of system matrices and Lyapunov matrices is decoupled. The controller that makes sure the closed-loop system is stochastically stable and with prescribed H_∞ performance is designed. The sufficient condition for the controller design is given in the form of bilinear matrix inequalities (BMIs), which can be solved using linear matrix inequalities along with line search. Nowdays, robot plays an important role in industrial production and our daily lives. So, the research of robot has attracted lots of interests.^32–34

The control strategy is based on the static output feedback control, and through analyzing the static output data, the systems are controlled. It is known that the static output feedback is more cheaper and reliable to put into application. Meanwhile, the designed controller do not need to work in the mode which is synchronous with the system mode, so it has a wider range of adaptability than normal controller. The designed controller is also resilient, that is to say, it has a stronger robustness. In spirit of the application of advanced control strategy to robot system^35–38 and the mentioned reasons, the controller designed in this article can be used in many actual situations, such as robot control, aircraft such as considered in Gai et al., ³⁹ and traffic control.

This article is organized as follows. The discrete-time MJLSs and the resilient asynchronous H_∞ static output feedback controller is formulated, and several essential definitions and lemmas are given in section “Preliminaries and problem formulation.” Based on these lemmas and definitions, the controller designed for the closed-loop system not only makes sure the system has a prescribe H_∞ performance index but also is stochastically stable in section “Main result.” In section “Numerical example,” numerical example is given. The conclusion of this article is given in section “Conclusion.”

Notation: In this article, we use commonly used notations. Given two matrices $C$ and $D$ that are symmetric and real, the notations $C\ge D$ or $C\le D$ denote $C-D$ is positively or negatively defined. There is a matrix $Z$ . $Z^T$ denotes its transpose, $Z^{-1}$ denotes its inverse (when it exists), and $Z^{\perp }$ denotes a matrix that $Z^{\perp }\times Z=0$ . $0$ represents zero matrix of compatible dimensions. The $a\times b$ real matrices are represented by ${\mathbb{R}}^{a\times b}.$ Block-diagonal matrix is denoted by $\mathrm{diag}\{·\}$ . The space of Euclidean in the dimensional of $p$ is denoted by ${\mathbb{R}}^p$ . The notation $\mathrm{He}(Y)$ stands for $Y+Y^T$ .

Preliminaries and problem formulation

On a probability space $(\Omega ,\mathcal{F},\mathcal{P})$ , establish the following discrete-time MJLSs

$$\{\begin{array}{c}x(k+1)=A(r_k)x(k)+B(r_k)u(k)+B^{\omega }(r_k)\omega (k)\hfill \\ \multicolumn{1}{c}{z(k)=C^z(r_k)x(k)+D(r_k)u(k)+D^{\omega }(r_k)\omega (k)}\\ \multicolumn{1}{c}{y(k)=C(r_k)x(k)}\end{array}$$

(1)

where $x(k)\in {\mathbb{R}}^n$ is the state of the system, $\omega (k)\in {\mathbb{R}}^p$ is the input of distrubance which belongs to $l_2[0,\infty )$ , $u(k)\in {\mathbb{R}}^m$ is the control input, $z(k)\in {\mathbb{R}}^q$ is the targeted output, and $y(k)\in {\mathbb{R}}^r$ is the measured output. The system matrices are denoted by $A(r(k))$ , $B(r(k))$ , $B^{\omega }(r(k))$ , $C^z(r(k))$ , $D(r(k))$ , $D^{\omega }(r(k))$ , and $C(r(k))$ , and the system matrices are given and have proper dimensions. $r(k)$ is the stochastic jumping process $\{r(k),k\ge 0\}$ . The Markov chain is represented by $r(k)$ . The discrete-time Markov chain used in this article is right-continuous. The values of it constitute a finite set ${\mathcal{L}}_r=\{1,2,\dots ,N\}$ with mode transition probabilities $\mathrm{Pr}\{r_{k+1}=j|r_k=i\}={\pi }_{\mathrm{ij}}$ , where ${\pi }_{\mathrm{ij}}\ge 0$ , $\forall i,j\in {\mathcal{L}}_r$ , and $\sum _{j=1}^N{\pi }_{\mathrm{ij}}=1$ for each mode $i$ . The transition probability matrix of system (1) can be written as

$$\pi =\left[\begin{array}{cccc}{\pi }_{11}\hfill & {\pi }_{12}\hfill & \cdots \hfill & {\pi }_{1N}\hfill \\ \multicolumn{1}{c}{{\pi }_{21}}& {\pi }_{22}\hfill & \cdots \hfill & {\pi }_{2N}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\pi }_{N1}}& {\pi }_{N2}\hfill & \cdots \hfill & {\pi }_{\mathrm{NN}}\hfill \end{array}\right]$$

With considering generality, we suggest that $C_i$ , $i\in {\mathcal{L}}_r$ are of full row rank, and then, we can have nonsingular transformation matrices $T_i$ , $i\in {\mathcal{L}}_r$ such that

$$C_i{T}_i=\left[\begin{array}{cc}I\hfill & 0\hfill \end{array}\right]$$

(2)

Remark 1

We can tell that for given $C_i$ , the corresponding $T_i$ are usually not unique. A particular $T_i$ can be given as

$$T_i=\left[\begin{array}{cc}{C}_i^T{(C_i{C}_i^T)}^{-1}\hfill & C_i^{\perp }\hfill \end{array}\right]$$

(3)

The following resilient asynchronous controller is presented

$$u(k)={\hat{K}}_{\theta (k)}y(k)$$

(4)

where

$${\hat{K}}_{\theta (k)}=K_{\theta (k)}+M_{\theta (k)}\mathrm{\Delta }(k)N_{\theta (k)}$$

(5)

$M_{\theta (k)}\in {\mathbb{R}}^{s\times v}$ and $N_{\theta (k)}\in {\mathbb{R}}^{v\times v}$ are known matrices and $\mathrm{\Delta }(k)\in {\mathbb{R}}^{v\times v}$ and satisfies $\mathrm{\Delta }(k{)}^T\mathrm{\Delta }(k)\le I$ . $y(k)$ denotes the controller input and $u(k)$ means the controller output. $K_{\theta (k)}$ are the controller gains that need to be designed. The parameter $\theta (k)\in {\mathcal{L}}_{\theta }=\{1,2,\dots ,S\}$ is a Markov chain. The transition probabilities between the modes of the controller is given as $\mathrm{Pr}\{{\theta }_{k+1}=n|{\theta }_k=m\}=v_{\mathrm{mn}}^{r(k+1)}$ , where $v_{\mathrm{mn}}^{r(k+1)}\ge 0$ , $\forall m,n\in {\mathcal{L}}_{\theta }$ , and $\sum _{n=1}^S{v}_{\mathrm{mn}}^{r(k+1)}=1$ , for each mode $m$ . The transition probabilities matrix of system (4) is given as

$$v^{r(k)}=\left[\begin{array}{cccc}{v}_{11}^{r(k)}\hfill & v_{12}^{r(k)}\hfill & \cdots \hfill & v_{1S}^{r(k)}\hfill \\ \multicolumn{1}{c}{v_{21}^{r(k)}}& v_{22}^{r(k)}\hfill & \cdots \hfill & v_{2S}^{r(k)}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{v_{S1}^{r(k)}}& v_{S2}^{r(k)}\hfill & \cdots \hfill & v_{\mathrm{SS}}^{r(k)}\hfill \end{array}\right]$$

Substituting equation (4) into equation (1), we can obtain the closed-loop system as

$$\{\begin{array}{c}{x}_{k+1}={\tilde{A}}_{i,m}{x}_k+B_i^{\omega }{\omega }_k\hfill \\ \multicolumn{1}{c}{z_k={\tilde{C}}_{i,m}{x}_k+D_i^{\omega }{\omega }_k}\end{array}$$

(6)

where for any $r_k=i\in {\mathcal{L}}_r$ , ${\theta }_k=m\in {\mathcal{L}}_{\theta }$ , and

$$\{\begin{array}{c}{\tilde{A}}_{i,m}=A_i+B_i{\hat{K}}_m{C}_i\hfill \\ \multicolumn{1}{c}{{\tilde{C}}_{i,m}=C_i^z+D_i{\hat{K}}_m{C}_i}\end{array}$$

(7)

To obtain the main objective, the following definitions are given.

Definition 1

When ${\omega }_k=0$ and the initial conditon $x_0\in {\mathbb{R}}^n,r_0\in {\mathcal{L}}_r,{\theta }_0\in {\mathcal{L}}_{\theta }$ is randomly given and the following inequality holds, the closed loop (equation (6)) is said to be stochastically stable

$$E\{\sum _{k=0}^{\infty }||x_k{||}^2|x_0,r_0,{\theta }_0\}<\infty$$

Definition 2

With a known scalar $\gamma >0$ , the closed-loop system in (equation (6)) not only has a prescribed H_∞ performance index but also is stochastically stable, and if it is stochastically stable under zero initial condition, the following inequality holds for all nonzero ${\omega }_k\in l_2[0,\infty )$

$$||z_k|{|}_{E_2}<\gamma ||{\omega }_k|{|}_2$$

The following lemmas will be used to obtain the main objective.

Lemma 1

Schur complement. For a given symmetry matrix

$$S=\left[\begin{array}{cc}{S}_{11}\hfill & S_{12}\hfill \\ \multicolumn{1}{c}{S_{21}}& S_{22}\hfill \end{array}\right]$$

where $S_{11}\in {\mathbb{R}}^{r\times r}$ and the following three conditions are equivalent

1. $S<0$

2. $S_{11}<0,S_{22}-S_{12}^T{S}_{11}^{-1}{S}_{12}<0$

3. $S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T<0$

Lemma 2

For matrices ${\epsilon }_1$ , ${\epsilon }_2$ , and ${\epsilon }_3$ with appropriate dimension and ${\epsilon }_1^T={\epsilon }_1$ , for all $\mathrm{\Delta }(k)$ satisfying $\mathrm{\Delta }(k{)}^T\mathrm{\Delta }(k)\le I$ , the following inequalities are equivalent ⁴⁰

1. ${\epsilon }_1+{\epsilon }_3\mathrm{\Delta }(k){\epsilon }_2+{\epsilon }_2^T\mathrm{\Delta }(k{)}^T{\epsilon }_3^T<0$

2. If and only if we can find a scalar $ϵ>0$ such that

$${\epsilon }_1+ϵ{\epsilon }_2^T{\epsilon }_2+\frac{1}{ϵ}{\epsilon }_3{\epsilon }_3^T<0$$

Lemma 3

Finsler’s Lemma: If there exist $\delta$ , $\eta$ , and $\vartheta$ with proper dimensions and $\eta$ are symmetric and $\mathrm{rank}(\vartheta )<n$ , the conditions (1) and (2) given as follows are equalient.

1. $\delta \eta {\delta }^T<0$ , when $\delta \ne 0$ and $\vartheta \delta =0$ ;

2. $\exists \chi \in {\mathbb{R}}^{n\times m}$ then $\eta +\chi \vartheta +{\vartheta }^T{\chi }^T<0$ .

Main result

Theorem

With a known scalars $\gamma >0$ , the closed-loop system in equation (5) is with a desired H_∞ performance index $\gamma$ and is stochastically stable. If a scalar $0<\alpha <1$ , a matrix $P_{i,m}$ which is positive definite and symmetric and matrices $G_m$ , $F_m$ , and $L_m$ with the structure exist

$$\begin{array}{c}{G}_m=\left[\begin{array}{cc}{G}_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{G_m^{21}}& G_m^{22}\hfill \end{array}\right]\hfill \\ F_m=\left[\begin{array}{cc}\lambda G_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{F_m^{21}}& F_m^{22}\hfill \end{array}\right]\hfill \\ L_m=\left[\begin{array}{cc}{L}_m^1\hfill & 0\hfill \end{array}\right]\hfill \end{array}$$

(8)

such that the inequality

$$\left[\begin{array}{ccc}{\mathrm{\Phi }}_{i,m}\hfill & {\mathrm{\Pi }}_{12}^T\hfill & \alpha {\mathrm{\Pi }}_{13}^T\hfill \\ \multicolumn{1}{c}{*}& -\alpha I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\alpha I\hfill \end{array}\right]<0$$

(9)

holds for all $r_k=i\in {\mathcal{L}}_r$ and ${\theta }_k=m\in {\mathcal{L}}_{\theta }$ , where

$$\begin{array}{c}{\mathrm{\Phi }}_{i,m}=\left[\begin{array}{cccc}{\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)\hfill & *\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{0}& -I\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\Theta }_{31}}& B_i^{\omega }\hfill & {\Theta }_{33}\hfill & *\hfill \\ \multicolumn{1}{c}{{\Theta }_{41}}& D_i^w\hfill & {\Theta }_{43}\hfill & -{\gamma }^{2}I\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\rho }_{i,m}=\sum _{j\in {\mathcal{L}}_r}{\pi }_{\mathrm{ij}}\sum _{n\in {\mathcal{L}}_{\theta }}{v}_{\mathrm{mn}}^{(j)}{P}_{j,n}}\\ \multicolumn{1}{c}{{\mathrm{\Pi }}_{12}=\left[\begin{array}{cccc}0\hfill & 0\hfill & M_m^T{B}_i^T\hfill & M_m^T{D}_i^T\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathrm{\Pi }}_{13}=\left[\begin{array}{cccc}{N}_m{C}_i{T}_i{G}_m{T}_i^T\hfill & 0\hfill & N_m{C}_i{T}_i{F}_m{T}_i^T\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\Theta }_{31}=A_i{T}_i{G}_m{T}_i^T+B_i{L}_m{T}_i^T-T_i{F}_m^T{T}_i^T}\\ \multicolumn{1}{c}{{\Theta }_{41}=C_i^z{T}_i{G}_m{T}_i^T+D_i{L}_m{T}_i^T}\\ \multicolumn{1}{c}{{\Theta }_{33}=\mathrm{He}(A_i{T}_i{F}_m{T}_i^T+\lambda B_i{L}_m{T}_i^T)-P_{i,m}}\\ \multicolumn{1}{c}{{\Theta }_{43}=C_i^z{T}_i{F}_m{T}_i^T+\lambda D_i{L}_m{T}_i^T}\end{array}$$

The controller gains in equation (5) can be obtained as

$$K_m=L_m^1(G_m^{11}{)}^{-1}$$

(10)

Proof

Using Schur complement in equation (9), the inequality is equivalent to

$${\mathrm{\Phi }}_{i,m}+\frac{1}{\alpha }{\mathrm{\Pi }}_{12}^T{\mathrm{\Pi }}_{12}+\alpha {\mathrm{\Pi }}_{13}{\mathrm{\Pi }}_{13}^T<0$$

(11)

According to Lemma 1, equation (10) is equivalent to

$${\mathrm{\Phi }}_{i,m}+{\mathrm{\Pi }}_{12}^T\mathrm{\Delta }(k){\mathrm{\Pi }}_{13}+{\mathrm{\Pi }}_{13}^T\mathrm{\Delta }(k{)}^T{\mathrm{\Pi }}_{12}<0$$

(12)

Based on the inequalities (9) and (12), the following inequality holds

$$\left[\begin{array}{cccc}{\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)\hfill & *\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{0}& -I\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\Xi }_{31}}& B_i^{\omega }\hfill & {\Xi }_{33}\hfill & *\hfill \\ \multicolumn{1}{c}{{\Xi }_{41}}& D_i^w\hfill & {\Xi }_{43}\hfill & -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(13)

where

$$\begin{array}{cc}\multicolumn{1}{c}{{\Xi }_{31}=}& A_i{T}_i{G}_m{T}_i^T+B_i{L}_m{T}_i^T-T_i{F}_m^T{T}_i^T\hfill \\ \multicolumn{1}{c}{}& +B_i{M}_m\mathrm{\Delta }(k)N_m{C}_i{T}_i{G}_m{T}_i^T\hfill \end{array}$$

(14)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Xi }_{41}=}& C_i^z{T}_i{G}_m{T}_i^T+D_i{L}_m{T}_i^T\hfill \\ \multicolumn{1}{c}{}& +D_i{M}_m\mathrm{\Delta }(k)N_m{C}_i{T}_i{G}_m{T}_i^T\hfill \end{array}$$

(15)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Xi }_{33}=}& \mathrm{He}(A_i{T}_i{F}_m{T}_i^T+\lambda B_i{L}_m{T}_i^T\hfill \\ \multicolumn{1}{c}{}& +B_i{M}_m\mathrm{\Delta }(k)N_m{C}_i{T}_i{F}_m{T}_i^T)-P_{i,m}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Xi }_{43}=}& C_i^z{T}_i{F}_m{T}_i^T+\lambda D_i{L}_m{T}_i^T\hfill \\ \multicolumn{1}{c}{}& +D_i{M}_m\mathrm{\Delta }(k)N_m{C}_i{T}_i{F}_m{T}_i^T\hfill \end{array}$$

(17)

We can have the following equalities from equations (2), (8), and (10)

$$\begin{array}{c}{A}_i{T}_i{G}_m{T}_i^T+B_i{L}_m{T}_i^T\hfill \\ \multicolumn{1}{c}{=A_i{T}_i{G}_m{T}_i^T+B_i\left[\begin{array}{cc}{L}_m^1\hfill & 0\hfill \end{array}\right]T_i^T}\\ \multicolumn{1}{c}{=A_i{T}_i{G}_m{T}_i^T+B_i\left[\begin{array}{cc}{K}_m{G}_m^{11}\hfill & 0\hfill \end{array}\right]T_i^T}\\ \multicolumn{1}{c}{=A_i{T}_i{G}_m{T}_i^T+B_i\left[\begin{array}{cc}{K}_m\hfill & 0\hfill \end{array}\right]\left[\begin{array}{cc}{G}_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{G_m^{21}}& G_m^{22}\hfill \end{array}\right]T_i^T}\\ \multicolumn{1}{c}{=A_i{T}_i{G}_m{T}_i^T+B_i{K}_m\left[\begin{array}{cc}I\hfill & 0\hfill \end{array}\right]G_m{T}_i^T}\\ \multicolumn{1}{c}{=A_i{T}_i{G}_m{T}_i^T+B_i{K}_m{C}_i{T}_i{G}_m{T}_i^T}\\ \multicolumn{1}{c}{=(A_i+B_i{K}_m{C}_i)T_i{G}_m{T}_i^T}\end{array}$$

(18)

It is easy to know that equation (14) is equivalent to the following equality from equations (5), (7), and (18)

$${\Xi }_{31}={\tilde{A}}_{i,m}{T}_i{G}_m{T}_i^T-T_i{F}_m^T{T}_i^T$$

(19)

Based on equations (2), (8), and (10), we obtain

$$\begin{array}{c}{A}_i{T}_i{F}_m{T}_i^T+\lambda B_i{L}_m{T}_i^T\hfill \\ =A_i{T}_i{F}_m{T}_i^T+\lambda B_i\left[\begin{array}{cc}{L}_m^1\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =A_i{T}_i{F}_m{T}_i^T+\lambda B_i\left[\begin{array}{cc}{K}_m{G}_m^{11}\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =A_i{T}_i{F}_m{T}_i^T+B_i\left[\begin{array}{cc}{K}_m\hfill & 0\hfill \end{array}\right]\left[\begin{array}{cc}\lambda G_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{F_m^{21}}& F_m^{22}\hfill \end{array}\right]T_i^T\hfill \\ =A_i{T}_i{F}_m{T}_i^T+B_i{K}_m\left[\begin{array}{cc}I\hfill & 0\hfill \end{array}\right]F_m{T}_i^T\hfill \\ =A_i{T}_i{F}_m{T}_i^T+B_i{K}_m{C}_i{T}_i{F}_m{T}_i^T\hfill \\ =(A_i+B_i{K}_m{C}_i)T_i{F}_m{T}_i^T\hfill \end{array}$$

(20)

We can know that equation (16) is equivalent to equation (21) based on equations (5), (7), and (20)

$${\Xi }_{33}=\mathrm{He}({\tilde{A}}_{i,m}{T}_i{F}_m{T}_i^T)-P_{i,m}$$

(21)

From equations (2), (8), and (10), the following equality holds

$$\begin{array}{c}{C}_i^z{T}_i{G}_m{T}_i^T+D_i{L}_m{T}_i^T\hfill \\ =C_i^z{T}_i{G}_m{T}_i^T+D_i\left[\begin{array}{cc}{L}_m^1\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i\left[\begin{array}{cc}{K}_m{G}_m^{11}\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i\left[\begin{array}{cc}{K}_m\hfill & 0\hfill \end{array}\right]\left[\begin{array}{cc}{G}_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{G_m^{21}}& G_m^{22}\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i{K}_m\left[\begin{array}{cc}I\hfill & 0\hfill \end{array}\right]G_m{T}_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i{K}_m{C}_i{T}_i{G}_m{T}_i^T\hfill \\ =(C_i^z+D_i{K}_m{C}_i)T_i{G}_m{T}_i^T\hfill \end{array}$$

(22)

Which is equivalent to the following equality based on equations (5), (7), and (22)

$${\Xi }_{41}={\tilde{C}}_{i,m}{T}_i{G}_m{T}_i^T$$

(23)

Similarly, from equations (2), (8), and (10), the following equality holds

$$\begin{array}{c}{C}_i^z{T}_i{F}_m{T}_i^T+\lambda D_i{L}_m{T}_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+\lambda D_i\left[\begin{array}{cc}{L}_m^1\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+\lambda D_i\left[\begin{array}{cc}{K}_m{G}_m^{11}\hfill & 0\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i\left[\begin{array}{cc}{K}_m\hfill & 0\hfill \end{array}\right]\left[\begin{array}{cc}\lambda G_m^{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{F_m^{21}}& F_m^{22}\hfill \end{array}\right]T_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i{K}_m\left[\begin{array}{cc}I\hfill & 0\hfill \end{array}\right]F_m{T}_i^T\hfill \\ =C_i^z{T}_i{F}_m{T}_i^T+D_i{K}_m{C}_i{T}_i{F}_m{T}_i^T\hfill \\ =(C_i^z+D_i{K}_m{C}_i)T_i{F}_m{T}_i^T\hfill \end{array}$$

(24)

Based on equations (5), (7), and (24), equation (24) is equivalent to the following equality

$${\Xi }_{43}={\tilde{C}}_{i,m}{T}_i{F}_m{T}_i^T$$

(25)

Substituting equations (19), (21), (23), and (25) into equation (13), we can have

$$\left[\begin{array}{cc}{\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)\hfill & *\hfill \\ \multicolumn{1}{c}{0}& -I\hfill \\ \multicolumn{1}{c}{{\tilde{A}}_{i,m}{T}_i{G}_m{T}_i^T-T_i{F}_m^T{T}^T}& B_i^{\omega }\hfill \\ \multicolumn{1}{c}{{\tilde{C}}_{i,m}{T}_i{G}_m{T}_i^T}& D_i^w\hfill \end{array}\begin{array}{cc}*\hfill & *\hfill \\ \multicolumn{1}{c}{*}& *\hfill \\ \multicolumn{1}{c}{\mathrm{He}({\tilde{A}}_{i,m}{T}_i{F}_m{T}_i^T)-P_{i,m}}& *\hfill \\ \multicolumn{1}{c}{{\tilde{C}}_{i,m}{T}_i{F}_m{T}_i^T}& -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(26)

We can have the Lyapunov function as

$$V(x_k,k,r_k,{\theta }_k)=x_k^T{P}_{\phi (r_k,{\theta }_k)}{x}_k$$

where $\phi (r_k,{\theta }_k)$ taking values in $(i,m)$ and $\forall i\times m\in {\mathcal{L}}_r\times {\mathcal{L}}_{\theta }$ .

The conditional probability is computed as follows

$$\begin{array}{c}\mathrm{Pr}\{{\theta }_{k+1}=n,r_{k+1}=j|{\theta }_k=m,r_k=i\}\hfill \\ \multicolumn{1}{c}{=\mathrm{Pr}\{{\theta }_{k+1}=n|{\theta }_k=m,r_{k+1}=j,r_k=i\}}\\ \multicolumn{1}{c}{\times \mathrm{Pr}\{r_{k+1}=j|{\theta }_k=m,r_k=i\}}\\ \multicolumn{1}{c}{=v_{\mathrm{mn}}^{(j)}{\pi }_{i,j}}\end{array}$$

Then for $r_k=i$ and ${\theta }_k=m$ , we have

$$\begin{array}{c}E\{\mathrm{\Delta }V(x_k,k,i,m)\}\hfill \\ \multicolumn{1}{c}{=E\{V(x_{k+1},k+1,r_{k+1},{\theta }_{k+1})\}-V(x_k,k,i,m)}\\ \multicolumn{1}{c}{=x_{k+1}^T\left[\sum _{j\in {\mathcal{L}}_r,n\in {\mathcal{L}}_{\theta }}{P}_{j,n}\mathrm{\Lambda }\right]x_{k+1}-x_k^T{P}_{i,m}{x}_k}\\ \multicolumn{1}{c}{=x_{k+1}^T{\rho }_{i,m}{x}_{k+1}-x_k^T{P}_{i,m}{x}_k}\end{array}$$

(27)

where

$$\begin{array}{c}\mathrm{\Lambda }=\mathrm{Pr}\{{\theta }_{k+1}=n,r_{k+1}=j|{\theta }_k=m,r_k=i\}\hfill \\ {\rho }_{i,m}=\sum _{j\in {\mathcal{L}}_r}{\pi }_{\mathrm{ij}}\sum _{n\in {\mathcal{L}}_{\theta }}{v}_{\mathrm{mn}}^{(j)}{P}_{j,n}\hfill \end{array}$$

Pre- and post-multiplying

$$\left[\begin{array}{cccc}{T}_i\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & T_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & I\hfill \end{array}\right]$$

(28)

and its transpose to equation (26), then we can have

$$\left[\begin{array}{c}{\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)\hfill \\ \multicolumn{1}{c}{{\tilde{A}}_{i,m}{T}_i{G}_m{T}_i^T-T_i{F}_m^T{T}^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{{\tilde{C}}_{i,m}{T}_i{G}_m{T}_i^T}\end{array}\begin{array}{ccc}*\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{\mathrm{He}({\tilde{A}}_{i,m}{T}_i{F}_m{T}_i^T)-P_{i,m}}& *\hfill & *\hfill \\ \multicolumn{1}{c}{{(B_i^{\omega })}^T}& -I\hfill & *\hfill \\ \multicolumn{1}{c}{{\tilde{C}}_{i,m}{T}_i{F}_m{T}_i^T}& D_i^w\hfill & -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(29)

From equation (29), it is easy to obtain

$$\left[\begin{array}{c}{\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)\hfill \\ \multicolumn{1}{c}{{\tilde{A}}_{i,m}{T}_i{G}_m{T}_i^T-T_i{F}_m^T{T}_i^T}\end{array}\begin{array}{c}*\hfill \\ \multicolumn{1}{c}{\mathrm{He}({\tilde{A}}_{i,m}{T}_i{F}_m{T}_i^T)-P_{i,m}}\end{array}\right]<0$$

(30)

which can be rewritten as

$$\begin{array}{c}\left[\begin{array}{cc}{\rho }_{i,m}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -P_{i,m}\hfill \end{array}\right]\hfill \\ +\mathrm{He}\left\{\left[\begin{array}{c}{T}_i{G}_m^T{T}_i^T\hfill \\ \multicolumn{1}{c}{T_i{F}_m^T{T}_i^T}\end{array}\right]\left[\begin{array}{cc}-I\hfill & {\tilde{A}}_{i,m}^T\hfill \end{array}\right]\right\}\hfill \\ <0\hfill \end{array}$$

(31)

When ${\omega }_k=0$ , consider the dual system of equation (6)

$$x_{k+1}={\tilde{A}}_{i,m}^T{x}_k$$

which is equivalent to

$$\left[\begin{array}{cc}-I\hfill & {\tilde{A}}_{i,m}^T\hfill \end{array}\right]\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{x_k}\end{array}\right]=0$$

(32)

Based on equations (31) and (32), using Finsler’s lemma, we can have

$${\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{x_k}\end{array}\right]}^T\left[\begin{array}{cc}{\rho }_{i,m}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -P_{i,m}\hfill \end{array}\right]\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{x_k}\end{array}\right]<0$$

which is equivalent to

$$x_{k+1}^T{\rho }_{i,m}{x}_{k+1}-x_k^T{P}_{i,m}{x}_k<0$$

(33)

Using equations (27) and (33), we have that

$$E\{\mathrm{\Delta }V(x_k,k,i,m)\}<0$$

(34)

It shows that

$$E\{\sum _{k=0}^{\infty }||x_k{||}^2|x_0,r_0,{\theta }_0\}<0$$

(35)

Hence, according to Definition 1, system (6) is stochastically stable.

Moreover, from equation (29), it can be seen that

$${\rho }_{i,m}-\mathrm{He}(T_i{G}_m{T}_i^T)<0$$

(36)

which is equivalent to

$$T_i^{-1}{\rho }_{i,m}{T}_i^{-T}-G_m-G_m^T<0$$

(37)

From equation (37), we have

$$-G_m-G_m^T<-T_i^{-1}{\rho }_{i,m}{T}_i^{-T}<0$$

(38)

which means $G_m$ is invertible. Then, $K_m{G}_m^{11}=L_m^1$ admits the solution (10).

From equation (26), we can see it also is equivalent to

$$\mathrm{diag}\left\{\begin{array}{cccc}{\rho }_{i,m}\hfill & I\hfill & -P_{i,m}\hfill & -{\gamma }^{2}I\hfill \end{array}\right\}+\mathrm{He}({\xi }_{i,m}{\sigma }_{i,m})<0$$

(39)

where

$$\begin{array}{c}{\xi }_{i,m}=\left[\begin{array}{cc}{T}_i{G}_m^T{T}_i^T\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \\ \multicolumn{1}{c}{T_i{F}_m^T{T}_i^T}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\sigma }_{i,m}=\left[\begin{array}{cccc}-I\hfill & 0\hfill & {\tilde{A}}_{i,m}^T\hfill & {\tilde{C}}_{i,m}^T\hfill \\ \multicolumn{1}{c}{0}& -I\hfill & {(B_i^{\omega })}^T\hfill & {(D_i^w)}^T\hfill \end{array}\right]}\end{array}$$

Consider the dual system of equation (6), when ${\omega }_k\ne 0$

$$\{\begin{array}{c}\multicolumn{1}{c}{x_{k+1}={\tilde{A}}_{i,m}^T{x}_k+{\tilde{C}}_{i,m}^T{\omega }_k}\\ \multicolumn{1}{c}{z_k=(B_i^{\omega }{)}^T{x}_k+{(D_i^w)}^T{\omega }_k}\end{array}$$

(40)

which can be rewritten as

$${\sigma }_{i,m}\eta (k)=0$$

(41)

where

$$\eta (k)=\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{z_k}\\ \multicolumn{1}{c}{x_k}\\ \multicolumn{1}{c}{{\omega }_k}\end{array}\right]$$

From equations (39) and (41), by Finsler’s lemma, we can obtain

$${\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{z_k}\\ \multicolumn{1}{c}{x_k}\\ \multicolumn{1}{c}{{\omega }_k}\end{array}\right]}^T\left[\begin{array}{cccc}{\rho }_{i,m}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -P_{i,m}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -{\gamma }^{2}I\hfill \end{array}\right]\left[\begin{array}{c}{x}_{k+1}\hfill \\ \multicolumn{1}{c}{z_k}\\ \multicolumn{1}{c}{x_k}\\ \multicolumn{1}{c}{{\omega }_k}\end{array}\right]<0$$

(42)

Equation (42) is equivalent to

$$x_{k+1}^T{\rho }_{i,m}{x}_{k+1}-x_k^T{P}_{i,m}{x}_k+z_k{z}_k^T-{\gamma }^2{\omega }_k^T{\omega }_k<0$$

(43)

From equations (27) and (43), we can have

$$E\{\mathrm{\Delta }V(x_k,k,i,m)+z_k{z}_k^T-{\gamma }^2{\omega }_k^T{\omega }_k\}<0$$

(44)

Then, we have $||z_k|{|}_{E_2}<\gamma ||{\omega }_k|{|}_2$ .

According to definition 2, system (6) is said to not only have a prescribed H_∞ performance index but also is stochastically stable.

Thus, the proof is completed.

Remark 2

The BMIs in the Theorem have the characters of non-convexity. Due to its non-convexity, it is difficult to solve in the math. Although using the linear matrix inequalities (LMIs) solver and line search, we can get the suboptimal results of the problem in the following form. The method has been used in Zhong et al. ⁴¹ The method can also be used to solve the nonlinear problem such as. ⁴²

Problem

$$\gamma (ϵ)=\underset{\{\gamma ,P_{i,m},G_{i,m},F_{i,m},L_{i,m}\}\in \mathrm{\Phi }(ϵ)}{inf}\gamma$$

where $\mathrm{\Phi }(ϵ)$ means all the possible solution of the BMI in the Theorem. If $ϵ$ is given, the set $\mathrm{\Phi }(ϵ)$ can be obtained from the BMI. So, we can use the method of line search to get the result, such that the problem can be solved using the Theorem.

Numerical example

In this section, we will have some numerical example to ensure the Theorem. Consider the discrete-time MJLSs (equation (1)) with two modes given as follows

Mode 1

$$\begin{array}{c}{A}_1=\left[\begin{array}{ccc}-0.51\hfill & -0.81\hfill & -0.009\hfill \\ \multicolumn{1}{c}{-0.65}& -0.5\hfill & -0.81\hfill \\ \multicolumn{1}{c}{0.04}& 0.43\hfill & -0.2\hfill \end{array}\right]\hfill \\ B_1=\left[\begin{array}{cc}0.19\hfill & -0.44\hfill \\ \multicolumn{1}{c}{-0.79}& 0.36\hfill \\ \multicolumn{1}{c}{0.53}& -0.97\hfill \end{array}\right]\hfill \\ B_1^{\omega }=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right],C_1=\left[\begin{array}{ccc}1\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ C_1^z=\left[\begin{array}{ccc}0.01\hfill & -0.7\hfill & -0.35\hfill \end{array}\right]\hfill \\ D_1=\left[\begin{array}{cc}0.02\hfill & 0\hfill \end{array}\right],D_1^{\omega }=0.1\hfill \\ M_{m1}=\left[\begin{array}{c}0.04\hfill \\ \multicolumn{1}{c}{0.6}\end{array}\right],N_{m1}=-0.05\hfill \end{array}$$

Mode 2

$$\begin{array}{c}{A}_2=\left[\begin{array}{ccc}0.19\hfill & -0.28\hfill & 0.95\hfill \\ \multicolumn{1}{c}{0.4}& 0.93\hfill & -0.41\hfill \\ \multicolumn{1}{c}{-0.65}& -0.5\hfill & 0.78\hfill \end{array}\right]\hfill \\ B_2=\left[\begin{array}{cc}-0.44\hfill & 0.24\hfill \\ \multicolumn{1}{c}{0.87}& -0.46\hfill \\ \multicolumn{1}{c}{-0.09}& -0.816\hfill \end{array}\right]\hfill \\ B_2^{\omega }=\left[\begin{array}{c}0.3\hfill \\ \multicolumn{1}{c}{-0.2}\\ \multicolumn{1}{c}{0.2}\end{array}\right],C_2=\left[\begin{array}{ccc}1\hfill & 1\hfill & 0\hfill \end{array}\right]\hfill \\ C_2^z=\left[\begin{array}{ccc}0.04\hfill & -0.28\hfill & 0.23\hfill \end{array}\right]\hfill \\ D_2=\left[\begin{array}{cc}0.08\hfill & -0.5\hfill \end{array}\right],D_2^{\omega }=0.2\hfill \\ M_{m2}=\left[\begin{array}{c}0.3\hfill \\ \multicolumn{1}{c}{0.2}\end{array}\right],N_{m2}=-0.04\hfill \end{array}$$

According to equation (3) and with the $C_i$ given as mentioned, we can have

$$\begin{array}{c}{T}_1=\left[\begin{array}{ccc}0.5\hfill & 1\hfill & -1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 1\hfill \\ \multicolumn{1}{c}{0.5}& -1\hfill & 1\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{T_2=\left[\begin{array}{ccc}0.5\hfill & 1\hfill & -1\hfill \\ \multicolumn{1}{c}{0.5}& -1\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 1\hfill \end{array}\right]}\end{array}$$

Next, the TPs $\pi$ of system (1) is given as follows

$$\pi =\left[\begin{array}{cc}0.4\hfill & 0.6\hfill \\ \multicolumn{1}{c}{0.3}& 0.7\hfill \end{array}\right]$$

and the TPs of Markov chains $r(k)$ and $\theta (k)$ are as follows

$$v^1=\left[\begin{array}{cc}0.7\hfill & 0.3\hfill \\ \multicolumn{1}{c}{0.7}& 0.3\hfill \end{array}\right],v^2=\left[\begin{array}{cc}0.2\hfill & 0.8\hfill \\ \multicolumn{1}{c}{0.2}& 0.8\hfill \end{array}\right]$$

Solving the problem by giving $\alpha (0<\alpha <1)$ , the suboptimal H_∞ performance index can be obtained as shown in Figure 1. From Figure 1, it can be observed that if we want to get the minimum value of the H_∞ performance index $\gamma$ , we should chose ${\alpha }^{*}=0.98$ , and meanwhile, ${\gamma }^2=0.1681$ .

Figure 1.

The suboptimal H_∞ performance ${\gamma }^{*}$ varying with $\alpha$ .

So, let $\alpha =0.98$ and ${\gamma }^2=0.158$ , with $\lambda =0.19$ , by solving the LMIs given in the Theorem, we can get the controller gains as follows

$$K_1=\left[\begin{array}{c}-3.3937\hfill \\ \multicolumn{1}{c}{0.9169}\end{array}\right],K_2=\left[\begin{array}{c}-3.4373\hfill \\ \multicolumn{1}{c}{0.9397}\end{array}\right]$$

Given the initial state of the system as follows

$$x(0)=\left[\begin{array}{c}0.9\hfill \\ \multicolumn{1}{c}{-1.2}\\ \multicolumn{1}{c}{0.1}\end{array}\right]$$

Figure 2 shows the jumping of Markov chains. The actual controller gains are ${\hat{K}}_{\theta (k)}=K_{\theta (k)}+M_{\theta (k)}\mathrm{\Delta }(k)N_{\theta (k)}$ with $\mathrm{\Delta }(k)=\sin (k)$ . In this simulation, the following disturbances are considered $w(k)=0.5\exp (-0.1k)$ . Figure 3 shows the controlled output of the closed-loop system, Figure 4 shows the trajectories of the states.

Figure 2.

The jumping conditions of the Markov chains.

Figure 3.

The trajectory of the controlled output.

Figure 4.

The trajectories of the system states.

Conclusion

Resilient asynchronous H_∞ static output feedback control for a set of discrete-time MJLSs has been designed and makes sure the system not only has a prescribed H_∞ performance index but also is stochastically stable. By Finsler’s lemma, and with the help of two sets of slack variables, the interconnection between Lyapunov matrices and system matrices is decoupled. The BMIs are proposed as the sufficient conditions for the design of controller, which can be solved using LMIs along with line search. The designed controller improves the robustness of the closed-loop systems and is able to work when the jump between the system mode and the control mode is asynchronous. In the fields of robot control and aircraft and traffic control, the control strategy proposed in this article can be used.