Output feedback asynchronous control for singular Markovian jump systems

Abstract

This paper is concerned with the problem of output feedback asynchronous control for singular Markovian jump systems (SMJSs). The dynamic output feedback controller (DOFC) modes do not need to be synchronized with the system modes and a hidden Markov model (HMM) is adopted to characterize this phenomenon. Firstly, sufficient novel conditions are derived to guarantee that the closed-loop system is stochastically admissible (SA) with $H_{\infty }$ performance. Secondly, an approach is established to design the DOFC gains. Finally, numerical examples are presented to show the effectiveness of the proposed approach.

Introduction

Singular systems, which can describe many physical systems, such as circuit systems and economic systems, ¹ have drawn extensive attention. Singular system is composed of differential equation and algebraic equation and the state systems can be obtained when the matrix $E$ of singular systems is selected as $E=I$ , which means that the state systems can be seen as a specific case of singular systems. Moreover, compared with state systems, the study of singular systems is more complicated because of the consideration of regularity and causality or non-impulsiveness simultaneously. Therefore, the research on singular systems is more complicated. Due to its wider applications, a lot of attention has been paid to the problem of admissibility analysis, controller design and security control for singular systems.^2–5 Such as, the admissibility analysis issue was studied for discrete nonlinear fuzzy singular systems in Chen et al., ² where a novel piecewise Lyapunov function was established to obtain a relaxed stability criterion. Zhang et al. ³ discussed the problem of security control for continuous singular systems subject to cyber-attacks, in which a dynamic event-triggered scheme is adopted to save network resources.

A large amount of existing literature is based on the assumption that the states of the system are fully available. However, this assumption is very difficult to satisfy in practice. To overcome this obstacle, the method to achieve the stability of the system through the output information of the system is proposed. The method contains DOF control, observer-based output feedback (OBOF) control and static output feedback (SOF) control. Thus, the output feedback control problem of networked control systems (NCSs) has been studied by many researchers.^6–10 The event-based consensus control problem was addressed for multi-agent systems under switching topologies by using the OBOF control approach in. ⁶ In Zhang et al., ⁷ the DOFC design problem was discussed for parameter-varying singular systems under inexact parameters and a sufficient and necessary criterion was established for the solvability of the admissible problem.

In practice, the parameters and structure of many network control systems (NCSs) are subject to random changes due to complex environments. MJSs can be regarded as a powerful tool to characterize the changeable model of the NCSs since it can capture the abrupt mode changes of the NCSs. Because of the great application potential of MJSs, it has received a lot of attention and fruitful outcomes associated with the controller design and sliding mode control (SMC) problem of MJSs have been established.^11–15 In Li et al., ¹¹ the guaranteed cost control issue was discussed for neutral-type MJSs by using the DOF technique, where the time delay phenomenon was considered. The authors ¹² studied the security SMC issue for nonlinear MJSs, where an adaptive SMC method was established to eliminate the impact of injection attacks on the system. Sun et al. ¹³ discussed the DOF control problem for SMJSs by variable elimination technique. One can find that the asynchronous control problem was not solved because when solving the controller, the mode of system is dependent on the mode of controller, that is, $A_{\mathrm{pc}}=P_i^{-1}{\overline{A}}_{\mathrm{pc}}$ .

The above-mentioned results about MJSs assumed that the controller modes must be consistent with the system modes. Unfortunately, this assumption is almost impossible to satisfy because of the existence of adverse factors, such as time-delay and data dropouts. To overcome this problem, the asynchronous control problem was developed for MJSs and some results have emerged.^16–20 In Liu et al., ¹⁶ the problem of asynchronous output feedback was studied for MJSs, where an output quantization method was adopted to achieve the effective use of the network resources. In Zhang et al., ¹⁷ the asynchronous filter design problem was discussed for nonlinear continuous SMJSs with event-triggered method, in which a novel approach was given to solve the filter gains. Li et al. ¹⁸ focused on the issue of DOF sliding mode asynchronous control design for continuous SMJSs, but the conditions provided by the author cannot guarantee that the system is regular and impulse-free. In our work, a new condition is given to ensure that the system is regular and impulse-free.

Based on the above discussions, the asynchronous control issue is discussed for SMJSs by using the DOF technique in this paper. This paper has the following contributions:

• The asynchronous problem is considered and a HMM is adopted to reveal the asynchronous phenomenon of the system and the DOFC.

• In contrast to Li et al., ¹⁸ some novel conditions for the stochastic admissibility with $H_{\infty }$ performance of the closed-loop system are established.

• Unlike the existing results in Sun et al., ¹³ a novel method is proposed to solve the controller gains, where a new nonsingular matrix is introduced.

The rest of this paper is structured as follows. The introduction is narrated in section II. Section III gives the main results. Simulation examples are given to evaluate the effectiveness of the proposed approach in Section IV. Finally, Section V summarizes this paper.

Notations: ${\mathbb{R}}^a$ and ${\mathbb{R}}^{p\times q}$ show the Euclidean space with $a$ dimensional and the set of $p\times q$ real matrices, respectively. $W>0$ indicates $W$ is positive define and sym $(W)$ denotes $W+W^T$ . ${\mathcal{L}}_2[0,\infty )$ represents the space of square integrable vectors.

Problem formulation

Consider the following SMJSs

$$\{\begin{array}{c}E\stackrel{·}{\eta }(t)=A(r_t)\eta (t)+B(r_t)u(t)+G(r_t)\nu (t)\hfill \\ \multicolumn{1}{c}{\theta (t)=C(r_t)\eta (t)}\\ \multicolumn{1}{c}{\vartheta (t)=L(r_t)\eta (t)}\end{array}$$ (1)

where $\eta (t)\in {\mathbb{R}}^{\eta }$ means the state, $u(t)\in {\mathbb{R}}^u$ denotes the input, $\theta (t)\in {\mathbb{R}}^{\theta }$ expresses the measured output, $\vartheta (t)\in {\mathbb{R}}^{\vartheta }$ stands for the controlled output, $\nu (t)\in {\mathbb{R}}^r$ is the external disturbance belonging to ${\mathcal{L}}_2[0,\infty )$ . $E,A(r_t),B(r_t),C(r_t)$ , and $G(r_t)$ are known constant matrices and rank $E=\varphi <n$ . $\{(r_t),t\ge 0\}$ denotes a right-continuous Markov process taking values in a finite set ${\mathbb{T}}_1=\{1,2,\cdots ,e\}$ with TPM $\mathrm{\Pi }=\{{\pi }_{\mathrm{ij}}\},i,j\in {\mathbb{T}}_1$ given by

$$\begin{array}{c}\mathrm{Pr}\{r(t+\delta h)=j|r(t)=i\}=\{\begin{array}{cc}{\pi }_{\mathrm{ij}}\delta h+o(\delta h),\hfill & i\ne j\hfill \\ \multicolumn{1}{c}{1+{\pi }_{\mathrm{ii}}\delta h+o(\delta h),}& i=j\hfill \end{array}\hfill \end{array}$$ (2)

where $\delta h>0,\underset{\delta h\to 0}{lim}\frac{o(\delta h)}{\delta h}=0,{\pi }_{\mathrm{ij}}\ge 0(j\ne i)$ is the transition rate from mode $i$ to mode $j$ and ${\pi }_{\mathrm{ii}}=-\sum _{j=1,j\ne i}^e{\pi }_{\mathrm{ij}}$ for all $i\in {\mathbb{T}}_1$ .

The following asynchronous DOFC is considered

$$\{\begin{array}{c}E{\stackrel{·}{\eta }}_c(t)=A{({\varsigma }_t)}_c{\eta }_c(t)+B{({\varsigma }_t)}_c\theta (t)\hfill \\ \multicolumn{1}{c}{u(t)=C{({\varsigma }_t)}_c\eta (t)+D{({\varsigma }_t)}_c\theta (t)}\end{array}$$ (3)

where $x_c(t)\in {\mathbb{R}}^n$ is the controller state, $A({\varsigma }_t{)}_c,B({\varsigma }_t{)}_c,C({\varsigma }_t{)}_c$ , and $D({\varsigma }_t{)}_c$ are the controller gains, which are to be determined. $\{\varsigma (t)\}\in {\mathbb{T}}_2=\{1,2,\cdots ,f\}$ denotes a Markov chain. The TPM $\mathrm{\Pi }=\{{\chi }_{\mathrm{ip}}\}$ is defined as

$$\mathrm{Pr}\{\varsigma (t)=p|r(t)=i\}={\chi }_{\mathrm{ip}}$$ (4)

where ${\chi }_{\mathrm{ip}}\in [0,1]$ and $\sum _{p=1}^f{\chi }_{\mathrm{ip}}=1$ .

Combining (1) and (3), for $r(t)=i$ and $\varsigma (t)=p$ , the closed-loop system can be get

$$\{\begin{array}{c}\overline{E}\stackrel{·}{\overline{\eta }}(t)={\overline{A}}_{\mathrm{ip}}\overline{\eta }(t)+{\overline{G}}_i\nu (t)\hfill \\ \multicolumn{1}{c}{z(t)={\overline{L}}_i\overline{\eta }(t)}\end{array}$$ (5)

where

$$\begin{array}{c}\overline{\eta }(t)=\left[\begin{array}{c}\eta (t)\hfill \\ \multicolumn{1}{c}{{\eta }_c(t)}\end{array}\right],\overline{E}=\left[\begin{array}{cc}E\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& E\hfill \end{array}\right],{\overline{G}}_i=\left[\begin{array}{c}{G}_i\hfill \\ \multicolumn{1}{c}{0}\end{array}\right],\hfill \\ {\overline{A}}_{\mathrm{ip}}=\left[\begin{array}{cc}{A}_i+B_i{D}_{\mathrm{pc}}{C}_i\hfill & B_i{C}_{\mathrm{pc}}\hfill \\ \multicolumn{1}{c}{B_{\mathrm{pc}}{C}_i}& A_{\mathrm{pc}}\hfill \end{array}\right],{\overline{L}}_i=\left[\begin{array}{cc}{L}_i\hfill & 0\hfill \end{array}\right].\hfill \end{array}$$

• Remark 1. In this paper, our aim is to discuss the asynchronous control problem for SMJSs. The analysis is more complex because the systems (1) and the DOFC (3) are non-synchronous. Besides, the synchronization phenomenon can be regarded as a special case when ${\chi }_{\mathrm{ii}}=1$ and ${\mathcal{T}}_1={\mathcal{T}}_2$ .

• Definition 1. ¹ The SMJSs (1) with $u(t)=0,\nu (t)=0$ is said to be

• (i)  regular if det $(\mathrm{zE}-A(r_t))\not\equiv 0$ ;

• (ii) impulse-free if deg $[\det (\mathrm{zE}-A(r_t))]=\mathrm{rank}(\mathcal{E})$ ;

• (iii) stochastically stable (SS) if there exists $\psi (i,x)>0$ such that

$$\begin{array}{c}\underset{i\to \infty }{lim}\mathbf{E}\{{\int }_0^i|\eta (\delta {)|}^{2}d\delta \}<\psi (m,x);\hfill \end{array}$$

• (iv) SA if (i)-(iii) are satisfied.

The main objective of this paper is to design the DOFC (3) such that

1. The system (5) is SA;

2. Under zero initial condition, the following condition is satisfied:

$${\int }_0^{\infty }{\vartheta }^T(s)\vartheta (s)\mathrm{ds}\le {\gamma }^2{\int }_0^{\infty }{\nu }^T(s)\nu (s)\mathrm{ds}.$$ (6)

Main results

In this section, some new conditions are given to assure that the system (5) is SA with $H_{\infty }$ performance.

• Theorem 1. For given positive constant $\gamma$ , the system (5) is SA with a desired ${\mathcal{H}}_{\infty }$ performance, if there exist nonsingular matrix $P_i$ such that

$${\overline{E}}^T{P}_i=P_i^T\overline{E}\ge 0$$ (7)

$${\mathrm{\Gamma }}_i=P_i^T{\overline{A}}_{\mathrm{ip}}+{\overline{A}}_{\mathrm{ip}}^T{P}_i+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{\overline{E}}^T{P}_j<0$$ (8)

$${\mathrm{\Psi }}_i=\left[\begin{array}{ccc}{\mathrm{\Theta }}_1\hfill & P_i^T{\overline{G}}_i\hfill & {\overline{L}}_i^T\hfill \\ \multicolumn{1}{c}{*}& -{\gamma }^{2}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -I\hfill \end{array}\right]<0$$ (9)

where

$${\mathrm{\Theta }}_1=\sum _{p=1}^f{\chi }_{\mathrm{ip}}(P_i^T{\overline{A}}_{\mathrm{ip}}+{\overline{A}}_{\mathrm{ip}}^T{P}_i+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{\overline{E}}^T{P}_j).$$

• Proof. First of all, we show that the system (5) is regular and impulse-free.

Since rank $(\overline{E})=2\varphi \le 2n$ , there exist two nonsingular matrices $M$ and $N$ , such that

$$\begin{array}{c}M{\overline{A}}_{\mathrm{ip}}N=\left[\begin{array}{cc}{\overline{A}}_{\mathrm{ip}11}\hfill & {\overline{A}}_{\mathrm{ip}12}\hfill \\ \multicolumn{1}{c}{{\overline{A}}_{\mathrm{ip}21}}& {\overline{A}}_{\mathrm{ip}22}\hfill \end{array}\right],\hfill \\ M\overline{E}N=\left[\begin{array}{cc}{I}_{2\varphi }\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \\ M^{-T}{P}_{i}N=\left[\begin{array}{cc}{\overline{P}}_{i11}\hfill & {\overline{P}}_{i12}\hfill \\ \multicolumn{1}{c}{{\overline{P}}_{i21}}& {\overline{P}}_{i22}\hfill \end{array}\right].\hfill \end{array}$$ (10)

From (7), one has ${\overline{P}}_{i12}=0.$

Then, pre- and post-multiplying (8) by $N^T$ and $N$ , one has

$$\left[\begin{array}{cc}★\hfill & ★\hfill \\ \multicolumn{1}{c}{★}& {\overline{P}}_{i22}^T{\overline{A}}_{\mathrm{ip}22}+{\overline{A}}_{\mathrm{ip}22}^T{\overline{P}}_{i22}\hfill \end{array}\right]<0.$$ (11)

This implies that ${\overline{A}}_{\mathrm{ip}22}$ is nonsingular. Thus, the system (5) is regular and impulse free.

Construct the following Lyapunov functional:

$$V(t)={\overline{\eta }}^T(t){\overline{E}}^T{P}_i\overline{\eta }(t).$$ (12)

Let $\mathfrak{L}$ be an infinitesimal operator. One has

$$\begin{array}{cc}\mathfrak{L}V(t)=\hfill & 2{\overline{\eta }}^T(t)P_i^T\overline{E}\stackrel{·}{\overline{\eta }}(t)+{\overline{\eta }}^T(t){\overline{E}}^T\left(\sum _{j=1}^e{\pi }_{\mathrm{ij}}{P}_j\right)\overline{\eta }(t).\hfill \end{array}$$ (13)

Combining (5) and (13), it yields

$$\mathfrak{L}V(t)+{\vartheta }^T(t)\vartheta (t)-{\gamma }^2{\nu }^T(t)\nu (t)\le {\xi }^T(t){\mathrm{\Phi }}_i\xi (t)$$ (14)

where ${\xi }^T(t)=[{\overline{\eta }}^T(t){\nu }^T(t)],$ ${\mathrm{\Phi }}_i={\mathrm{\Theta }}_{11}+{\mathrm{\Theta }}_2^T{\mathrm{\Theta }}_2$ and ${\mathrm{\Theta }}_{11}=\left[\begin{array}{cc}{\mathrm{\Theta }}_1\hfill & P_i^T{\overline{G}}_i\hfill \\ \multicolumn{1}{c}{*}& -{\gamma }^{2}I\hfill \end{array}\right],{\mathrm{\Theta }}_2^T=[{\overline{L}}_{i}0{]}^T.$

By using Schur complement and (9), one can obtain

$$\mathfrak{L}V(t)+{\vartheta }^T(t)\vartheta (t)-{\gamma }^2{\nu }^T(t)\nu (t)<0.$$ (15)

Under zero initial condition, integrating both sides of (15) from $0$ to $t$ and letting $t\to \infty$ , one has

$${\int }_0^{\infty }{\vartheta }^T(s)\vartheta (s)\mathrm{ds}\le {\gamma }^2{\int }_0^{\infty }{\nu }^T(s)\nu (s)\mathrm{ds}.$$ (16)

When $\nu (t)=0$ , it is known that ${\vartheta }^T(t)\vartheta (t)\ge 0$ , one has

$$\mathfrak{L}V(t)<0.$$ (17)

Then, the system (5) is SA. The proof is completed.□

• Remark 2. In Theorem 1, the condition (8) is very important. However, this condition is ignored. In fact, the condition (8) can ensure that $\overline{E}\stackrel{·}{\overline{\eta }}(t)={\overline{A}}_{\mathrm{ip}}\overline{\eta }(t)+{\overline{G}}_i\nu (t)$ is regular and impulse-free. However, the condition (9) can ensure that $\overline{E}\stackrel{·}{\overline{\eta }}(t)=\sum _{p=1}^f{\chi }_{\mathrm{ip}}{\overline{A}}_{\mathrm{ip}}\overline{\eta }(t)+{\overline{G}}_i\nu (t)$ is regular and impulse-free.

Next, a novel approach is developed to solve DOFC.

• Theorem 2. For given constants $d_{i1},d_{i2}$ and $d_{i1}\ne d_{i2}$ , the system (5) is SA with $H_{\infty }$ performance $\gamma$ , if there exist nonsingular matrices $U_i,L_i,Y_i$ and matrices ${\overline{A}}_{\mathrm{pc}},{\overline{B}}_{\mathrm{pc}},$ ${\overline{C}}_{\mathrm{pc}},{\overline{D}}_{\mathrm{pc}}$ such that

$$E^T{U}_i=U_i^{T}E\ge 0,E^T{L}_i=L_i^{T}E\ge 0$$ (18)

$$\left[\begin{array}{cccc}{\theta }_{11}\hfill & {\theta }_{12}\hfill & {\theta }_{13}\hfill & {\theta }_{14}\hfill \\ \multicolumn{1}{c}{*}& {\theta }_{22}\hfill & {\theta }_{23}\hfill & {\theta }_{24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -2d_{i1}I\hfill & -Y-d_{i2}I\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -Y-Y^T\hfill \end{array}\right]<0$$ (19)

$$\left[\begin{array}{cccccc}{\hat{\theta }}_{11}\hfill & {\hat{\theta }}_{12}\hfill & {\hat{\theta }}_{13}\hfill & {\hat{\theta }}_{14}\hfill & d_{i1}{G}_i\hfill & L_i^T\hfill \\ \multicolumn{1}{c}{*}& {\hat{\theta }}_{22}\hfill & {\hat{\theta }}_{23}\hfill & {\hat{\theta }}_{24}\hfill & d_{i2}{G}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -2d_{i1}I\hfill & -Y-d_{i2}I\hfill & d_{i1}{G}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -Y-Y^T\hfill & d_{i2}{G}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -{\gamma }^{2}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -I\hfill \end{array}\right]<0$$ (20)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\theta }_{11}=\mathrm{sym}(d_{i1}{A}_i+d_{i1}{B}_i{\overline{D}}_{\mathrm{pc}}{C}_i+{\overline{B}}_{\mathrm{pc}}{C}_i)+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{U}_j,}\\ \multicolumn{1}{c}{{\theta }_{12}=d_{i1}{B}_i{C}_{\mathrm{pc}}+d_{i2}{A}_i^T+{\overline{A}}_{\mathrm{pc}}+d_{i2}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T,}\\ \multicolumn{1}{c}{{\theta }_{13}=U_i^T-d_{i1}I+d_{i1}{A}_i^T+d_{i1}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T,}\\ \multicolumn{1}{c}{{\theta }_{14}=d_{i2}{A}_i^T-Y+d_{i2}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T,}\\ \multicolumn{1}{c}{{\theta }_{22}=\mathrm{sym}(d_{i2}{B}_i{\overline{C}}_{\mathrm{pc}}+{\overline{A}}_{\mathrm{pc}})+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{L}_j,}\\ \multicolumn{1}{c}{{\theta }_{23}=d_{i1}{\overline{C}}_{\mathrm{pc}}^T{B}_i^T+{\overline{A}}_{\mathrm{pc}}^T-d_{i2}I,}\\ \multicolumn{1}{c}{{\theta }_{24}=L_i^T-Y+d_{i1}{\overline{C}}_{\mathrm{pc}}^T{B}_i^T+{\overline{A}}_{\mathrm{pc}}^T,}\\ \multicolumn{1}{c}{{\hat{\theta }}_{11}=\mathrm{sym}\left[\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i1}{A}_i+d_{i1}{B}_i{\overline{D}}_{\mathrm{pc}}{C}_i+{\overline{B}}_{\mathrm{pc}}{C}_i)\right]}\\ +\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{U}_j,\hfill \\ \multicolumn{1}{c}{{\hat{\theta }}_{12}=\sum _{p=1}^f{\chi }_{\mathrm{ip}}({\overline{A}}_{\mathrm{pc}}+d_{i2}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T)}\\ +d_{i1}{B}_i{C}_{\mathrm{pc}}+d_{i2}{A}_i^T,\hfill \\ \multicolumn{1}{c}{{\hat{\theta }}_{13}=\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i1}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T)}\\ +U_i^T-d_{i1}I+d_{i1}{A}_i^T,\hfill \\ \multicolumn{1}{c}{{\hat{\theta }}_{14}=\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i2}{C}_i^T{\overline{D}}_{\mathrm{pc}}^T{B}_i^T+C_i^T{\overline{B}}_{\mathrm{pc}}^T)+d_{i2}{A}_i^T-Y,}\\ \multicolumn{1}{c}{{\hat{\theta }}_{22}=\mathrm{sym}\left[\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i2}{B}_i{\overline{C}}_{\mathrm{pc}}+{\overline{A}}_{\mathrm{pc}})\right]+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{L}_j,}\\ \multicolumn{1}{c}{{\hat{\theta }}_{23}=\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i1}{\overline{C}}_{\mathrm{pc}}^T{B}_i^T+{\overline{A}}_{\mathrm{pc}}^T)-d_{i2}I,}\\ \multicolumn{1}{c}{{\hat{\theta }}_{24}=L_i^T-Y+\sum _{p=1}^f{\chi }_{\mathrm{ip}}(d_{i1}{\overline{C}}_{\mathrm{pc}}^T{B}_i^T+{\overline{A}}_{\mathrm{pc}}^T).}\end{array}$$

Moreover, the DOFC is given as

$$\begin{array}{c}{A}_{\mathrm{pc}}=Y^{-1}{\overline{A}}_{\mathrm{pc}},C_{\mathrm{pc}}={\overline{C}}_{\mathrm{pc}},\hfill \\ B_{\mathrm{pc}}=Y^{-1}{\overline{B}}_{\mathrm{pc}},D_{\mathrm{pc}}={\overline{D}}_{\mathrm{pc}}.\hfill \end{array}$$ (21)

• Proof. By decomposing ${\mathrm{\Psi }}_i$ , one has

$${\mathrm{\Psi }}_i={\Sigma }_i^T{\mathrm{\Upsilon }}_i{\Sigma }_i<0$$ (22)

where $J_i$ is a nonsingular matrix and

$${\Sigma }_i=\left[\begin{array}{ccc}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\sum _{p=1}^f{\chi }_{\mathrm{ip}}{\overline{A}}_{\mathrm{ip}}}& {\overline{G}}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & I\hfill \end{array}\right],$$ (23)

$${\mathrm{\Upsilon }}_i=\left[\begin{array}{cccc}{\mathrm{\Xi }}_1\hfill & {\mathrm{\Xi }}_2\hfill & J_i{\overline{G}}_i\hfill & {\overline{L}}_i^T\hfill \\ \multicolumn{1}{c}{*}& -J_i-J_i^T\hfill & J_i{\overline{G}}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }^{2}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -I\hfill \end{array}\right]$$ (24)

$${\mathrm{\Xi }}_1=\sum _{p=1}^f{\chi }_{\mathrm{ip}}(J_i{\overline{A}}_{\mathrm{ip}}+{\overline{A}}_{\mathrm{ip}}^T{J}_i^T)+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{P}_j,$$

$${\mathrm{\Xi }}_2=P_i^T-J_i+\sum _{p=1}^f{\chi }_{\mathrm{ip}}{\overline{A}}_{\mathrm{ip}}^T{J}_i^T.$$

Hence, ${\mathrm{\Psi }}_i<0$ holds if ${\mathrm{\Upsilon }}_i<0.$

Similar to the above process, by decomposing (8), one has

$${\mathrm{\Gamma }}_i={\hat{\Sigma }}_i^T{\hat{\mathrm{\Upsilon }}}_i{\hat{\Sigma }}_i<0$$ (25)

where

$${\Sigma }_i=\left[\begin{array}{c}I\hfill \\ \multicolumn{1}{c}{{\overline{A}}_{\mathrm{ip}}}\end{array}\right],$$ (26)

$${\hat{\mathrm{\Upsilon }}}_i=\left[\begin{array}{cc}{\hat{\mathrm{\Xi }}}_1\hfill & {\hat{\mathrm{\Xi }}}_2\hfill \\ \multicolumn{1}{c}{*}& -J_i-J_i^T\hfill \end{array}\right],$$ (27)

$${\hat{\mathrm{\Xi }}}_1=J_i{\overline{A}}_{\mathrm{ip}}+{\overline{A}}_{\mathrm{ip}}^T{J}_i^T+\sum _{j=1}^e{\pi }_{\mathrm{ij}}{E}^T{P}_j,$$

$${\hat{\mathrm{\Xi }}}_2=P_i^T-J_i+{\overline{A}}_{\mathrm{ip}}^T{J}_i^T.$$

Hence, ${\mathrm{\Gamma }}_i<0$ holds if ${\hat{\mathrm{\Upsilon }}}_i<0.$

Define

$$\begin{array}{c}{P}_i=\left[\begin{array}{cc}{U}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& L_i\hfill \end{array}\right],{\overline{A}}_{\mathrm{pc}}=Y{A}_{\mathrm{pc}},{\overline{C}}_{\mathrm{pc}}=C_{\mathrm{pc}},\hfill \\ J_i=\left[\begin{array}{cc}{d}_{i1}I\hfill & Y\hfill \\ \multicolumn{1}{c}{d_{i2}I}& Y\hfill \end{array}\right],{\overline{B}}_{\mathrm{pc}}=Y{B}_{\mathrm{pc}},{\overline{D}}_{\mathrm{pc}}=D_{\mathrm{pc}},\hfill \\ d_{i1}\ne d_{i2}.\hfill \end{array}$$ (28)

Thus, substituting (28) into (24) and (27), one can obtain (20) and (19).

Moreover, from (18), one has

$$\begin{array}{c}{\overline{E}}^T{P}_i=P_i^T\overline{E}\ge 0.\hfill \end{array}$$

The proof is completed. □

• Remark 3. In Theorem 2, a new nonsingular matrix $J_i$ is introduced to solve the DOFC gains. In the common approach, the controller gain is given as $A_{\mathrm{pc}}=P_i^{-1}{\overline{A}}_{\mathrm{pc}}.$ When $p=i=1$ , one can obtain $A_{1c}=P_1^{-1}{\overline{A}}_{1c}.$ When $p=1,i=2$ , one can also obtain $A_{1c}=P_2^{-1}{\overline{A}}_{1c}.$ This method is unreasonable because two different controllers $A_{1c}$ are obtained.

Simulation

This section illustrates the validity of the proposed method through two examples.

• Example 1. Consider the system (1) with the following parameters

$$\begin{array}{cc}\multicolumn{1}{c}{E}& =\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],A_1=\left[\begin{array}{cc}0.9\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& -5\hfill \end{array}\right],B_1=\left[\begin{array}{c}0.3\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right],\hfill \\ \multicolumn{1}{c}{C_1}& =\left[\begin{array}{cc}0.3\hfill & 0.1\hfill \end{array}\right],G_1=\left[\begin{array}{c}0.2\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right],\hfill \\ \multicolumn{1}{c}{L_1}& =\left[\begin{array}{cc}0.2\hfill & 0.1\hfill \end{array}\right],A_2=\left[\begin{array}{cc}0.7\hfill & 0.1\hfill \\ \multicolumn{1}{c}{1.5}& -5\hfill \end{array}\right],B_2=\left[\begin{array}{c}0.2\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right],\hfill \\ \multicolumn{1}{c}{C_2}& =\left[\begin{array}{cc}0.2\hfill & 0.2\hfill \end{array}\right],G_2=\left[\begin{array}{c}0.15\hfill \\ \multicolumn{1}{c}{0.15}\end{array}\right],L_2=\left[\begin{array}{cc}0.1\hfill & 0.2\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{\mathrm{\Lambda }}& =\left[\begin{array}{cc}-0.8\hfill & 0.8\hfill \\ \multicolumn{1}{c}{0.2}& -0.2\hfill \end{array}\right],\mathrm{\Pi }=\left[\begin{array}{cc}0.7\hfill & 0.3\hfill \\ \multicolumn{1}{c}{0.5}& 0.5\hfill \end{array}\right].\hfill \end{array}$$

Choose $d_{11}=d_{21}=2,d_{12}=d_{22}=-2,$ and the disturbance $\omega (t)=e^{-0.35t}.$ By Theorem 2, one has 4 cm

$$\begin{array}{cc}\multicolumn{1}{c}{A_{1c}}& =\left[\begin{array}{cc}-0.8789\hfill & -0.0381\hfill \\ \multicolumn{1}{c}{0.0550}& -1.1515\hfill \end{array}\right],B_{c1}=\left[\begin{array}{c}-1.5490\hfill \\ \multicolumn{1}{c}{0.8901}\end{array}\right],\hfill \\ \multicolumn{1}{c}{A_{c2}}& =\left[\begin{array}{cc}-0.9318\hfill & 0.0568\hfill \\ \multicolumn{1}{c}{0.0568}& -1.1247\hfill \end{array}\right],B_{c2}=\left[\begin{array}{c}-1.4015\hfill \\ \multicolumn{1}{c}{0.9016}\end{array}\right],\hfill \\ \multicolumn{1}{c}{C_{c1}}& =\left[\begin{array}{cc}4.3946\hfill & 0.9035\hfill \end{array}\right]{,}_{c1}=-35.8275,\hfill \\ \multicolumn{1}{c}{C_{c2}}& =\left[\begin{array}{cc}4.5816\hfill & 2.4918\hfill \end{array}\right]{,}_{c2}=-42.6335.\hfill \end{array}$$

Supposes $\overline{\eta }(0)=[1.20.31.20.3{]}^T.$ Figures 1–4 show the simulation results. Figures 1 and 2 depict the trajectories of the open-loop system and the closed-loop system, respectively. One can find that the open-loop system is unstable and the closed-loop system is SA. Figures 3 and 4 show the modes of the system and the controller, respectively.

• Example 2. An oil catalytic cracking process is considered, ²¹ which is depicted as follows:

$$\{\begin{array}{c}{\stackrel{·}{\xi }}_1(t)=T_{11}{\xi }_1(t)+T_{12}{\xi }_2(t)+T_{13}u(t)+T_{14}\nu (t)\hfill \\ \multicolumn{1}{c}{0=T_{21}{\xi }_1(t)+T_{22}{\xi }_2(t)+T_{23}u(t)+T_{24}\nu (t)}\end{array}$$

where ${\xi }_1(t)$ stands for blower capacity or regenerate temperature, ${\xi }_2(t)$ denotes the vector of policy, administration or business benefits, $\nu (t)$ is the disturbance, $u(t)$ represents the regulation value. The system parameters are given as

$$\begin{array}{cc}\multicolumn{1}{c}{E}& =\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],A_1=\left[\begin{array}{cc}1\hfill & 2\hfill \\ \multicolumn{1}{c}{0.5}& -1\hfill \end{array}\right],B_1=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\right],\hfill \\ \multicolumn{1}{c}{G_1}& =\left[\begin{array}{c}0.2\hfill \\ \multicolumn{1}{c}{-0.1}\end{array}\right],L_1=\left[\begin{array}{cc}0.2\hfill & 0.1\hfill \end{array}\right],C_1=\left[\begin{array}{cc}1\hfill & 0.1\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{A_2}& =\left[\begin{array}{cc}1\hfill & 1.5\hfill \\ \multicolumn{1}{c}{1}& -0.7\hfill \end{array}\right],B_2=\left[\begin{array}{c}1.2\hfill \\ \multicolumn{1}{c}{1}\end{array}\right],G_2=\left[\begin{array}{c}0.2\hfill \\ \multicolumn{1}{c}{-0.15}\end{array}\right],\hfill \\ \multicolumn{1}{c}{L_2}& =\left[\begin{array}{cc}0.1\hfill & 0.2\hfill \end{array}\right],C_2=\left[\begin{array}{cc}0.8\hfill & 0.2\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{\mathrm{\Lambda }}& =\left[\begin{array}{cc}-0.5\hfill & 0.5\hfill \\ \multicolumn{1}{c}{0.3}& -0.3\hfill \end{array}\right],\mathrm{\Pi }=\left[\begin{array}{cc}0.6\hfill & 0.4\hfill \\ \multicolumn{1}{c}{0.3}& 0.7\hfill \end{array}\right].\hfill \end{array}$$

Figure 1.

Trajectories of the open-loop system of example 1.

Figure 2.

Trajectories of the closed-loop system of example 1.

Figure 3.

System modes of example 1.

Figure 4.

Controller modes of example 1.

Assume that $d_{11}=d_{21}=2,d_{12}=d_{22}=-2,$ and the disturbance $\omega (t)=e^{-0.5t}.$ By Theorem 2, the DOFC is given as

$$\begin{array}{cc}\multicolumn{1}{c}{A_{1c}}& =\left[\begin{array}{cc}-1.8685\hfill & -0.36478\hfill \\ \multicolumn{1}{c}{-0.18947}& -1.1883\hfill \end{array}\right],B_{c1}=\left[\begin{array}{c}-0.12017\hfill \\ \multicolumn{1}{c}{-0.87036}\end{array}\right],\hfill \\ \multicolumn{1}{c}{A_{c2}}& =\left[\begin{array}{cc}-1.8968\hfill & -0.41808\hfill \\ \multicolumn{1}{c}{-0.41345}& -1.2698\hfill \end{array}\right],B_{c2}=\left[\begin{array}{c}0.09061\hfill \\ \multicolumn{1}{c}{-0.8834}\end{array}\right],\hfill \\ \multicolumn{1}{c}{C_{c1}}& =\left[\begin{array}{cc}3.0214\hfill & 1.2979\hfill \end{array}\right],D_{c1}=-2.9971,\hfill \\ \multicolumn{1}{c}{C_{c2}}& =\left[\begin{array}{cc}1.5275\hfill & 0.36894\hfill \end{array}\right],D_{c2}=-2.9769.\hfill \end{array}$$

Supposes $\overline{\eta }(0)=[20.520.5{]}^T.$ Figures 5–7 show the simulation results. Figure 5 depicts the trajectories of the closed-loop system. Figures 6 and 7 show the modes of the system and the controller, respectively. These results illustrate that the proposed approach is effective.

Figure 5.

Trajectories of the closed-loop system of example 2.

Figure 6.

System modes of example 2.

Figure 7.

Controller modes of example 2.

Conclusions

In this paper, the problem of DOF asynchronous control has been investigated for continuous-time SMJSs. A HMM is applied to characterize the non-synchronization between the system and the DOFC. Some new conditions are given which guarantee the stochastic admissibility as well as the desired performance of the system. Moreover, a method is established to design the DOFC gains. At last, the effectiveness of the proposed method is demonstrated by two examples. In the future, we will further discuss the problem of DOF asynchronous control for semi-SMJSs with actuator saturation.