Dissipativity-based asynchronous control for discrete-time singular Markov jump systems with multiplicative noises

Abstract

The singular systems, which could widely describe more general systems and present traits of physical features, are discussed in this study. Taking the fact that noises always exist in the state and output measurement of one singular system into consideration, which may cause some errors and decrease system performance, this article devotes itself to the dissipative control for discrete-time singular Markov jump systems (SMJSs) with multiplicative noises. To deal with the asynchronous phenomena between the system modes and the controller modes, a set of Markov chains are constructed. To make sure the closed-loop singular system is dissipative, a set of sufficient conditions are derived based on the linear matrix inequalities, and then the asynchronous controller is designed to ensure that SMJSs are stochastically admissible and strictly dissipative. Finally, a simulation example is carried out to verify the correctness of the derived theorem. The designed asynchronous controller improves the robustness of the controller and overcomes the asynchronous phenomenon. This control method can be applied in the fields of robot control system.

Introduction

Many dynamic systems subjected to abrupt change of environment and failure of system following the rule of Markov jump process, containing stochastic process and Markov chain-based constrained modes, are called Markov jump systems, which can be used to describe real systems with sudden change or modes switching among structure and parameters. According to the characteristics of the systems, Markov jump systems can be divided into two types including singular ones and nonsingular ones. Recently, the study of nonsingular ones is maturing, ^{1 –5} and the singular ones have been an attractive research topic for its wide application on diverse aspects about robot systems, control theory, fault tolerance, and nonlinear systems.

Singular systems, which also called descriptor systems, semistate systems or differential-algebraic systems, possess stronger capacity of maintaining system physical characteristics than nonsingular systems. Up to now, there are some related reports about singular systems, ^{6 –8} in which the detailed features of singular systems are presented. After analysis and summary, it is observed that the main and vital problem for singular systems is the stability, which is more complex than nonsingular systems due to the fact that the regularity and impulse elimination of the system need to be dealt. Thus, some researchers study the stability and admissibility of singular Markov jump system (SMJS). ^{9 –12} For SMJSs, the problem of regularity and stability is analyzed by Chvez-Fuentes et al. ⁹ In the study by Zhang et al., ¹⁰ the slide mode control method is adopted to study the SMJSs. The admissibility performance is investigated by Gao et al. ¹¹ for discrete-time SMJSs. For SMJSs with time-varying switchings, the problem of stabilization is researched by Wang et al. ¹² In the study by Maand Zhang, ¹³ the $H_{\infty }$ control theory is introduced to study the SMJSs and analyze the stability. The method of passivity-based control theory have been put forward to design controller for Markov jump systems and SMJSs. ^14,15 So far, many researchers mainly focus on passivity and $H_{\infty }$ control for Markov jump systems with time-delay, multiplicative noises, packet dropouts, and so on.

Dissipative theory is a systematic theoretical concept based on generalized system energy, which provides a more useful method for the stability analysis and controller design of dynamical system. Up to now, it has been applied in many fields such as control theory and system theory. With the consideration that the dissipative control is more representative, whose performance takes account of the advantages of both $H_{\infty }$ and passivity theory. Therefore, the dissipative control theory is excepted to be adopted to the SMJSs. The dissipative theory is firstly introduced, ^16,17 in which the basic notions of dissipativeness are presented. Meanwhile, how to utilize the theory to analyze the stability of linear system is also discussed. Subsequently, the relationships among passivity, positive realness, and dissipativeness for linear systems are analyzed and presented by Kottenstette et al. ¹⁸ Due to the remarkable performance of the dissipative theory, it has been effectively applied in a large number of subjects, including control engineering, circuit, network, and so on. Especially there are many valuable results in designing an output feedback controller and state feedback controller. ^{19 –22} In the study by Chen et al., ²³ the finite-time dissipative controller has been designed for stochastic interval systems with time-delay and Markovian switching. In the study by Zhang et al., ²⁴ resilient dissipative dynamic output feedback controller is considered for uncertain Markov jump Lur’e systems. The reliable dissipative control problem for Markov jump systems based on an event-triggered sampling information scheme is studied by Shen et al. ²⁵ However, most papers are based on a common assumption that modes of the controller/filter are completely synchronous to the modes of systems.

In fact, the asynchronous phenomena between the system modes and the controller modes commonly exist in practical systems. For example, asynchronous phenomena usually occur in networked control systems, when the system mode could not follow the controller mode for the limited transmission capacity of the network systems. Therefore, the study of the issue of asynchronous control seems much more useful and meaningful. The issue of energy-to-peak state estimation for Markov jump RNNs was investigated by Zhang et al. ²⁶ Zhanget al. ²⁷ studied the asynchronous filtering problem of discrete-time switched linear systems with average dwell time. In the study by Wu et al., ²⁸ the problem of asynchronous $l_2-l_{\infty }$ filtering for discrete-time stochastic Markov jump systems was considered. In the study by Zhanget al., ²⁹ the synchronization and state estimation problem has been considered for hierarchical hybrid neural networks with time-varying delays. Finite-time asynchronous $H_{\infty }$ control problem for Markov jump systems was studied by Chen et al. ³⁰

The parameter uncertainties are existing in many practical systems, such as fuzzy systems and nonlinear systems, which may affect the stability of systems. In this article, we consider the multiplicative noises as the source of the modeling uncertainties. However, research on dissipative control of singular systems with asynchronous phenomenon are scarcely reported. Therefore, it is vital and significative for us to design an asynchronous controller for discrete-time SMJSs with multiplicative noises. A little contribution is provided on dealing this issue, which motivates to carry forward this work. Nowadays, many control strategies are applied in robot systems, ^{31 –33} the controller designed in this article could be used in various of systems such as robot control, traffic control, and aircraft.

In this article, the asynchronous dissipative control for discrete-time SMJSs with multiplicative noises is investigated. Most researchers have studied the nonsingular systems, without loss of generality, the singular system will be investigated in this article. To deal with the asynchronous phenomenon, two correlated Markov chains are constructed to describe the asynchronous features between the system modes and the controller modes. Asynchronous controller is designed to ensure that the resulting closed-loop system is stochastically admissible and strictly dissipative. The main contributions of this article are as follows:

The designed asynchronous controller is more representative than the synchronous one for the controller works asynchronously with the system mode.

The singular systems is considered and dissipative control theory, which could reduce to $H_{\infty }$ performance and passivity performance is adopted to address the issue.

The asynchronous dissipative state feedback controller is designed by given certain values for Q, S, and R and the simulation results illustrate the correctness of the derived dissipative conditions.

The rest of the article is organized as follows. The primary problems and some main Lemmas are stated in second section. The main results are derived in the third section, which provides sufficient conditions for stochastically admissible and strictly dissipative conditions. In the fourth section, an example is set to demonstrate the availability of the derived theorems. Conclusions are derived in the fifth section.

Notation

Throughout this article, the notations are standard. ${ℝ}^n$ represents the n-dimensional Euclidean space. ${ℝ}^{n\times n}$ means $n\times n$ real matrices. Block-diagonal matrices are denoted by $\text{diag}\left\{\mathrm{...}\right\}$ . The superscript “T” stands for the transpose. $\left(\Lambda ,\mathcal{F},\mathcal{P}\right)$ denotes a probability space. The notation $||\cdot ||$ denotes the Euclidean vector norm. $X>0$ means that the matrix is symmetric positive-definite. An identity matrix with appropriate dimensions is denoted by I. The symbol asterisk $(\ast )$ represents a matrix induced by the symmetry. $\mathbb{E}\{\cdot \}$ denotes the mathematical expectation. The notation $l_2\left[0,\infty \right]$ which is a bounded set $\left[0,\infty \right)$ represents square summable sequences. $\text{cov}\left(W\right)$ is the covariance matrix of W.

Problem formulation

Considering the problem of designing an asynchronous controller for discrete-time SMJs with multiplicative noises, the state space equation is described as

$\{\begin{array}{c}Ex\left(k+1\right)=A_{\theta \left(k\right)}x\left(k\right)+B_{\theta \left(k\right)}u\left(k\right)+C_{\theta \left(k\right)}v\left(k\right)\\ y\left(k\right)=D_{\theta \left(k\right)}x\left(k\right)+F_{\theta \left(k\right)}u\left(k\right)+H_{\theta \left(k\right)}v\left(k\right)\end{array}$

1

where $A_{\theta \left(k\right)}=A_{\theta \left(k\right).0}+A_{\theta \left(k\right).1}{\omega }^x\left(k\right)$ , $D_{\theta \left(k\right)}=D_{\theta \left(k\right).0}+D_{\theta \left(k\right).1}{\omega }^y\left(k\right)$ , and $x\left(k\right)\in {ℝ}^n$ is the system state vector, $u\left(k\right)\in {ℝ}^m$ is the control input, $y\left(k\right)\in {ℝ}^q$ is the system output, the disturbance signal $v\left(k\right)\in {ℝ}^p$ is input vector that belongs to $l_2\left[0,\infty \right]$ . The matrix $E\in {ℝ}^{n\times n}$ is singular with $\text{rank}\left(E\right)<n$ . ${\omega }^x\left(k\right)$ and ${\omega }^y\left(k\right)$ represent the state multiplicative noises and measurement multiplicative noises, respectively. The matrices with compatible dimensions $A_{\theta \left(k\right).0},A_{\theta \left(k\right).1},D_{\theta \left(k\right).0},D_{\theta \left(k\right).1},B_{\theta \left(k\right)},C_{\theta \left(k\right)},F_{\theta \left(k\right)},H_{\theta \left(k\right)}$ are constant and real.

Assumption 1

The state- and measurement-multiplicative noises are zero-mean sequences of independent random variables with variance equal to 1.

$\mathbb{E}\left\{{\omega }^x\left(k\right){\omega }^y\left(k\right)\right\}=\rho$ with $\rho \in \left[-1,1\right]$ , which means that ${\omega }^{\text{x}}\left(\text{k}\right)$ $\text{and}$ ${\omega }^{\text{y}}\left(\text{k}\right)$ are correlated.

$\{v(k)\}$ is a zero-mean sequence of independent random vectors with $\mathbb{E}\{v(k)v(k{)}^{\prime }\}=1.$

$v\left(k\right)$ is independent of $x_0,{\omega }^x\left(k\right),{\omega }^y\left(k\right)$ , and $\theta \left(k\right)$ .

The set $\mathcal{N}=\left\{1,2,\mathrm{...},N\right\}$ is finite and positive. The parameter $\left\{\theta \left(k\right),k\in Z^{+}\right\}$ represents a Markov chain, which takes value in the set $\mathcal{N}$ Hence, the system mode transition probability matrix $\Omega =\left\{{\pi }_{ij}\right\}$ can be defined by

$\text{Pr}\left\{\theta \left(k+1\right)=j|\theta \left(k\right)=i\right\}={\pi }_{ij}$

2

for all $i\in \mathcal{N},$ $j\in \mathcal{N},$ satisfying $0\le {\pi }_{ij}\le 1$ and ${\displaystyle {\sum }_{j=1}^N}{\pi }_{ij}=1$ . We will set a $\left(\theta \left(k\right),\delta \left(k\right),\Omega ,\Pi \right)$ to construct a correlated Markov model to show the basic features of the asynchronous phenomenon between the system modes and the controller modes in this article.

The finite set $\mathcal{M}=\left\{1,2,\mathrm{...},M\right\}$ is introduced, the parameter $\left\{\delta \left(k\right),k\in Z^{+}\right\}$ stands for a Markov chain, which takes value in the set $\mathcal{M}$ Therefore, the controller mode transition probability matrix ${\Pi }^{\theta \left(k+1\right)}=\left\{{\chi }_{\text{pq}}^{\theta \left(k+1\right)}\right\}$ can be defined by

$\text{Pr}\left\{\delta \left(k+1\right)=q|\delta \left(k\right)=p\right\}={\chi }_{\text{pq}}^{\theta \left(k+1\right)}$

3

for all $p,q\in \mathcal{M}$ , satisfying $0\le {\chi }_{\text{pq}}^{\theta \left(k+1\right)}\le 1$ and ${\displaystyle {\sum }_{q=1}^M}{\chi }_{\text{pq}}^{\theta \left(k+1\right)}=1$ . Assuming the Markov chain $\theta \left(k\right)$ is independent on ${\Im }_{k-1}=\sigma \left\{\delta \left(1\right),\delta \left(2\right)\mathrm{,...},\delta \left(k-1\right)\right\}$ , where ${\Im }_{k-1}$ is a $\sigma$ -algebra generated by $\left\{\delta \left(1\right),\delta \left(2\right)\mathrm{,...},\delta \left(k-1\right)\right\}$ .

Remark 1

From the definition, ^34,35 it can be obtained that for the same $\theta \left(k+1\right)$ , the transition ${\Pi }^{\theta \left(k+1\right)}$ is time-varying but invariant. The chain $\delta \left(k\right)$ is nonhomogeneous but finite piecewise homogeneous, and the Markov chain $\theta \left(k\right)$ is homogeneous. Hence, the investigated system is a kind of finite piecewise homogeneous Markov jump system.

For such a system, an asynchronous controller is designed as follows:

$u\left(k\right)=K\left(\delta \left(k\right)\right)x\left(k\right)$

4

Remark 2

Obviously, when the set $\mathcal{M}=\left\{1\right\}$ , which means the controller has one mode. The designed asynchronous controller degenerated to the mode-independent controller. Especially when $\mathcal{N}=\left\{1\right\}$ , the controller is synchronous, which means that the asynchronous controller contains both the mode-independent controller and the mode-dependent controller. In practice, system modes from the plant cannot be always synchronous with modes received by controller. Thus, the asynchronous controller is designed in this article.

From system (1), we can get that the noises existing in the state and measurement vectors are one-dimensional, so a linear transformation approach is used to deal with those noises.

Lemma 1

From the literature, ³⁴ considering a random vector L, the linear transformation method is introduced as follows. Given a random n-vector T of the following form

$T=HL+e$

5

where e is a n-vector and H is a $n\times p$ matrix. If any random n-vector T is finite variance, for some $p\le n$ , there exist a vector b and a random p-vector L so that $\text{cov}\left(L\right)=I_p$ , then the above equality holds.

By Lemma 1, it is supposed that $\varpi \left(k\right)$ is a two-dimensional random vector of null-mean and $\text{cov}\left(\varpi \left(k\right)\right)=1$ , the following equation holds

$\left[\begin{array}{c}{\omega }^x\left(k\right)\\ {\omega }^y\left(k\right)\end{array}\right]=\left[\begin{array}{c}{\Gamma }^x\\ {\Gamma }^y\end{array}\right]\varpi \left(k\right)$

6

where $\varpi \left(k\right)={\left[\begin{array}{cc}{\varpi }_1\left(k\right)& {\varpi }_2\left(k\right)\end{array}\right]}^T$ , ${\Gamma }^x=\left[\begin{array}{cc}{\gamma }_{11}^x& {\gamma }_{12}^x\end{array}\right]$ , and ${\Gamma }^y=\left[\begin{array}{cc}{\gamma }_{11}^y& {\gamma }_{12}^y\end{array}\right]$ .

Through the linear transformation, the following result is derived

$\{\begin{array}{c}{A}_{\theta \left(k\right).1}{\omega }^x\left(k\right)={\displaystyle {\sum }_{s=1}^2{\overline{A}}_{\theta \left(k\right)\mathrm{.}s}{\varpi }_s\left(k\right)}\\ D_{\theta \left(k\right).1}{\omega }^y\left(k\right)={\displaystyle {\sum }_{s=1}^2{\overline{D}}_{\theta \left(k\right)\mathrm{.}s}{\varpi }_s\left(k\right)}\end{array}$

7

where ${\overline{A}}_{\theta \left(k\right)\mathrm{.}s}=A_{\theta \left(k\right).1}{\gamma }_{1s}^x$ and ${\overline{D}}_{\theta \left(k\right)\mathrm{.}s}=D_{\theta \left(k\right).1}{\gamma }_{1s}^y$ .

To study the performance of stochastically admissible and strictly dissipative, the closed-loop system can be rewritten as

$\{\begin{array}{c}Ex\left(k+1\right)=\left\{A_{\theta \left(k\right).0}+{\displaystyle {\sum }_{s=1}^2{\overline{A}}_{\theta \left(k\right)\mathrm{.}s}{\varpi }_s\left(k\right)}\right\}x\left(k\right)\\ +B_{\theta \left(k\right)}u\left(k\right)+C_{\theta \left(k\right)}v\left(k\right)\\ y\left(k\right)=\left\{D_{\theta \left(k\right).0}+{\displaystyle {\sum }_{s=1}^2{\overline{D}}_{\theta \left(k\right)\mathrm{.}s}{\varpi }_s\left(k\right)}\right\}x\left(k\right)\\ +F_{\theta \left(k\right)}u\left(k\right)+H_{\theta \left(k\right)}v\left(k\right)\end{array}$

8

For notational simplicity, replacing $\theta \left(k\right)$ by i and the nominal system of equation (1) can be formulated as

$Ex\left(k+1\right)=A_{i.0}x\left(k\right)$

9

Definition 1

System (9) is regarded as regular, for any $i\in \mathbb{N},$ $s\in ℝ,$ $\text{det}\left(sE-A_{i\mathrm{,0}}\right)$ is not identically zero.

System (9) is regarded as causal, for any $i\in \mathbb{N},$ $s\in ℝ,$ $\text{deg}\left(\text{det}\left(sE-A_{i.0}\right)\right)=\text{rank}\left(E\right)$ .

System (9) is called stochastically stable, if there exists a positive scalar $Ϝ\left(x_{0,}{\theta }_0\right),$ such that

$\mathbb{E}\left\{{\displaystyle \sum _{k=0}^{\infty }}||x\left(k\right){||}^2|x_{0,}{\theta }_0\right\}\le Ϝ\left(x_{0,}{\theta }_0\right)$

10

where $x\left(k\right)$ is the solution of the system (7) at time k with initial values $x_{0,}{\theta }_0$ .

System (9) is regarded as stochastically admissible, if the conditions 1–3 are satisfied.

To ensure that the discrete-time SMJs is strictly dissipative, the dissipative control theory is introduced as follows. Defining the energy supply function:

$\aleph \left(v,y,T\right)={〈y,Qy〉}_T+2{〈y,Sv〉}_T+{〈v,Rv〉}_T$

11

where ${〈a,b〉}_T={\displaystyle {\sum }_{k=0}^T}{a}^T\left(k\right)b\left(k\right)$ . The real matrices Q, R, and S are given with proper dimensions, where Q is negative semidefinite matrix and R and S are symmetric matrices .

For any $T\ge 0$ , if there exists a positive scalar α, the following inequality holds

$\aleph \left(v,y,T\right)\ge \alpha {〈v,v〉}_T$

12

system (8) under zero initial state condition with $u\left(k\right)=0$ is called strictly dissipative.

Lemma 2

(Schur Complement) The following symmetric and constant matrices are given

$N=\left[\begin{array}{cc}{N}_{11}& N_{12}\\ N_{21}& N_{22}\end{array}\right]$

13

where $N_{11}=N_{11}^T$ , $N_{\text{22}}\text{=}{N}_{\text{22}}^{\text{T}}$ , and $N_{21}=N_{12}^T$ , then the following conditions are equivalent:

$N<0$ ;

$N_{11}<0$ , $N_{22}-N_{12}^T{N}_{11}^{-1}{N}_{12}<0$ ;

$N_{22}<0$ , $N_{11}-N_{12}{N}_{22}^{-1}{N}_{12}^T<0$ .

Lemma 3

From the literature, ³⁶ the matrices P, U, and V are known with compatible dimensions and $U=U^T$ . If there exist a positive scalar λ, satisfying $\lambda I+U>0$ and $-P^{T}V-V^{T}P-V^{T}UV\le P^T{(\lambda I+U)}^{-1}P+\lambda V^{T}V$ When $U>0,$ then λ could be taken as 0.

Main results

In this subsection, the two theorems are derived through dissipative control theory and linear matrix inequalities (LMIs) technique. In theorem 1, the sufficient condition of open-loop system with zero initial status is obtained. Then in theorem 2, an asynchronous state feedback controller is designed and the sufficient condition of closed-loop is derived to ensure the SMJs with multiplicative noises are stochastically admissible and strictly dissipative.

Theorem 1

Giving the real matrices Q, S, and R, where Q is negative semidefinite matrix and R, S are symmetric matrices. System (8) under zero initial status with $u\left(k\right)=0$ is regarded as stochastically admissible and strictly dissipative, if there exists a positive scalar $\alpha >0$ and symmetric matrices $P_{ip}>0,$ the following LMIs hold

${\Psi }_1=\left[\begin{array}{cccc}{\Psi }_{11}& {\Psi }_{12}& {\Psi }_{13}& {\Psi }_{14}\\ \ast & {\Psi }_{22}& {\Psi }_{23}& 0\\ \ast & \ast & {\Psi }_{33}& 0\\ \ast & \ast & \ast & {\Psi }_{44}\end{array}\right]<0$

14

where

$\begin{array}{l}{\Psi }_{11}=A_{i.0}^T{\chi }_{ip}{A}_{i.0}+{\overline{A}}_{i\mathrm{.}s}^T{\chi }_{ip}{\overline{A}}_{i\mathrm{.}s}-E^T{P}_{ip}E\\ {\Psi }_{12}=A_{i.0}^T{\chi }_{ip}{C}_i-D_{i\mathrm{.}o}^{T}S,{\Psi }_{13}=D_{i.0}^{T}Q\\ {\Psi }_{14}={\overline{D}}_{i\mathrm{.}s}^{T}Q,{\Psi }_{23}=H_i^{T}Q\\ {\Psi }_{22}=C_i^T{\chi }_{ip}{C}_i-2H_i^{T}S-R+\alpha I\\ {\Psi }_{33}=Q,{\Psi }_{44}=Q\mathrm{.}\end{array}$

with ${\chi }_{ip}={\displaystyle {\sum }_{j=1}^N}{\displaystyle {\sum }_{q=1}^M}{\pi }_{ij}{\chi }_{pq}^j{P}_{jq}$ , ${\overline{A}}_{i\mathrm{.}s}^T={\overline{A}}_{i.1}^T+{\overline{A}}_{i.2}^T$ , and ${\overline{D}}_{i\mathrm{.}s}^T={\overline{D}}_{i.1}^T+{\overline{D}}_{i.2}^T$ .

Proof

Firstly, the regular and casual of the closed-loop system should be proved. We know that ${\Psi }_1<0,$ for system (8)

${\overline{\Psi }}_{11}={\Psi }_{11}-D_{i.0}^{T}Q{D}_{i.0}-{\overline{D}}_{i\mathrm{.}s}^{T}Q{\overline{D}}_{i\mathrm{.}s}$

15

and ${\overline{\Psi }}_{11}<0$ . For $\text{rank}(E)=r\le n$ , there exists two nonsingular $M,N\in {ℝ}^{n\times n}$ matrices such that

$\begin{array}{c}\text{MEN}=\left[\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right],M{A}_{i\mathrm{.}p}N=\left[\begin{array}{cc}{A}_{i.0}^1& A_{i.0}^2\\ A_{i.0}^3& A_{i.0}^4\end{array}\right]\\ N^{-T}{P}_{ip}{N}^{-1}=\left[\begin{array}{cc}{P}_{ip}^1& P_{ip}^2\\ P_{ip}^3& P_{ip}^4\end{array}\right],R{N}^{-1}=\left[\begin{array}{cc}{R}_1& R_2\end{array}\right]\end{array}$

Giving positive definite matrix $P_{ip}$ and $E^{T}R=0.$ Premultiplying and postmultiplying (15) by $M^T$ and $M,$ we can obtain

$\left[\begin{array}{cc}\ast & \ast \\ \ast & {(A_{i.0}^4)}^T{R}_2^T{\chi }_{ip}{R}_2{A}_{i.0}^4\end{array}\right]<0$

16

where ∗ means that the matrix elements that we do not need. Obviously, ${(A_{i.0}^4)}^T{R}_2^T{\chi }_{ip}{R}_2{A}_{i.0}^4<0$ is satisfied and $A_{i.0}^4$ is nonsingular. The regular and casual of the closed-loop system is proved.

Considering the following Lyapunov function of system (8) with $u\left(k\right)=0$

$V\left(k,x\left(k\right),\theta \left(k\right),\delta \left(k\right)\right)=x^T\left(k\right)E^T{P}_{\theta \left(k\right)\delta \left(k\right)}Ex\left(k\right)$

17

Let

$\begin{array}{l}\mathbb{E}\left(\Delta V\left(k\right)\right)\\ =\mathbb{E}[V(k+1,x\left(k+1\right),\theta \left(k+1\right),\delta \left(k+1\right)|(k,x(k),\\ \theta \left(k\right)=i,\delta \left(k\right)=p)]-\mathbb{E}\left[V\left(k,x\left(k\right),\theta \left(k\right),\delta \left(k\right)\right)\right]\end{array}$

then

$\begin{array}{l}\mathbb{E}\left(\Delta V\left(k\right)\right)=\mathbb{E}{x}^T\left(k+1\right)E^T{P}_{\theta \left(k+1\right)\delta \left(k+1\right)}Ex\left(k+1\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-x^T\left(k\right)E^T{P}_{ip}Ex\left(k\right)\end{array}$

It can be found that

$\text{Pr}\left\{\theta \left(k+1\right)=j|\theta \left(k\right)=i,\delta \left(k\right)=p\right\}={\pi }_{ij}$

and

$\begin{array}{l}\hspace{1em}\text{Pr}\{\theta \left(k+1\right)=j,\delta \left(k+1\right)=q|\theta (k)=i\mathrm{.}\delta \left(k\right)=p\}\\ =\text{Pr}\{\delta \left(k+1\right)=q|\theta \left(k\right)=i,\delta \left(k\right)=p,\theta \left(k+1\right)=j\}\\ \hspace{1em}\times \text{Pr}\{\theta \left(k+1\right)=j|\theta \left(k\right)=i,\delta \left(k\right)=p\}\hspace{0.17em}\\ ={\pi }_{ij}{\chi }_{pq}^j\end{array}$

then

$\begin{array}{l}\hspace{1em}\mathbb{E}\left(\Delta V\left(k\right)\right)\\ =\mathbb{E}[x^T\left(k+1\right)E^T{P}_{\theta \left(k+1\right)\delta \left(k+1\right)}Ex\left(k+1\right)\\ \hspace{1em}-x^T\left(k\right)E^T{P}_{ip}Ex\left(k\right)]\\ =\mathbb{E}[(x^T\left(k+1\right)E^T{\chi }_{ip}Ex(k+1)\\ \hspace{1em}-x^T\left(k\right)E^T{P}_{ip}Ex\left(k\right)]\end{array}$

Considering the following performance index

$\begin{array}{l}J={\displaystyle \sum _{s=0}^{k-1}}\{\mathbb{E}(\alpha v^T\left(s\right)v\left(s\right)-y^T\left(s\right)Qy\left(s\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}-2y^T\left(s\right)Sv\left(s\right)-v^T\left(s\right)Rv\left(s\right))\}\\ \le {\displaystyle \sum _{s=0}^{k-1}}\{\mathbb{E}(\Delta V\left(s\right)+\alpha v^T\left(s\right)v\left(s\right)-y^T\left(s\right)Qy\left(s\right)\\ \hspace{1em}-2y^T\left(s\right)Sv\left(s\right)-v^T\left(s\right)Rv\left(s\right))\}\\ ={\displaystyle \sum _{s=0}^{k-1}}{\eta }^T\left(s\right)\Xi \eta \left(s\right)\end{array}$

18

where

$\Xi =\left[\begin{array}{cc}{\Xi }_{11}& {\Xi }_{12}\\ \ast & {\Xi }_{22}\end{array}\right]$

$\begin{array}{l}{\Xi }_{11}=A_{i.0}^T{\chi }_{ip}{A}_{i.0}+{\overline{A}}_{i\mathrm{.}s}^T{\chi }_{ip}{\overline{A}}_{i\mathrm{.}s}-E^T{P}_{ip}E\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-D_{i.0}^{T}Q{D}_{i.0}-{\overline{D}}_{i\mathrm{.}s}^{T}Q{\overline{D}}_{i\mathrm{.}s}\\ {\Xi }_{12}=A_{i.0}^T{\chi }_{ip}{C}_i-D_{i\mathrm{.}o}^{T}S-D_{i.0}^{T}Q{H}_i\\ {\Xi }_{22}=C_i^T{\chi }_{ip}{C}_i-2H_i^{T}S+\alpha I-R-H_i^{T}Q{H}_i\end{array}$

with $\eta \left(s\right)={\left[\begin{array}{cc}{x}^T\left(s\right)& v^T\left(s\right)\end{array}\right]}^T$ , ${\overline{A}}_{i\mathrm{.}s}^T={\overline{A}}_{i.1}^T+{\overline{A}}_{i.2}^T$ , and ${\overline{D}}_{i\mathrm{.}s}^T={\overline{D}}_{i.1}^T+{\overline{D}}_{i.2}^T$ .

According to lemma 2

$\begin{array}{l}\Xi =\left[\begin{array}{cc}{\Phi }_{11}& {\Phi }_{12}\\ \ast & {\Phi }_{22}\end{array}\right]-\left[\begin{array}{c}{\overline{D}}_{i\mathrm{.}s}^{T}Q\\ 0\end{array}\right]Q^{-1}\left[\begin{array}{cc}Q{\overline{D}}_{i\mathrm{.}s}& 0\end{array}\right]\\ \hspace{1em}\hspace{1em}-\left[\begin{array}{c}{D}_{i.0}^{T}Q\\ H_i^{T}Q\end{array}\right]Q^{-1}\left[\begin{array}{cc}Q{D}_{i.0}& Q{H}_i\end{array}\right]\end{array}$

19

where

$\begin{array}{c}{\Phi }_{11}=A_{i.0}^T{\chi }_{ip}{A}_{i.0}+{\overline{A}}_{i\mathrm{.}s}^T{\chi }_{ip}{\overline{A}}_{i\mathrm{.}s}-E^T{P}_{ip}E\\ {\Phi }_{12}=A_{i.0}^T{\chi }_{ip}{C}_i-D_{i\mathrm{.}o}^{T}S\\ {\Phi }_{22}=C_i^T{\chi }_{ip}{C}_i-2H_i^{T}S-R+\alpha I\end{array}$

According to equation (19), under the condition that $u\left(k\right)=0$ , we can get ${\Psi }_1<0$ which implies $\mathbb{J}<0$ .

Then the stochastically stable should be proved. We know that $\Delta V\left(k\right)-y^T\left(k\right)Qy\left(k\right)-2y^T\left(k\right)Sv\left(k\right)-v^T\left(k\right)Rv\left(k\right)<0$ and $Q<0$ . When $v(k)=0$ , $\Delta V\left(k\right)\le -c||x(k{)||}^2$ holds for $c>0$ . Therefore, the singular system (8) is stochastically stable. The LMIs (11) are derived. This completes the proof.

Finally, the asynchronous state feedback controller is designed as $u\left(k\right)=K\left(\delta \left(k\right)\right)x\left(k\right)$ . For notational simplicity, $u\left(k\right)$ can be rewritten as $u\left(k\right)=K_{p}x\left(k\right)$ . So the system is described as follows:

$\{\begin{array}{c}Ex\left(k+1\right)=\left\{A_{i.0}+B_i{K}_p+{\displaystyle {\sum }_{s=1}^2{\overline{A}}_{i\mathrm{.}s}{\varpi }_s\left(k\right)}\right\}x\left(k\right)\\ +C_{i}v\left(k\right)\\ y\left(k\right)=\left\{D_{i.0}+F_i{K}_p+{\displaystyle {\sum }_{s=1}^2{\overline{D}}_{i\mathrm{.}s}{\varpi }_s\left(k\right)}\right\}x\left(k\right)\\ +H_{i}v\left(k\right)\end{array}$

20

To investigate the stochastically admissible and strictly dissipative performance of closed-loop system (14), based on theorem 1, the following theorem can be obtained.

Theorem 2

Given the real matrices Q, S, and R, Q is negative semidefinite matrix and R and S are symmetric matrices. The asynchronous state feedback controller is designed as $u\left(k\right)=K_{p}x\left(k\right)$ . The closed-loop system under zero initial status is regarded as stochastically admissible and strictly dissipative, if there exist symmetric matrices ${\widehat{P}}_{ip}>0$ , $L_p$ , ${\overline{K}}_p$ and positive scalars $\alpha ,\beta >0$ , satisfying the following LMIs:

${\Theta }_2=\left[\begin{array}{ccccccc}{\Theta }_{11}& {\Theta }_{12}& {\Theta }_{13}& {\Theta }_{14}& {\Theta }_{15}& {\Theta }_{16}& {\Theta }_{17}\\ \ast & {\Theta }_{22}& {\Theta }_{23}& 0& {\Theta }_{25}& 0& 0\\ \ast & \ast & {\Theta }_{33}& 0& 0& 0& 0\\ \ast & \ast & \ast & {\Theta }_{44}& 0& 0& 0\\ \ast & \ast & \ast & \ast & {\Theta }_{55}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & {\Theta }_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\Theta }_{77}\end{array}\right]<0$

21

$\begin{array}{l}{\Theta }_{11}={\text{EL}}_p+L_p^T{E}^T+\beta I,\\ {\Theta }_{12}=-L_p^T{D}_{i\mathrm{.}o}^{T}S-{\overline{K}}_p^T{F}_i^{T}S\\ {\Theta }_{13}=L_p^T{D}_{i\mathrm{.}o}^{T}Q+{\overline{K}}_p^T{F}_i^{T}Q,{\Theta }_{14}=L_p^T{\overline{D}}_{i\mathrm{.}s}^{T}Q\\ {\Theta }_{15}=L_p^T{A}_{i.0}^T+{\overline{K}}_p^T{B}_i^T,{\Theta }_{16}=L_p^T{\overline{A}}_{i\mathrm{.}s}^T,{\Theta }_{17}=L_p\\ {\Theta }_{22}=-2H_i^{T}S-R+\alpha I,{\Theta }_{23}=H_i^{T}Q\\ {\Theta }_{25}=C_i^T,{\Theta }_{33}=Q,{\Theta }_{44}=Q,{\Theta }_{77}=-{\widehat{P}}_{ip}\\ {\Theta }_{55}={\widehat{\chi }}_{ip}-L_p-L_p^T,{\Theta }_{66}={\widehat{\chi }}_{ip}-L_p-L_p^T\end{array}$

where $\widehat{\chi }$ ${}_{ip}={\displaystyle {\sum }_{j=1}^N}{\displaystyle {\sum }_{q=1}^M}{\pi }_{ij}{\chi }_{pq}^j{\widehat{P}}_{jq}$ , ${\widehat{\text{P}}}_{ip}{\text{=L}}_p^T{P}_{ip}{L}_p$ , and $K_p={\overline{K}}_p{L}_p^{-1}\mathrm{.}$

Proof

We note $A_{i\mathrm{.}p}=A_{i.0}+B_i{K}_p$ and $D_{i\mathrm{.}p}=D_{i.0}+F_i{K}_p$ .

According to theorem 1, it is easy to get

$\Gamma =\left[\begin{array}{cccc}{\Gamma }_{11}& {\Gamma }_{12}& D_{i\mathrm{.}p}^{T}Q& {\overline{D}}_{i\mathrm{.}s}^{T}Q\\ \ast & {\Gamma }_{22}& H_i^{T}Q& 0\\ \ast & \ast & Q& 0\\ \ast & \ast & \ast & Q\end{array}\right]<0$

22

where ${\Gamma }_{11}=A_{i\mathrm{.}p}^T{\chi }_{ip}{A}_{i\mathrm{.}p}+{\overline{A}}_{i\mathrm{.}s}^T{\chi }_{ip}{\overline{A}}_{i\mathrm{.}s}-E^T{P}_{ip}E$ , ${\Gamma }_{12}=A_{i\mathrm{.}p}^T{\chi }_{ip}{C}_i-D_{i\mathrm{.}p}^{T}S$ , ${\Gamma }_{22}=C_i^T{\chi }_{ip}{C}_i-2H_i^{T}S+\alpha I-R\mathrm{.}$

By lemma 2, the following inequality is derived.

$\Phi =\left[\begin{array}{cccccc}{\Phi }_{11}& {\Phi }_{12}& {\Phi }_{13}& {\overline{D}}_{i\mathrm{.}s}^{T}Q& {\Phi }_{15}& {\overline{A}}_{i\mathrm{.}s}^T\\ \ast & {\Phi }_{22}& H_i^{T}Q& 0& C_i^T& 0\\ \ast & \ast & Q& 0& 0& 0\\ \ast & \ast & \ast & Q& 0& 0\\ \ast & \ast & \ast & \ast & {\Phi }_{55}& 0\\ \ast & \ast & \ast & \ast & \ast & {\Phi }_{66}\end{array}\right]<0$

23

where

$\begin{array}{l}{\Phi }_{11}=-E^T{P}_{ip}E,{\Phi }_{12}=-D_{i\mathrm{.}o}^{T}S-K_p^T{F}_i^{T}S\\ {\Phi }_{22}=-2H_i^{T}S-R+\alpha I,{\Phi }_{13}=D_{i\mathrm{.}o}^{T}Q+K_p^T{F}_i^{T}Q\\ {\Phi }_{15}=A_{i.0}^T+K_p^T{B}_i^T,{\Phi }_{55}={\Phi }_{66}=-{\chi }_{ip}^{-1}\mathrm{.}\end{array}$

By premultiplying and postmultiplying equation (23) via $\text{diag}\left\{L_p^T,I,I,I,I,I\right\}$ and its transpose, the following inequality is obtained

${\widehat{\Phi }}_1=\left[\begin{array}{cccccc}{\widehat{\Phi }}_{11}& {\Phi }_{12}& {\Phi }_{13}& {\Phi }_{14}& {\Phi }_{15}& {\overline{A}}_{i\mathrm{.}s}^T\\ \ast & {\Phi }_{22}& H_i^{T}Q& 0& C_i^T& 0\\ \ast & \ast & Q& 0& 0& 0\\ \ast & \ast & \ast & Q& 0& 0\\ \ast & \ast & \ast & \ast & {\Phi }_{55}& 0\\ \ast & \ast & \ast & \ast & \ast & {\Phi }_{66}\end{array}\right]<0$

24

$\begin{array}{l}{\widehat{\Phi }}_{11}=L_p^T{E}^T\mathrm{PipEL},{\Phi }_{12}=-L_p^T{D}_{i\mathrm{.}o}^{T}S-L_p^T{K}_p^T{F}_i^{T}S\\ {\Phi }_{13}=L_p^T{D}_{i\mathrm{.}o}^{T}Q+L_p^T{K}_p^T{F}_i^{T}Q,{\Phi }_{14}={\overline{D}}_{i\mathrm{.}s}^{T}Q\\ {\Phi }_{22}=-2H_i^{T}S-R+\alpha I,{\Phi }_{15}=L_p^T{A}_{i.0}^T+L_p^T{K}_p^T{B}_i^T\\ {\Phi }_{55}=-L_p{\widehat{\chi }}_{ip}^{-1}{L}_p^T,{\Phi }_{66}=-L_p{\widehat{\chi }}_{ip}^{-1}{L}_p^T.\end{array}$

As well, from $\left({\widehat{\chi }}_{ip}-L_p\right){\widehat{\chi }}_{ip}^{-1}\left({\widehat{\chi }}_{ip}-L_p^T\right)={\widehat{\chi }}_{ip}-L_p-L_p^T+L_p{\widehat{\chi }}_{ip}^{-1}{L}_p^T\ge 0$ , the following inequality can be obtained

$-L_p{\widehat{\chi }}_{ip}^{-1}{L}_p^T\le {\widehat{\chi }}_{ip}-L_p-L_p^T$

25

From lemma 3, there exists a scalar $\beta >0$ , $\beta I+P_{ip}>0$ .

$-L_p^T{E}^T{P}_{ip}E{L}_p\le L_p^T{E}^T+E{L}_p+P_{ip}{}^{-1}+\beta I$ . By Schur complement and equation (19), the following inequality holds

${\widehat{\Phi }}_2=\left[\begin{array}{ccccccc}{\stackrel{⌣}{\Phi }}_{11}& {\Phi }_{12}& {\Phi }_{13}& {\Phi }_{14}& {\Phi }_{15}& {\overline{A}}_{i\mathrm{.}s}^T& I\\ \ast & {\Phi }_{22}& H_i^{T}Q& 0& C_i^T& 0& 0\\ \ast & \ast & Q& 0& 0& 0& 0\\ \ast & \ast & \ast & Q& 0& 0& 0\\ \ast & \ast & \ast & \ast & {\stackrel{⌣}{\Phi }}_{55}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & {\stackrel{⌣}{\Phi }}_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\Phi }_{77}\end{array}\right]$

26

$\begin{array}{l}{\stackrel{⌣}{\Phi }}_{11}=L_p^T{E}^T+E{L}_p+\beta I,\\ {\Phi }_{12}=-L_p^T{D}_{i\mathrm{.}o}^{T}S-L_p^T{K}_p^T{F}_i^{T}S\\ {\Phi }_{13}=L_p^T{D}_{i\mathrm{.}o}^{T}Q+L_p^T{K}_p^T{F}_i^{T}Q,{\stackrel{⌣}{\Phi }}_{55}={\widehat{\chi }}_{ip}-L_p-L_p^T\\ {\Phi }_{22}=-2H_i^{T}S-R+\alpha I,{\Phi }_{15}=L_p^T{A}_{i.0}^T+L_p^T{K}_p^T{B}_i^T\\ {\stackrel{⌣}{\Phi }}_{66}={\widehat{\chi }}_{ip}-L_p-L_p^T,{\Phi }_{77}=-P_{ip}\mathrm{.}{\Phi }_{14}={\overline{D}}_{i\mathrm{.}s}^{T}Q\end{array}$

Premultiplying and postmultiplying equation (20) by $\text{diag}\{I,I,I,I,I,I,L_p^T\}$ and its transpose, respectively. Therefore, we can get that ${\Theta }_2<0$ , equation (21) is derived. The asynchronous controller gains can be calculated as $K_p={\overline{K}}_p{L}_p^{-1}$ . This completes the proof.

Remark 3

In theorem 2, the singular matrix E of closed-loop system can suddenly change with system modes. In practical plants, when system modes change, fast and slow subsystem of system always change. Such kind of model contains more general situations. Therefore, the singular system has more useful and general meaning in real applications.

An illustrative example

In this section, a simulation example is put to demonstrate the effectiveness of the derived theorems and designed asynchronous controller. Given the following parameters of the discrete-time SMJS as:

$\begin{array}{c}{A}_{\mathrm{1,0}}=\left[\begin{array}{ccc}0.18& 0.22& 0.45\\ 0.78& 0.37& 0.45\\ 0.14& 0.23& 0.56\end{array}\right],E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]\\ A_{\mathrm{1,1}}=\left[\begin{array}{ccc}0.16& 0.14& 0.25\\ 0.19& 0.46& 0.23\\ 0.18& 0.65& 0.66\end{array}\right],B_1=\left[\begin{array}{c}0.3\\ 0.7\\ -0.6\end{array}\right]\\ A_{\mathrm{2,1}}=\left[\begin{array}{ccc}0.58& 0.25& 0.17\\ 0.16& 0.37& 0.37\\ 0.58& 0.26& 0.96\end{array}\right],B_2=\left[\begin{array}{c}0.5\\ 0.4\\ -0.5\end{array}\right]\\ A_{\mathrm{2,0}}=\left[\begin{array}{ccc}0.24& 0.24& 0.63\\ 0.23& 0.56& 0.17\\ 0.04& 0.85& 0.36\end{array}\right],I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ D_{\mathrm{1,0}}=\left[\begin{array}{ccc}0.38& 0.23& 0.45\end{array}\right],C_1={\left[\begin{array}{ccc}1& 1& 1\end{array}\right]}^T\\ D_{\mathrm{1,1}}=\left[\begin{array}{ccc}0.38& 0.23& 0.45\end{array}\right],C_2={\left[\begin{array}{ccc}1& 0& 1\end{array}\right]}^T\\ D_{\mathrm{2,0}}=\left[\begin{array}{ccc}0.15& 0.32& 0.49\end{array}\right],H_1=0.1\\ D_{\mathrm{2,1}}=\left[\begin{array}{ccc}0.68& 0.92& 0.35\end{array}\right],H_2=0.2\\ F_1=\mathrm{0.4, }{F}_2=0.8,p=\mathrm{0.5.}\\ Q=-\mathrm{0.5, }R=2,S=4\end{array}$

Considering the state- and measurement-multiplicative noises are one-dimensional, equation (6) is seen as the following equation

$\left[\begin{array}{c}{\omega }^x\left(k\right)\\ {\omega }^y\left(k\right)\end{array}\right]=\left[\begin{array}{cc}\sqrt{1-p^2}& p\\ 0& 1\end{array}\right]\overline{\omega }\left(k\right)$

where $\overline{\omega }\left(k\right)={\left[\begin{array}{cc}{\overline{\omega }}_{1\left(k\right)}^T& {\overline{\omega }}_{2\left(k\right)}^T\end{array}\right]}^T$ , $\mathbb{E}\left({\omega }^x\left(k\right){\omega }^y\left(k\right)\right)=p\mathrm{.}$ By lemma 1 and equation (7), we obtain that ${\overline{A}}_{i\mathrm{,1}}=\sqrt{1-p^2}{A}_{i\mathrm{,1}},{\overline{A}}_{i\mathrm{,2}}=p{A}_{i\mathrm{,1}},{\overline{D}}_{i\mathrm{,1}}=0,{\overline{D}}_{i\mathrm{,2}}=D_{i\mathrm{,1}}$ . The disturbance is assumed to be $v\left(k\right)=0.5\text{exp}(-0.1k)$ . In this example, the transition matrix of Markov chain $\theta \left(k\right)$ is taken as $\Omega =\left[\begin{array}{cc}0.4& 0.6\\ 0.3& 0.7\end{array}\right]$ , and the transition probabilities matrices of Markov chain $\delta \left(k\right)$ are considered as

${\Pi }^1=\left[\begin{array}{cc}0.7& 0.3\\ 0.7& 0.3\end{array}\right],\hspace{0.17em}{\Pi }^2=\left[\begin{array}{cc}0.2& 0.8\\ 0.2& 0.8\end{array}\right]\hspace{0.17em}$

which implies that the asynchronous controller has two modes. The objective is to design an asynchronous controller for the above stated SMJS, such that the closed-loop system is stochastically admissible and strictly dissipative. By solving the LMIs feasibility problem in equation (21), the best value of t could be calculated as $t_{\text{min}}=-3.7909$ , and the dissipative performance index is $\mathrm{3.3232.}$ Hence, the derived LMIs is proved to be solvable. The controller parameters can be calculated as

$\begin{array}{l}{K}_1=\left[\begin{array}{ccc}-0.3889& -0.3991& -0.1247\end{array}\right]\\ K_2=\left[\begin{array}{ccc}-0.4330& -0.4598& -0.1193\end{array}\right]\end{array}$

Let the initial state of the system equals to be $x\left(0\right)={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$ . The Markov jumping conditions of system and controller modes are presented in Figure 1. The state- and measurement-multiplicative noises ${\omega }^x\left(k\right)$ and ${\omega }^y\left(k\right)$ are randomly generated as shown in Figure 2. Figure 3 shows the controlled output of the closed-loop system. The trajectories of the state are given in Figure 4. From Figures 3 and 4, the state and output responses are presented with the corresponding control input. The designed asynchronous controller ensures that the state responses tend to zero with time passing by. Our developed technique is proven effective by these results. For Markov jump systems, it can be applied in the single-link robot arm system. Because the mass of payload and the moment of inertia is vary while the system mode of Markov jump systems can describe such characteristics.

Figure 1.

The jumping process of Markov chains.

Figure 2.

Multiplicative noises ${\omega }^x\left(k\right)$ and ${\omega }^y\left(k\right)$ .

Figure 3.

The trajectory of the controlled output.

Figure 4.

The trajectories of the system states.

To demonstrate the effective and correctness of the designed asynchronous controller, the designed controller is applied to study the problem of positioning in robot domain. Figure 5 shows the robot arm used in our experiments which has six degrees of freedom. To make the positioning results more intuitive, the checkerboard is chosen to get the corner position.

In this experiment, the industrial camera is mounted on the flange at the end of the manipulator and the end piece is a mechanical device that can be precisely positioned.

On PC, according to the information extracted from the industrial cameras, the designed control algorithm combined with the localization algorithm can calculated the control quantity in real time for the manipulator. The motion of the manipulator is controlled by the upper computer program to complete the positioning experiment.

The location coordinates of the corners and the actual corners are extracted.

The actual corner coordinates and the location of the corner coordinate information are displayed in matlab.

Figure 5.

Six-axis manipulator.

In this experiment, eight corner points of known coordinates were selected with unit (mm). The designed controller algorithm is applied to the positioning experiment. The corresponding positioning results are shown in Figure 6, which exhibits the actual corner position coordinates and the location coordinates of corner position. From Figure 6, it is concluded that the designed asynchronous controller is effective and can be applied in robot domain.

Figure 6.

Positioning result.

Conclusions

The problem of asynchronous dissipative control for singular discrete-time Markov jump system with multiplicative noises is investigated by Lyapunov function approach together with LMIs technique in this article. The asynchronous jumpings between system and controller modes are concerned, and the constructed Markov model efficiently model such jumpings. To reflect more practical plant, the state and output multiplicative noises are introduced and a linear transformation technique is applied to deal with these multiplicative noises. To overcome the problem that the controller modes are asynchronous to the system modes, the asynchronous dissipative controller has been proposed to improve the robustness of the controller. The sufficient conditions are given which guarantee the closed-loop system is stochastically admissible and strictly dissipative. Meanwhile, an example is taken, and simulation results verify the correctness of the derived theorems and designed controller. The control method proposed in this article can be used in the area of robot control and traffic control. It should be pointed out that our future research topics would be asynchronous dissipative control for more complex systems. For example, time delays including random delay, randomly occurring unideal measurements, system with partial transition probabilities and so on. What’s more, the mainly direction of our research is to use the Markov jump systems with dissipative control theory to study the related issues of robot arm systems.