Output Feedback Robust H∞ Control for Discrete 2D Switched Systems

The two-dimensional (2-D) system has a wide range of applications in different fields, including satellite meteorological maps, process control, and digital filtering. Therefore, the research on the stability of 2-D systems is of great significance. Considering that multiple systems exist in switching and alternating work in the actual production process, but the system itself often has external perturbation and interference. To solve the above problems, this paper investigates the output feedback robust H∞ stabilization for a class of discrete-time 2-D switched systems, which the Roesser model with uncertainties represents. First, sufficient conditions for exponential stability are derived via the average dwell time method, when the system’s interference and external input are zero. Furthermore, in the case of introducing the external interference, the weighted robust H∞ disturbance attenuation performance of the underlying system is further analyzed. An output feedback controller is then proposed to guarantee that the resulting closed-loop system is exponentially stable and has a prescribed disturbance attenuation level γ. All theorems mentioned in the article will also be given in the form of linear matrix inequalities (LMI). Finally, a numerical example is given, which takes two uncertain values respectively and solves the output feedback controller’s parameters by the theorem proposed in the paper. According to the required controller parameter values, the validity of the theorem proposed in the article is compared and verified by simulation.

models such as the Roesser model and Fornasini-Marchesini model can represent 2-D systems, and the stability issues concerning these two models can be found [8,9].
On the other hand, considerable interest has been devoted to the research of switched systems during the recent decades. A switched system comprises a family of subsystems described by continuous or discretetime dynamics and a switching law that specifies the active subsystem at each instant of time. The switching strategy improves control performance [10][11][12][13] and arise many engineering applications, such as in motor engine control, constrained robotics, and satellite image control systems [14]. As far as timedependent switching is concerned, the average dwell time (ADT) switching is employed in most references owing to its flexibility [15,16].
However, perturbations and uncertainties widely exist in practical systems. In some cases, the perturbations can be merged into the disturbance, which can be bounded in the appropriate norms. The main advantage of robust H ∞ control is that its performance specification considers the system's worstcase performance in terms of energy gain. This is more appropriate for system robustness analysis and robust control under modeling disturbances than other performance specifications. Recently, the problems of robust H ∞ control and filtering for 2-D systems have been studied by many researchers [17][18][19][20], and so do the same problems of switched systems [21][22][23]. However, to the best of our knowledge, the output feedback robust H ∞ control problem of 2-D switched systems in the Roesser model with uncertainties has not yet been thoroughly investigated, which motivates this present study.
In this paper, we confine our attention to the robust H ∞ control problem of discrete 2-D switched systems described by the Roesser model with uncertainties. The main theoretical contributions are threefold: (1) We contribute to the development of stabilization for a class of 2-D switched systems that are exponentially stable, which the Roesser model with uncertainties represents. (2) A sufficient condition is presented to ensure a 2-D switched system's exponential stability at a given disturbance attenuation level robust weighted H ∞ . (3) Based on the above two points, this article further designs the output feedback controller of the closed-loop system. The paper is organized as follows. According to the current research results, we first study the exponential stability of 2-D switched systems described by the Roesser model with uncertainties. Further, when the system contains perturbations, we analyze the robust H ∞ performance index of the system. Finally, we design an output feedback controller for the open-loop system, and an example is given to illustrate the effectiveness of the proposed method.

Notation
The following notations are used throughout the paper: the superscript "T" denotes the transpose, and the notation X ≥ Y (X > Y) means that matrix X − Y is positive semi-definite (positive definite, respectively). Á k k denotes the Euclidean norm. I represents the identity matrix. Diag a i f g denotes a diagonal matrix with the diagonal elements a i , i = 1, 2,…,n. X -1 denotes the inverse of X. R n denotes the n dimensional vector. Z + represents the set of all non-negative integers. The l 2 norm of a 2-D signal w(i,j) is given by

Problem Formulation and Preliminaries
The uncertain Roesser model for a 2-D switched system G : u ! y is given by the following state equation: where G d k;l ð Þ k; l ð Þis an unknown uncertain matrix and satisfies the norm bounded condition G Td k;l ð Þ k; l ð ÞG d k;l ð Þ k; l ð Þ I, and where x h k; l ð Þ 2 R n 1 and x v k; l ð Þ 2 R n 2 denote the system horizontal state and vertical state, respectively. Furthermore, x k; l ð Þ is the whole state in R n with n ¼ n 1 þ n 2 , and w k; l ð Þ 2 R q is the interference input which belongs to w k; l ð Þ 2 l 2 0; 1 ½ Þ; 0; 1 ½ Þ f g . u k; l ð Þ 2 R p and z k; l ð Þ 2 R q are control input and control output respectively; k and l are integers in Z.
The boundary condition satisfies: It is easy to know from the above formula X 0 ð Þ k k 2 , 1.
Remark 1. "In this paper, it is assumed that switching occurs only at each sampling point of k or l. The switching sequence can be described as with k k ; l k ð Þdenoting the kth switching instantly. It should be noted that the value of d k; l ð Þ only depends on k+l [24,25].
Remark 2. From Definition 1, it is easy to see that when j is given, P kþl¼j x k; l ð Þ k k 2 r will be bounded, and P kþl¼D x k; l ð Þ k k 2 r will tend to be zero exponentially as D goes to infinity, which also means that x k; l ð Þ k kwill tend to be zero exponentially. (1) when w k; l ð Þ=0, system (1) is asymptotically stable or exponentially stable; (2) under the zero-boundary condition, we have where 0 <f< 1 and the l 2 -norm of 2D discrete signal z k; l ð Þ and w k; l ð Þ are defined as holds for given N 0 ! 0 and s a ! 0, then the constant s a is called the average dwell time and N 0 is the chatter bound [27].
, where S 11 and S 22 are square matrices, the following conditions are equivalent [28].
Assuming that x 2 R p ; y 2 R q and U ; V ; W are a suitable dimension matrix, then inequality Proof.

Exponential Stability Analysis
This section focuses on the exponential stability analysis of the 2D switched systems. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable. Theorem 1. Consider 2D discrete switched system (1) with w k; l ð Þ ¼ 0, for a given positive constant f < 1, if there exist a set of positive-definite symmetric matrices P i 2 R nÂn ; i 2 N and two positive scalars e 1 ; e 2 , such that where l ≥ 1 satisfies Proof. Without loss of generality, we assume that the ith subsystem is active. For the ith subsystem, we consider the following Lyapunov function candidate: where P i is an n Â n positive-definite matrix for any i 2 N, and thus Using Lemma 1 to (8), we can get the equivalent inequality as follows: Then using Lemma 2 to (13), we can get Using Lemma 1 to (14), we can get Form (15), we know The equality holds only if It follows from (16) Now, let n = N d j; D ð Þ denote the switching number of d Á ð Þ on an interval j; D ½ Þ, and let m kÀnþ1 , m kÀnþ2 , Á Á Á , m kÀ1 , m k denote the switching points of d Á ð Þ over the interval j; D ½ Þ, thus, for Using (11) and (12), at switching instant m k ¼ k þ l, we have Also, according to Definition 3, it follows that Therefore, the following inequality can be obtained easily: Combining (20), inequality (21) can be written as follows: There exist two positive constants a and b (a By Definition 1, we know that if À ln l s a À ln f >0, that is s a . s Ã a ¼ ln l À ln f , the 2D discrete switched system is exponentially stable.
The proof is completed.
Remark 3. Note that when l ¼ 1 in (10), (11) turns out to be P i ¼ P j ; 8i; j 2 N . In this case, we have s a . s Ã a ¼ ln l À ln f , which means that the switching signal can be arbitrary.

Robust H ∞ Performance Analysis
This section focuses on the Robust H ∞ stabilization problem for a class of discrete-time 2D switched systems represented by a Roesser model with uncertainties. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable and has a prescribed weighted H ∞ disturbance attenuation level c.
Theorem 2. For given positive scalars c and f < 1, there exist symmetric and positive-definite matrices P i 2 R nÂn ; i 2 N, and two positive scalars e 1 ; e 2 ,such that , 0 Then, 2D switched system (1) is exponentially stable and has a prescribed weighted H ∞ disturbance attenuation level c for any switching signals with average dwell time satisfying (10), where μ ≥ 1 satisfies (11).
Proof. It is an obvious fact that (24) implies that inequality (8) holds. By Lemma 2, we can find that system (1) is exponentially stable when w k; l ð Þ= 0. Now we are able to prove that system (1) has a prescribed weighted H ∞ performance γ for any nonzero w k; l ð Þ 2 l 2 0; 1 ½ Þ; 0; 1 ½ Þ f g .
To establish the weighted H ∞ performance, we choose the same Lyapunov functional candidate as in (12) for the system (1). Following the proof line of Theorem 1, we can get Using Lemma 1 to (26), we can get the equivalent inequality as follows: Pre-and post-multiplying (24) by diag I I P i ð Þ À1 I I I È É , we obtain , 0 Using Lemma 1 to (28), we can get Then using Lemma 2, we find that (29) is equivalent to (27).
Thus it can be obtained from (24) that Then we have Summing up both sides of (31) from (D-1) to 0 with respect to l and 0 to (D-1) with respect to k, respectively, and applying the zero-boundary condition, one gets Under the zero-initial condition, we have X Multiplying both sides of (35) by l ÀN d 0; D ð Þ , we can get the following inequality: Noting N d 0; k þ l ð Þ kþl s a , and using (10), we have Thus According to Definition 3, we can see that system (1) is exponentially stable and has a prescribed weighted robust H ∞ disturbance attenuation level γ.
The proof is completed.

Robust H ∞ Control Problem
This subsection will deal with the Robust H ∞ control problem of 2D switched systems via dynamic output feedback. Our purpose is to design a dynamic output feedback controller such that the closed-loop system is exponentially stable and has a specified robust weighted H ∞ disturbance attenuation level γ.
Consider the following discrete 2D switched systems in the Roesser model with uncertainties: where x k; l ð Þ 2 R n ; w k; l ð Þ 2 R n w ; u k; l ð Þ 2 R u ; z k; l ð Þ 2 R z and y k; l ð Þ 2 R y are, respectively, the state, the disturbance input, the control input, the controlled output, and the measurement output of the plant, k and l are integers in Z þ .
i and D 21 i with i 2 N are constant matrices with appropriate dimensions. We do not assume the disturbance input signal's statistics w k; l ð Þ other than that its energy is bounded, i.e., w k k 2 , 1.
Introduce the following output feedback controller of order n c : where The closed-loop system consisting of the plant (39) and the controller (40) is of the form For the closed-loop system (41), we state the 2D Robust H ∞ control problem as finding a 2D dynamic output feedback controller of the form in (40) for the 2D systems (39) such that the closed-loop system (41) has a specified weighted robust H ∞ disturbance attenuation level γ. The controller design procedure is provided in the following theorem.  , 0 Pre-and post-multiplying (47) by diag X i ð Þ À1 I X i ð Þ À1 I I 1 " e 1 I n o leads to , 0 (48)

Conclusions
This paper has investigated the problems of stability and weighted robust H 1 disturbance attenuation performance analyses for 2D discrete switched systems described by the Roesser model with uncertainties. An exponential stability criterion is obtained via the average dwell time method. Some sufficient conditions for the existence of weighted robust H 1 disturbance attenuation level c for the considered system are derived from LMIs. Besides, a 2D output feedback controller is designed to solve the robust H 1 control problem. Finally, an example is also given to illustrate the applicability of the proposed results. The future work will be associated with the following directions: 1) stability analysis and stabilization for nonlinear continuous-time descriptor semi-Markov jump systems; and 2) finite-time stabilization for nonlinear discrete-time singular Markov jump systems with piecewise-constant transition probabilities subjected to average dwell time.
Data Availability: The authors declare that they have no conflicts of interest to report regarding the present study.

Conflicts of Interest:
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.