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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 961870, 12 pages

http://dx.doi.org/10.1155/2013/961870

## Stabilization and Controller Design of 2D Discrete Switched Systems with State Delays under Asynchronous Switching

^{1}School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China^{2}Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 22 February 2013; Accepted 14 April 2013

Academic Editor: Jun Hu

Copyright © 2013 Shipei Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the problem of robust stabilization for a class of uncertain two-dimensional (2D) discrete switched systems with state delays under asynchronous switching. The asynchronous switching here means that the switching instants of the controller experience delays with respect to those of the system. The parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee the exponential stability. The dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.

#### 1. Introduction

Two-dimensional (2D) systems have attracted considerable attention for several decades due to their numerous applications in many areas, such as multidimensional digital filtering, linear image processing, signal processing and process control [1–3]. It is well known that 2D systems can be represented by different models such as Roesser model, Fornasini-Marchesini model, and Attasi model. The issues of stability analysis and control synthesis of these systems have been studied in [4–8]. Considering that time delays frequently occur in practical systems and are often the source of instability, many authors have devoted their energies to studying time-delay systems. Recently, many results on delay systems have been reported in the literature. For example, the delay-fractional approach has been utilized to deal with discrete time-delay systems in [9–11]. The stability of 2D discrete systems with state delays has been investigated in [12–16].

On the other hand, because of their wide applications in many fields, such as mechanical systems, automotive industry, aircraft and air traffic control, and switched power converters, switched systems have also received considerable attention during the past few decades. A switched system is a hybrid system which consists of a finite number of continuous-time or discrete-time subsystems and a switching signal specifying the switch between these subsystems. The stability and stabilization problems have been extensively studied in [17–25].

In many modelling problems of physical processes, a 2D switching representation is needed. One can cite a 2D physically based model for advanced power bipolar devices [26] and heat flux switching, and modulating in a thermal transistor [27]. This class of systems can correspond to 2D state space or 2D time space switched systems. Recently, there are a few reports on 2D switched systems. Benzaouia et al. firstly studied the stabilization problem of 2D discrete switched systems with arbitrary switching sequences in [28, 29]. By using the common Lyapunov function method and multiple Lyapunov functions method, two different sufficient conditions for the existence of state feedback controllers were proposed. In [30], the authors first extended the concept of average dwell time to 2D switched systems and designed a switching signal to guarantee the exponential stability of delay-free 2D switched systems. It should be pointed out that a very common assumption in [30] is that the controllers are switched synchronously with the switching of system modes, which is quite unpractical. As stated in [31, 32], there inevitably exists asynchronous switching in actual operation that is, the switching instants of the controllers exceed or lag behind those of system modes. Thus, it is necessary to consider asynchronous switching for efficient control design. Some results on the control synthesis for switched systems under asynchronous switching have been proposed in [33–36]. However, to the best of our knowledge, the stabilization problem for 2D switched systems under asynchronous switching has not been yet investigated to date, especially for 2D switched systems with state delays, which motivates our present study.

In this paper, we are interested in designing a stabilizing controller for 2D discrete switched delayed systems represented by a model of Roesser type under asynchronous switching such that the corresponding closed-loop systems are exponentially stable. The dwell time approach is utilized for the stability analysis and controller design. The main contributions of this paper can be summarized as follows: (i) the asynchronous stabilization problem is for the first time addressed in the paper; (ii) an exponential stability criterion is established for 2D switched systems with state delays; and (iii) an asynchronous switching controller design scheme is proposed to guarantee the exponential stability of the resulting closed-loop system.

This paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the dwell time approach, stability and stabilization for 2D discrete switched systems with state delays are addressed. Then, a sufficient condition for the existence of a stabilizing controller for such 2D discrete switched systems under asynchronous switching is derived in terms of a set of matrix inequalities. A numerical example is provided to illustrate the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

*Notations*. Throughout this paper, the superscript ‘‘’’ denotes the transpose, and the notation means that the matrix is positive semidefinite (positive definite, resp.). denotes the Euclidean norm. represents the identity matrix with an appropriate dimension. is the identity matrix with dimension and is the identity matrix with dimension. denotes the diagonal matrix with the diagonal elements , and . denotes the inverse of . The asterisk in a matrix is used to denote the term that is induced by symmetry. The set of all nonnegative integers is represented by .

#### 2. Problem Formulation and Preliminaries

Consider the following uncertain 2D discrete switched systems with state delays: where and are the horizontal state and the vertical state, respectively, is the whole state in with , and is the control input. and are integers in . is the switching signal. is the number of subsystems, and means that the th subsystem is activated. and are constant delays along horizontal and vertical directions, respectively. and () are uncertain real-valued matrices with appropriate dimensions and are assumed to be of the form with where matrices , , , , , , , , , , , , , , , are constant matrices. () is an unknown matrix representing parameter uncertainty and satisfies The boundary conditions are given by where and are positive integers, and and are given vectors.

In this paper, it is assumed that (1) at each time only one subsystem is active; (2) the switching signal is not known a priori, but its value is available at each sampling period; (3) the switch occurs only at each sampling point of or . The switching sequence can be described as follows: where denotes the th switching instant. It should be noted that the value of only depends on (see [29, 30]).

However, in actual operation, there inevitably exists asynchronous switching between the controller and the system. Without loss of generality, we only consider the case where the switching instants of the controller experience delays with respect to those of the system. Let denote the switching signal of the controller. Denoting , , , then the switching points of the controller can be described as where and represent the delayed period along horizontal and vertical directions, respectively. is said to be the mismatched period.

*Remark 1. *Similar to the one-dimensional switched system case [33–36], the mismatched period guarantees that there always exists a period in which the controller and the system operate synchronously. This period is said to be the matched period in the later section.

*Remark 2. *If there is only one subsystem in system (1), it will degenerate to the following 2D system in Roesser model with state delays [12]:

*Definition 3. *System (1) is said to be exponentially stable under if, for a given , there exist positive constants and , such that
holds for all , where

*Remark 4. *From Definition 3, it is easy to see that when is given, will be bounded and will tend to be zero exponentially as goes to infinity, which implies that tends to be zero.

*Definition 5. *Let denote the th switching point and denote the th switching point. Denote , , , then is called the dwell time.

*Definition 6 (see [30]). *For any , let denote the switching number of on the interval . If
holds for given and , then the constant is called the average dwell time and is the chatter bound.

Lemma 7 (see [37]). *For a given matrix , where and are square matrices, the following conditions are equivalent:*(i)*,*(ii)*, ,*(iii)*, .*

Lemma 8 (see [38]). *Let , , , and be real matrices of appropriate dimensions with satisfying , then for all ,
**
if and only if there exists a scalar such that
*

#### 3. Main results

##### 3.1. Stability Analysis

In this section, we first focus on the stability analysis for system (8).

Lemma 9. *Consider system (8) with the boundary conditions (5), suppose that there exists a function . For a given positive constant , if there exist positive definite symmetric matrices and with appropriate dimensions, such that the following inequality holds:
**
where , then along the trajectory of systems (8), the following inequality holds for any :
**
where and .*

*Proof. *See appendix for the detailed proof, it is omitted here.

*Remark 10. *Lemma 9 provides a method for the estimation of the function which will be used to design the controller for system (1) under asynchronous switching. It is worth pointing out that when , (15) presents the decay estimation of the function , and when , (15) shows the growth estimation of the function .

*Remark 11. *It is noted that the block diagonal matrices and are often chosen as the matrices for Lyapunov functional analysis of 2D systems by the Roesser model in the existing literature (see, e.g., [12, 13, 15]). This is because 2D systems in Roesser model may be unstable when the block diagonal matrices are not chosen, which has been shown in the literature [4, 5].

##### 3.2. Controller Design

Consider system (1), under the following asynchronous switching controller: where and , the corresponding closed-loop system is given by

Without loss of generality, we denote and , then due to the existence of asynchronous switching, we can obtain from (7) that

In many actual applications, it is always difficult to verify each asynchronous period in advance, but the maximal asynchronous period can be easily predicted offline. Let denote the maximal asynchronous period, then we can get the following result.

Theorem 12. *Consider system (1), for given positive constants and , if there exist positive definite symmetric matrices , , , and with appropriate dimensions, and positive scalars and such that, for , , the following inequalities hold,
**where , then under the following switching controller:
**
and the following average dwell time scheme:
**
the resulting closed-loop system (17) is exponentially stable, where , , and satisfies
*

*Proof. *See the appendix.

*Remark 13. *In Theorem 12, we propose a sufficient condition for the existence of a state feedback controller such that the resulting closed-loop system (17) is exponentially stable. It is worth noting that this condition is obtained by using the average dwell time approach. Here, plays a key role in controlling the rate of decaying of the system in the matched period, and plays a key role in controlling the rate of increasing of the system in the mismatched period. From (22), we know that the value of is proportional to the value of , which also means that it needs much larger dwell time to guarantee the stability of the considered system when there exists much larger asynchronous period.

*Remark 14. *It is noticed that (19) and (20) are mutually dependent. Therefore, we can firstly solve (19) to obtain the solution of matrices , , and . Then, (20) can be transformed into the LMI by substituting into it. By adjusting the parameters and , we can find a feasible solution of , , , , and such that (19) and (20) hold.

The procedure of the controller design for system (1) can be given as follows.

*Step 1. *Given system matrices and positive scalar , and by solving LMI (19), we can get the feasible solution of matrices , , and and positive scalar then, the controller can be obtained from (21).

*Step 2. *Substituting into (20), we can find the feasible solution of , , and such that (20) holds by adjusting the parameter .

*Step 3. *From (23), we can obtain and satisfying .

*Step 4. *Taking the value of , we can compute the value of by (22).

*Remark 15. *From the procedure above, it can be seen that the proposed method is feasible. We can find the desire controller and switching signal according to the procedure. However, we would like to point out that there still exists the conservatism to some extent for this method because (19) and (20) are mutually dependent, which brings about the increase of the complex computation. The result can be improved by adopting the method presented in [31, 32].

When , system (17) degenerates to the following delay-free system:
Then, we can get the following result.

Corollary 16. *Consider system (1) with , for given positive constants and , if there exist positive definite symmetric matrices and with appropriate dimensions, and positive scalars and such that, for , , the following inequalities hold:
**
then under the following switching controller:
**
and the following average dwell time scheme:
**
the resulting closed-loop system (24) is exponentially stable, where satisfies
*

Furthermore, it should also be noted that if the criterion in Theorem 12 is satisfied when , which means that the controller and the subsystem are synchronous, in other words, the results presented in Theorem 12 can be reduced to the synchronous case, then we can obtain the following corollary.

Corollary 17. *Consider system (1) under synchronous switching, for a given positive scalar , if there exist positive definite symmetric matrices , , and , with appropriate dimensions, and a positive scalar , such that, for , the following inequality holds:
**
then under the following controller:
**
and the following average dwell time scheme:
**
the resulting closed-loop system is exponentially stable, where satisfies
*

*Remark 18. *In [30], by using the average dwell time approach, a criterion of exponential stability for a class of 2D discrete delay-free switched systems is developed. However, the focus of our work is on stability analysis and controller design under asynchronous switching, which is different from [30], and this is also the major contribution of our work. In fact, if we let and do not consider the uncertainties and state delays, then the closed-loop system (24) is the same as (36) in [30]. In this case, Corollary 17 can be reduced to Theorem 2 in [30].

#### 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach.

Consider system (1) with the following parameters: The boundary conditions are given as follows: where the state dimensions are and .

Take and , then solving (19) in Theorem 12 gives rise to By (21), and can be obtained as follows: Substituting and into (20), and solving it, we get the following solution: Then, from (23), we can get that and . Taking , it is easy to obtain from (22) that . Choosing , the trajectories of the states and are shown in Figures 1 and 2, respectively. The system switching signal and the controller switching signal are shown in Figure 3. One can see that the states of the closed-loop system converge to zero under the asynchronous switching. This demonstrates the effectiveness of the proposed approach.

#### 5. Conclusions

This paper has investigated the problem of stabilization for a class of 2D discrete switched systems with constant state delays under asynchronous switching. A state feedback controller is proposed to stabilize such system, and the dwell time approach is utilized for the stability analysis and controller design. A sufficient condition for the existence of such controller is formulated in terms of a set of LMIs. An example is also given to illustrate the applicability of the proposed approach. Our future work will focus on extending the proposed design method to other problems such as robust control for 2D discrete switched systems with time-varying delays and fractional uncertainties under asynchronous switching.

#### Appendix

*Proof of Lemma 9. *Consider the following Lyapunov-Krasovskii functional candidate:
where
Along the trajectory of system (8), we have
It follows that
where
Applying Lemma 7, it can be obtained from (14) that
For simplicity, we denote
Thus, it is easy to get that
Notice that for any nonnegative integer , it holds that ; then summing up both sides of (A.9) from to with respect to and to with respect to , for any nonnegative integer , one gets

The proof is completed.

*Proof of Theorem 12. *When , the closed-loop system (17) can be written as
For the system, we consider the following Lyapunov function candidate:
where
By Lemma 9, one gets that if there exist positive definite symmetric matrices and with appropriate dimensions, such that
holds, then the following inequality holds for any :

When , the closed-loop system (17) can be written as
Consider the following Lyapunov function candidate:
where
Similarly, by Lemma 9, we get that if there exist positive definite symmetric matrices and with appropriate dimensions, such that
holds, then the following inequality holds for any :
Consider the following piecewise Lyapunov functional candidate for system (17):

Now, let denote the switch number of on the interval , and let denote the switching points of over the interval ; then, the switching points of can be denoted as follows:
Denoting , , then we can get from (23) and (A.21) that
where and satisfy the following conditions:
where is a sufficient small positive constant.

When , we have, for ,
When , we can also get that
From (A.25) and (A.26), we can obtain
According to Definition 6, one has
From condition (22), the following inequality can be obtained:
Thus, it is easy to get the following inequality:
It follows that
where
Thus, it can be obtained from (22) that the closed-loop system (17) is exponentially stable.

Denote and , then it is easy to get and . Using to pre- and postmultiply the left of (A.14), respectively, and applying Lemma 7, it follows that (A.33) and (A.14) are equivalent:
where .

By substituting (2) into (A.33), we can get the following inequality
where
By Lemma 8, we get
Applying Lemma 7 again, we obtain that (A.33) holds if (19) is satisfied.

Similarly, substitute (2) into (A.19) and denote and , then using to pre- and postmultiply the left of (A.19), respectively, and applying Lemmas 7 and 8, it is easy to get that (A.19) holds if (20) is satisfied.

The proof is completed.

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120.

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