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Stability and -gain analysis for 2D discrete switched systems with time-varying delays in the second FM model
Advances in Difference Equations volume 2013, Article number: 56 (2013)
This paper is concerned with the problems of stability and -gain analysis for 2D (two-dimensional) discrete switched systems with time-varying delays described by the second FM state-space model. Firstly, we introduce the definition of the average dwell time for a 2D discrete switched system, which is an extension of the ‘average dwell time’ concept of a 1D (one-dimensional) switched system. Secondly, based on the average dwell time approach, delay-dependent sufficient conditions for the existence of the exponential stability for the 2D discrete switched system are derived and -gain performance for the considered system is also analyzed. All the obtained results are formulated in a set of LMIs (linear matrix inequalities). Finally, a numerical example is given to illustrate the effectiveness of the proposed results.
2D (Two-dimensional) systems, which are a class of multi-dimensional systems, have received considerable attention over the past few decades due to their wide applications in many areas such as multi-dimensional digital filtering, linear image processing, signal processing, and process control [1–3]. The 2D system theory is frequently used as an analysis tool to solve some problems, e.g., iterative learning control [4, 5] and repetitive process control [6, 7]. The problems on realization, stability analysis, stabilization, filter design, and so on for 2D or n D systems have attracted a great deal of interest by many researchers. Xu et al. [8, 9] investigated the realization of 2D systems, and the problems of stability and stabilization for these systems were studied extensively in [10–17]. The observer and filter design problems have also been considered in [18–20].
It is known that modeling uncertainties and disturbances are unavoidable in practical systems, and it is important to investigate the problems of control and robust stabilization of 2D systems. Recently, many results on control for 2D systems have been presented in [21–23]. Because time delays frequently occur in practical systems and are often the source of instability, the control problem for a class of 2D discrete systems with state delays has also been investigated in [24, 25].
On the other hand, since switched systems have numerous applications in many fields, such as mechanical systems, automotive industry, switched power converters, this class of systems has also attracted considerable attention during the past several decades [26–33]. Recently, some approaches have been applied widely to deal with these systems; see, for example, [26–33] and references cited therein. As stated in [34, 35], in many modeling problems of physical processes, a 2D switching representation is needed. One can cite a 2D physically based model for advanced power bipolar devices and heat flux switching and modulating in a thermal transistor. At present, there have been a few reports on 2D discrete switched systems. Benzaouia et al.  firstly considered 2D switched systems with arbitrary switching sequences, and the process of switch is considered as a Markovian jumping one. In addition, the stabilization problem of discrete 2D switched systems was also studied in . In , we extended the concept of average dwell time in 1D switched systems to 2D switched systems and designed a switching rule to guarantee the exponential stability of 2D switched delay-free systems. However, to the best of our knowledge, no works have considered the disturbance attenuation property of 2D switched systems to date. Moreover, because of the complicated behavior caused by the interaction between the continuous dynamics and discrete switching, the problem of disturbance attenuation performance analysis for 2D switched systems is more difficult to study, and the methods proposed in [21–25] cannot be directly applied to such systems. This motivates the present study.
In this paper, we are interested in investigating the issues of the exponential stability and -gain analysis for 2D discrete switched systems with time-varying delays represented by the second FM model. The main contributions of this paper can be summarized as follows: (i) Based on the average dwell time approach, a delay-dependent exponential stability criterion for such systems is obtained and formulated in terms of LMIs (linear matrix inequalities); (ii) The Lyapunov-Krasovskii function with exponential term, which is different from the previous ones, is constructed to investigate the stability of the considered systems; and (iii) In order to investigate the disturbance attenuation property of the considered systems, we for the first time introduce the concept of -gain for a 2D switched system, which is an extension of the -gain performance index in the 1D case. The -gain performance index can characterize the disturbance attenuation property of the underlying systems, and then, based on the established stability results, delay-dependent sufficient conditions for the existence of -gain performance are derived in terms of LMIs, which can be easily verified by using some standard numerical software. The proposed method can also be applied to non-switched 2D discrete linear systems.
This paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the average dwell time approach, delay-dependent sufficient conditions for the existence of the exponential stability and -gain property are 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 ‘T’ denotes the transpose, and the notation () means that matrix is positive semi-definite (positive definite, respectively). denotes the Euclidean norm. I represents an identity matrix with an appropriate dimension. denotes a diagonal matrix with the diagonal elements , . denotes the inverse of X. 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 . The norm of a 2D signal is given by
and belongs to if .
2 Problem formulation and preliminaries
Consider the following 2D discrete switched systems with time-varying delays described by the second FM model:
where is the state vector, is the noise input which belongs to , is the controlled output. i and j are integers in . is the switching signal. N is the number of subsystems. , , denotes that the k th subsystem is active. , , , , , , , and are constant matrices with appropriate dimensions. and are delays along horizontal and vertical directions, respectively. We assume that and satisfy
where , , , and denote the lower and upper delay bounds along horizontal and vertical directions, respectively.
In this paper, it is assumed that the switch occurs only at each sampling point of i or j. The switching sequence can be described as
Remark 1 If there is only one subsystem in system (1), it will degenerate to the following 2D system:
Therefore, the addressed system (1) can be viewed as an extension of 2D time-varying delays systems to switched systems.
For system (1), we consider a finite set of initial conditions, that is, there exist positive integers and such that
where and are positive integers, and are given vectors.
Definition 1 System (1) with is said to be exponentially stable under the switching signal if for a given , there exist positive constants c and ξ such that
holds for all , where
Remark 2 From Definition 1, it is easy to see that when Z is given, will be bounded and will tend to be zero exponentially as D goes to infinity, which also means will tend to be zero exponentially.
Definition 2 
For any , let denote the switching number of the switching signal on an interval . If
holds for given , , then the constant is called the average dwell time and is the chatter bound.
Remark 3 Definition 2 is an extension of the ‘average dwell time’ concept in a 1D switched system, which can be seen in . In what follows, based on the extended average dwell time concept, we will investigate the problems of stability and -gain analysis for a 2D discrete switched system with time-varying delays. It should be noted that we have studied the problems of stability analysis and stabilization of delay-free 2D switched systems using the average dwell time approach in .
Remark 4 Similar to the 1D switched system case, Definition 2 means that if there exists a positive number such that the switching signal has the average dwell time property, the average time interval between consecutive switching is at least . The average dwell time method is used to restrict the switching number of the switching signal during a time interval such that the stability or other performances of the system can be guaranteed.
Definition 3 Consider 2D discrete switched system (1). For a given scalar , system (1) is said to have -gain γ under the switching signal if it satisfies the following conditions: (1)
When , system (1) is asymptotically stable;
Under the zero boundary condition, it holds that
Remark 5 It is not difficult to see that Definition 3 is an extension of the -gain performance index in the 1D case. γ characterizes the disturbance attenuation performance. The smaller γ is, the better performance is.
Definition 4 Consider 2D discrete switched system (1). For a given scalar , system (1) is said to have weighted -gain γ under the switching signal if it satisfies the following conditions:
(1) When , system (1) is asymptotically stable;
(2) Under the zero boundary condition, it holds that
Remark 6 Similar to the 1D switched system case, Definition 4 means that system (1) can also have disturbances attenuation properties when it satisfies conditions (1) and (2) in Definition 4.
Lemma 1 Consider 2D discrete switched system (1). Suppose that there exist a series of functions () and two positive scalars and for which the following inequality holds:
if there exists a number for which along with the solution of system (1) satisfies the inequality
and such that
then 2D discrete switched system (1) is exponentially stable for every switching signal with the average dwell time scheme
Proof Let denote the switch number of switching on an interval , and let denote the switching points of over the interval . Denoting , , it follows from (8) and (9) that
According to Definition 2, one obtains
Then from (12), we have
From (7), we get
Therefore, according to Definition 1, system (1) is exponentially stable under the average dwell time scheme (10). □
3 Main results
3.1 Stability analysis
Theorem 1 Consider system (1) with . For a given positive constant , if there exist positive definite symmetric matrices , , , , , , , , and matrices , , , , and with appropriate dimensions, , such that
hold, then under the average dwell time scheme
where and satisfies
the system is exponentially stable.
Proof See the Appendix for the detailed proof, it is omitted here. □
Remark 7 In Theorem 1, we propose a sufficient condition for the existence of exponential stability for the considered 2D discrete switched system (1). It is worth noting that this condition is obtained by using the average dwell time approach.
3.2 -gain performance analysis
Theorem 2 Consider system (1). For given positive constants γ and , if there exist positive definite symmetric matrices , , , , , , , , and matrices , , , , and with appropriate dimensions, , such that (16) and the following inequality
hold, then under the average dwell time scheme (17), the system is exponentially stable and has weighted -gain γ.
Proof It is easy to get that (15) can be deduced from (19), and according to Theorem 1, we can obtain that system (1) is exponentially stable.
Now we are in a position to consider the -gain performance of system (1) under the zero boundary condition. Following the proof line of Theorem 1, we get the following relationship for the k th subsystem:
Then by the Schur complement lemma, we can obtain from (16) and (19) that
Summing up both sides of (20) from to 0 with respect to j and 0 to with respect to i and applying the zero boundary condition, one gets
Under the zero initial condition, it holds that
Thus we have
Multiplying the both sides of (23) by , we get the following inequality:
Note that , then using (17), we have
It follows that
According to Definition 4, we obtain that system (1) is exponentially stable and has weighted -gain γ. The proof is completed. □
Remark 8 We would like to stress that the -gain performance analysis problem of 2D discrete switched systems is firstly considered in the paper. Although some results on -gain performance analysis of 2D systems have been obtained in [21–25], the existing methods proposed in these papers cannot be directly applied to 2D switched systems. In Theorem 2, sufficient conditions for the existence of -gain performance for system (1) are derived in terms of a set of LMIs.
Remark 9 As for the applicability of Theorem 2, it is easy to see that a larger α and a larger γ will be favorable to the feasibility of matrix inequality (19), while a smaller α is more expectable to decrease , and a smaller γ means the better performance of the system. Thus for the first time, we can chose a smaller α and a smaller γ, and then, by adjusting the values of α and γ, we can find a feasible solution.
Remark 10 It is noticed that when in , (18) turns out to be , , , , , , , , and thus a common Lyapunov function exists for all subsystems. Then from (23) we get
Summing D from 2 to ∞, we get
Therefore, -gain is achieved for switched system (1) under arbitrary switching. We state the fact in the following corollary.
Corollary 1 Consider system (1). For given positive constants γ and , if there exist positive definite symmetric matrices , , , , , , , , and matrices , , , , and with appropriate dimensions such that (16) and (19) hold for , then the 2D discrete switched system (1) achieves the -gain under arbitrary switching signals.
4 Numerical example
In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (1) with parameters as follows:
According to Remark 9, we can firstly take and , then solving (16) and (19) gives rise to the following solution:
and . It can be seen from Figures 1-3 that the system is asymptotically stable. Furthermore, when the boundary condition is zero, by computing, we get and , and it satisfies condition (2) in Definition 4. Therefore, it can be observed that the system has weighted -gain . This demonstrates the effectiveness of the proposed approach.
This paper has investigated the problems of stability and -gain analysis for 2D discrete switched systems with time-varying delays described by the second FM model. A delay-dependent exponential stability criterion is obtained via the average dwell time approach. Then some sufficient conditions for the existence of weighted -gain for the considered system are derived in terms of LMIs. Finally, an example is also given to illustrate the applicability of the proposed results. Our future work will focus on extending the proposed results to other kinds of 2D discrete switched systems such as 2D discrete switched stochastic systems or 2D discrete switched nonlinear systems.
Proof of Theorem 1 It is assumed that the k th subsystem is active on the interval , and the l th subsystem is active on the interval . Now we consider the Lyapunov function candidate for the k th subsystem
Then we have
For simplicity, we denote
Now let us discuss the case where . It follows from (28) that
Notice that the following equations hold for any matrices , , , and with appropriate dimensions:
On the other hand, for any matrices and , the following equations also hold:
Thus it follows from (15)-(16) that
When , we have . Then summing up both sides of (37) from to 0 with respect to j and 0 to with respect to i, one gets
Thus (8) can be directly obtained. Moreover, by (28), we can find two positive scalars and such that (7) holds, where
In addition, (9) can be deduced from (18), thus by Lemma 1, we can conclude that 2D discrete switched system (1) is exponentially stable. □
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This research was supported by the National Natural Science Foundation of China under Grant Nos. 60974027 and 61273120.
The authors declare that they have no competing interests.
SH carried out the main results of this article and drafted the manuscript. ZX directed the study and helped with the inspection. All the authors read and approved the final manuscript.
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Huang, S., Xiang, Z. Stability and -gain analysis for 2D discrete switched systems with time-varying delays in the second FM model. Adv Differ Equ 2013, 56 (2013). https://doi.org/10.1186/1687-1847-2013-56
- 2D systems
- switched systems
- time-varying delays
- average dwell time
- linear matrix inequality