Abstract
This paper is concerned with the problem of exponential stability and model reduction of a class of switched discrete-time systems with state time-varying delay. Some subsystems can be unstable. Based on the average dwell time technique and Lyapunov-Krasovskii functional (LKF) approach, sufficient conditions for exponential stability with performance of such systems are derived in terms of linear matrix inequalities (LMIs). For the high-order systems, sufficient conditions for the existence of reduced-order model are derived in terms of LMIs. Moreover, the error system is guaranteed to be exponentially stable and an error performance is guaranteed. Numerical examples are also given to demonstrate the effectiveness and reduced conservatism of the obtained results.
1. Introduction
Switched systems belong to a special class of hybrid control systems, which comprises a collection of subsystems described by dynamics differential or difference equations, together with a switching law that specifies the switching rule among the subsystems. Due to the theoretical development as well as practical applications, analysis and synthesis of switched systems have recently gained considerable attention [1–4].
Furthermore, the time-delay phenomenon is frequently encountered in a variety of industrial and engineering systems [5–7], for instance, chemical process, long distance transmission line, communication networks, and so forth. Moreover, time-delay is a predominant source of the poor performance and instability. In the last two decades, there has been increasing interest in the stability analysis for the systems; see, for example, [8, 9] and the references cited there in. For switched delay systems, due to the impact of time-delays, the behavior of switched delay systems is usually much more complicated than that of switched systems or delay systems [10, 11].
The average dwell time (ADT) technique [12] and multiple Lyapunov function approach [13] are two powerful and effective tools for studying the problems of stability for switched systems under controlled switching. By applying ADT scheme, the disturbance attenuation properties of time-controlled switched systems are investigated in [14]. In [15], the exponential stability and -gain of switched delay systems are studied by using ADT approach. Furthermore, based on ADT method, in [16–18] the stability of switched systems with stable and unstable subsystems co existing was considered. Using the ADT scheme, switching design for exponential stability was proposed in [19] for a class of switched discrete-time constant time-delay systems. By using the multiple Lyapunov function approach and ADT technique, the literature [20] studied the problem of state feedback stabilization of a class of discrete-time switched singular systems with time-varying state delay under asynchronous switching. However, many free weighing matrices were introduced, which made the stability result complicated. In [11], the problem of stabilization and robust control via ADT method switching for discrete switched system with time-delay was considered. However, the procedures given in [11] could not be applied to the case of asynchronous switching or the case of switching delay systems with stable and unstable subsystems co existing. This motivates the present study.
On another research front line, it is well known that mathematical modeling of physical systems often results in complex high-order models. However, this causes the great difficulties in analysis and synthesis of the systems. Therefore, in practical applications it is desirable to replace high-order models with reduced-order ones for reducing the computational complexities in some given criteria without incurring much loss of performance or information. The purpose of model reduction is to obtain a lower-order system which approximates a high-order system according to certain criterion. Recently, much attention has been focused on the model reduction problem [21–25]. Many important results have been reported, which involve various efficient approaches, such as the balanced truncation method [24], the optimal Hanker norm reduction method [25], the cone complementarily linearization method [26], and sequential linear programming matrix method [27]. In terms of LMIs with inverse constraints or other non convex conditions, the model reduction of the discrete-time context has been investigated in [28, 29]. However, it is difficult to obtain the numerical solutions. In [30], the existence conditions for model reduction for discrete-time uncertain switched systems are derived in terms of strict LMIs by using switched Lyapunov function method. However, time delays are not taken into account. In the literature [31], a novel idea to approximate the original time-delay system by a reduced time-delay model has been proposed recently. However, the unstable subsystems are not taken into account.
Motivated by the preceding discussion, the main contributions of this paper are highlighted as follows. The problem of exponential stability and model reduction for a class of switched linear discrete-time systems with time-varying delay have been investigated. To lessen the computation complexity and to reduce the conservatism, new discrete LKF are constructed and the delay interval is divided into two unequal subintervals by the delay decomposition method. The switching law is given by ADT scheme, such that even if one or more subsystem is unstable the overall switched system still can be stable. For the high-order systems, sufficient conditions for the existence of the desired reduced-order model are derived in terms of strict LMIs, which can be easily solved by using MATLAB LMI control toolbox. Finally, numerical examples are given to show the effectiveness of the proposed methods.
The remainder of this paper is structured as follows. In Section 2, the problem formulation and some preliminaries are introduced. In Section 3, the main results are presented on the exponential stability of switched discrete-time systems with time-varying delay. In Section 4, the main results on the model reduction for the high-order systems are presented. Numerical examples are given in Section 5. The last section concludes the work.
Notations. We use standard notations throughout the paper. stands for the minimal (maximum) eigenvalue of . is the transpose of the matrix . The relation means that the matrix is positive (negative) definite. denotes the Euclidian-norm of the vector . represents the -dimensional real Euclidean space. is the set of all real matrices. stands for a block-diagonal matrix. In symmetric block matrices or long matrix expressions, we use an asterisk “” to represent a term that is induced by symmetry. denotes the identity matrix.
2. Problem Description and Preliminaries
Consider a class of switched linear discrete-time systems with time-varying state delay of the form where denotes the system state, is the measured output, is the disturbance input vector which belongs to . is a vector-valued initial function. The switching signal (denoting for simplicity) is a piecewise constant function and depends on time. means that the th subsystem is activated. is the number of subsystems. The system matrices , and are a set of known real matrices with appropriate dimensions.
For a given finite positive integer , is time-varying delay and satisfies the following condition To facilitate theoretical development, we introduce the following definitions and lemmas.
Definition 1 (see [19]). The system (1) with disturbance input is said to be exponentially stable if there exist a switching function and positive number such that every solution of the system satisfies for any initial conditions . is the decay coefficient, is the decay rate, and .
Definition 2 (see [11]). Consider the system (1) with the following conditions.(1)With , the system (1) is exponentially stable with convergence rate .(2)The performance is guaranteed for all nonzero and a prescribed under the zero condition.
In the above conditions, the system (1) is exponentially stabilizable with performance and convergence rate .
Here characterizes the disturbance attenuation performance. The smaller the is, the better the performance is.
Definition 3 (see [12]). For a switching signal and any , let denote the number of switching of over . If for any given and , we have , then and are called the ADT and the chatter bound, respectively.
Lemma 4 (see [9]). For any matrix , integers , vector function , then Here
Lemma 5 (Schur complement [32]). Let be given matrices such that . Then
The aim of this paper is to find a class of time-based switching signals for the discrete-time switched time-delay systems (1), whose subsystem is not necessarily stable, to guarantee the system to be exponentially stable. For a high-order system, we are interested in constructing a reduced-order switched system to approximate the system.
3. Stability Analysis
With the preliminaries given in the previous section we are ready to state the exponential stability and performance of switched systems (1). To obtain the exponential stability of switched systems (1), we construct following discrete LKF: Here where , are symmetric positive definite matrices with appropriate dimensions, , and integer and are given constants.
Remark 6. In order to derive less conservative criteria than the existing ones, the delay interval is divided into two unequal subintervals: and , where is a tuning parameter. The information about can be taken into account. This plays a vital role in deriving less conservative results. Thus, for any , we have or .
Firstly, we will provide a decay estimation of the system LKF in (7) along the state trajectory of switched system (1) without disturbance input (i.e., ).
Lemma 7. Given constants , and , if there exist some symmetric positive definite matrices , such that the following LMIs hold: where Then, by means of LKF (7), along the trajectory of the systems (1) without disturbance input, one has
Proof. Let us choose the system LKF (7). Define where Therefore, the following equality holds along the solution of (1): For any , we have or .(1) If , it gets So (17) could be From Lemma 4, we have where Combining (13)–(22), it yields where Multiplying (9) both from left and right by , by Schur Complement, further, considering (24), one can infer that (12) holds. (2) If , it gets One obtains Similarly, it is easy to get that If (10) holds, by Schur Complement, then we have (12). This completes the proof.
Remark 8. Our LKF does not include free-weighing matrices as in previous investigations, and this may lead to reduce the computational complexity and get less conservation results.
Remark 9. In order to get less conservative results, the delay interval can be divided into much more subintervals. However, when the number of dipartite numbers increases, the matrix formulation becomes more complex and the time-consuming grows bigger.
Now we have the following theorem.
Theorem 10. If there exist some constants and positive definite symmetric matrices , and such that (9), (10), and the following inequalities hold: Then, the switched system (1) with and ADT satisfies which is exponentially stable.
Proof. By Lemma 7, we have
Therefore,
There exists , such that
We let denote the switching times of in , and let be the switching number of in , by (31) and (32), one obtains
where .
By , for all , we know that there exists . Let ; from (33), one obtains
By Definition 2, for any , it follows that
By the system LKF (7), there always exist two positive constants such that
where
Therefore,
If the average dwell time satisfies for , then the switched system (1) with is exponentially stable with stability degree.
Remark 11. The case implies the asymptotic stability.
The following theorem provides exponential stability analysis with performance of the system (1).
Theorem 12. For given constants , and , if there exist positive definite symmetric matrices and such that (29) and the following LMIs hold: where Then, the system (1) with average dwell time satisfies which is globally exponentially stable with convergence rate and performance .
Proof. Choose the LKF (7); the result is carried out by using the techniques employed for proving Lemma 7 and Theorem 10. If , by (24), we have where If , by (28), we have Combining (39) and (40), by Schur Complement, one can obtain Let we have By Definition 2 and Theorem 10, it is sufficient to show that for any nonzero . Combining (35) and (47), it can be shown that Under the zero initial condition, we have Combining (48), we have Now, we consider Exchanging the double-sum region, by and , one can easily get which means that . Then, by Definition 2, the system (1) with average dwell time satisfies which is globally exponentially stable with convergence rate and performance . This completes the proof.
If there exist some unstable subsystems in the switched system (1) with , in this case, we need to estimate the growth rate of the system LKF in (7) along the state trajectory of switched system (1). And the corresponding . By using the techniques employed for proving Lemma 7, one can easily obtain the following Lemma.
Lemma 13. Given constants , and , if there exist some symmetric positive definite matrices , such that the following LMIs hold: where Then, by means of LKF (7), along the trajectory of the systems (1) without disturbance input, one has
Remark 14. The proof of Lemma 13 is similar to that of Lemma 7 and is thus omitted here. Based on Lemmas 7 and 13, one can easily design the stabilizing switching law to guarantee the system (1) with to be exponentially stable, although some subsystems are unstable.
Without loss of generality, we can assume that is the set of all unstable subsystems and is the set of all stable subsystems. For simplicity, the LKF (7) is defined as . Choose the LKF , for the stable subsystem and choose the LKF , for the unstable subsystem. Then, we have the following conclusion.
Theorem 15. If there exist some constants , , , and positive definite symmetric matrices and such that Lemmas 7 and 13 and the following LMIs hold: Then, the switched system (1) with and the average dwell time satisfies , , which is exponentially stable.
Proof. Consider the following LKF candidate: By Lemmas 7 and 13, we have Let be the total activity time in which all subsystems satisfied on the interval and the total activity time in which all subsystems satisfied on the interval . By using the techniques employed for proving Theorem 10, combining (56) and (58), we derive that where By , , one obtains So we have By Definition 2, for any , it follows that By , we have . Moreover, the overall system is exponentially stable. This completes the proof.
Remark 16. From the proof of Theorem 15, one can see that the obtained exponential stability for the switched system (1) with is exponential stable with stability degree. In order to get a free decay rate, we can replace the condition , / by , , , ; then the switched system (1) with is exponentially stable with stability degree.
Theorem 17. For given constants , , , , , if there exist positive definite symmetric matrices and such that (56), (39), (40), and the following LMIs hold: and , , and the average dwell time satisfies ; then the switched system (1) is exponentially stable and with performance .
Remark 18. The proof of Theorem 17 is similar to that of Theorems 12 and 15 and is thus omitted here.
4. Model Reduction
In this section, we will approximate system (1) by a reduced-order switched system described by where is the state vector of the reduced-order system with and is the output of reduced-order system. , and are the matrices with compatible dimensions to be determined. The system (65) is assumed to be switched synchronously by switching signal in system (1).
Augmenting the model of system (1) to include the states of (65), we can obtain the error system as follows: Here
The following theorem gives a sufficient condition for the existence of an admissible reduced-order model (65) for system (1).
Theorem 19. Given constants , , , , and , if there exist some symmetric positive definite matrices and matrices such that the following LMIs hold
Then system (66) with the average dwell time satisfies which is exponentially stable with an norm bound .
Here
Furthermore, if a feasible solution to the above LMIs (68), (69), and (70) exists, then the system matrices of an admissible reduced-order model in the form of (65) are given by
Proof. Consider the following LKF for the switched system (66):
Here
where and , are symmetric positive definite matrices with appropriate dimensions; integer and are given constants.
By using the techniques employed for proving Lemma 7, one can easily obtain the result. Calculate the difference of in (73) along the state trajectory of system (66). (1) If , it gets
where
For any appropriately dimensioned matrices and nonsingular matrices , we have
Thus
If (68) holds, we have
where
Let
Multiplying (79) both from left and right by , by Schur Complement, further, considering (75), one can infer
Similarly, for the case of , the fact that (69) holds means that (82) is true. Set
we have
Let be the number of switching times in . From (84) and (70), we can obtain
Assume the zero disturbances input to the state equation of system (66). By Definition 2, for any , it follows that
From , one obtains . There exist , , such that
Here
Therefore
If the average dwell time satisfies , then the switched system (66) is exponentially stable with stability degree. For any nonzero , under zero initial condition, combining (68), (69), (70), (85), and (89), one can easily obtain
Therefore . This completes the proof.
Remark 20. Recently, authors in [30, 31] have studied the problem of model reduction for discrete-time switched systems. In those papers, time delays are not taken into account. However, in most of the cases in engineering problems, there always exist unknown time-varying delays; moreover, the case of stable and unstable subsystems co exists. Motivated by this, in this paper, we discussed the problem of model reduction for switched linear discrete-time systems with time-varying delays via delay decomposition approach [10–12]. Accordingly, numerical results are given for time-varying delay cases.
If there exist some unstable subsystems in the switched system (1), we have the following conclusion.
Theorem 21. Given constants , , , , , and , if there exist some symmetric positive definite matrices and matrices such that (68), (69), (70), and the following LMIs hold:
And , ; then system (66) with the average dwell time satisfies which is exponentially stable with an norm bound . Furthermore, if a feasible solution to the above LMIs (68), (69), (70), and (91) exists, then the system matrices of an admissible reduced-order model in the form of (65) are given by (72).
Here,
Remark 22. The proof of Theorem 21 is carried out by using the techniques employed in the previous section and is thus omitted here.
5. Examples
In this section, we consider some numerical examples to illustrate the benefits of our results.
Example 1 (see [20]). Consider the discrete-time switched system (1) with and the following parameters: For this system, we choose and . Applying Theorem 10, by solving the LMIs (9) and (10) and (29), we can obtain the allowable delay upper bound . It is reported, with decay rate , that the upper bound can be obtained as 14 in [19] and 16 in [20]. Therefore, the result in this brief can indeed provide larger delay bounds than the results in [19, 20]. This supports the effectiveness of the proposed idea in Theorem 10 in reducing the conservatism of stability criteria.
Example 2. Consider the discrete-time switched system (1) with and parameters as follows: It is easy to check that the is unstable. In this case, we need to find a class of switching signals to guarantee the overall switched system to be exponentially stable. Set and , according to Theorem 15 and by solving the LMIs (9), (10), (53), and (29), set ; we have and . Choosing , we have / and . The simulation result of the switched system is shown in Figure 1, where the initial condition and the switching law is shown in Figure 2. It can be seen from Figure 1 that the designed switching signals are effective although one subsystem is unstable. However, the results in [20] cannot find any feasible solution to guarantee the exponential stability of system (1).
Example 3 (see [31]). Consider the system (1) with parameters as follows: When the decay rate is fixed, the maximum value of the time-delay and the minimum value of the performance index can computed by solving the LMIs (68)–(70) procedure in Theorem 19, which is listed in Table 1 via different methods. Here, we choose . Assume that decay rate ; we can compute the maximum value of allowed delay and the minimum value of the performance index . From ADT , we have . When and , we can compute the minimum value of performance index . On the other hand, assume that maximum allowed delay and performance index ; we can compute the minimum value of the decay rate and .
Let ; here, we are interested in designing a -order () system (65) and choose the ADT switching signals such that the model error system (66) is exponentially stable with norm bound . By solving the corresponding LMIs (68)–(70) procedure in Theorem 19. For comparison with [31], we set the delay , and the following reduced-order models can be given.
Third Order Model
Second Order Model
First Order Model To illustrate the model reduction performances of the obtained reduced-order models, let the initial condition be zero; the exogenous input is given as . The output errors between the original system and the corresponding three reduced models obtained in this paper (shown by the blue line) and the literature [31] (shown by the red line) are displayed in Figures 3, 4, and 5. The switching signal is shown in Figure 6. The simulation result of the switched system is shown in Figures 3–5. It can be seen from Figures 3–5 that the output errors between the original system and the reduced-order models obtained in this paper are smaller than that in [31].
6. Conclusions
The problem of exponential stability with performance and model reduction for a class of switched linear discrete-time systems with time-varying delay have been investigated in this paper. The switching law is given by ADT technique, such that even if one or more subsystem is unstable the overall switched system can still be exponentially stable. Sufficient conditions for the existence of the desired reduced-order model are derived and formulated in terms of strict LMIs. By solving the LMIs, the system of reduced-order model can be obtained, which also provides an gain for the error system between the original system and the reduced-order model. Finally, numerical examples are provided to illustrate the effectiveness and less conservativeness of the proposed method. A potential extension of this method to nonlinear case deserves further research.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grant no. 61104064 and no. 61174038. The authors declare that there is no conflict of interests regarding the publication of this paper.