Abstract
The problem of finite-time boundedness for a class of switched linear systems with time-varying delay and external disturbance is investigated. First of all, the multiply Lyapunov function of the system is constructed. Then, based on the Jensen inequality approach and the average dwell time method, the sufficient conditions which guarantee the system is finite-time bounded are given. Finally, an example is employed to verify the validity of the proposed method.
1. Introduction
The switched system is a special kind of hybrid dynamic system, composed of a family of subsystems and a switching law specifying the switches between subsystems [1, 2]. The fact that the structure and working mechanism of the switched system are more complex than general systems leads to that the switched system possesses much richer dynamic characteristics. The switched systems are widely applied in engineering practice, such as power system control, robot control, network control, and so forth [3–9].
In practice, switched systems are commonly subjected to time-delay and external disturbance. Due to their significant impact on the performances of switched systems, many scholars have been attracted to investigate the problem. Sun et al. analyzed the asymptotic stability of the switched linear system with time-delay perturbation by using common Lyapunov function and multiple Lyapunov function [3]. Lu and Zhao also investigated the asymptotic stability for switched linear systems with time-delay and proposed an effective method which can direct researchers to choose an appropriate switching law to make sure the system is asymptotic stable [10]. Zhao and Zhang studied the stability of the switched system with time-varying delays based on the average dwell time and time-delay decomposition approaches [11]. For switched systems with time-varying delay, Lian et al. utilized the Lyapunov-Krasovskii function method to design H infinity filter [12]. For switched systems affected by the nonlinear impact and disturbance, Sun used transfer matrix estimation and Gronwall inequality methods to design a feedback law stabilizing system [13]. For the switched system with fixed time-delay and norm bounded disturbance, Lin et al. proposed the finite-time boundedness concept and a method to judge whether the system is finite-time bounded [14].
Up to now, to the best of the authors’ knowledge, there are a few papers concerning the finite-time boundedness problem of switched system. For switched systems with time-varying delay and external disturbance, the problem has not yet been discussed by any literature. However, in practical engineering, the time-delays are generally changeable over time, not fixed. In addition, many practical systems are just required that their state trajectories are bounded over a fixed interval. In other words, those systems may be unstable. On the contrary, although some systems are asymptotically stable, they cannot meet the application requirements because of their large transient state amplitudes. Considering the wide application of switched systems with time-varying delay and the requirements for transient behaviors in engineering fields, it is a significant task to investigate finite-time boundedness for switched linear systems with time-varying delay and external disturbance. The main contributions in this paper are listed as follows. (1) For the convenience of processing, a concise definition on the finite-time boundedness is proposed for the switched system. (2) Sufficient conditions of finite-time boundedness for switched linear systems with time-varying delay and external disturbance are given.
2. Preliminaries and Problem Formulation
Consider the following switched linear system with time-varying delay and external disturbance: where is state variable and is the switching law which is a piecewise continuous function with which means the switched system is consisted of subsystems. The th subsystem is activated when , , and are constant matrices. represents time-varying delay. stands for external disturbance. is the continuous vector-valued initial function on denotes the derivative of is a positive constant.
For the convenience of subsequent processing, assume that the system (1) satisfies the following assumptions.
Assumption 1 (see [14]). The value of external disturbance changes over time, but it satisfies
Assumption 2. For the time-varying delay, the following inequalities hold: where and are positive constants.
Assumption 3. The system state variable does not “jump” at switching instant, that is to say the state trajectory is continuous. In addition, the switching number of is finite in a limited time interval which implies that the frequency of switching signal is not infinite.
Definition 4 (see [15]). For , let denote the switching number of over . If holds for and an integer , then is called average dwell time.
Definition 5. For a given four positive constants , , , , and a switching signal , if then the system (1) is said to be finite-time bounded. Where , without loss of generality, specify .
Remark 6. Definition 5 implies that if the system (1) is finite-time bounded, the state remains within the prescribed bound in the fixed interval. Notice that finite-time boundedness is different from asymptotic stability. The system which is finite-time bounded may not be asymptotically stable while a system is asymptotically stable does not mean it is finite-time bounded either. In a word, there is no necessary relation between them.
Remark 7. The definition of finite-time boundedness in this paper is much more concise than that in [14]. However, they are consistent in essence. By using the definition in this paper, some complex matrix transformations can be avoided in the subsequent mathematical processing.
3. Main Result
Theorem 8. For system (1), for all and for all , assume there exists symmetric positive matrixes , , , , , , and and positive constants , such that
If the average dwell time satisfies
then system (1) is finite-time bounded, where
The left of inequality (6) is a symmetric matrix. Thus, the symmetric terms are denoted by “*”. represents the maximum eigenvalue of .
Proof. Construct the multiply Lyapunov function as follows:
Calculate the derivatives of , and as
Furthermore, it follows that
Due to the Jensen inequality, inequality (15) holds
By (14) and (15), we obtain
From (11), (12), (13), and (16), it is easy to get
According to the definition of finite-time boundedness, the rest of the proof will be divided into two steps. Under the given conditions, we need to prove that and , respectively.
(i) We will prove that holds for all on .
By (6) and (17), inequality (18) holds
Since , inequality (18) can be transformed into
Let stand for the instant of the th switching.
Integrating from to on both sides of (19), it follows that
Notice that , , , , , , , and and the continuity of , hence (21) holds
where denotes the instant just before .
It is easy to see
Then (23) is obtained
Assume the switching number of over is . (24) is obtained via the iterative calculation
Thus, it follows that
On the other hand, since and , we have
Applying the above inequality to (27), we get
With respect to in (29), it is processed as follows:
Applying known mathematical relationships to (30), (31) can be obtained as
Inequality (32) is obtained via (29) and (31) as
According to the definition of , inequality (33) holds
Then the following holds based on (32) and (33):
(ii) Next, will be demonstrated.
By (7), we have
On the other hand, due to , there exist the following mathematical relations:
Combining (38) and (39), we get .
By (i) and (ii), the system (1) satisfies the definition of finite-time boundedness under given conditions. This completes the proof of Theorem 8.
Remark 9. Notice that (6) is not a linear matrix inequality. Thus, it cannot be directly solved via LMI toolbox. Before solving (6), the inequality can be transformed to a linear matrix inequality by specifying the value of .
4. A Numerical Example
An example is presented to illustrate Theorem 8. Consider
Let , and , then and .
Solving (6) leads to feasible solutions that
Further, we get that and . The simulation of the numerical example is performed and its results are shown in Figures 1 and 2. From Figure 1, one can get that holds. From Figure 2, it is easily found that the value of remains within for . So, the system is indeed finite-time bounded over .
(a)
(b)
5. Conclusion
(1) For the switched linear system, a new definition on finite-time boundedness is proposed which can reduce some complex matrix calculations.
(2) Under given conditions, the sufficient conditions which guarantee the system is finite-time bounded are given for the switched linear system with time-varying delay and external disturbance.
(3) In the future study, a challenging research topic is how to ensure the switched system with time-varying delay remains finite-time bounded for any switching signal.
Conflict of Interests
The authors (Yanke Zhong and Tefang Chen) declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grant no. 61273158.