Abstract
This paper focuses on the problem of state estimation for certain switched time-varying systems with time-varying delay and nonlinear disturbance. By using an integral inequality technique and a method used in positive systems, we have established several explicit criteria for state estimation of the system, which reduce to stability criteria for some particular cases. The involved nonlinear disturbance of the system takes more general form including both the internal disturbance and the external disturbance. Three numerical examples are also given to verify the validity of the obtained theoretical results.
1. Introduction
Switched system is a particular hybrid system containing a number of subsystems and a switching signal. Each subsystem is usually described by a definite differential equation or difference equation. For switched systems, the issue of stability plays a key role in system analysis. As a result, stability of switched systems has received considerable attention during the past several decades owing to its extensive applications in automotive engine control system [1], chemical process control system [2], multiagent systems [3, 4], and so on.
Some basic problems in stability and design of switching systems were put forward by Liberzon and Morse [5]. Later, there are several very important monographs devoted to the stability analysis and design of switched systems; e.g., see Liberzon [6] and Sun and Ge [7]. There are also many interesting results for stability of switched systems in [8–15]. In most of the existing references, the Lyapunov-Krasovskii functional method was most commonly used for switched time-invariant systems. It seems to us that little attention has been paid to the stability of switched time-varying systems. Recently, by using a positive system method, exponential stability of switched time-varying systems with delay and nonlinear disturbance was investigated in [16, 17].
Integral inequality plays an important role in qualitative analysis of delay systems [18–22, 22–25]. For example, stabilization of switched systems with impulsive effects and disturbances was studied in [18] by using the Gronwall integral inequality. By introducing a generalized Gronwall-Bellman inequality, the authors established stability criteria under arbitrary switching for switched systems with general nonlinear disturbances in [21]. Later, the main results in [21] were extended to switched delay systems with nonlinear disturbances in [25]. The same method was also applied to study a class of switched delay systems in [23], where global exponential stability criteria for the system were established.
Note that time delay has attracted much attention in the theory analysis of switched systems due to its detrimental effects on system performance such as oscillation [24, 26–30] and stability [31–36]. Inspired by the work in [18, 25], we will use a delay integral inequality technique and a method developed in positive systems [37, 38] to study the problem of state estimation for a class of switched time-varying systems with time-varying delay and nonlinear disturbance. The main contributions of this paper are as follows: unlike most existing results in the literature, all the subsystems considered in this paper are time-varying; explicit global (local) state estimation criteria will be established for the cases when the nonlinear disturbance satisfies linear and nonlinear growth conditions, respectively; the nonlinear disturbance of the system has more general form which contains both the internal state disturbance and the external input disturbance, and hence it contains some cases in the literature.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and preliminaries that are essential for deriving the main results of this paper. Section 3 then focuses on establishing explicit state estimation criteria for the system. Simulations are given to illustrate the main results in Section 4. Finally, conclusions are drawn in Section 5.
2. Problem Statements and Preliminaries
In the sequel, denote by the set of -dimensional real matrices, and denote by the -dimensional Euclidean space with the vector norm , where for . Set and . For two vectors and , we write if for . For a matrix , we write if for .
Now, we consider the following switched time-varying system of the formwhere is the state, is the switching signal which is a piecewise constant function, and are continuous matrix functions for and , is the time-varying delay satisfying , is a constant, is a continuous vector function such that system (1) has a unique solution for each initial condition , , and is a continuous vector function.
We need the following assumptions for establishing the main results of this paper.
() There exist continuous functions and for , constant matrices and , , such that if , if , and for .
() There exist continuous functions and for , constant matrices and , , an -dimensional vector function , and a constant such that where the first two parts in the right-hand side of above inequality are interpreted as the internal state disturbances, and the third part is defined as the external input disturbance.
The following two lemmas play a crucial role in the state estimation of system (1).
Lemma 1 (see [39]). Assume that is a constant, and are nonnegative continuous functions defined on , and satisfies the following integral inequality Then we have
Lemma 2 (see [21]). Assume that , , and are nonnegative continuous functions defined on , and satisfies the following integral inequality where and are constants. If then we have
3. Main Results
We first study the case of in Assumption ().
Theorem 3. Assume that () and () with hold. If there exists an -dimensional vector such thatand then all solutions of system (1) satisfywhere, , , , are the th entry of vectors , , , and , respectively.
Proof. Let Without loss of generality, assume that . Denote by the right derivative of along the trajectory of system (1). We get from assumptions () and () that Therefore, According to definitions of for , we derive from the above inequality that Multiply the above inequality by and let Then we have Integrating it from to , we obtain Setwhere is defined as in Theorem 3. Since is a monotone nondecreasing function on , we get Consequently, Combining this and Lemma 1, it implies that By using the definition of , we have Therefore, (10) holds. This completes the proof of Theorem 3.
Remark 4. For the particular case when , we get from Theorem 3 that all solutions of system (1) satisfy It implies that system (1) is globally exponentially stable if .
Next, we consider the case of in Assumption ().
Theorem 5. Assume that () and () with hold. If there exists an -dimensional vector such that (8) holds,andthen the corresponding solutions of system (1) satisfywhere, , , and are defined as in Theorem 3.
Proof. Let . Following the same discussion in Theorem 3, we have By using the definitions of for and the basic inequality for and , we can derive from the above inequality that By multiplying the above inequality by and letting , we obtain Integrating it from to , we get Set where is defined as in Theorem 5. It can be seen that is a monotone nondecreasing function on and for . Therefore, That is, where and are defined as in Theorem 5. Note that (27) holds. We conclude from Lemma 2 that It implies that (28) holds true. This completes the proof of Theorem 5.
Remark 6. Note that Theorem 5 gives a local state estimation result of system (1). That is, the state estimation (28) only holds for the solutions of system (1) satisfying (27). Let . If and , we have that condition (27) holds and the corresponding solutions of system (1) satisfy That is, system (1) is locally exponentially stable.
Finally, we introduce the following assumption on the nonlinear disturbance
() There exist continuous functions and for , an -dimensional vector function , and constants , , such that where is an -dimensional vector.
Similar to the proof of Theorem 5, the following result is immediate. Hence, we omit its proof.
Theorem 7. Assume that () holds and . If there exists an -dimensional vector such that (8), (26), (), and (27) hold, where and are replaced by and , respectively, then the corresponding solutions of system (1) satisfy (28).
Remark 8. Note that condition (27) is always valid for . Therefore, Theorem 7 gives a global state estimation result for system (1). For the particular case when and , a straightforward computation yields that all solutions of system (1) satisfy where , , , , and are defined as in Theorems 3 and 7. Consequently, all solutions of system (1) are bounded if .
4. Numerical Examples
In this section, three examples are given to illustrate the main results.
Example 1. Consider system (1) with , , A straightforward computation yieldsChoosing , then for , , , , . Note that , and . By Theorem 3 and Remark 4, system (1) is exponentially stable. Figure 1 shows the state trajectories of the system, where and the switching signal is defined as in Figure 2.
Example 2. Consider system (1) with , , A straightforward computation yields thatChoosing , it can be seen that for , , , , , . Note that and . By Theorem 5 and Remark 6, system (1) is exponentially stable if . Figure 3 shows the state trajectories of the system, where the switching signal is defined as in Figure 2.
Example 3. Consider system (1) with , , Based on a direct computation, we get If we choose , we have that for , , , . Note that , , and () holds for appropriate constants and . By Theorem 7 and Remark 8, we conclude that each solution of system (1) is bounded. Figure 4 shows the state trajectories of the system, where the switching signal is defined as in Figure 2.
5. Conclusions
The problem of state estimation for switched time-varying systems with time-varying delay and nonlinear disturbance has been discussed in this paper. When the nonlinear disturbance satisfies linear and nonlinear growth conditions, explicit global (local) state estimation criteria have been established. For some particular cases, exponential stability and boundness of the system are taken into consideration. The method used in this paper is mainly based on the integral inequality technique. Finally, three numerical examples demonstrate the effectiveness of our main results.
Data Availability
We claim that all the data supporting the conclusions of the study has been contained in the paper. The readers can directly access it.
Conflicts of Interest
We declare that there are no conflicts of interest regarding this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under grants nos. 61873110, 61473133, and 11671227.