Stabilization of switched positive system with impulse and marginally stable subsystems: A mode-dependent dwell time method
Introduction
Positive system is a system which has the special property that nonnegative initial value leads to nonnegative state trajectory [1], [2]. Positive system has attracted an increasing attention during last decades due to the numerous applications in congestions control [3], chemical processes [4], and power electronics [5]. In practice, switching phenomenon often occurs in a family of positive subsystems [6], [7], [8]. Therefore, stability of SPLS has been widely studied in recent years [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].
Impulse phenomenon is common in many fields. It means that a physical quantity changes suddenly in a short time and returns to its original state quickly. Impulses which contribute to dynamics of systems have been applied to many practical problems, such as satellite orbit transfer, electronic engineering, stability and synchronization of chaotic secure communication systems, etc. In the past few years, the control and analysis of impulsive systems has become one of the research hot spots and developed rapidly [19], [20], [21], [22], [23], [24], [25].
For stability of the SPLS, there are two important issues worthy of studying. One is asymptotic stability under arbitrary switching, which is investigated based on the common LCLF method, see, e.g., [26], [27], [28], [29], [30], [31], [32]. Another stability issue for the SPLS is to design appropriate switching laws such that the PSLS remains asymptotic stability [33], [34], which is also called the stabilization of SPLS. A piecewise LCLF was first introduced in [35], [36] to get the stabilization criterion for SPLSs with ADT switching. By virtue of mode-dependent ADT method, exponential stability of the SPLS was discussed in [37]. Stability and stabilization of the SPLS under dwell time switching was addressed in [38], [39]. When each subsystem is only stable in the Lyapunov sense (or marginally stable), time-dependent switching laws were determined to ensure that the SPLS is asymptotically (or exponentially) stable in [40], [41], [42], [43], [44], [45]. Note that strong restrictions Aiξ⪯0 or for the case of continuous-time SPLS, and Aiξ⪯ξ or for the case of discrete-time SPLS were imposed on subsystem matrices in [40], [41], [42], [43], [44], [45], where the corresponding notations can be found in Section 2.
It is evident that all the above research on stability analysis aims at time invariant SPLSs. Few people study the stabilization of time-varying SPLS by virtue of time-dependent switching control signals. Motivated by the work in [46], [47] where all subsystems are assumed to be of asymptotic stability, we will use a piecewise weak LCLF method to determine appropriate MDT such that the time-invariant SPLS with impulse and marginally stable subsystems is asymptotically stable under the chosen switching. Then, by virtue of the comparison principle, the main results of this article are generalized to the time-varying SPLS.
The contributions of this article can be summarized as follows. Firstly, a piecewise weak LCLF method is proposed, which does not impose the same constraints on subsystem matrices as that given in [40], [41], [42], [43], [44], [45], and hence the theoretical results could be used for some models that are not contained by known ones. Secondly, unlike the references [46], [47], each subsystem of the involved switched system is only marginally stable. Moreover, both the cases of impulse and time-varying subsystem are taken into consideration. Finally, the definition of MDT is introduced in this article, which includes the usual dwell time as a special case and has the obvious advantage over it.
The organization of this article is listed as follows. Section 2 reviews some preliminaries that will be required in deriving the main results. In Sections 3 and 4, by introducing the piecewise weak LCLF, explicit criteria for the stabilization of the SPLS with MDT switching are established. Two illustrative examples are worked out in Section 5 to validate the effectiveness of the proposed results. Finally, Section 6 concludes this article.
Section snippets
Problem formulation
In this article, let and be the n-dimensional real vector set and the n × n-dimensional real matrix set, respectively. Given two vectors denote u≻v (or v≺u) if ui > vi, u≽v (or v⪯u) if ui ≥ vi, where ui and vi are the ith entries of u and v, i ∈ {1, 2, ⋅⋅⋅, n}. Given two matrices denote X⪰Y (or Y⪯X) if xij ≥ yij, where xij and yij are the ijth entries of X and Y, i, j ∈ {1, 2, ⋅⋅⋅, n}. is called to be nonnegative if X⪰0. A Metzler matrix is a square matrix whose
Stabilization of the continuous-time SPLS
We first design a time-dependent switching control lawsuch that SPLS (2.4) is asymptotically stable for any initial state x(0)⪰0, where tk > 0, are contiguous switching points satisfying is defined to be mode-dependent dwell time (MDT) of the ikth subsystem.
To study the stabilization of SPLS (2.4), the following Assumptions are first given. Assumption 1 fji(x)⪰0 for x⪰0, and there are matrices Fji ∈ Rn × n, Fji⪰0 satisfying fji(x)⪯Fjix, x⪰0, i, j ∈ {1, 2,
Stabilization of the discrete-time SPLS
Our main purpose is to find the time-dependent switching control lawsuch that the discrete-time SPLS (2.6) is asymptotically stable for any initial state x(0)⪰0, where tk > 0, are switching points satisfying . is called MDT of the ikth subsystem.
The following assumption is required. Assumption 3 Ai⪰0 for i ∈ {1, 2, ⋅⋅⋅, m}. Obviously, if Assumption 1 and Assumption 3 hold, Lemma 3 yields that system (2.6) is positive. Now, we can get the
Numerical examples
Now we will use the obtained results to solve the stabilization problem in the following two examples. Example 1 Consider the SPLS (2.4) with marginally stable subsystemsand the impulse whereNote that both A1 and A2 are not Hurwitz. So, the results in [46] is not applicable. In addition, we can not find a column vector ξ≻0 satisfying Aiξ⪯0 (or ξTAi⪯0) for . Therefore, all the results
Conclusions
The article has discussed the stabilization problem of the SPLS with impulse and marginally stable subsystems. Both the continuous time case and the discrete time case are taken into account. By proposing the piecewise weak LCLF method and MDT method, time-dependent switching control laws are determined under which the SPLS is asymptotically stable. Moreover, the theoretical results are also applied to the time-varying SPLS by virtue of the comparison principle. Two numerical examples show that
Acknowledgment
The authors thank the reviewers for their helpful comments on this paper. The reported study was supported by the National Natural Science Foundation of China under Grant 61873110 and the Foundation of Taishan Scholar of Shandong Province under Grant ts20190938.
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