Finite-time stabilization of continuous-time switched positive delayed systems

https://doi.org/10.1016/j.jfranklin.2021.04.022Get rights and content

Abstract

This paper deals with the problems of finite-time stability and stabilization for continuous-time switched positive linear time-delay systems under mode-dependent average dwell time switching signals. First, finite-time stability conditions are established by constructing an multiple piecewise copositive Lyapunov–Krasovskii functional. Then, finite-time stabilization is achieved by designing a state-feedback controller in the form of linear programming. This numerical construction approach proposed for controller cancels the restriction of the multiple piecewise copositive Lyapunov–Krasovskii functional on controllers, which can decrease the conservatism. Finally, two numerical examples are given to show the advantages of our methods.

Introduction

In recent years, due to the characteristics of the state value and output signal always remaining non-negativity whenever non-negative initial conditions and non-negative input signal, the related problems of positive system are sought after by more scholars such as observer design [1], [2], stability analysis [3], [4] and stabilization [5]. Switched systems as a relatively simple kind of hybrid dynamic systems are composed of a series of continuous or discrete subsystems and the switching rule that governs the switching between them [6], [7], [8]. Due to the hybrid dynamic features, the controller design and stability analysis for switched systems are more difficult, challenging, and also more important. To name only a few, in [9] stability and L2-gain analysis results have been obtained for discrete-time switched systems by proposing a multiple discontinuous Lyapunov function in both nonlinear and linear context. Then adaptive neural control schemes have been designed for switched nonlinear systems in [10], [11]. Switched positive systems consist of a family of positive subsystems and a switching signal, specifying when and how the switching among the various subsystems happens. Recently, switched positive systems have attracted more and more scholars’ attention in control communities [12], [13], [14], [15], [16]. To name a few, in [17], sufficient conditions of finite-time stability and finite-time boundedness have been constructed by using the multiple linear copositive Lyapunov function approach. Then in [18], stabilisation problem of switched positive systems with actuator saturation has been investigated under two cases of time-dependent switching and state-dependent switching. On the other hand, the phenomenon of time delay is universal in the actual system. Therefore, many scholars have carried out researches on switched positive systems with time delays and some results have been achieved [19], [20], [21], [22], [23]. In [24], a novel design approach has been proposed for the observation gain matrix of switched positive time-delay systems, which can reduce the computational complexity of the numerical conditions. In [25], the notion of absolute exponential L1 stability has been first introduced for switched nonlinear positive systems by constructing a nonlinear Lyapunov–Krasovskii functional. Then a novel multiple discontinuous copositive Lyapunov–Krasovskii functional approach has been developed in [26] and the stability conditions have been established for switched positive time delay systems by a linear programming approach under mode-dependent average dwell time switching.

Stability has always been a hot topic in control field. In some cases, we need to pay more attention to the behavior of a system over a fixed period of time. That is to say, the state of the system needs to remain within a threshold during a given period of time. The requirement is called finite time stability. Obviously, it is necessary to study finite time stability because it has wide application value in practical applications, such as actuator saturation. Recently, the concept of finite time stability has been reviewed with the help of the tools of linear matrix inequalities, which appeals to us to find computable conditions that guarantee finite time stability [27], [28]. A static output-feedback controllers has been designed in [29] for switched positive delayed systems by constructing multiple linear copositive Lyapunov functions, and a sufficient condition has been derived to ensure that the closed-loop system is positive and finite-time bounded. Then in [20], the finite-time control has been discussed for a class of discrete impulsive switched positive time-delay systems under asynchronous switching.

Motivated by the above illustrations, this article considers the problem of finite-time stabilization for continuous-time switched positive delayed systems under mode-dependent average dwell time (MDADT) switching. First, sufficient conditions are established for the finite-time stability of the corresponding system. Furthermore, a multiple piecewise copositive Lyapunov function is introduced, and a lower bound on MDADT can be achieved. Compared with the traditional copositive Lyapunov–Krasovskii functional, our approach proposed in this paper can provide more flexibility. Then, by designing a state-feedback controller, finite-time stabilization conditions are obatined for the corresponding system. By constructing a combinatorial copositive Lyapunov–Krasovskii functional, a numerical form of controller is proposed explicitly. Compared with the existing literature [28], [30], [31], our numerical form of controller can have lower complexity and be easy to compute.

This paper is organized as follows. Section gives some necessary concepts and definitions of continuous-time switched positive delayed systems. In Section 3, sufficient conditions of finite-time stability and stabilization are established for the corresponding system with MDADT switching. Numerical simulations are given in Section 4 to show the efficiency and validity of the results. Finally, the paper is concluded in Section 5.

Notations: For a real matrix A, AT denotes its transpose, A0(A0) means that all elements of matrix A are positive (i.e. aij>0)(non-negative, i.e. aij0). R, Rn and Rn×n denote the filed of real numbers, n-dimensional Euclidean space, and the space of n×n matrix with real entries, respectively. R+n stands for the non-negative orthant in Rn. Denote by N and K the sets of nonnegative numbers and natural numbers, respectively. I denotes identity matrix with an appropriate dimension. Let 1n=(1,,1)T with n entries and 1n(i)=(0,,0,1,0,,0)T with the ith entry being 1. Given aij>=0, ij, A=[aij]n×n is called a Metzler matrix. p={0,1,,Gp1}, where Gp denotes the discontinuous segments of Lyapunov function during the pth subsystem is activated.

Section snippets

System description and preliminaries

Consider the following switched positive linear time-delay system:x˙(t)=Aσ(t)x(t)+Adσ(t)x(th)+Bσ(t)u(t),x(θ)=ϕ(θ),θ[h,0],where xRn is the system state, h denotes the constant delay with h0and σ(t) represents a switching signal which is a piecewise constant function from the right of time and takes its values in the finite set S={1,2,,n}, where n>1 is the number of subsystems. When t[tk,tk+1), we say the σ(tk)th subsystem is active and therefore the trajectory xt of system (1)

Main results

In this paper, we aim to establish finite-time stability conditions for system (1). Based on the multiple piecewise copositive Lyapunov–Krasovskii functional (MPCLKF) approach proposed in [23], we first analyze the problem of finite-time stability and then we establish the stabilization conditions for system (1).

Illustrative examples

In this section, two examples are given to illustrate the effectiveness of theoretical findings.

Conclusions

This paper has investigated the problem of the finite-time stabilization for switched positive delayed systems. First, some sufficient conditions of finite-time stability have been developed for the corresponding system by using LP method. Then by designing a switched state-feedback controller, the finite-time stabilization can be achieved. By proposing a combinational linear copositive Lyapunov–Krasovskii functional method, it provides a possibility for the numerical form of the controller.

CRediT authorship contribution statement

Ning Xu: Writing - original draft. Yun Chen: Writing - review & editing. Anke Xue: Software. Guangdeng Zong: Data curation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported by the the National Natural Science Foundation of China under [grants number 61973102].

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