Stability, L1-gain and control synthesis for positive switched systems with time-varying delay

doi:10.1016/j.nahs.2013.01.001

Nonlinear Analysis: Hybrid Systems

Volume 9, August 2013, Pages 9-17

https://doi.org/10.1016/j.nahs.2013.01.001 Get rights and content

Abstract

Exponential stability, $L_{1}$ -gain performance and controller design problems for a class of positive switched systems with time-varying delay are investigated in this paper. First, by constructing an appropriate co-positive type Lyapunov–Krasovskii functional, sufficient conditions for the exponential stability are developed by using the average dwell time approach. Then, the weighted $L_{1}$ -gain performance is investigated for the system considered. Based on the results obtained, an effective method is proposed for the construction of a stabilizing feedback controller with $L_{1}$ -gain property. All the results are formulated as a set of linear matrix inequalities (LMIs) and therefore can be easily implemented. Finally, the theoretical results obtained are demonstrated by a numerical example.

Introduction

A switched system is a type of hybrid dynamical system that consists of a number of subsystems and a switching signal, which defines a specific subsystem being activated during a certain interval. As a special class of hybrid system, many dynamical systems can be modeled as switched systems [1], [2], [3]. Recently, the importance of positive switched systems, whose states and outputs are non-negative whenever the initial conditions and inputs are non-negative, has been highlighted and investigated by many researchers due to their broad applications in communication systems [4], [5], the viral mutation dynamics under drug treatment [6], formation flying [7], and system theory [8], [9], [10], [11], [12]. A positive switched system means a switched system in which each subsystem is itself a positive system. It should be noted that, although switched systems have been studied in much recent control engineering and mathematics literature, there are still many open questions relating to positive switched systems.

It is well known that the reaction of real-world systems to exogenous signals is always not instantaneous, and is affected by certain time delays, for example long-distance transportation systems, hydraulic pressure systems, network control systems, and so on. Time delay is frequently a source of instability: it often causes undesirable performance in feedback systems such as chaos [13], [14], and it even causes a system to become out of control. Although many results have been reported for time-delay systems [15], [16], [17], [18], only recently have positive switched systems with time delay been investigated [19], [20].

It should be noted that many previous results on positive switched system mainly focus on stability and stabilization [21], [22]. It is well known that the traditional Lyapunov–Krasovskii functional may give conservative stability conditions for positive systems, as it fails to take account of the fact that the trajectories are naturally constrained to the positive orthant, so, when the stability of positive systems is considered, it is natural to apply a linear co-positive Lyapunov function. In addition, positive switched systems with disturbances are commonly found in practice. And due to the non-negative property, it would be natural to evaluate the size of positive systems via the $L_{1}$ -gain (i.e., the $L_{1}$ -induced norm) in terms of the input and output signals. Thus, stability and $L_{1}$ -gain analysis problems have become interesting issues for disturbed positive switched systems. Some results on $L_{1}$ -gain analysis have been reported for positive systems [23], [24], and positive switched systems [25], [26]. Unfortunately, little attempt has been made to investigate the issue of $L_{1}$ -gain analysis for positive switched systems with time delay, which motivates the present research.

The main contributions of this paper are three-fold: (1) by constructing an appropriate co-positive type Lyapunov–Krasovskii functional, sufficient conditions for the exponential stability are proposed by using the average dwell time approach; (2) weighted $L_{1}$ -gain performance analysis for positive switched systems with time-varying delay is performed for the first time; (3) the desired controller is proposed under which exponential stability of a closed-loop system with weighted $L_{1}$ -gain performance is obtained.

The remainder of this paper is organized as follows. In Section 2, the system formulation and some necessary lemmas are given. Section 3 is devoted to deriving the results on stability, $L_{1}$ -gain analysis and controller design. An example is provided to illustrate the feasibility of the obtained results in Section 4. Concluding remarks are given in Section 5.

Notation: In this paper, $A \underline{≻} 0 (\underline{≺} 0)$ means that all entries of matrix $A$ are non-negative (non-positive); $A ≻ 0 (≺ 0)$ means that all entries of $A$ are positive (negative); $A ≻ B (A \underline{≻} B)$ means that $A - B ≻ 0 (A - B \underline{≻} 0)$ ; $A^{T}$ means the transpose of matrix $A$ . $R (R^{+})$ is the set of all real (positive real) numbers; $R^{n} (R_{+}^{n})$ is $n$ -dimensional real (positive) vector space; $R^{n \times k}$ is the set of all real matrices of dimension $(n \times k)$ ; $Z_{+}$ refers to the set of all positive integers. $‖ x ‖ = \sum_{k = 1}^{n} | x_{k} |$ , where $x_{k}$ is the $k$ th element of $x \in R^{n}$ .

Section snippets

Problem statements and preliminaries

Consider the following switched linear systems with time-varying delay: ${\begin{matrix} \dot{x} (t) = A_{σ (t)} x (t) + A_{d σ (t)} x (t - d (t)) + B_{σ (t)} w (t), \\ x_{θ} = ϕ (θ), θ \in [- τ, 0], \\ z (t) = C_{σ (t)} x (t) + D_{σ (t)} w (t), \end{matrix}$ where $x (t) \in R^{n}$ and $z (t) \in R^{q}$ denote the state and controlled output, respectively; $w (t) \in R^{w}$ is the disturbance input, which belongs to $L_{1} [0, \infty)$ ; $σ (t) : [0, \infty) \to \underline{M} = {1, 2, \dots, m}$ is the switching signal, with $m$ being the number of subsystems, which depends on time $t$ or state $x (t)$ ; $A_{p}, A_{d p}, B_{p}, C_{p}$ and $D_{p}$ are constant matrices with appropriate dimensions, and

Main results

This section will focus on the problem of stability analysis and control synthesis for positive switched systems with time-varying delay.

Numerical example

Consider positive switched system (3) with two subsystems described by $A_{1} = [\begin{matrix} - 3 & 8 \\ 1 & - 3 \end{matrix}], A_{d 1} = [\begin{matrix} 0.1 & 0.1 \\ 0.0 & 0.0 \end{matrix}], B_{1} = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}],$ $C_{1} = [\begin{matrix} 0.2 & 0.3 \end{matrix}], D_{1} = [0.4], E_{1} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}],$ $A_{2} = [\begin{matrix} - 2 & 4 \\ 9 & - 5 \end{matrix}], A_{d 2} = [\begin{matrix} 0.1 & 0.1 \\ 0 & 0 \end{matrix}], B_{2} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}],$ $C_{2} = [\begin{matrix} 0.1 & 0.2 \end{matrix}], D_{2} = [0.2], E_{2} = [\begin{matrix} 0.1 \\ 0.3 \end{matrix}] .$

Let $d (t) = 0.5 | sin (0.2 t) |$ , and take $λ = 0.1, γ = 0.6$ . Solving the matrix inequalities in Theorem 3 gives rise to $v_{1} = [\begin{matrix} 0.7596 \\ 0.7596 \end{matrix}], v_{2} = [\begin{matrix} 0.7071 \\ 0.7729 \end{matrix}], υ_{1} = [\begin{matrix} 0.8386 \\ 0.8386 \end{matrix}], υ_{2} = [\begin{matrix} 0.8386 \\ 0.8386 \end{matrix}],$ $ϑ_{1} = [\begin{matrix} 0.8386 \\ 0.8386 \end{matrix}], ϑ_{2} = [\begin{matrix} 0.8386 \\ 0.8386 \end{matrix}], ς_{1} = [\begin{matrix} 0.1966 \\ 0.1966 \end{matrix}], ς_{2} = [\begin{matrix} 0.1914 \\ 0.1914 \end{matrix}],$ and the state feedback gain

Conclusions

In this paper we have investigated the problem of stability analysis and $L_{1}$ -gain performance for a class of positive switched systems with time-varying delay. By constructing an appropriate co-positive type Lyapunov–Krasovskii functional and using the average dwell time approach, sufficient conditions for the exponential stability and the weighted $L_{1}$ -gain performance have been proposed. An effective method has been proposed for the construction of a state feedback controller which not only

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant Nos. 60974027 and 61273120.

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View full text

Stability, L1-gain and control synthesis for positive switched systems with time-varying delay

Abstract

Introduction

Section snippets

Problem statements and preliminaries

Main results

Numerical example

Conclusions

Acknowledgment

Internat. J. Engrg. Sci.

Chaos Solitons Fractals

Hybrid dynamical systems

IEEE Control Syst. Mag.

Switching in Systems and Control

Switched and Impulsive Systems: Analysis, Design and Applications

A positive systems model of TCP-like congestion control: asymptotic results

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Internat. J. Robust Nonlinear Control

Coordination of groups of mobile autonomous agents using nearest neighbor rules

IEEE Trans. Automat. Control

The choice of the forms of Lyapunov functions for a positive 2D Roesser model

Int. J. Appl. Math. Comput. Sci.

A realization problem for positive continuous-time systems with reduced numbers of delays

Int. J. Appl. Math. Comput. Sci.

Stability, $L_{1}$ -gain and control synthesis for positive switched systems with time-varying delay

The choice of the forms of Lyapunov functions for a positive 2D Roesse $r$ model