Lyapunov–Krasovskii functionals for predictor feedback control of linear systems with multiple input delays

Abstract

This paper is concerned with the Lyapunov–Krasovskii functional construction of linear control systems with multiple input delays. By transforming the predictor feedback control systems into a delay-free linear system with external inputs, a Lyapunov–Krasovskii functional is constructed in terms of a set of linear matrix inequalities (LMIs). It is shown that the solvability of this set of LMIs is equivalent to the asymptotic stability of the delay-free linear system induced from the predictor feedback control system. The proposed Lyapunov–Krasovskii functional is also found to be an ISS Lyapunov–Krasovskii functional for the predictor feedback control systems. An example is worked out to validate the effectiveness of the proposed method.

Introduction

Dynamic systems with time delays have received more and more attention because time-delay systems have many applications in practice engineering such as network control system, chemical process control, population model for instance (see, for example, [3], [24], [31], [34] and the references therein). Since the stability and stabilization problems of time-delay systems are very important in both theory and practice, considerable research efforts were focused on them and many important related results have been obtained in the literature (see [5], [6], [11], [15], [21], [23], [28], [40] and the references therein). Nevertheless there are still many unsolved problems in the stability analysis and stabilization of time-delay systems because of their infinite dimensional nature.

In the existing results regarding stability and stabilization of time-delay systems, most of them deal with linear systems with state delays (see, for example, [10], [19], [30], [38]) while relatively few results are available for control problems of linear systems with input delays (for example, [33], [42]). To deal with the stabilization of control systems with input delays, there are basically two efficient design methods, namely, memory controllers design by predictor feedback and memoryless controllers design adopted from delay-free systems. Memoryless controllers have been utilized by many researchers, for example, [4], [12]. The advantage of this kind of methods is that they are very easy to implement. However, this kind of methods may fail if the delays are too large. In contrast, the predictor feedback that was originated by Mayne [26] can allow arbitrarily large input delays. The basic idea of this approach is to transform the delay system into an equivalent delay-free system for which any conventional design approaches are applicable. This method has been widely investigated in the literature and has received renewed interest in recent years (see, for example, [1], [25], [41]).

It is well known that in the stability analysis of time-delay systems, Lyapunov–Krasovskii functional is one of the most important tools. Many results about the stability and stabilization of the time-delay systems can be derived by using Lyapunov–Krasovskii functionals (see [22], [39] and the references therein). To analyse the exponential stability of the linear predictor feedback system with a single time-varying input delay, a time-varying Lyapunov–Krasovskii functional was constructed based on the backstepping method for partial differential equations in [13]. By using a transformation of the actuator states, the Lyapunov–Krasovskii functional of the linear predictor feedback system with distributed input delays was obtained in [2]. The linear systems with pointwise and distributed input delays was considered in [27] and the ISS Lyapunov–Krasovskii functionals defined by Pepe and Jiang [29] was constructed for linear systems with pointwise and distributed input delays. The proposed Lyapunov–Krasovskii functionals are helpful in the analysis of the ISS property of the closed-loop time-delay systems. In recent years, many advanced Lyapunov–Krasovskii functionals have been established to the analysis and design for many classes of complex dynamic systems with time delays, for example, dissipative analysis for neural networks with time-varying delays [18], stability analysis of static neural networks with interval time-varying delays [17], stability analysis of linear systems with two delays of overlapping ranges [14], and stability analysis of neural networks with time-varying delays [16]. For more related papers, see [7], [8], [20], [35], [36], [37] and the references therein.

In this paper, we will study the exponential stability of linear systems with multiple input delays by constructing suitable Lyapunov–Krasovskii functionals. To this end, we first transform the closed-loop time-delay system with predictor feedback into an equivalent delay-free system. Then, based on the delay-free linear system, a Lyapunov–Krasovskii functional guaranteeing the exponential stability of the original time-delay system is proposed in terms of the solvability of a set of linear matrix inequalities (LMIs). It is proven that the set of LMIs provided is always solvable provided the delay-free linear system is asymptotically stable. At the same time, we show that the obtained Lyapunov–Krasovskii functional is also an ISS Lyapunov–Krasovskii functional for the predictor feedback control system. A numerical example is worked out to show the effectiveness of the proposed approach. The merit of the proposed approach is that the construction of the Lyapunov–Krasovskii functional does not need the backstepping method for partial differential equation as used in [13] and is simply based on some LMIs which can be efficiently solved by the existing software package.

The rest of this paper is organized as follows. The problem formulation and some preliminary results, including the definition of the exponential stability, Lyapunov–Krasovskii functionals for the predictor feedback control systems and some useful lemmas, are given in Section 2. Section 3 contains the main results of this paper. A numerical example is worked out in Section 4 to indicate how to compute a Lyapunov–Krasovskii functionals of the predictor feedback control system. Finally, Section 5 concludes the paper.

Notation: For any given matrix $A\in {\mathbf{R}}^{n\times m}$, we respectively use $A^{\mathrm{T}}$ and $\lambda \left(A\right)$ to denote its transpose and eigenvalue set (when A is square). For a positive definite matrix P, the symbol ${\lambda }_{\min }\left(P\right)$ and ${\lambda }_{\max }\left(P\right)$ denote respectively its minimal and maximal eigenvalues. For two positive integers p and q, we use $\mathbf{I}\left[p\text{,}q\right]$ to present the set $\{p\text{,}p+1\text{,}\dots q\}$. The symbol $\left|·\right|$ refers to the Euclidean norm. Let $\tau >0$ is a given real number. ${\mathcal{C}}_{n\text{,}\tau }=\mathbf{C}\left(\left[-\tau \text{,}0\right]\text{,}{\mathbf{R}}^n\right)$ present the Banach space of continuous functions mapping the interval $\left[-\tau \text{,}0\right]$ into ${\mathbf{R}}^n$ with the topology of uniform convergence.