Robust $H_2$ control of continuous-time Markov jump linear systems

Abstract

This paper is concerned with the problem of designing robust $H_2$ state-feedback controllers for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty. Based on the parameter-dependent Lyapunov function approach, a new method for designing robust $H_2$ controllers is presented in terms of solutions to a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed design method.

Introduction

In control engineering, there exists a wide class of dynamic systems subject to random abrupt variations in their structures. This is in part due to the fact that very often dynamic systems are inherently vulnerable to component failures or repairs, abrupt variations of the operating point of nonlinear plant, etc. Markovian jump linear systems (MJLSs) are the suitable mathematical models to represent the class of systems. MJLSs belong to the category of hybrid systems with finite operation models. Each operation mode corresponds to some dynamic system, and the model transitions from one to another are governed by a Markov process. The stability analysis, stabilization, optimal control and $H_{\infty }$ control of MJLSs have been widely studied, see Fang and Loparo (2002), do Val and Costa (2005) and Zhang, Huang, and Lam (2003) and the references therein. In Rami and Ghaoui (1996), linear matrix inequality (LMI) method is applied to compute the optimal state-feedback control law for MJLSs. Recently, the stability and the stabilizability of a class of nonlinear systems with Markovian jump and Wiener process have been theoretically addressed by Boukas (2005). Moreover, output-feedback control problems are studied in de Farias, Geromel, do Val, and Costa (2000) and do Val, Geromel, and Goncalves (2002), respectively by dynamic output feedback and incomplete observations of the Markov state.

Robust control problem for MJLSs against parameter uncertainties of systems has been one of the most challenging problems and received considerable attention from control engineers and scientists, see Ghaoui and Rami (1996), Shi and Boukas (1997), and the references therein. In Mahmoud and Shi (2002) and Shi, Boukas, and Agarwal (1999) robust $H_{\infty }$ state-feedback controller design approaches for discrete-time MJLSs are given, respectively, using Riccati equation approach and LMI technique. For continuous-time MJLSs subject to norm-bounded parameter uncertainties in system matrices, linear quadratic (LQ) control synthesis of MJLSs is formulated by means of LMIs in Bernard, Dufour, and Bertrand (1997) and $H_{\infty }$ controller design is studied in de Farias, Geromel, and do Val (2002), whereas MJLSs subject to polytopic-type parameter uncertainty have been treated in, e.g., Boukas and Liu (2000) and Pakshin and Mitrofanov (2005). With respect to the uncertainty on the transition Markov probability matrix of MJLSs, stability analysis and synthesis are respectively considered in Costa and Marques (1998), Costa, do Val, and Geromel (1999) and Karan, Shi, and Kaya (2006). For the parameter uncertainty on both the system matrices and the mode transition rate matrix, a sufficient condition for designing robust state-feedback controllers is proposed in Xiong, Lam, Gao, and Ho (2005). Most of the above-mentioned results are based on a single Lyapunov function, i.e., independent of uncertainty parameters. These methods, referred to as uncertainty-independent, are numerically appealing. However, since a common Lyapunov function is used to ensure stability for all admissible uncertainties, they can be quite conservative. In order to reduce the conservatism, by introducing slack variables to separating system matrix and Lyapunov matrix and based on a parameter-dependent Lyapunov function approach, stability analysis and stabilization of discrete-time MJLSs subject to polytopic-type parameter uncertainty are given in de Souza (2006). For continuous case, de Souza and Coutinho (2006) also give a relaxing condition based on parameter-dependent Lyapunov function for stability analysis. However, by the authors’ knowledge, the control synthesis for continuous-time MJLSs has not been fully addressed based on parameter-dependent Lyapunov function approach. For continuous-time linear systems with polytopic-type parameter uncertainty, a parameter-dependent Lyapunov function approach to stability analysis and control synthesis is given in Apkarian, Tuan, and Bernussou (2001), which is a continuous-time version of the results for discrete-time case in de Oliveira, Bernussou, and Geromel (1999). Differently from the discrete case, one cannot prove that the parameter-dependent Lyapunov function approach generalizes the one based on a single Lyapunov function. But, the parameter-dependent Lyapunov function approach potentially is less conservative because more slacking variables are involved in the LMI formulation. In this paper, a new technique of improving the one given in Apkarian et al. (2001) is developed by adding more slacking variables, and exploited to derive an LMI-based approach to robust $H_2$ control synthesis for continuous MJLSs via parameter-dependent Lyapunov functions. A numerical example is given to illustrate the effectiveness of the proposed approach.

This paper is organized as follows. Section 2 presents system description and problem statement. In Section 3, an LMI-based robust $H_2$ controller design method is presented. In contrast to the existing results, the effectiveness of the proposed method is illustrated by a numerical example in Section 4. Finally, in Section 5, conclusions are given.