Robust H reliable control for uncertain switched systems with circular disk pole constraints

https://doi.org/10.1016/j.jfranklin.2012.12.027Get rights and content

Abstract

This paper addresses the issue of optimal robust H reliable control with circular disk pole constraints for switched systems with actuator faults and arbitrary switching rules. The design method of the state feedback controller is proposed, which guarantees that the robust H performance is minimum and the closed poles are located in a specified circular disk. The corresponding parameters of the controller are obtained by using the linear matrix inequalities (LMIs) optimization. Finally, a simulation example is provided to validate the effectiveness of the proposed approach.

Introduction

Hybrid dynamic systems are the systems that interact both with continuous-time and discrete-time sub-systems together [1], [2]. The fault-tolerant control has been developed well [3], [4], [5]. For a class of uncertain nonlinear systems with faults ranging over a finite cover, a unique scheme is designed to simultaneously perform fault isolation and fault-tolerant control [3]. If a stochastic system is of faults by switching diffusion processes (SDP), the fault tolerability analysis in the sense of input-to-state stability of overall SDP would be entirely accomplished [4]. It is worth mentioning that the fault-tolerant control of hybrid systems was systematically given in [5]. H control is also a vital field in system control, and it is widely noted [6], [7]. As we all know, switched system is a class of hybrid systems. Switched systems have gained considerable attention [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Based on the multiple Lyapunov functions method, the problems of stability, L2-gain analysis and H control for switched systems were resolved by Zhao and Hill [10]. Earlier, the robust H control was proposed [11], [12]. The difference between [11], [12] is that the main method of [11] is not the multiple Lyapunov functions method but the LMI approach. As for the finite-time case of switched nonlinear discrete-time systems, several sufficient conditions were proposed to solve the H finite-time boundedness [13]. The robust H control with circular disk pole constraints for uncertain switched systems was proposed for the first time in [14], using LMI. The work of [14] was based on the bounded real lemma which was proposed in [15]. A single sliding surface with an H disturbance attenuation level was constructed in [16], and it satisfied the robust stabilization which was confined by the reduced-order equivalent sliding motion. Based on dynamic output feedback, the H control problem for switched systems was solved in [17]. The H model reduction for a class of discrete-time switched systems was concerned by Gao et al. [18], and the robustness was addressed by Zhang et al. [19]. For a class of switched systems, Zhang and Gao [20] developed the asynchronously switched control by using the average dwell time method. As for time delay switched network systems, the H optimal control was solved in [21]. For discrete-time switched systems, the switching law was designed for global exponential stability in [22]. It is generally known that average dwell time (ADT) switching is of flexibility and effectiveness in switched systems. Based on the ADT approach, the asynchronous H filter for a class of linear discrete-time switched systems was designed [23]. Recently, there are many works on switched systems appeared [24], [25], [26], [27]. For a class of switched neural network systems with time-delays, a state estimation L2L filter was designed for the first time in [24]. By using the ADT approach, the exponential stability and stabilization was studied for a class of discrete-time nonlinear switched systems with time delays in [25]. For the switching T–S fuzzy model of two-wheeled mobile robots, the guaranteed cost control was studied, while the state-feedback controller was designed [26]. In control, the static output feedback control rarely appeared. By using static output feedback, the stabilization problem of a class of linear discrete-time switched systems was settled [27]. Markovian switching problems are also hot issues in control. Based on the proportional and derivative sliding mode observer technique, the fault-tolerant control problem for a class of nonlinear Markovian jump systems with state delays was studied in [28].

As for regional pole assignment, a circular disk pole assignment approach was proposed in [29], in terms of the Riccati equation. LMI region was proposed by Chilali and Gahinet [30] for the first time. It follows from the definition of LMI region that circular disk is an LMI region, so it is possible to place all the poles of a system in a circular disk by using the approach in [30]. For a class of power systems with discrete actuator failures, the reliable circular disk pole assignment and guaranteed-cost reliable control were concerned in [31]. Mansouri et al. [14] applied this theory to the H control of uncertain switched systems for the first time. It is known that a linear system is called D stability if all the poles lie in an LMI region D. Here, the circular disk is considered.

In this paper, we focus on the robust H reliable control problem for uncertain switched systems having the following properties: (1) the uncertainties are considered in the input matrix, which is different from most of the existing results that consider uncertainties in the system matrix only, e.g. [32], [33]; (2) the switching rule is arbitrary; (3) each model may have the actuator faults; and (4) all the poles of the system are required to be placed in a specified circular disk. The main contributions of this paper are as follows:

Firstly, a novel circular disk pole assignment approach is proposed, which goes together with a bounded real lemma that can guarantee the specified H performance and D stability of the closed-loop system. By placing the closed-loop poles, the time and frequency of oscillation are settled, and the dynamical response assigning bounds for overshoot is improved.

Secondly, a novel method dealing with a class of the uncertain matrices multiplication phenomena is proposed, which has broad applicable prospects. For example, we can apply it to flight control and electrical system, in which both uncertainties and faults exist. The method is emerged to guarantee the robustness and fault-tolerability of the system.

Thirdly, a sufficient condition for the existence of a switched state feedback controller that achieves robust H reliable control goal is provided in terms of LMIs, and the corresponding optimal problem is also settled.

The rest of the paper is organized as follows. System model, problem formulation and several lemmas are given in Section 2. The main lemmas and the solution of the problem are proposed in Section 3. To illustrate the effectiveness of the proposed method, a simulation is presented in Section 4.

Section snippets

Problem formulation

Consider the following uncertain switched system with actuator faults:x˙(t)=(Aσ+ΔAσ)x(t)+(Bσ+ΔBσ)uf(t)+B1σw(t),z(t)=Cσx(t)+Dσw(t),where σ(t):R+N¯={1,2,,N} is a switching rule, x(t)Rn is the state vector, uf(t)Rm is the control input vector of actuator fault, w(t)Rp is the disturbance input vector, z(t)Rτ is the regulated output vector, AiRn×n, BiRn×m, B1i(t)Rn×p, CiRτ×n, and DiRτ×p are known constant matrices, ΔAi and ΔBi are uncertain matrices with the following forms: (ΔAiΔBi)=EiHi(

Main results

Lemma 4

Let ARn×n and D(q,r) be a circular disk in complex plane with center (q,0) and radius r. Then, λ(A)D(q,r) if and only if there exists a positive definite symmetric matrix XRn×n such that(q2r2)X+qAX+qXATXATAXX<0.

Proof

The circular disk D(q,r) can be described as follows: D(q,r)={zC:|z+q|<r}.Let x=Rez and y=Imz, then z+z¯=2x, zz¯=2y. We have|z+q|<r|x+iy+q|<r(x+q)2+y2<r2x2+y2+2qx+q2r2<0q2r2+q(z+z¯)+zz¯<0q2r2+qz+qz¯z¯z1<0.i.e. q2r2001+zq010+z¯q010T<0.Let L=q2r2001and M=q010,then L is

Numerical example

Consider the uncertain switched system of the system (1) with N=2 in the fault-free case as follows [11]: A1=110010012,A2=201110001,B1=200.11.21000.50.5,B2=0.40.120.70000.20.1,B11=0.2500,B12=000.25,C1=[010],C2=[100],D1=D2=0,E1=000.200.200.200,E2=0.10000.20000.1,H1=sint000cost000sint,H2=cost000sint000cost,G11=0.51000.50000.5,G12=0.30000.20000.4,G21=0.10000.30.1000.4,G22=0.4000.20.3000.10.2.The actuator fault model is described as Eq. (3), with Fd1=0.40000.50000.3,Fu1=1.20001.40001.1,

Conclusions

The robust H reliable control with circular disk pole constraints problem has been solved for uncertain switched systems with actuator faults, and the corresponding optimal problem has also been worked out. In terms of LMI, we have designed the optimal switched state feedback controller which can guarantee the H performance is minimized for all uncertainties and admissible actuator faults.

It is important to note that there are still many questions of settling the uncertain matrices

Acknowledgement

The work is supported by the National Natural Science Foundations of China under Grant No. 61034005, No. 61104116 and No. 61273171, the Doctoral Fund of Ministry of Education of China under Grant No. 20113218110011, and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References (37)

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