Elsevier

Signal Processing

Volume 93, Issue 7, July 2013, Pages 1804-1812
Signal Processing

Non-fragile H2 reliable control for switched linear systems with actuator faults

https://doi.org/10.1016/j.sigpro.2013.01.011Get rights and content

Abstract

This paper focuses on the issue of non-fragile H2 reliable control for switched linear systems with actuator faults and circular disk pole constraints. The state feedback and the switching law are designed to guarantee that the closed-loop H2 performance is less than a specified scalar and all closed-loop poles are located in a specified circular disk, in terms of linear matrix inequalities (LMIs). An example is provided to illustrate the effectiveness of the proposed method.

Highlights

► A new method of describing the continuous fault matrix is adopted. ► A new method settling non-fragile reliable control is proposed. ► H2 control and pole assignment are combined together. ► The state-dependent switching law is designed for H2 control.

Introduction

Switched systems, a vital class of hybrid systems, have received more and more attentions during the past decade [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. The basic developments of switched systems are located in three basic problems [1]: asymptotical stability of the switched system with arbitrary switching, stability for certain useful classes of switching sequences, and construction of asymptotically stabilizing switching signals for a switched system. Almost all the control problems of switched systems are developed on the basis of the basic problems. H control problem of switched systems is solved in [2]. Based on the multiple storage function approach, the passivity theory and stability theory of switched systems are provided for the first time in [3]. The work in [4] can be extended to switched systems. H filtering [5], [6], [7], stabilization [8], [9], [10], fault-tolerant control [11], [12] and fault recoverability analysis [13] for switched systems have received good developments. By means of linear matrix inequality (LMI) approach and average dwell time technique, the fault detection problem is solved [14] for a discrete-time switched linear system with time-varying delays. The robust fault detection problem is settled [15] for a class of uncertain switched linear time-delay systems with arbitrary switching signal. The delay-depend exponential H control problems for continuous- and discrete-time switched systems are settled in [16] and [17], respectively. Based on reduced-order observer, the delay-deoendent H control is concerned in [18]. For a class of discrete-time nonlinear switched systems, the exponential stability and stabilization problems are solved in [19], by using LMI approach and Lyapunov functional technique. Based on switching generalized gramians, the switching Petrov–Galerkin projection is determined for the model reduction of switched systems [20]. By using observer technique, the state estimation and sliding mode control (SMC) problems are settled for a class of uncertain switched systems [21]. As another class of hybird systems, Markovian system is also concerned in the SMC area, recently [22], [23], [24]. For a class of discrete-time linear networked control systems with Markovian jump, the stabilization problem is investigated by using interactive LMI approach [22]. By using the proportional and derivative sliding mode observer technique, the fault-tolerant control problem for a class of nonlinear Markovian jump systems with sensor fault is concerned in [23]. For a class of Markovian jump singular time-delay systems, the problems of L2 control and SMC are solved in [24] by using LMI approach.

Only a few results are devoted to fault tolerant control of switched linear systems. The finite-time stability of switched systems with actuator discrete faults is concerned in [25], by using average dwell-time approach. For mixed-fault model, the integrity of the switched systems with actuator faults is analysed in [26]. The issue of fault estimator design for switched systems is discussed in [27], by using LMI approach.

H2 control is an important problem in the control field. For non-switched systems, H2 control has been developed well [28]. But it is rarely reported for switched systems [29], [30], so is non-fragile control. Non-fragile control is to design a controller which is not sensitive to the disturbance in the feedback. There are many research results about non-fragile control, e.g. [31], [32], [33] and the references therein.

Now we focus on pole assignment. Pole assignment is closely related to D-stability. If all the poles of the system matrix lie in a specified region D, the system is so-called D-stable [34]. The point is that the region D must be a subregion of the asymptotically stable region of the system. The universal pole assignment is proposed in [34] for the first time, and it also promotes the development of pole assignment greatly.

Based on the statement above, we can see that there is still much room for mobility development in developing H2 control for switched systems. In many practical systems, such as power systems and flight control systems, an important measure of peak values that produced by finite energy content signals is H2 norm, so that H2 control is widely concerned. However, the non-fragile H2 control problem of switched linear systems with actuator faults has not been solved. For a class of switched linear systems with actuator faults, the non-fragile H2 reliable control is concerned in this paper.

This paper is organized as follows. Section 2 formulates the main purpose of this paper and prepares some useful theories. Section 3 proposes a new approach of settling a class of matrix multiply phenomenon, and then provides an approach to design the switching signal and the non-fragile reliable controller which can guarantee that the closed-loop system is of a specified H2 performance and all the closed-loop poles lie in a specified circular disk. Finally, the effectiveness and practicality of this method are shown by a practical example in Section 4.

Section snippets

Problem statement and preliminaries

Consider the following switched linear system with actuator faults:x˙(t)=Aσx(t)+B1σw(t)+B2σuσf(t),z(t)=Cσx(t)+Dσuσf(t),where x(t)Rn is the state; uif(t)Rmi is the control input vector of actuator fault for the i-th mode of the system; w(t)Rl is the exogenous disturbance input; z(t)Rp is the regulated output. σ(t):R+N¯={1,2,,N} is a switching rule to be designed. Furthermore, σ(t)=i implies that the i-th subsystem is activated, and AiRn×n, B1iRn×l, B2iRn×mi, CiRp×n and DiRp×mi(iN¯)

Main results

Lemma 4

Let B, E, X and Y be matrices of appropriate dimensions, and H be an uncertain matrix such that HTHI, and Σ be a diagonal uncertain matrix satisfying ΣTΣ=Σ2I. Then, there exist a positive definite diagonal matrix U and a positive scalar ε such thatUεEET>0andBΣY+YTΣTBT+BΣEHX+XTHTETΣTBTBUBT+ε1XTX+YT(UεEET)1Y.

Proof

It follows from Lemma 1 that there exists a positive scalar ε such thatBΣEHX+XTHTETΣTBTε1XTX+εBΣEETΣTBT.

Obviously, there exists a positive definite diagonal matrix U satisfies the

Example

Consider the linear switched system of the system (1) with N=3 in the fault-free case as follows [36]:A1=100.86300001.1605.43005.48806.590035.893024.815115.160,A2=100.86300001.1605.43005.48806.590035.893024.8151.160,A3=74.59739.034009.2448.050004.0423.0131.160000020.160,B11=B12=B13=0.08300000.33300,B21=1.1430000009.500,B22=B23=1.143000,C1=10000000100,C2=C3=100000000.

The parameters of perturbation of the controller (2) are given byE1=0.310.14,E2=E3=0.6,H1=cost,H2=H3=sint,G1

Conclusions

Non-fragile H2 reliable control with circular disk pole constraints problem has been solved for switched systems with actuator faults based on state feedback. In terms of LMI, we have designed the switched state feedback controller which can guarantee the H2 performance is smaller than a specified scalar for all uncertainties in the state feedback and admissible actuator faults. The effectiveness of the proposed result has been shown in the given example.

Our future work is to extend the method

Acknowledgement

The work is supported by the National Natural Science Foundations of China under Grant Nos. 61034005, 61104116 and 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)

  • G. Yang et al.

    Non-fragile H filter design for linear continuous-time systems

    Automatica

    (2008)
  • L. Liu et al.

    H non-fragile observer-based sliding mode control for uncertain time-delay systems

    Journal of the Franklin Institute

    (2010)
  • X. Chang et al.

    Non-fragile design for nonlinear continuous-time systems with D stability constraints

    Signal Processing

    (2012)
  • Y.S. Lee et al.

    Delay-dependent robust H control for uncertain systems with a state-delay

    Automatica

    (2004)
  • V.F. Montagner et al.

    State feedback control of switched linear systemsan LMI approach

    Journal of Computational and Applied Mathematics

    (2006)
  • D. Liberzon et al.

    Basic problems in stability and design of switched systems

    IEEE Control Systems Magazine

    (1999)
  • D. Wang et al.

    Delay-dependent exponential H filtering for discrete-time switched delay systems

    International Journal of Robust and Nonlinear Control

    (2012)
  • H. Yang et al.

    Stabilization of switched nonlinear systems with all unstable modesapplications to multi-agent systems

    IEEE Transactions on Automatic Control

    (2011)
  • Cited by (0)

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