Non-fragile controller design for discrete descriptor systems

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Abstract

This paper is concerned with the design problem of non-fragile controller for discrete descriptor systems. The controller gain variations are assumed to be time-invariant and norm-bounded appearing in the controller coefficients. Sufficient conditions for the existence of a static state feedback controller with additive uncertainty and multiplicative uncertainty are obtained in terms of a set of matrix inequalities, respectively. With these conditions, the problems of non-fragile control for discrete descriptor systems are solved. Explicit expressions of desired non-fragile state feedback controllers are also given, and no matrix decomposition is required. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.

Introduction

It is well known that descriptor systems (also known as singular systems, implicit systems, generalized state-space systems, differential-algebraic systems) can naturally describe the dynamics of many practical systems such as power systems, economical systems, robotic systems, chemical processes, etc. In recent years, the problems of stability analysis and stabilization for descriptor systems have attracted a lot of attention and significant advances have been made on these topics (see, e.g. [1], and the references therein). Compared with state-space systems, descriptor systems are more complicated yet richer structures. It has been shown that the study of robust control problem of descriptor systems is much more difficult than that for state-space systems since descriptor systems require not only stability, but also regularity and impulse immunity (for continuous singular systems) and causality (for discrete-time singular systems) simultaneously. Based on a matrix inequality approach or a generalized algebraic Riccati equation approach, the robust control problem for descriptor systems was studied in [2], [3], [4], [5], [6], [7], [8], [9], [10], respectively, and desired robust controllers were designed.

It is known that control systems are designed for robustness with respect to plant parameters, or designed for the optimization of a single performance measure; many designed control systems require accurate controllers. Thus, when implementing a desired controller, all of the controller coefficients are required to be with the exact values as those to be designed, which, however, is not always possible in practical applications since finite word length and round-off errors in numerical computations by computers are frequently encountered. Therefore, it is necessary that the designed controller should be able to tolerate some uncertainty in parameters. Since controller fragility problem has to be considered when implementing a designed controller in practical applications, the non-fragile control problem has been investigated in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. For state-space systems, several recent research works have been devoted to the design problem of non-fragile robust control [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Most of these are derived based on a Riccati matrix equation approach and an LMI approach [29]. The design problem of non-fragile robust controllers of continuous-time descriptor systems was investigated in [26], [27], [28].

In this paper, we deal with the non-fragile control problem for discrete-time descriptor systems. The controller gain uncertainties under consideration are supposed to be time-invariant but norm-bounded. The problem to be addressed is the design of a state feedback controller, which is subject to norm-bounded uncertainty, such that for all admissible uncertainties the resulting closed-loop system is regular, causal and stable. Sufficient conditions for the solvability of the non-fragile control problem for discrete-time descriptor systems are obtained, respectively, for the cases with additive and multiplicative controller uncertainties. The desired controllers can be constructed by solving certain matrix inequalities.

Notation: Throughout this paper, for real symmetric matrices X and Y, the notation XY (respectively, X>Y) means that the matrix XY is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimensions. The superscript “T” represents the transpose. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

Section snippets

Problem formulation and definitions

Consider the following discrete-time descriptor system:Ex(k+1)=Ax(k)+Bu(k),where x(k)∈Rn is the state; u(k)∈Rm is the control input; E, A and B are known real constant matrices with appropriate dimensions and rank En.

The uncontrolled discrete-time descriptor system of Eq. (1) can be written asEx(k+1)=Ax(k).

Throughout this paper, we shall adopt the following concepts.

Definition 1

  • (i)

    The discrete-time descriptor system (2) is said to be regular if det(zEA) is not identically zero.

  • (ii)

    The discrete-time descriptor

Main results

We first introduce the following preliminary results which will be used in the proof of our main results.

Lemma 1

(Xu et al. [3]). Given matrices Ω, Γ and Ξ of appropriate dimensions and with Ω symmetrical, then Ω+ΓF(σ)Ξ+(ΓF(σ)Ξ)T<0.for all F(σ) satisfyingF(σ)F(σ)TIif and only if there exists a scalar ε>0 such thatΞ+εΓΓT+ε-1ΞTΞ<0.

Lemma 2

(Xu and Lam [8]). The discrete-time descriptor system (2) is admissible, if and only if there exist a matrix P>0 and a matrix Q such thatATPA-ETPE+QSTA+ATSQT<0,where SRn×(nr)

Numerical example

In this section, we provide an example to demonstrate the applicability of the proposed method.

Consider a discrete-time singular system described in Eq. (1) with parameters as follows:E=(100.5211000),A=(2.40.21.241.52000),B=(011100121),Ma=(0.10.2-0.1),Na=(0.10.20).

In this example, we assume the uncertain matrix Fa(σ)=sin(σ). It is easy to see that the nominal discrete-singular system is not regular, non-causal, and unstable. The purpose of this example is the design of a state feedback control

Conclusion

In this paper, we have considered the non-fragile controller design problem of discrete-time descriptor systems with state feedback controller uncertainties. Sufficient conditions for the existence of non-fragile state feedback controllers with additive and multiplicative uncertainties have been proposed, respectively. When these conditions are feasible, explicit expressions of a desired state feedback controller have been given. An illustrative example has shown the applicability and validity

Acknowledgement

This work was supported by a grant from National Laboratory of Space Intelligent Control.

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