Robust D-stability analysis for linear uncertain discrete singular systems with state delay

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Applied Mathematics Letters19 (2006) 197–205

www.elsevier.com/locate/aml

Robust D-stability analysis for linear uncertain discrete singular

systems with state delay

Shinn-Horng Chen

a

, Jyh-Horng Chou

b,∗

a_{Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road,}

Kaohsiung 807, Taiwan, ROC

b_{Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology,}

1 University Road, Yenchao, Kaohsiung 824, Taiwan, ROC

Received 21 May 2004; received in revised form 6 May 2005; accepted 12 May 2005

Abstract

In this work, by using the maximum modulus principle and the spectral radii of matrices, a new robust D-stability (i.e., robust eigenvalue clustering in a specified circular region) condition is proposed to ensure that, for all admissible structured parameter uncertainties, the linear discrete singular system with state delay is regular, causal and D-stable. The proposed criterion is mathematically proved to be less conservative than the existing one reported very recently.

Keywords: D-stability robustness; Discrete singular systems; Structured parameter uncertainties; State delay

1. Introduction

Linear discrete singular delay systems can be found in many practical applications, such as electrical networks, large-scale systems, constrained robots, and economical systems [1]. On the other hand, to ensure both stability robustness and a certain performance robustness, it is important to guarantee that the eigenvalues of a linear time-invariant system under parameter perturbations remain in a specified region. Therefore, very recently, Xu and his associates [2,3] studied the robust D-stability (i.e., robust

∗_{Corresponding author. Tel.: +886 7 6011000x2218; fax: +886 7 6011066.}

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198 S.-H. Chen, J.-H. Chou / Applied Mathematics Letters 19 (2006) 197–205

eigenvalue clustering in a specified circular region) problem of a linear discrete singular delay system with structured (elemental) parameter uncertainties. Here it should be emphasized that the robust D-stability analysis of linear uncertain discrete singular delay systems should consider not only the D-stability robustness but also system regularity and causality simultaneously. Under the assumption that the linear discrete nominal singular system, E x(k+1) = Ax(k) denoted by (E, A), is regular, causal and D-stable, by using the pulse-response sequence matrix approach of Chou [4], Xu and Lam [3] presented the robust D-stability condition, which is suitable for more general cases than that previously considered by Xu et al. [2], to guarantee the D-stability robustness of the following linear discrete singular time-delay system denoted by(E, A + ˜A) with structured parameter uncertainties:

E x(k + 1) = (A + A)x(k) + (Ad+ Ad)x(k − τ), E x(0) = Ex0, (1)

where E ∈ Rn×n, A ∈ Rn×n, Ad ∈ Rn×n, x(k) ∈ Rn,A and Ad denote the n× n time-invariant

structured (elemental) parameter uncertain matrices, andτ ≥ 1 is a known positive integer representing the time delay. Here the matrix E may be a singular matrix with rank(E) ≤ n. In many applications, the matrix E is a structural information matrix rather than a parameter matrix, i.e., the elements of E contain only structural information regarding the problem considered. Suppose the uncertain matricesA and

Ad are bounded by the following inequalities:

|A| ≤ U and |Ad| ≤ Ud, (2)

where U and Ud are given nonnegative constant matrices and represent highly structured information.

The purpose of this work is, by using the maximum modulus principle [5] and the spectral radii of matrices, to propose another new approach for studying the robust D-stability problem of the linear discrete uncertain singular delay system(E, A + ˜A) in (1) under the same assumption as in the work by Xu and Lam [3]. This work is organized as follows. The main result is presented in Section 2. By mathematical analysis, a comparison between the degree of conservativism of the two sets of sufficient conditions which are, respectively, proposed in this work and by Xu and Lam [3] is given inSection 3. A numerical example is also given for illustration in this section. Finally,Section 4offers some conclusions.

2. Main result

In this work, as in the work by Xu and Lam [3], we consider the linear discrete uncertain singular delay system(E, A + ˜A) in (1) under the assumption that the linear discrete nominal singular system

(E, A) is regular, causal and D-stable. Our problem is to determine the condition such that, under the

aforementioned assumption, the linear discrete uncertain singular delay system (E, A + ˜A) is still regular, causal and D-stable.

Before we analyze the D-stability robustness of the linear discrete uncertain singular delay system

(E, A + ˜A), the following definition and lemmas need to be introduced.

Definition ([6,7]). The linear discrete singular system E x(k +1) = Ax(k) is termed regular and causal

if det(zE − A) is not identically zero and if rank(E) = degree of det(zE − A) in the z-plane, where

E ∈ Rn×n, A∈ Rn×n, x(k) ∈ Rn, and rank(E) ≤ n.

Lemma 1 ([6,7]). The linear discrete singular system(E, A) is said to be stable, regular and causal if and only if the following two conditions are satisfied:(i) all the eigenvalues of det(zE − A) = 0 lie inside the unit circle of the z-plane, and(ii) (zE − A)−1is proper.

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204 S.-H. Chen, J.-H. Chou / Applied Mathematics Letters 19 (2006) 197–205 |A| ≤  00.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01   , and |Ad| ≤  00.01 0.01 0.01 0.01 0.01 0 0.01 0.01   .

All the finite eigenvalues of the linear discrete nominal singular system (E, A) in Eq. (18) are 0.2 ± ˜j0.5292 which are located inside a specified circular region D(0.2, 0.8) centered at 0.2 + ˜j0 with radius 0.8.

By using the sufficient condition in (13) proposed by Xu and Lam [3], we have

r[ f−1˜T (U + δ−1(|Ad| + Ud))] = 1.2116 < 1.

Then, no conclusion can be drawn. That is, the sufficient condition of Xu and Lam [3] cannot be applied in this example.

Now, applying the sufficient condition in (5), we have

r[|J|U] = 0.01 < 1,

and

r[ f−1H˜(U + γ−1(|Ad| + Ud))] = 0.9894 < 1.

Therefore, the proposed sufficient condition in (5) is satisfied. This implies that the linear discrete uncertain singular time-delay system (18) is still regular and causal, and has all its finite eigenvalues retained within the same circular region as the linear discrete nominal singular system(E, A) does. It is obvious that the proposed sufficient condition (5) can overcome the conservatism of the sufficient condition (13) given by Xu and Lam [3]. Note that this fact that the proposed sufficient condition (5) is less conservative than that given by Xu and Lam [3] has been proved by mathematical analysis in this section.

4. Conclusions

Under the assumptions that the linear discrete singular time-delay system is regular and causal, and has all its finite eigenvalues lying inside a specified circular region, a new sufficient condition has been proposed for preserving the assumed properties when the structured (elemental) parameter uncertainties are added into the linear discrete singular time-delay system. When all the finite eigenvalues lie inside the unit circle of the z-plane, the proposed criterion will become the stability robustness criterion. By mathematical analysis, the proposed criterion has been proved to be less conservative than that presented by Xu and Lam [3].

Acknowledgement

This work was supported by the National Science Council, Taiwan, Republic of China, under grant number NSC92-2213-E151-006.

References

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