Stability robustness of linear output feedback systems with both time-varying structured and unstructured parameter uncertainties as well as delayed perturbations

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Journal of the Franklin Institute 342 (2005) 213–234

Stability robustness of linear output feedback

systems with both time-varying structured and

unstructured parameter uncertainties as well as

delayed perturbations

Shinn-Horng Chen

a,

, Jyh-Horng Chou

b

, Liang-An Zheng

a

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

Abstract

This paper investigates the stability robustness of linear output feedback systems with both time-varying structured (elemental) and unstructured (norm-bounded) parameter uncertain-ties as well as delayed perturbations by directly considering the mixed quadratically coupled uncertainties in the problem formulation. Based on the Lyapunov approach and some essential properties of matrix measures, two new sufficient conditions are proposed for ensuring that the linear output feedback systems with delayed perturbations as well as both time-varying structured and unstructured parameter uncertainties are asymptotically stable. The corresponding stable region, that is obtained by using the proposed sufficient conditions, in the parameter space is not necessarily symmetric with respect to the origin of the parameter space. Two numerical examples are given to illustrate the application of the presented sufficient conditions, and for the case of only considering both the delayed perturbations and time-varying structured parameter uncertainties, it can be shown that the results proposed in this paper are better than the existing one reported in the literature.

Keywords: Stability robustness; Linear output feedback systems; Time-varying structured and unstructured uncertainties; Delayed perturbations

www.elsevier.com/locate/jfranklin

Corresponding author. Tel.: +886 7 3814526x5326; fax: +886 7 3831373. E-mail address: shchen@cc.kuas.edu.tw (S.-H. Chen).

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1. Introduction

In general, a mathematical description is only an approximation of the actual physical system and deals with ﬁxed nominal parameters. Usually, these parameters are not known exactly due to imperfect identiﬁcation or measurement, aging of components and/or changes in environmental conditions. Thus, it is almost impossible to get an exact model for the system due to the existence of various parameter uncertainties. Moreover, time-delay is commonly encountered in various engineering systems, such as chemical processes, long transmission lines, pneumatic systems, hydraulic systems, electric networks, and so forth. Here, we consider linear uncertain delay systems with time-varying uncertain parameters in the system matrix, input matrix and output matrix. Because the output feedback controller design is usually based on the nominal values of these matrices, it is interesting to know whether the closed-loop system remains asymptotically stable in the presence of delayed perturbations and time-varying uncertain parameters. Applying those

previous robust stability analysis results[1–8] to solve this problemis not easy, in

that after output feedback, there will be coupled terms of parameters in the closed-loop system matrix because of the uncertain parameters in both input and output

matrices [9]. Though we may regard these coupled terms as new independent

parameters if we insist on using previous robust stability analysis results, Su and

Fong[9]and Tseng et al.[10]have showed that a conservative analysis conclusion

may be reached. Therefore, Tseng et al. [10] applied the structured singular value

technique to solve the robust stability problemof linear state delayed systems with constant output feedback in the presence of time-varying uncertain parameters by directly considering the coupled terms in the problem formulation. Here it should be noticed that only the article of Tseng et al.[10] has studied the robust stability of linear systems with quadratically coupled structured uncertainties and delayed perturbations. That is, research on the stability robustness of linear output feedback systems with time-varying uncertain parameters and delayed perturbations by directly considering the coupled terms in the problem formulation is considerably rare and almost embryonic.

On the other hand, it is well-known that an approximate system model is always used in practice and sometimes the approximation error should be covered by introducing both structured (elemental) and unstructured (norm-bounded)

para-meter uncertainties in control system analysis and design [11]. That is, it is not

unusual that at times we have to deal with a system simultaneously consisting of two parts: one part has only structured parameter uncertainties, and the other part has unstructured parameter uncertainties. But, to the authors’ best knowledge, none of the research works published in the literature proposes any robust stability criterion to study the problemof stability robustness for linear output feedback systems with both time-varying structured and unstructured parameter uncertainties as well as delayed perturbations by directly considering the mixed quadratically coupled uncertainties (i.e., quadratically coupled structured, quadratically coupled unstruc-tured, and coupled structured–unstructured uncertainties appearing together) in the problemformulation.

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Therefore, the purpose of this paper is to investigate the robust stability pro-blem of linear output feedback systems with both time-varying structured and unstructured parameter uncertainties as well as delayed perturbations by directly considering the mixed quadratically coupled uncertainties in the

pro-blemformulation. The main results are presented in Section 2. In Section 3,

two numerical examples are given to illustrate the application of the proposed sufﬁcient conditions. For the case of linear output feedback systems only subject to both structured parameter uncertainties and delayed perturbations, a

conservatismcomparison between the criteria, which are proposed in this paper

and by Tseng et al. [10], is also given in this section. Finally, Section 4 offers

some conclusions.

2. Robust stability analysis

Consider the linear uncertain systemwith the state-space model _ xðtÞ ¼ AxðtÞ þX h l¼1 Xm i¼1 kiðtÞAlixðt tlÞ þBuðtÞ þ Xh l¼1 Xm i¼1 kiðtÞBliuðt tlÞ; (1) yðtÞ ¼ CxðtÞ; (2)

where x 2 Rn is the state vector, y 2 Rp is the output vector, u 2 Rq is the input

vector, and A ¼ A0þ Xm i¼1 kiðtÞAiþ eAðtÞ; B ¼ B0þ Xm i¼1 kiðtÞBiþ eBðtÞ; C ¼ C0þ Xm i¼1 kiðtÞCiþ eCðtÞ ð3Þ

are the systemmatrices, kiðtÞ is the ith time-varying uncertain parameter, and m is

the number of independent time-varying uncertain parameters. The parameters tlX0 ðl ¼ 1; 2;. . . ; hÞ are scalar numbers representing delays in the state, and the

scalar number h is the number of delays. The matrices Ai; Bi; Ci; Ali and Bli of

suitable dimensions are the structural inﬂuence matrices of kiðtÞ; and are assumed to

be known. The time-varying unstructured uncertain matrices eAðtÞ; eBðtÞ and eCðtÞ are assumed to be bounded, i.e.,

k eAðtÞkpeb₁; k eBðtÞkpeb₂ and k eCðtÞkpeb₃; (4)

where eb₁; eb₂; and eb₃ are non-negative real constant number, and k k denotes any matrix norm.

Any dynamic controllers of order w for the system(1) and (2) can be viewed

as a static output feedback gain of an augmented system with dimension n þ w[9],

so, in this paper, we only discuss the static output feedback gain controllers. Thus, the closed-loop systemequations of the linear uncertain systemcan be

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expressed as _ xðtÞ ¼ A0þB0KC0þ Xm i¼1 kiðtÞðAiþB0KCiþBiKC0Þ " þ X m i¼1 Xm j¼1 kiðtÞkjðtÞBiKCj þ eAðtÞ þ B0K eCðtÞ þ eBðtÞKC0þ eBðtÞK eCðtÞ þX m i¼1 kiðtÞðBiK eCðtÞ þ eBðtÞKCiÞ # xðtÞ þ X h l¼1 Xm i¼1 kiðtÞðAliþBliKC0Þ þ Xm i¼1 Xm j¼1 kiðtÞkjðtÞBliKCj " # xðt tlÞ þ X h l¼1 Xm i¼1 kiðtÞBliK eCðtÞxðt tlÞ ¼ A þX m i¼1 kiðtÞEiþ Xm i¼1 Xm j¼1 kiðtÞkjðtÞEijþF ðtÞ " # xðtÞ þ X h l¼1 Xm i¼1 kiðtÞDliþ Xm i¼1 Xm j¼1 kiðtÞkjðtÞDlijþFlðtÞ " # xðt tlÞ; ð5Þ

where K denotes the output feedback gain matrix, A ¼ A0þB0KC0; Ei¼AiþB0KCiþBiKC0; Eij¼BiKCj; Dli¼AliþBliKC0; Dlij ¼BliKCj; F ðtÞ ¼ eAðtÞ þ B0K eCðtÞ þ eBðtÞKC0þ eBðtÞK eCðtÞ þX m i¼1 kiðtÞðBiK eCðtÞ þ eBðtÞKCiÞ ð6Þ and FlðtÞ ¼ Xm i¼1 kiðtÞBliK eCðtÞ: (7)

Before we analyze the robust stability problemof the linear closed-loop uncertain system in Eq. (5), the following deﬁnition and lemmas need to be introduced ﬁrst.

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Deﬁnition (Desoer and Vidyasagar [12]). The measure of a matrix W 2 Cn n is deﬁned as mðW Þ ¼ lim y!0 ðkI þ yW k 1Þ y ;

where k k is the induced matrix norm on Cn n:

Lemma 1 (Desoer and Vidyasagar[12]). The matrix measures of the matrices W and

V, mðW Þ and mðV Þ are well defined for any norm and have the following properties: (i) mðI Þ ¼ 1; for the identity matrix I;

(ii) kW kp mðWÞpReðlðWÞÞpmðWÞpkWk; for any norm k k and any matrix

W 2 Cn n;

(iii) mðW þ V ÞpmðWÞ þ mðVÞ; for any two matrices W; V 2 Cn n;

(iv) mðgW Þ ¼ gmðW Þ; for any matrix W 2 Cn nand any non-negative real number g;

where lðW Þ denotes any eigenvalue of W, and ReðlðW ÞÞ denotes the real part of lðW Þ:

Lemma 2 (Chen et al. [13]). For any go0 and any matrix W 2 Cn n; mðgW Þ ¼

gmðW Þ:

In this paper, we assume an output feedback gain matrix K has been previously designed to make A a stable matrix; therefore there exists a symmetric positive deﬁnite matrix P that is the unique solution of the Lyapunov equation

ATP þ PA ¼ 2Q (8)

for some symmetric positive deﬁnite matrix Q: In what follows, we present two new sufﬁcient criteria for ensuring that the linear closed-loop uncertain systemin (5) remains asymptotically stable.

Theorem 1. Assume that matrix A in Eq. (5) is stable; the linear closed-loop uncertain system (5) remains asymptotically stable if the following condition is satisfied:

Xm i¼1 kiðtÞfiþ Xm i¼1 Xm j¼1 kiðtÞkjðtÞfijþ Xm i¼1 Xm j¼1 Xm r¼1 kiðtÞkjðtÞkrðtÞfijr þ X m i¼1 Xm j¼1 Xm r¼1 Xm s¼1 kiðtÞkjðtÞkrðtÞksðtÞfijrsþ2 b þ Xh l¼1 b_l ! kPkkH1ko1;ð9Þ

where P ¼ P þ bI is a symmetric positive definite matrix, b is a real number, Dl ðl ¼

1; 2;. . . ; hÞ are any symmetric positive definite matrices, H ¼ 2Q 2bA_sPh_l¼1Dl;

As¼ ðA þ A

T

Þ=2; and

Pi¼PEiþETiP; (10)

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fi¼ mðH1=2PiH1=2Þ; for kiðtÞX0; mðH1=2PiH1=2Þ; for kiðtÞo0; ( (12) f_ij¼ m H1=2 PijþP h l¼1 PDliD 1 l DTljP H1=2 ; for kiðtÞkjðtÞX0; m H1=2 _P ijþP h l¼1 PDliD 1 l DTljP H1=2 ; for kiðtÞkjðtÞo0; 8 > > > < > > > : (13) f_ijr¼ m P h l¼1 H1=2_PðD liD 1 l DTljr þDlijD 1 l DTlrÞPH1=2 ; for kiðtÞkjðtÞkrðtÞX0; m P h l¼1 H1=2PðDliD 1 l D T ljr þDlijD 1 l DTlrÞPH1=2 ; for kiðtÞkjðtÞkrðtÞo0; 8 > > > > > > > > > > > < > > > > > > > > > > > : (14) fijrs¼ m P h l¼1 H1=2PDlijD 1 l DTlrsPH1=2 ; for kiðtÞkjðtÞkrðtÞksðtÞX0; m P h l¼1 H1=2PDlijD 1 l DTlrsPH1=2 ; for kiðtÞkjðtÞkrðtÞksðtÞo0; 8 > > > < > > > : (15) b ¼ eb₁þ eb₂kKC0k þ eb3kB0Kk þ eb2eb3kKk þ X m i¼1 jkiðtÞjðeb3kBiKk þ eb2kKCikÞ ð16Þ and b_l¼X m i¼1 jkiðtÞjkBliKkeb3: (17)

Proof. See Appendix A. &

Theorem 2. Assume that matrix A in (5) is stable; the linear closed-loop system (5) remains asymptotically stable if the following condition is satisfied:

Xm i¼1 kiðtÞjiþ Xm i¼1 k2_iðtÞmðQ1=2Q_iiQ1=2Þ þ X m i¼1 Xm j¼2 j4i kiðtÞkjðtÞjijþ2 b þ Xh l¼1 b_l ! kPkkQ1ko1; ð18Þ

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where j_i¼ mðQ 1=2_Q iQ1=2Þ; for kiðtÞX0; mðQ1=2Q_iQ1=2Þ; for kiðtÞo0; ( (19) j_ij¼ mðQ 1=2_Q ijQ1=2þQ1=2QjiQ1=2Þ; for kiðtÞkjðtÞX0; mðQ1=2Q_ijQ1=2Q1=2Q_jiQ1=2Þ; for kiðtÞkjðtÞo0; ( (20) and b and bl are given in Eqs. (16) and (17), respectively, in which

Q ¼ H 0 0 0 0 D1 0 0 0 0 D2 0 0 0 0 Dh 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ; (21a) Q_i¼ Pi PD1i PD2i PDhi DT 1iP 0 0 0 DT_2iP 0 0 0 DT_hiP 0 0 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 (21b) and Q_ij¼

Pij PD1ij PD2ij PDhij

DT_1ijP 0 0 0 DT_2ijP 0 0 0 DT_hijP 0 0 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 : (21c)

Proof. See Appendix B. &

Remark 1. As long as either of the sufﬁcient condition (9) or (18) is satisﬁed, we can

guarantee that the linear closed-loop system(5) is asymptotically stable. The

proposed sufﬁcient conditions (9) and (18) can give the explicit relationship of the bounds on kiðtÞ ði ¼ 1; 2; . . . ; mÞ and ebj ðj ¼ 1; 2; 3Þ for preserving stability

robust-ness. Besides, the bounds, those are obtained by using these proposed sufﬁcient conditions, on kiðtÞ and ebjare not necessarily symmetric with respect to the origin of

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Remark 2. If we do not consider delayed perturbations in Eq. (1) (i.e., if we only

consider the delay-free system _xðtÞ ¼ AxðtÞ þ BuðtÞ and yðtÞ ¼ CxðtÞ), then the

sufﬁcient condition (9) becomes those given by Chen and Chou[14]. Therefore, the

result given by Chen and Chou[14]may be viewed as a special case of the proposed

sufﬁcient condition (9). That is, the proposed sufﬁcient condition (9) is the

generalized version of the result given by Chen and Chou[14]. Chen and Chou[14]

have shown that their sufﬁcient condition is less conservative than those proposed by

Su and Fong [9] and Tseng et al.[15]. This generalized version can be applied to

some wider application examples as those given in Examples 1 and 2 to which the

results of Su and Fong [9], Tseng et al. [15] and Chen and Chou [14] cannot be

applied to.

3. Illustrative examples

In this section, two numerical examples are given for illustrating the application of the proposed sufﬁcient conditions and making a conservatism comparison between the proposed sufﬁcient criteria and those of Tseng et al.[10].

Example 1. Consider the following linear uncertain system: _ xðtÞ ¼ AxðtÞ þX 2 i¼1 kiðtÞA1ixðt t1Þ þBuðtÞ; (22) yðtÞ ¼ CxðtÞ (23)

with time-varying uncertain parameters kiðtÞ ði ¼ 1; 2Þ and controlled by the output

feedback gain K ¼ 1; where

A ¼ 1 1 0 2 þk1ðtÞ 5 2 8 3 ; B ¼ 1 3 þk1ðtÞ 1 0 ; C ¼ ½0 1 þ k1ðtÞ½3 1 þ k2ðtÞ½1 0; A11¼ 1 1 1 0 ; A12 ¼ 0 1 1 1 ; k1ðtÞ 2 ½0:21 0:14 and k2ðtÞ 2 ½0:176 0:138:

It is seen that the closed-loop systembecomes _

xðtÞ ¼ AcxðtÞ þ

X2

i¼1

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where

Ac¼A þ k1ðtÞE1þk2ðtÞE2þk21ðtÞE11þk1ðtÞk2ðtÞE12

¼ 1 0 0 1 " # þk1ðtÞ 2 0 1 0 " # þk2ðtÞ 1 0 3 0 " # þk2₁ðtÞ 3 1 0 0 " # þk1ðtÞk2ðtÞ 1 0 0 0 " # :

By adopting the method of Tseng et al.[10]and the software of Matlab Toolbox

for structured singular value, we can obtain jk1ðtÞj50:1295

and

jk2ðtÞj50:1295:

Thus, we cannot reach any conclusion for guaranteeing the robust stability. That is, the robust stability condition of Tseng et al.[10]cannot be applied in this example. Now, applying the sufﬁcient condition (9) with the 2-norm-based matrix measure and taking Q ¼ I2; D1¼1:5I2; and b ¼ 2:7; we have

(i) X2 i¼1 kiðtÞfiþ X2 i¼1 X2 j¼1 kiðtÞkjðtÞfijp0:9985o1; for k1ðtÞ 2 ½0 0:14 and k2ðtÞ 2 ½0 0:138; (ii) X2 i¼1 kiðtÞfiþ X2 i¼1 X2 j¼1 kiðtÞkjðtÞfijp0:5790o1; for k1ðtÞ 2 ½0:21 0 and k2ðtÞ 2 ½0 0:138; (iii) X2 i¼1 kiðtÞfiþ X2 i¼1 X2 j¼1 kiðtÞkjðtÞfijp0:7764o1; for k1ðtÞ 2 ½0:21 0 and k2ðtÞ 2 ½0:176 0; (iv) X2 i¼1 kiðtÞfiþ X2 i¼1 X2 j¼1 kiðtÞkjðtÞfijp0:7785o1; for k1ðtÞ 2 ½0 0:14 and k2ðtÞ 2 ½0:176 0:

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p 1 þX m i¼1 kiðtÞjiþ Xm i¼1 k2_iðtÞmðQ1=2Q_iiQ1=2Þ þX m i¼1 Xm j¼2 j4i kiðtÞkjðtÞjij þ2 b þ rX h l¼1 b1 ! kPk kQ1k: ðB:2Þ

The inequality (18) implies

1o 1 X m i¼1 kiðtÞjiþ Xm i¼1 k2_iðtÞmðQ1=2Q_iiQ1=2Þ þX m i¼1 Xm j¼1 kiðtÞkjðtÞjij ( þ2bkPk kQ1k !) 2X h l¼1 b_lkPk kQ1k ! , ; ðB:3Þ

where it should be noticed that 2Ph_l¼1b_lkPk kQ1k40: This implies that, if the inequality (18) is satisﬁed, then there exist a r with

1oro 1 X m i¼1 kiðtÞjiþ Xm i¼1 k2_iðtÞmðQ1=2Q_iiQ1=2Þ þX m i¼1 Xm j¼1 kiðtÞkjðtÞjij ( þ2bkPk kQ1k !) 2 X h l¼1 b_lkPk kQ1k ! , ; ðB:4Þ such that Xm i¼1 kiðtÞjiþ Xm i¼1 k2_iðtÞmðQ1=2Q_iiQ1=2Þ þX m i¼1 Xm j¼2 j4i kiðtÞkjðtÞjij þ2 b þ rX h l¼1 b_l ! kPk kQ1ko1: ðB:5Þ

This implies that _V ðt; xtÞo0: So, according to the Razumikhin-type theorem of Wu

[16], we have the stated result. Thus, the proof is completed. &

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