Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

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Volume 2012, Article ID 825416,14pages doi:10.1155/2012/825416

Research Article

Robust Local Regularity and Controllability of

Uncertain TS Fuzzy Descriptor Systems

Shinn-Horng Chen,

1

_{Wen-Hsien Ho,}

2

_{and Jyh-Horng Chou}

1, 3

1_{Department of Mechanical/Electrical Engineering, National Kaohsiung University of Applied Sciences,}

415 Chien-Kung Road, Kaohsiung 807, Taiwan

2_{Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University,}

100 Shi-Chuan 1st Road, Kaohsiung 807, Taiwan

3_{Institute of System Information and Control, National Kaohsiung First University of Science and}

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

Correspondence should be addressed to Jyh-Horng Chou,choujh@nkfust.edu.tw Received 2 October 2012; Accepted 28 October 2012

Academic Editor: Jen Chih Yao

Copyrightq 2012 Shinn-Horng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The robust local regularity and controllability problem for the Takagi-SugenoTS fuzzy descriptor systems is studied in this paper. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on parameter uncertainties for preserving the assumed properties. An example is given to illustrate the application of the proposed sufficient condition.

1. Introduction

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno 1, known as the TS fuzzy model, is a successful approach for dealing with

the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to the nonlinear control systemse.g., 2–19 and references therein.

Descriptor systems represent a much wider class of systems than the standard systems20.

In recent years, some researcherse.g., 4–6,8,21–28 and references therein have studied

the design issue of the fuzzy parallel-distributed-compensation PDC controllers for each fuzzy rule of the TS fuzzy descriptor systems. Both regularity and controllability are actually two very important properties of descriptor systems with control inputs29. So, before the

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systems, it is necessary to consider both properties of local regularity and controllability for each fuzzy rule23. However, both regularity and controllability of the TS fuzzy systems are

not considered by those mentioned-above researchers before the fuzzy PDC controllers are designed. Therefore, it is meaningful to further study the criterion that the local regularity and controllability for each fuzzy rule of the TS fuzzy descriptor systems hold30.

On the other hand, in fact, in many cases it is very diﬃcult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters. These parametric uncertainties may destroy the local regularity and controllability properties of the TS fuzzy descriptor systems. But, to the authors’ best knowledge, there is no literature to study the issue of robust local regularity and controllability for the uncertain TS fuzzy descriptor systems.

The purpose of this paper is to present an approach for investigating the robust local regularity and controllability problem of the TS fuzzy descriptor systems with structured parameter uncertainties. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on structured parameter uncertainties for preserving the assumed properties. A numerical example is given in this paper to illustrate the application of the proposed sufficient criterion.

2. Robust Local Regularity and Controllability Analysis

Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of premise part and the linear dynamic model with parametric uncertainties of consequent part in the exact TS fuzzy control model with parametric uncertainties can be derived from the given nonlinear control model with parametric uncertainties5. The TS

continuous-time fuzzy descriptor system with parametric uncertainties for the nonlinear con-trol system with structured parametric uncertainties can be obtained as the following form:

Ri: IF z1is Mi1and . . . and zgis Mig,

then Ei˙xt Ai ΔAixt Bi ΔBiut,

2.1

or the uncertain discrete-time TS fuzzy descriptor system can be described by

Ri: IF z1is Mi1and . . . and zgis Mig,

then Eixk 1 Ai ΔAixk Bi ΔBiuk,

2.2

with the initial state vector x0, where Rii 1, 2, . . . , N denotes the ith implication, N is the

number of fuzzy rules, xt x1t, x2t, . . . , xntT and xk x1k, x2k, . . . , xnkT

denote the n-dimensional state vectors, ut u1t, u2t, . . . , uptT and uk 

u1k, u2k, . . . , upkT denote the p-dimensional input vectors, zii 1, 2, . . . , g are

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n× p consequent constant matrices, ΔAi and ΔBii 1, 2, . . . , N are, respectively, the

parametric uncertain matrices existing in the system matrices Aiand the input matrices Biof

the consequent part of the ith rule due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters, and Miji 1, 2, . . . , N and j 1, 2, . . . , g

are the fuzzy sets. Here the matrices Ei i 1, 2, . . . , N may be singular matrices with

rankEi ≤ n i 1, 2, . . . , N. In many applications, the matrices Ei i 1, 2, . . . , N

are the structure information matrices; rather than parameter matrices, that is, the elements of Eii 1, 2, . . . , N contain only structure information regarding the problem

considered.

In many interesting problemse.g., plant uncertainties, constant output feedback with uncertainty in the gain matrix, we have only a small number of uncertain parameters, but these uncertain parameters may enter into many entries of the system and input matrices 31,32. Therefore, in this paper, we suppose that the parametric uncertain matrices ΔAiand

ΔBitake the forms

ΔAi m k1 εikAik, ΔBi m k1 εikBik, 2.3

where εiki 1, 2, . . . , N and k 1, 2, . . . , m are the elemental parametric uncertainties, and

Aik and Biki 1, 2, . . . , N and k 1, 2, . . . , m are, respectively, the given n × n and n × p

constant matrices which are prescribed a priori to denote the linearly dependent information on the elemental parametric uncertainties εik.

In this paper, for the uncertain TS fuzzy descriptor system in2.1 or 2.2, each

fuzzy-rule-nominal model Ei˙xt Aixt Biut or Eixk 1 Aixk Biuk, which

is denoted by {Ei, Ai, Bi}, is assumed to be regular and controllable. Due to inevitable

uncertainties, each fuzzy-rule-nominal model {Ei, Ai, Bi} is perturbed into the

fuzzy-rule-uncertain model {Ei, Ai  ΔAi, Bi  ΔBi}. Our problem is to determine the conditions

such that each fuzzy-uncertain model {Ei, Ai  ΔAi, Bi  ΔBi} for the uncertain TS fuzzy

descriptor system 2.1 or 2.2 is robustly locally regular and controllable. Before we

investigate the robust properties of regularity and controllability for the uncertain TS fuzzy descriptor system2.1 or 2.2, the following definitions and lemmas need to be introduced

first.

Definition 2.1see 33. The measure of a matrix W ∈ Cn×nis defined as

μW≡ lim

θ→ 0

_{I θW}_{− 1}

θ ,

2.4

where · is the induced matrix norm on Cn×n.

Definition 2.2see 34. The system {Ei, Ai, Bi} is called controllable, if for any t1 > 0 or

k1 > 0, x0 ∈ Rn, and w∈ Rn, there exists a control input ut or uk such that xt1 w

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Definition 2.3. The uncertain TS fuzzy descriptor system in2.1 or 2.2 is locally regular, if

each fuzzy-rule-uncertain model{Ei, Ai ΔAi, Bi ΔBi} i 1, 2, . . . , N is regular.

Definition 2.4. The uncertain TS fuzzy descriptor system in 2.1 or 2.2 is locally

controllable, if each fuzzy-rule-uncertain model{Ei, Ai ΔAi, Bi ΔBi} i 1, 2, . . . , N is

controllable.

Lemma 2.5 see 34. The system {Ei, Ai, Bi} is regular if and only if rankEni Bdi n2, where

Eni∈ Rn 2_×n and Edi∈ Rn 2_×n2 are given by Eni ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ei 0 · · · 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Edi ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Ai Ei Ai · · · · · · Ei Ai ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . 2.5

Lemma 2.6 see 29,35. Suppose that the system {Ei, Ai, Bi} is regular. The system {Ei, Ai, Bi}

is controllable if and only if rankEdi Ebi n2and rankEi Bi n, where Edi∈ Rn

2_×n2

is given in2.5 and Ebi diag{Bi, Bi, . . . , Bi} ∈ Rn

2_×np

.

Lemma 2.7 see 33. The matrix measures of the matrices W and V , namely, μW and μV , are

well defined for any norm and have the following properties:

i μ±I ±1, for the identity matrix I;

ii −W ≤ −μ−W ≤ ReλW ≤ μW ≤ W, for any norm · and any matrix

W ∈ Cn×n;

iii μW V ≤ μW μV , for any two matrices W, V ∈ Cn×n_;

iv μγW γμW, for any matrix W ∈ Cn×n_{and any non-negative real number γ,}

where λW denotes any eigenvalue of W, and ReλW denotes the real part of λW.

Lemma 2.8. For any γ < 0 and any matrix W ∈ Cn×n_{, μ}_{γW −γμ−W.}

Proof. This lemma can be immediately obtained from the propertyiv in Lemma2.7.

Lemma 2.9. Let N ∈ Cn×n_{. If μ}_{−N < 1, then detI N / 0.}

Proof. From the propertyii in Lemma2.7and since μ−N < 1, we can get that ReλN ≥

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Now, let the singular value decompositions of Ri  Eni Edi, Qi  Edi Ebi, and Pi Ei Bi be, respectively, Ri Ui Si 0n2_×nV_iH, 2.6 Qi Uri Sri 0n2_×np V_riH, 2.7 Pi  Uci Sci 0n2_×q V_ciH, 2.8 where Ui ∈ Rn 2_×n2 and Vi ∈ Rn 2_n×n2_n

are the unitary matrices, Si diag{σi1, σi2, . . . , σin2},

and σi1≥ σi2≥ · · · ≥ σin2> 0 are the singular values of R_i; U_ri∈ Rn 2_×n2

and Vri ∈ Rn

2_np×n2_np

are the unitary matrices, Sri  diag{σri1, σri2, . . . , σrin2} and σ_ri1 ≥ σ_ri2 ≥ · · · ≥ σ_rin2 > 0 are

the singular values of Qi; Uci ∈ Rn×n and Vci ∈ Rnp×np are the unitary matrices, Sci

diag{σci1, σci2, . . . , σcin} and σci1 ≥ σci2 ≥ · · · ≥ σcin > 0 are the singular values of Pi; ViH, VriH,

and VH

ci denote, respectively, the complex-conjugate transposes of the matrices Vi, Vri, and

Vci.

In what follows, with the preceding definitions and lemmas, we present a suﬃcient criterion for ensuring that the uncertain TS fuzzy descriptor system in2.1 or 2.2 remains

locally regular and controllable.

Theorem 2.10. Suppose that the each fuzzy-rule-nominal descriptor system {Ei, Ai, Bi} is regular

and controllable. The uncertain TS fuzzy descriptor system in2.1 (or 2.2) is still locally regular

and controllable (i.e., each fuzzy-rule-uncertain descriptor system{Ei, Ai ΔAi, Bi ΔBi} remains

regular and controllable), if the following conditions simultaneously hold

m k1 εikϕik < 1, 2.9a m k1 εikθik< 1, 2.9b m k1 εikφik < 1, 2.9c where i 1, 2, . . . , N, and k 1, 2, . . . , m: ϕik ⎧ ⎪ ⎨ ⎪ ⎩ μ−S−1_i U_iHRikViIn2, 0_n2_×nT , for εik≥ 0, −μSi−1UH_i RikViIn2, 0_n2_×nT , for εik< 0, Rik 0n2_×n R_ik ∈ Rn2_×n2_n , Rik  diag{Aik, . . . , Aik} ∈ Rn 2_×n2 , θik ⎧ ⎪ ⎨ ⎪ ⎩ μ−S−1_riUH riQikVri In2, 0_n2_×npT , for εik≥ 0, −μS−1_riUH riQikVri In2, 0_n2_×npT , for εik< 0,

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Qik ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Aik Aik · · · Aik Bik Bik · · · Bik ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ Rn2_×n2_np , φik ⎧ ⎪ ⎨ ⎪ ⎩ μ−S−1_ciUH ciPikVci In, 0n×p T , for εik≥ 0, −μS−1_ciUH ciPikVci In, 0n×pT , for εik< 0, Pik 0n×n Bik ∈ Rn×np_, 2.10

the matrices Si, Ui,Vi, Sri, Uri,Vri, Sci, Uci, and Vcii 1, 2, . . . , N are, respectively, defined in

2.6–2.8, and In2denotes the n2× n2identity matrix.

Proof. Firstly, we show the regularity. Since each fuzzy-rule-nominal descriptor system

{Ei, Ai, Bi} i 1, 2, . . . , N is regular, then, from Lemma 2.5, we can get that the matrix

Ri  Eni Edi ∈ Rn

2_×n2_n

has full row ranki.e., rankRi n2. With the uncertain matrices

Ai ΔAiand Bi ΔBi, each fuzzy-rule-uncertain descriptor system{Ei, Ai ΔAi, Bi ΔBi}

is regular if and only if

Ri Ri m k1 εikRik 2.11

has full row rank, where Rik 0n2_×n R_ik ∈ Rn2×n2nand R_ik diag{A_ik, . . . , A_ik} ∈ Rn2×n2.

It is known that rankRi

rankS−1_i UH

i RiVi

. 2.12

Thus, instead of rank Ri, we can discuss the rank of

In2, 0_n2_×n

m

k1

εikRik, 2.13

where Rik  S−1i UHi RikVi, for i  1, 2, . . . , N and k 1, 2, . . . , m. Since a matrix has at least

rank n2_{if it has at least one nonsingular n}2_×n2_{submatrix, a suﬃcient condition for the matrix}

in2.13 to have rank n2_{is the nonsingularity of}

Li In2

m

k1

εikRik, 2.14

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