Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

The robust local regularity and controllability problem for the Takagi-Sugeno (cid:2) TS (cid:3) fuzzy descriptor systems is studied in this paper. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a su ﬃ cient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed su ﬃ cient 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 su ﬃ cient condition.


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-fuzzymodel-based approach to the nonlinear control systems e.g., 2-19 and references therein . Descriptor systems represent a much wider class of systems than the standard systems 20 . In recent years, some researchers e.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 inputs 29 . So, before the design of the fuzzy PDC controllers in the corresponding rule of the TS fuzzy descriptor 2 Journal of Applied Mathematics systems, it is necessary to consider both properties of local regularity and controllability for each fuzzy rule 23 . 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 hold 30 . 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.

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 uncertainties 5 . The TS continuous-time fuzzy descriptor system with parametric uncertainties for the nonlinear control system with structured parametric uncertainties can be obtained as the following form: 2.1 or the uncertain discrete-time TS fuzzy descriptor system can be described by with the initial state vector x 0 , where R i i 1, 2, . . . , N denotes the ith implication, N is the number of fuzzy rules, x t x 1 t , x 2 t , . . . , x n t T and x k x 1 k , x 2 k , . . . , x n k T denote the n-dimensional state vectors, u t u 1 t , u 2 t , . . . , u p t T and u k u 1 k , u 2 k , . . . , u p k T denote the p-dimensional input vectors, z i i 1, 2, . . . , g are the premise variables, E i , A i , and B i i 1, 2, . . . , N are, respectively, the n × n, n × n and n × p consequent constant matrices, ΔA i and ΔB i i 1, 2, . . . , N are, respectively, the parametric uncertain matrices existing in the system matrices A i and the input matrices B i of the consequent part of the ith rule due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters, and M ij i 1, 2, . . . , N and j 1, 2, . . . , g are the fuzzy sets. Here the matrices E i i 1, 2, . . . , N may be singular matrices with rank E i ≤ n i 1, 2, . . . , N . In many applications, the matrices E i i 1, 2, . . . , N are the structure information matrices; rather than parameter matrices, that is, the elements of E i i 1, 2, . . . , N contain only structure information regarding the problem considered.
In many interesting problems e.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 ΔA i and ΔB i take the forms where ε ik i 1, 2, . . . , N and k 1, 2, . . . , m are the elemental parametric uncertainties, and A ik and B ik i 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 ε ik .
In this paper, for the uncertain TS fuzzy descriptor system in 2.1 or 2.2 , each fuzzy-rule-nominal model is assumed to be regular and controllable. Due to inevitable uncertainties, each fuzzy-rule-nominal model Our problem is to determine the conditions such that each fuzzy-uncertain model {E i , A i ΔA i , B i ΔB i } 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 system 2.1 or 2.2 , the following definitions and lemmas need to be introduced first.
where · is the induced matrix norm on C n×n .
x 0 ∈ R n , and w ∈ R n , there exists a control input u t or u k such that x t 1 w or x k 1 w .
In what follows, with the preceding definitions and lemmas, we present a sufficient criterion for ensuring that the uncertain TS fuzzy descriptor system in 2.1 or 2.2 remains locally regular and controllable.
Journal of Applied Mathematics has full row rank, where R ik 0 n 2 ×n R ik ∈ R n 2 × n 2 n and R ik diag{A ik , . . . , A ik } ∈ R n 2 ×n 2 .

It is known that rank
Thus, instead of rank R i , we can discuss the rank of for i 1, 2, . . . , N and k  1, 2, . . . , m. Since a matrix has at least rank n 2 if it has at least one nonsingular n 2 ×n 2 submatrix, a sufficient condition for the matrix in 2.13 to have rank n 2 is the nonsingularity of 1, 2, . . . , N and k 1, 2, . . . , m .

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Using the properties in Lemmas 2.7 and 2.8 and from 2.9a , we get ε ik ϕ ik < 1.

2.15
From Lemma 2.9, we have that Hence, the matrix L i in 2.14 is nonsingular. That is, the matrix R i in 2.11 has full row rank n 2 . Thus, from the Lemma 2.5, the regularity of each fuzzy-rule-uncertain descriptor and P ik 0 n×n B ik ∈ R n× n p .

Journal of Applied Mathematics
It is known that Thus, instead of rank Q i , we can discuss the rank of where Q ik S −1 ri U H ri Q ik V ri , for i 1, 2, . . . , N and k 1, 2, . . . , m. Since a matrix has at least rank n 2 if it has at least one nonsingular n 2 ×n 2 submatrix, a sufficient condition for the matrix in 2.21 to have rank n 2 is the nonsingularity of where Q ik S −1 ri U H ri Q ik V ri I n 2 , 0 n 2 ×np T for i 1, 2, . . . , N and k 1, 2, . . . , m .
Applying the properties in Lemmas 2.7 and 2.8 and from 2.9b , we get  N and k  1, 2, . . . , m for preserving both regularity and controllability. In addition, the bounds, that are obtained by using the proposed sufficient conditions, on ε ik are not necessarily symmetric with respect to the origin of the parameter space regarding ε ik i 1, 2, . . . , N and k 1, 2, . . . , m . Remark 2.12. This paper studies the problem of robust local regularity and controllability analysis. If the proposed conditions in 2.9a -2.9c are satisfied, each rule of the uncertain TS fuzzy descriptor system {E i , A i ΔA i , B i ΔB i } is guaranteed to be robustly locally regular and controllable. This implies that, in the fuzzy PDC controller design, if the proposed conditions in 2.9a -2.9c are satisfied, the PDC controller of each fuzzy rule can control every state variable in the corresponding rule of the uncertain TS fuzzy descriptor system {E i , A i ΔA i , B i ΔB i }. However, here, it should be noticed that although the PDC controller of each control rule can control every state variable in the corresponding rule under the presented conditions being held, the PDC controller gains should be determined using global design criteria that are needed to guarantee the global stability and control performance 5 , where many useful global design criteria have been proposed by some researchers e.g., 4-6, 8, and 21-28 and references therein .

Illustrative Example
Consider a two-rule fuzzy descriptor system as that considered by Wang et al. 21 . The TS fuzzy descriptor system with the elemental parametric uncertainties is described by