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The robust local regularity and controllability problem for the Takagi-Sugeno (TS) 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.

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno [

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.

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 [

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 [

In this paper, for the uncertain TS fuzzy descriptor system in (

The measure of a matrix

The system

The uncertain TS fuzzy descriptor system in (

The uncertain TS fuzzy descriptor system in (

The system

Suppose that the system

The matrix measures of the matrices

For any

This lemma can be immediately obtained from the property (iv) in Lemma

Let

From the property (ii) in Lemma

Now, let the singular value decompositions of

In what follows, with the preceding definitions and lemmas, we present a sufficient criterion for ensuring that the uncertain TS fuzzy descriptor system in (

Suppose that the each fuzzy-rule-nominal descriptor system

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

Using the properties in Lemmas

Next, we show the controllability. Since each fuzzy-rule-nominal descriptor system

It is known that

Applying the properties in Lemmas

And then, it is also known that

Adopting the properties in Lemmas

The proposed sufficient conditions in (

This paper studies the problem of robust local regularity and controllability analysis. If the proposed conditions in (

Consider a two-rule fuzzy descriptor system as that considered by Wang et al. [

Now, applying the sufficient conditions in (

for the fuzzy rule 1:

From the results in (

The robust local regularity and controllability problem for the uncertain TS fuzzy descriptor systems has been investigated. The rank preservation problem for robust local regularity and controllability of the uncertain TS fuzzy descriptor systems is converted to the nonsingularity analysis problem. Under the assumption that each fuzzy rule of the nominal TS fuzzy descriptor system has the full row rank for its related regularity and controllability matrices, a sufficient criterion has been proposed to preserve the assumed properties when the elemental parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient conditions in (

This work was in part supported by the National Science Council, Taiwan, under Grants nos. NSC 100-2221-E-151-009, NSC 101-2221-E-151-076, and NSC 101-2320-B-037-022.