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Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 197845, 14 pages

http://dx.doi.org/10.1155/2013/197845

## Robust Fuzzy Output Feedback Control for a Class of Nonlinear Uncertain Systems with Mixed Time Delays

Electronic and Information Engineering College, Henan University of Science and Technology, Luoyang 471023, China

Received 25 June 2013; Accepted 18 August 2013

Academic Editor: Baoyong Zhang

Copyright © 2013 Xiaona Song and Shanzhong Liu. 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.

#### Abstract

This paper studies the problem of delay-dependent robust output feedback control for a class of uncertain fuzzy neutral systems with both discrete and distributed delays. The system is described by a state-space Takagi-Sugeno fuzzy model with distributed delays and norm-bounded parameter uncertainties. The purpose is to design a fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop fuzzy neutral system and satisfies an norm bound constraint for all admissible uncertainties. In terms of linear matrix inequalities, sufficient conditions for the solvability of this problem are presented. Finally, a numerical example is included to demonstrate the effectiveness of the proposed method.

#### 1. Introduction

Fuzzy control, as a promising way to approach nonlinear control problems, has had an impact on the control community [1–4]. Furthermore, the Takagi-Sugeno () fuzzy dynamic model [5–9] is nonlinear system described by fuzzy IF-THEN rules which give local linear representations of the underlying systems [10, 11]. It has been shown that such models can describe a wide class of nonlinear systems. Hence it is important to investigate their stability analysis and controller design problems, and in the past two decades many stability and control issues related to the fuzzy systems have been studied; see, for example, [12–15] and the references cited therein.

On the other hand, time delays exist commonly in dynamic systems due to measurement, transmission, transport lags, and so forth [16], which have been generally regarded as a main source of instability and poor performance. Thus the analysis of time delay systems and controller design for them is very important [17–20]. Recently, fuzzy systems with time delays have attracted a great deal of interests. For example, in [21], the stability analysis and stabilization problems for fuzzy delay systems were considered, and state feedback fuzzy controllers and fuzzy observers were designed. The robust control problem for fuzzy systems with time delays was investigated in [22, 23], and state feedback fuzzy controllers were designed; the corresponding results for the discrete case can be found in [24, 25], while in [26, 27], the robust output feedback controllers were designed for the continuous and discrete fuzzy time-delay systems, respectively.

Quite recently, fuzzy time-delay systems of neutral type were introduced in [28], where both the stabilization and control problems were studied. It should be point out that distributed delays were not taken into account. While in [29], authors considered the problems of robust stabilization and robust control for uncertain fuzzy neutral systems with both discrete and distributed time delays. However, the results in [28, 29] were all delay-independent. It is known that delay-dependent results are less conservative than delay-independent ones, especially in the case when the size of the delay is small. On the other hand, these obtained results, however, are mainly dealt with through a state feedback controller design method that requires all state variables to be available. In many cases, this condition is too restrictive. So it is meaningful to investigate the output feedback control method. To the best of our knowledge, so far, there are no results of delay-dependent robust output feedback control for uncertain fuzzy neutral systems with both discrete and distributed delays. This motives the present studies.

In this paper, we consider the delay-dependent robust output feedback control problem for fuzzy neutral systems with both discrete and distributed delays. The system to be considered is described by a state-space fuzzy model with mixed delays and norm-bounded parameter uncertainties. The distributed delays are assumed to appear in the state equation, and the uncertainties are allowed to be time varying but norm bounded. The aim of this paper is to design a full-order fuzzy dynamic output feedback controller such that the resulting closed-loop system is robustly asymptotically stable while satisfying an norm condition with a prescribed level irrespective of the parameter uncertainties. A sufficient condition for the solvability of this problem is proposed in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired output feedback controller is also given.

*Notation.* Throughout this paper, for real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). is an identity matrix with appropriate dimension. is the set of natural numbers. refers to the space of square summable infinite vector sequences. stands for the usual norm. The notation represents the transpose of the matrix . Matrices, if not explicitly stated, are assumed to have compatible dimensions. “*” is used as an ellipsis for terms induced by symmetry.

#### 2. Problem Formulation

A continuous fuzzy neutral model with distributed delays and parameter uncertainties can be described by the following.

Plant Rule : if is and *⋯* and is , then
where is the fuzzy set, is the number of IF-THEN rules, and are the premise variables. Throughout this paper, it is assumed that the premise variables do not depend on control variables; is the state; is the control input; is the measured output; is the controlled output; is the noise signal; are integers representing the time delay of the fuzzy systems; ; , , , , , , , , , , , and are known real constant matrices; , , , , , and are real-valued unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form
where , , , , , , and are known real constant matrices and are unknown time-varying matrix function satisfying
The parameter uncertainties , , , , , and are said to be admissible if both (2) and (3) hold.

Then the final output of the fuzzy neutral system is inferred as follows: where in which is the grade of membership of in . Then, it can be seen that for all . Therefore, for all , Now, by the parallel distributed compensation (PDC), the following full-order fuzzy dynamic output feedback controller for the fuzzy neutral system in (4) is considered.

Control Rule : if is and *⋯* and is , then
where is the controller state and , , and are matrices to be determined later. Then, the overall fuzzy output feedback controller is given by
From (4) and (11), one can obtain the closed-loop system as
where
Then, the problem of robust fuzzy control problem to be addressed in the previous is formulated as follows: given an uncertain distributed delay fuzzy neutral system in (4) and a scalar , determine a dynamic output feedback fuzzy controller in the form of (8) and (9) such that the closed-loop system in (13) and (14) is robustly asymptotically stable when and
under zero-initial conditions for any nonzero and all admissible uncertainties.

#### 3. Main Results

In this section, an LMI approach will be developed to solve the problem of robust output feedback control of uncertain distributed delay fuzzy neutral systems formulated in the previous section. We first give the following results which will be used in the proof of our main results.

Lemma 1 (see [30]). *Let , , , , and be real matrices of appropriate dimensions with and satisfying . Then one has the following.*(1)*For any scalar and vectors , ,
*(2)*For any scalar such that ,
*

Lemma 2 (see [31]). *Given any matrices , , and with appropriate dimensions such that , then, one has
*

Theorem 3. *The uncertain fuzzy neutral delay system in (13) and (14) is asymptotically stable, and (16) is satisfied if there exist matrices , , , , , , and and scalars , , such that the following LMIs hold:**where
*

*Proof. *To establish the robust stability of the system in (13), we consider (13) with ; that is,
For this system, we define the following Lyapunov function candidate:
where
The time derivative of is given by
where
Now, by Lemma 1, it can be shown that
Therefore
This together with (26) implies
where
Then, there holds
It follows from (22) and Lemma 2 that
where
It also can be verified that
Hence, with the support of the above conditions, we have
where
for . On the other hand, by applying the Schur complement formula to (20), we have that for ,
and from this and (7), we have that for all . Therefore, the system in (13) is robustly asymptotically stable.

This completes the proof.

Next, we will establish the robust performance of the system in (13) and (14) under the zero initial condition. To this end, we introduce where . Noting the zero initial condition, it can be shown that for any nonzero and , where is defined in (23), and then we have Then, by noting (14) and using Lemma 2, we have It can be deduced that wherefor . Similar to the previous section, applying the Schur complement formula to the LMI in (20) results in , , which together with (39) gives for any nonzero and . Therefore, we have .

Now, we are in a position to present a solution to the robust output feedback control problem.

Theorem 4. *Consider the uncertain fuzzy neutral delay system in (1), and let be a prescribed constant scalar. The robust problem is solvable if there exist matrices , , , , , , and and scalars , , such that the following LMIs hold:**where
**
Furthermore a desired robust dynamic output feedback controller is given in the form of (11) with parameters as follows:
*

*Proof. *Applying the Schur complements formula to (44) and (45) and using Lemma 2 result in
whereNote that
Then, by this inequality, it follows from (49) that
whereNow, set
Then, by (46), it can be verified that . Noting the parameters in (48) and pre- and postmultiplying (52) by
and its transpose, respectively, we havewhere
Finally, by Theorem 3, the desired result follows immediately.

*Remark 5. *Theorem 4 provides a sufficient condition for the solvability of the robust output feedback control problem for uncertain fuzzy neutral systems with both discrete and distributed delays. It is worth pointing out that the result in Theorem 4 can be readily extended to the case with multiple delays.

*Remark 6. *In Theorem 4, if we set , , and then the results in [32] are included in our paper. Also, the controller design method in our paper can be the reference for designing the observer-based output feedback controllers and so forth.

#### 4. Simulation Example

In this section, we provide one example to illustrate the output feedback controller design approach developed in this paper.

The uncertain fuzzy system with distributed delays considered in this example is with two rules.

Plant Rule 1: if is , then

Plant Rule 2: if is , then