#### Abstract

This paper investigates the - filtering problem of T-S fuzzy systems with multiple time-varying delays. First, by the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on --disturbance attenuation is presented, in which both stability and prescribed - performance are required to be achieved for the filtering-error systems. Then, based on the condition, the full-order and reduced-order delay-dependent - filter design schemes for T-S fuzzy multiple time-varying delays systems are developed in terms of linear matrix inequality (LMI). Finally, an example is given to illustrate the effectiveness of the result.

#### 1. Introduction

Time delay arises frequently in many engineering areas of the real world, which is usually a source of instability. Therefore, the stability analysis and synthesis for time-delay system have been one of a most hot research area in the control community over the past years [1–9]. To research the nonlinear time-delay system, the scholars considered the Takagi-Sugeno (T-S) fuzzy time-delay model which is a kind of effective representation, and many analysis and synthesis methods for T-S fuzzy time-delay systems have been developed over the past years [10–13].

Since the state variables in control systems are not always available, filtering or state estimation of a dynamic system through an available measurement state is one of the fundamental problems in signal processing, communications, and control application [14–22]. There are many works that appeared to cope with the nonlinear filtering problem for T-S fuzzy systems with time delays [23–31]. For example, in [23, 24], a delay-dependent filter design via continuous-time T-S fuzzy model approach is proposed in terms of linear matrix inequalities. In [25], by using a basis-dependent Lyapunov function, a delay-dependent result on the performance of the discrete filtering error system is presented. Based on the similar Lyapunov function combined with Finslers Lemma, [26] researched the delay-dependent robust filtering problem for a class of uncertain discrete-time T-S fuzzy systems with interval-like time-varying state delay. Reference [27] proposed the delay-dependent approach to robust and - filtering for a class of uncertain nonlinear time-delayed systems. References [28, 29] investigated delay-dependent filter design problems for discrete-time T-S fuzzy time-delayed systems and continuous-time T-S fuzzy time-delayed systems, respectively, which were both based on a delay-dependent piecewise Lyapunov-Krasovskii functional.

Several filtering approaches for T-S fuzzy systems with multiple delays have been developed over the past few years [32–36]. For instance [32] studied the filter design problem for discrete-time T-S fuzzy systems with multiple time delays. In [33], a robust mixed filtering problem for continuous-time T-S fuzzy systems with multiple time-varying delays in state variables was addressed.

While the time-varying delay functions above mentioned were all assumed slow-varying (the derivative of delay function is less than one) or fast-varying (the derivative of delay function is unknown). Reference [34] dealt with the fuzzy filter design problem for discrete-time T-S fuzzy systems with multiple time delays in the state variables. Reference [35] introduced a decentralized fuzzy filter design for nonlinear interconnected systems with multiple constant delays via T-S fuzzy models. Reference [36] addressed the problem of - filter design for T-S fuzzy systems with multiple time-varying delays, but the derivative of delay functions must be less than one.

To the best of our knowledge, the problem of - filter design for T-S fuzzy systems with multiple time-varying delays has not been fully investigated in the literature. As is well known, time delays usually exist in many physical systems and result in unsatisfactory performance, and the derivative of delay function may vary from to . So, the research on T-S fuzzy systems with multiple time-varying delays is of great practical and theoretical significance. This motivates the research in this paper.

In summary, the purpose of this paper is to develop an - filter for T-S fuzzy systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on --disturbance attenuation is presented. Then, the full-order and reduced-order delay-dependent - filter design schemes for T-S fuzzy multiple time-varying-delays systems are developed in terms of LMI. The example illustrates the effectiveness of the result.

This paper is organized as follows. In Section 2, the T-S fuzzy model and corresponding filter are formulated. In Section 3 we give the sufficient condition to assure asymptotic stability and the - noise-attenuation level bound for the T-S fuzzy filtering-error systems. Based on the condition in Section 3, we present a stable fuzzy filter in terms of LMIs. Section 4 provides illustrative examples to demonstrate the effectiveness of the proposed method. Conclusions are given in Section 5.

*Notations*. The notations used throughout this paper are fairly standard. The superscript “” stands for matrix transpose, and the notation means that matrix is real symmetric and positive (or being positive semidefinite). and and are used to denote identity matrix and zero matrix with appropriate dimension, respectively. The notation in a symmetric matrix always denotes the symmetric block in the matrix. The parameter denotes a block-diagonal matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

#### 2. System Descriptions and Preliminaries

Consider the nonlinear system with multiple state delays that is described by the following T-S model.

##### 2.1. Plant Form

*Rule *. IF is , is , and is , then
where , and are the premise variables that are measurable, and each is a fuzzy set. is the state variables. is the measured output of the system. is the signal to be estimated. is the disturbance input. is the number of IF-THEN rules. Also is the time-varying delay in the state, and it is assumed that . That is, the derivative of time-varying delay function is continuous and bounded. is the number of time delays. is a vector-valued initial continuous function.

By using center-average defuzzifier, product inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model: with in which is the grade of membership of in . It is assumed that , , for all . Therefore, and for all . In this paper, we study the following filter form of order ( for full-order filter, and for reduced-order filter): where the are the filter parameters to be designed. Combining (2) and (4) and defining , , and , we have the filtering-error system: where for and

Before ending this section, we introduce the following definition, which will be used in the derivation of our main results.

*Definition 1 (- performance). *Given a scalar , the system (1) is said to be with - performance if the system (1) is asymptotically stable and the output satisfies
for all nonzero under zero-initial condition, where,

Here, we want to design a suitable filter (4) for the system (1) with a - performance.

#### 3. Main Results

In this section, the conditions to assure the system (1) asymptotically stable with - performance for the T-S fuzzy filtering-error systems are presented (Lemmas 2 and 3). Then, based on the conditions, a filter is given in terms of LMIs.

Lemma 2. *Given , if there exist common matrices , , , , , , , and , , and satisfying
**
where*

then the system (5) is asymptotically stable with an - performance .

*Proof. *Choose a Lyapunov-Krasovskii functional candidate as
where

The time derivative of along the solution of (5) is computed as follows:

Applying free-weighting matrix method [37], using the Newton-Leibniz formula that

and defining

we can know that

Consider the index
Then for any nonzero under zero-initial condition,

After substitution of into (16) with (5). and taking into consideration (19), one has from (14), (15), and (16) that

Applying the Schur complement to (22), we know that (9) guarantees , which implies that

On the other hand, using the Schur complement to (10), we can know that . Then it can be easily got that for all

Taking the supremum over yields for all nonzero .

Next, we prove the asymptotic stability of system (5) when . Choose a Lyapunov-Krasovskii functional as in (12), where , . It is easy to find that there exist two scalars and such that

Similar to the above deduction, we can know from (9) that the time derivative of along the solution of (5) with satisfies . This proves the asymptotic stability of system (5) with according to the same method of [19]. This completes the proof.

Lemma 2 is the sufficient condition for the - filter design which contains the coupled matrix variables in the matrix inequality. Using the decoupling technique as follows, we can transform Lemma 2 into another form.

Lemma 3. *Given , if there exist common matrices , , , , , , , and , , and , such that (9), (10) hold if and only if there exist matrices , , , , , , , and , , and such that the following inequalities hold:
**
where*

then the system (5) is asymptotically stable with an - performance .

*Proof. **Necessity*. Suppose (9), (10) hold. Partition as
where , , and is invertible. Let

We pre- and postmultiply and its transpose to (9) and (10), respectively, where

Apply the changes of variables such that

Then we obtain (26) and (27).*Sufficiency*. Suppose that (26) and (27) hold for , , , , , , , and , , and . Choose two matrices with and being invertible such that . Let and be defined as in (29) and (30). Then is concluded from (27). Pre- and postmultiply and to (26) and (27), respectively. We can get (9) and (10) with the changes of variables as

This completes the proof.

Theorem 4. *Given , if there exist common matrices , , , , , , , and , , and such that the following inequalities hold:
**then - filter parameters in (4) are given by
**
where*

*Proof. *By considering (6),we know that in (26) and the in (34) satisfy

Based on Lemmas 2 and 3, the - filter matrices are given by (33). Under the transformation , the filter matrices functions can be of the following forms:

Hence, the filter in (4) can be got by (36). This completes the proof.

#### 4. Simulation

In this section, we give a numerical example to illustrate the use of the present method. Consider the system of the form (1) with two plants () and two delays (), where

Here, we only consider the full-order filter design. First, we set , . Figure 1 shows the - gain bound got from Theorem 4 in pointwise manner, where the derivatives of time-varying delay function bound , . We can find that the maximum - gain is at the point , . The minimum - gain is at the point , .

Then, we set , . Figure 2 shows the - gain bound got from Theorem 4 in pointwise manner, where the time-varying delay function bound , . We can find that the maximum - gain is at the point , . The minimum - gain is at the point , .

Now, We choose both time-varying delays to be which gives , , , and , and we get a set of feasible solutions to Theorem 4 with the - gain and

Therefore, we can solve the corresponding filter from (36) as

To illustrate the performance of the designed filter, we assume zero initial condition and the disturbance as follows:

The simulation result of signal is given in Figure 3.

The resulting output -norm of the filtering-error system is about 0.15, while . Simulation result for the ratio of the output -norm to the disturbance -norm is 0.15, and with , , , and .

#### 5. Conclusion

The problem on - filter design has been addressed for a class of TCS fuzzy-model-based systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a sufficient condition for the existence of - filter, which stabilizes the T-S fuzzy-model-based filtering-error systems and guarantees a prescribed level on disturbance attenuation, has been obtained in terms of LMI form. The numerical example has shown the effectiveness of the proposed method. In addition, the basis-dependent Lyapunov-Krasovskii functional approach for filtering problems of T-S fuzzy delayed systems is also challenging, and could be our further work.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61203005), in part by Harbin Engineering University Central University Foundation Research Special Fund (HEUCFR1024), and in part by Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q12130) and Heilongjiang Province Natural Science Foundation (F201221).