State Estimation for Discrete-Time Takagi-Sugeno Fuzzy Systems with Time-Varying Delays

Abstract

The state estimation problem is investigated for discrete-time Takagi-Sugeno fuzzy systems with time-varying delays. By constructing appropriate Lyapunov-Krasovskii functionals and employing matrix inequality technique, a delay-dependent linear matrix inequalities (LMIs) criterion is developed to estimate the systems state with some observed output measurements such that the error-state system is globally asymptotically stable. An example with simulations is given to show the effectiveness of the proposed criterion.

Ting Lei 1 and Qiankun Song 2 and Yurong Liu 3, 4

Academic Editor:Zidong Wang

1, Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China
2, School of Management, Chongqing Jiaotong University, Chongqing 400074, China
3, Department of Mathematics, Yangzhou University, Yangzhou 225002, China
4, Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 25 August 2014; Accepted 10 September 2014; 2 March 2015

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.

1. Introduction

It is known that many physical systems and processes have severe nonlinearities, which bring difficulties to the analysis and synthesis of the underlying systems [1, 2]. As a powerful means to deal with complex nonlinear systems, fuzzy models have attracted rapidly growing attentions over the past decades [3]. Among various model-based fuzzy control approaches, Takagi-Sugeno (T-S) model has been an active topic due to the fact that it can combine the flexibility of fuzzy logic theory and fruitful linear system theory into a unified framework to approximate complex nonlinear systems [4]. In practice, time-delays often occur in many dynamic systems such as chemical processes, metallurgical processes, biological systems, and neural networks [5]. The existence of time-delays is usually a source of instability and poor performance [6]. Recently, many important results have been published on various analysis aspects for T-S models with time-delays. In particular, the stability, synchronization, and passivity have been intensively investigated; for example, see [7-27] and references therein.

On the other hand, state estimation is the process of assigning a value to an unknown system state variable based on measurements collected from the system [28]. Because of the complexity of large-scale systems, it is often the case that only partial information about the states of the nodes is available in the system outputs [29]. In order to understand the T-S fuzzy systems better, it becomes necessary to estimate the states of the nodes through available measurements [30].

Recently, some authors have studied the state estimation problem for T-S fuzzy systems with or without delays; for example, see [31-36] and references therein. In [31], a novel fuzzy-observer-design technique was developed for T-S fuzzy systems with unknown output disturbances. The system state and output disturbance were estimated asymptotically and simultaneously. In [32], the authors investigated the problem of robust [figure omitted; refer to PDF] state estimation for a class of multichannel networked nonlinear systems with multiple packet dropouts. The nonlinear plant was represented by T-S fuzzy-affine dynamic models with norm-bounded uncertainties, and stochastic variables with general probability distributions were adopted to characterize the data missing phenomenon in output channels. An admissible state estimator guaranteeing the stochastic stability of the resulting estimation error system with a prescribed [figure omitted; refer to PDF] disturbance attenuation level was designed. In [33], the authors proposed a novel fuzzy-based particle filter to reduce continuous state estimation errors due to failures in mode detection. It was fulfilled by considering a fuzzified contribution of each feasible mode in overall estimation. Two new resampling strategies were presented to tackle the degeneracy problem. A set of simulation test studies are conducted to extract the characteristic features and evaluate the performance of the proposed algorithm compared to observation and transition-based most likely modes tracking particle filter as one of the most meticulous proposed estimation algorithms. In [34], the T-S fuzzy model representation is extended to the state estimation of uncertain Markovian jumping Hopfield neural networks with mixed interval time-varying delays. Based on the Lyapunov-Krasovskii functional and stochastic analysis approach, several delay-dependent robust state estimators for such T-S fuzzy Markovian jumping Hopfield neural networks were achieved by solving a LMI. In [35], a design scheme for the state estimator for T-S fuzzy delayed Hopfield neural networks that uses strict output passivation of the error system was presented. Based on Lyapunov-Krasovskii functional, Jensens inequality, and linear matrix inequality (LMI) formulation, a delay-dependent criterion was proposed such that it makes the resulting estimation error system exponentially stable and passive from the input vector to the output error vector. In [36], the authors were concerned with the mixed [figure omitted; refer to PDF] and passivity based state estimation for a class of discrete-time fuzzy neural networks with the estimator gain change. Based on the Markovian system approach and linear matrix inequality technique, a new sufficient condition was derived such that the estimation error system was exponentially stable in the mean square sense. The estimator parameter was then determined by solving a set of LMIs.

It should be pointed out that most of aforementioned literatures on state estimation of T-S fuzzy systems have been concerned with the continuous-time systems only. To date, there have been very few results on the state estimation problems of discrete-time T-S fuzzy systems with delays [37]. In this paper, we continue to study the state estimation problems for discrete-time T-S fuzzy systems with time-varying delay. By employing appropriate Lyapunov-Krasovskii functionals and matrix inequality technique, we obtain a new delay-dependent LMI sufficient condition such that estimation errors are globally asymptotically stable for discrete-time T-S fuzzy systems with time-varying delay.

Notations . The notations are quite standard. Throughout this paper, [figure omitted; refer to PDF] represents the unitary matrix with appropriate dimensions; [figure omitted; refer to PDF] stands for the set of nonnegative integers; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote, respectively, the [figure omitted; refer to PDF] -dimensional Euclidean space and the set of all [figure omitted; refer to PDF] real matrices. The superscript " [figure omitted; refer to PDF] " denotes matrix transposition and the asterisk " [figure omitted; refer to PDF] " denotes the elements below the main diagonal of a symmetric block matrix. The notation [figure omitted; refer to PDF] (resp., [figure omitted; refer to PDF] ) means that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are symmetric matrices and that [figure omitted; refer to PDF] is positive semidefinite (resp., positive definite). [figure omitted; refer to PDF] is the Euclidean norm in [figure omitted; refer to PDF] . [figure omitted; refer to PDF] (resp., [figure omitted; refer to PDF] ) denotes the least (resp., largest) eigenvalue of symmetric matrix [figure omitted; refer to PDF] . For a positive constant [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the integer part of [figure omitted; refer to PDF] . For integers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the discrete interval given by [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes the set of all functions [figure omitted; refer to PDF] : [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes the difference of function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] . Matrices, if not explicitly specified, are assumed to have compatible dimensions.

2. Model Description and Preliminaries

In this section, we consider a discrete-time T-S fuzzy system with time-varying delay with the [figure omitted; refer to PDF] th rule formulated in the following form.

Plant Rule [figure omitted; refer to PDF] . IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , THEN [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the number of IF-THEN rules, where [figure omitted; refer to PDF] is the state vector; [figure omitted; refer to PDF] is a deterministic exogenous input; [figure omitted; refer to PDF] is the measurement output vector; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are system matrices with compatible dimensions; the positive integer [figure omitted; refer to PDF] corresponds to the transmission delay and satisfies [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are known integers.

Let [figure omitted; refer to PDF] be the normalized membership function of the inferred fuzzy set [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] being the grade of membership function of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . It is assumed that [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . And the T-S fuzzy model (1) can be represented as [figure omitted; refer to PDF]

For the discrete-time T-S fuzzy systems with time-varying delay (1), we construct the full-order state estimation as follows.

Estimator Rule [figure omitted; refer to PDF] . IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , THEN [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is an estimation of the state [figure omitted; refer to PDF] in model (1), [figure omitted; refer to PDF] is the output vector of the state estimator, and [figure omitted; refer to PDF] is the gain matrix of the state estimator to be designed. Using a standard fuzzy inference method, the fuzzy state estimator (4) is inferred as follows: [figure omitted; refer to PDF] Let the error state be [figure omitted; refer to PDF] ; then it follows from (3) and (5) that [figure omitted; refer to PDF]

Definition 1.

The system (5) is said to be a globally asymptotically state estimated system of the system (3) if the estimation error system (6) is globally asymptotically stable.

In obtaining the main result of this paper, the following lemma will be useful for the proof, which can be proved by using the methods in [38].

Lemma 2.

For any positive definite matrix [figure omitted; refer to PDF] , four integers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and a vector function [figure omitted; refer to PDF] , the following inequalities hold: [figure omitted; refer to PDF]

Remark 3.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; then the first inequality of Lemma 2 in this paper turns into the first inequality of Lemma 1 in [38]. If we let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , then the second inequality of Lemma 2 in this paper turns into the second inequality of Lemma 1 in [38]. So, Lemma 2 in this paper is more general than Lemma 1 in [38].

3. Main Results

In this section, we will establish our main criterion based on the LMI approach.

Theorem 4.

If there exist seven symmetric positive definite matrices [figure omitted; refer to PDF] [figure omitted; refer to PDF] and a matrix [figure omitted; refer to PDF] such that the following LMIs hold for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , then system (5) is a globally asymptotically state estimated system of system (3), and the estimator gain matrix [figure omitted; refer to PDF] can be designed as [figure omitted; refer to PDF]

Proof.

Defining [figure omitted; refer to PDF] , we consider the following Lyapunov-Krasovskii functional candidate for model (6) as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Calculating the difference of [figure omitted; refer to PDF] along the trajectories of model (6), we obtain that [figure omitted; refer to PDF] In deriving inequality (14), the conditions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have been used.

From the first inequality of Lemma 2, we have that [figure omitted; refer to PDF] From the second inequality of Lemma 2, we can get that [figure omitted; refer to PDF]

Denote [figure omitted; refer to PDF] . It follows from (9) to (18) that [figure omitted; refer to PDF]

From (8) and (19), we know that the system (5) is a globally asymptotically state estimated system of system (3). The proof is completed.

Remark 5.

In order to reduce the conservatism of the obtained result, the triple-summable term in [figure omitted; refer to PDF] is added, which has been used in literature [38].

4. Example

To verify the effectiveness of the theoretical result of this paper, we consider the following example.

Example 1.

Consider a T-S fuzzy system (1) with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the number of IF-THEN rules. The time-varying delay [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and the deterministic exogenous input [figure omitted; refer to PDF] . Other parameters are given as follows: [figure omitted; refer to PDF] The fuzzy membership functions are taken as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

By using the Matlab LMI Control Toolbox, we can find a solution to the LMIs in (8) as follows: [figure omitted; refer to PDF] Subsequently, we can obtain from [figure omitted; refer to PDF] that [figure omitted; refer to PDF]

According to Theorem 4, we know that the system (5) is a globally asymptotically state estimated system of system (3). The simulation results are shown in Figures 1 and 2, which demonstrate the effectiveness of the developed approach for the design of the state estimator for discrete-time Takagi-Sugeno fuzzy systems with time-varying delays.

Figure 1: The responses of the true state [figure omitted; refer to PDF] (red triangle) and its estimation [figure omitted; refer to PDF] (blue circle).

[figure omitted; refer to PDF]

Figure 2: The responses of the true state [figure omitted; refer to PDF] (red triangle) and its estimation [figure omitted; refer to PDF] (blue circle).

[figure omitted; refer to PDF]

5. Conclusions

In this paper, the state estimation problem has been investigated for discrete-time T-S fuzzy systems with time-varying delays. By employing Lyapunov functional method and the matrix inequality techniques, a delay-dependent LMIs criterion has been established to estimate the systems state with some observed output measurements such that the error-state system is globally asymptotically stable. An example has been provided to show the effectiveness of the proposed criterion.

Acknowledgments

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 61273021 and 11172247 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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