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Journal of Applied Mathematics
Volume 2012, Article ID 475728, 20 pages
http://dx.doi.org/10.1155/2012/475728
Research Article

Delay-Range-Dependent Stability Criteria for Takagi-Sugeno Fuzzy Systems with Fast Time-Varying Delays

Department of Automation Engineering, Institute of Mechatronoptic System, Chienkuo Technology University, Changhua 500, Taiwan

Received 23 April 2012; Revised 16 June 2012; Accepted 23 June 2012

Academic Editor: Reinaldo Martinez Palhares

Copyright © 2012 Pin-Lin 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

The problem of delay-range-dependent stability for T-S fuzzy system with interval time-varying delay is investigated. The constraint on the derivative of the time-varying delay is not required, which allows the time delay to be a fast time-varying function. By developing delay decomposition approach, integral inequalities approach (IIA), and Leibniz-Newton formula, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Simulation examples show resulting criteria outperform all existing ones in the literature. It is worth pointing out that our criteria are carried out more efficiently for computation and less conservatism of the proposed criteria.

1. Introduction

It is well known that time delay often appears in the dynamic systems, which is an important source of instability and degradation in the control performance. Fuzzy system in the form of Takagi-Sugeno (T-S) model has been paid considerable attention in the past two decades [1, 2]. It has been shown that the T-S model method gives an effective way to represent complex nonlinear systems by some simple local linear dynamic systems, and some analysis methods in the linear systems can be effectively extended to the T-S fuzzy systems. However, all the aforementioned criteria aim at time delay free T-S fuzzy systems. In practice, time delay, one of the instability sources in dynamical systems, is a common and complex phenomenon in many industrial and engineering systems such as chemical process, metallurgical processes, biological systems, rolling mill systems, and communication networks. As a result, stability analysis for T-S fuzzy systems with time delay is of great significance both in theory and in practice. Some approaches developed for general delay systems have been borrowed to deal with fuzzy systems with time delay. In recent years, the problems of stability and stabilization of the T-S fuzzy systems with time delay have attracted rapidly growing interests [321]. Among these references, great efforts have been focused on effective reduction of the conservation of the delayed T-S fuzzy model. Many effective methods, such as new bounding technique for cross-terms [7, 9], augmented Lyapunov functional method [7], and free-weighing matrix method [6, 10, 1720], have been proposed. We can see that the free-weighting matrix approach is used as a main tool to make the criteria less conservative in the literature, and only the lower and upper bounds of delay function (𝑡) are considered. When stability analysis of delayed systems is concerned, a very effective strategy is to apply the Gu’s Lyapunov-Krasovskii functional discretization technique [22]. However, this discretization technique has been developed for linear systems subject to constant time delay. Besides, it is very hard to extend the stability analysis conditions obtained via this technique to control design since several products between decision variables will be generated, leading to nonconvex formulations. Therefore, less-conservative conditions for stability and control of T-S fuzzy systems subject to uncertain time delay are proposed based on a fuzzy weighting-dependent Lyapunov function, the Gu discretization technique [22], and extra strategies to introduce slack matrix variables by [13, 23]. However, these results have conservatism to some extent, which exist room for further improvement.

The delay varying in an interval has strong application background, which commonly exists in many practical systems. The investigation for the systems with interval time-varying delay has caused considerable attention, see [13, 18, 2426] and the references therein. In [15], Lyapunov-Krasovskii function and augmented Lyapunov-Krasovskii function to construct uncorrelated augmented matrix (UAM) and to deal with cross terms in the UAM through improved Jessen’s inequality. An improved delay-dependent criterion is derived in [18] by constructing a new Lyapunov functional and using free-weighting matrices. In [24], a weighting delay method is used to deal with the stability of system with time varying delay. In [25, 26], by developing a delay decomposition approach, the integral interval [𝑡,𝑡] is decomposed into [𝑡,𝑡𝛼] and [𝑡𝛼,𝑡]. Since a tuning parameter 𝛼 is introduced, the information about 𝑥(𝑡𝛼) can be taken into full consideration; thus, the upper bound of 𝑡𝑡̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑s can be estimated more exactly no matter the delay derivative exists or not. However, it has been realized that too many free variables introduced in the free-weighting matrix method will complicate the system synthesis and consequently lead to a significant computational demand [7]. The problem of improving system performance while reducing the computational demand will be addressed in this paper.

The main contributions of this paper are highlighted as follows. (1) delay-dependent stability criteria are developed, which are an improvement over the latest results available from the open literature [3, 57, 9, 10, 12, 13, 15, 1721, 23, 27]; (2) theoretical proof is provided to show that the results in [6] are a special case of the results derived in this paper. The approach developed in this work uses the least number of unknown variables and consequently is the least mathematically complex and most computationally efficient. This implies that some redundant variables in the existing stability criteria can be removed while maintaining the efficiency of the stability conditions. With the present stability conditions, the computational burden is largely reduced; (3) since the delay decomposition approach is introduced in delay interval, it is clear that the stability results are based on the delay decomposition approach. When the positions of delay decomposition are varied, the stability results of the proposed criteria are also different. In order to obtain the optimal delay decomposition sequence, we proposed an implementation based on optimization methods.

Motivated by the above discussions, we propose new stability criteria for T-S fuzzy system with interval time-varying delay. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Since the delay terms are concerned more exactly, less conservative results are presented. Moreover, the restriction on the change rate of time-varying delays is relaxed in the proposed criteria. The proposed stability conditions are much less conservative and are more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.

2. Stability Analysis

Consider a T-S fuzzy system with a time-varying delay, which is represented by a T-S fuzzy model, composed of a set of fuzzy implications, and each implication is expressed by a linear system model. The ith rule of this T-S fuzzy model is of the following form:

Plant rule 𝑖 if 𝑧1(𝑡) is 𝑀𝑖1, and …, and 𝑧𝑝(𝑡) is 𝑀𝑖𝑝 theṅ𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐴di[]𝑥(𝑡(𝑡)),(2.1a)𝑥(𝑡)=𝜙(𝑡),𝑡,0,𝑖=1,2,...,𝑟,(2.1b)where 𝑧1(𝑡),𝑧2(𝑡),...,𝑧𝑝(𝑡) are the premise variables; 𝑀𝑖𝑗,𝑖=1,2,...,𝑟,𝑗=1,2,...,𝑝 are the fuzzy sets; 𝑥(𝑡)𝑅𝑛 is the state; 𝜙(𝑡) is a vector-valued initial condition; 𝐴𝑖 and 𝐴di are constant real matrices with appropriate dimensions; the scalars ris the number of if-then rules; time delay, (𝑡), is a time-varying delay. We will consider the following two cases for the time-varying delay.

Case 1. (𝑡) is a differentiable function satisfying 01(𝑡)2,||̇||(𝑡)𝑑,𝑡0.(2.2)

Case 2. (𝑡) is a differentiable function satisfying 01(𝑡)2,(2.3) where 1 and 2 are the lower and upper delay bounds, respectively; 1,2, and 𝑑 are constants. Here 1 the lower bound of delay may not be equal to 0, and when 𝑑=0 we have 1=2. Both Cases 1 and 2 have considered the upper and nonzero lower delay bounds of the interval time-varying delay. Case 1 is a special case of Case 2. If the time-varying delay is differentiable and 𝑑<1, one can obtain a less conservative result using Case 1 than that using Case 2.

By fuzzy blending, the overall fuzzy model is inferred as follows: ̇𝑥(𝑡)=𝑟𝑖=1𝑤𝑖𝐴(𝑧(𝑡))𝑖𝑥(𝑡)+𝐴di𝑥(𝑡(𝑡))𝑟𝑖=1𝑤𝑖=(𝑧(𝑡))𝑟𝑖=1𝜃𝑖𝐴(𝑧(𝑡))𝑖𝑥(𝑡)+𝐴di𝑥(𝑡(𝑡))=𝐴𝑥(𝑡)+𝐴𝑑𝑥(𝑡(𝑡))𝑥(𝑡)=𝑟𝑖=1𝜃𝑖(𝑧(𝑡))𝜙𝑖[],(𝑡),𝑡,0(2.4) where 𝑧=[𝑧1,𝑧2,...,𝑧𝑝];𝑤𝑖𝑅𝑝[0,1],𝑖=1,2,...,𝑟, is the membership function of the system with respect to the plant rule 𝑖; 𝜃𝑖(𝑧(𝑡))=𝑤𝑖(𝑧(𝑡))/𝑟𝑖=1𝑤𝑖(𝑧(𝑡)); 𝐴=𝑟𝑖=1𝜃𝑖(𝑧(𝑡))𝐴𝑖,  𝐴𝑑=𝑟𝑖=1𝜃𝑖(𝑧(𝑡))𝐴di, and It is assumed that 𝑤𝑖(𝑧(𝑡))0,𝑖=1,2,...,𝑟,  𝑟𝑖=1𝑤𝑖(𝑧(𝑡))0forall𝑡, so we have 𝜃𝑖(𝑧(𝑡))0,𝑟𝑖=1𝜃𝑖(𝑧(𝑡))=1.

In the following, we will develop some practically computable stability criteria for the system described (2.1a). The following lemmas are useful in deriving the criteria. First, we introduce the following technical Lemma 2.1 of integral inequality approach (IIA).

Lemma 2.1 (see [11]). For any positive semidefinite matrices 𝑋𝑋=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋330.(2.5) Then, one obtains 𝑡𝑡̇𝑥𝑇(𝑠)𝑋33̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝑥𝑇(𝑡)𝑥𝑇(𝑡)̇𝑥𝑇(𝑋𝑠)11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇230𝑥(𝑡)𝑥(𝑡)̇𝑥(𝑠)𝑑𝑠.(2.6)

Lemma 2.2 (see [28]). The following matrix inequality 𝑆𝑄(𝑥)𝑆(𝑥)𝑇(𝑥)𝑅(𝑥)<0,(2.7) where 𝑄(𝑥)=𝑄𝑇(𝑥),𝑅(𝑥)=𝑅𝑇(𝑥)and𝑆(𝑥) depend on affine on 𝑥 is equivalent to 𝑄𝑅(𝑥)<0,(𝑥)<0,𝑄(𝑥)𝑆(𝑥)𝑅1(𝑥)𝑆𝑇(𝑥)<0.(2.8)

In this paper, a new Lyapunov functional is constructed, which contains the information of the lower bound of delay 1 and upper bound 2.The following Theorem 2.3 presents a delay-range-dependent result in terms of LMIs and expresses the relationships between the terms of the Leibniz-Newton formula.

Theorem 2.3. Under Case 1, for given scalars 1,2,d, and 𝛼(0<𝛼<1), System (2.4) subject to (2.2) is asymptotically stable if there exist symmetry positive-definite matrices 𝑃=𝑃𝑇>0, 𝑄1=𝑄𝑇1>0, 𝑄2=𝑄𝑇2>0, 𝑄3=𝑄𝑇3>0, 𝑅1=𝑅𝑇1>0, 𝑅2=𝑅𝑇2>0, and positive semidefinite matrices 𝑋𝑋=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33𝑌0,𝑌=11𝑌12𝑌13𝑌𝑇12𝑌22𝑌23𝑌𝑇13𝑌𝑇23𝑌33𝑍0,𝑍=11𝑍12𝑍13𝑍𝑇12𝑍22𝑍23𝑍𝑇13𝑍𝑇23𝑍330(2.9) such that the following LMIs hold:Ξi=Ξ11Ξ120Ξ14Ξ15Ξ16ΞT12Ξ22Ξ23Ξ24Ξ25Ξ260ΞT23Ξ33Ξ000T14ΞT240Ξ44Ξ00T15ΞT2500Ξ550ΞT16ΞT26000Ξ66𝑅<0,(2.10a)1𝑋33𝑅0,(2.10b)2𝑌33𝑅0,(2.10c)2𝑍330,(2.10d)where Ξ11=𝐴𝑇𝑖𝑃+𝑃𝐴𝑖+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ξ12=𝑃𝐴di,Ξ14=𝛼𝛿𝑋12𝑋13+𝑋𝑇23,Ξ15=𝛼𝛿𝐴𝑇𝑖𝑅1,Ξ16=2𝐴𝛼𝛿𝑇𝑖𝑅2,Ξ22=1𝑑𝑄2+2𝑌𝛼𝛿22𝑌23𝑌𝑇23+2𝑍𝛼𝛿11+𝑍13+𝑍𝑇13,Ξ23=2𝑍𝛼𝛿12𝑍13+𝑍𝑇23,Ξ24=2𝑌𝛼𝛿𝑇12𝑌𝑇13+𝑌23,Ξ25=𝛼𝛿𝐴𝑇di𝑅1,Ξ26=2𝐴𝛼𝛿𝑇di𝑅2,Ξ33=𝑄3+2𝑍𝛼𝛿22𝑍23𝑍𝑇23,Ξ44=𝑄1+2𝑌𝛼𝛿11+𝑌13+𝑌𝑇13+𝛼𝛿𝑋22𝑋23𝑋𝑇23,Ξ55=𝛿𝑅1,Ξ66=2𝑅𝛼𝛿2.(2.11)

Proof. If we can proof that Theorem 2.3 holds for two cases, that is, 𝛼𝛿(𝑡)2 and 1(𝑡)𝛼𝛿, where 𝛼𝛿=(2+1)/2, then Theorem 2.3 is true.Case 1. When 𝛼𝛿(𝑡)2.
Construct a Lyapunov-Krasovskii functional candidate as 𝑉𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝛼𝛿𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡(𝑡)𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠+𝑡𝑡2𝑥𝑇(𝑠)𝑄3+𝑥(𝑠)𝑑𝑠0𝛼𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑑𝜃+𝛼𝛿2𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠𝑑𝜃.(2.12)
Calculating the derivative of (2.12) with respect to 𝑡>0 along the trajectories of (2.1a) and (2.1b) leads to ̇𝑉𝑥𝑡=𝑥𝑇(𝑡)𝑃𝐴+𝐴𝑇𝑃𝑥(𝑡)+𝑥𝑇(𝑡)𝑃𝐴𝑑𝑥(𝑡(𝑡))+𝑥𝑇(𝑡(𝑡))𝐴𝑇𝑑𝑃𝑥(𝑡)+𝑥𝑇(𝑄𝑡)1+𝑄2+𝑄3𝑥(𝑡)𝑥𝑇(𝑡𝛼𝛿)𝑄1𝑥(𝑡𝛼𝛿)𝑥𝑇̇𝑄(𝑡(𝑡))1(𝑡)2𝑥(𝑡(𝑡))𝑥𝑇𝑡2𝑄3𝑥𝑡2+̇𝑥𝑇(𝑡)𝛼𝛿𝑅1̇𝑥(𝑡)+̇𝑥𝑇(𝑡)2𝑅𝛼𝛿2̇𝑥(𝑡)𝑡𝑡𝛿̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑡𝛿𝑡2̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠𝑥𝑇(𝑡)𝑃𝐴+𝐴𝑇𝑃𝑥(𝑡)+𝑥𝑇(𝑡)𝑃𝐴𝑑𝑥(𝑡(𝑡))+𝑥𝑇(𝑡(𝑡))𝐴𝑇𝑑𝑃𝑥(𝑡)+𝑥𝑇𝑄(𝑡)1+𝑄2+𝑄3𝑥(𝑡)𝑥𝑇(𝑡𝛼𝛿)𝑄1𝑥(𝑡𝛼𝛿)𝑥𝑇(𝑡(𝑡))1𝑑𝑄2𝑥(𝑡(𝑡))𝑥𝑇𝑡2𝑄3𝑥𝑡2+̇𝑥𝑇(𝑡)𝛼𝛿𝑅1+2𝑅𝛼𝛿2̇𝑥(𝑡)𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡2̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠,(2.13) with the operator for the term ̇𝑥𝑇(𝑡)[𝛼𝛿𝑅1+(2𝛼𝛿)𝑅2]̇𝑥(𝑡) as follows: ̇𝑥𝑇(𝑡)𝛼𝛿𝑅1+2𝑅𝛼𝛿2=̇𝑥(𝑡)𝐴𝑥(𝑡)+𝐴𝑑𝑥(𝑡(𝑡))𝑇𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑥(𝑡)+𝐴𝑑𝑥(𝑡(𝑡))=𝑥𝑇(𝑡)𝐴𝑇𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑥(𝑡)+𝑥𝑇(𝑡)𝐴𝑇𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑑𝑥(𝑡(𝑡))+𝑥𝑇(𝑡(𝑡))𝐴𝑇𝑑𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑥(𝑡)+𝑥𝑇(𝑡(𝑡))𝐴𝑇𝑑𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑑𝑥(𝑡(𝑡)).(2.14)
Alternatively, the following equations are true: 𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡2̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠=𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡(𝑡)̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠𝑡(𝑡)𝑡2̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠=𝑡𝑡𝛼𝛿̇𝑥𝑇𝑅(𝑠)1𝑋33̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡(𝑡)̇𝑥𝑇𝑅(𝑠)2𝑌33̇𝑥(𝑠)𝑑𝑠𝑡(𝑡)𝑡2̇𝑥𝑇𝑅(𝑠)2𝑍33̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑋33̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡(𝑡)̇𝑥𝑇(𝑠)𝑌33̇𝑥(𝑠)𝑑𝑠𝑡(𝑡)𝑡2̇𝑥𝑇(𝑠)𝑍33̇𝑥(𝑠)𝑑𝑠.(2.15)
By utilizing Lemma 2.1 and the Leibniz-Newton formula, we have 𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑋33̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝛼𝛿𝑥𝑇(𝑡)𝑥𝑇(𝑡𝛼𝛿)̇𝑥𝑇(𝑋𝑠)11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇230𝑥(𝑡)𝑥(𝑡𝛼𝛿)̇𝑥(𝑠)𝑑𝑠𝑥𝑇(𝑡)𝛼𝛿𝑋11𝑥(𝑡)+𝑥𝑇(𝑡)𝛼𝛿𝑋12𝑥(𝑡𝛼𝛿)+𝑥𝑇(𝑡𝛼𝛿)𝛼𝛿𝑋𝑇12𝑥(𝑡)+𝑥𝑇(𝑡𝛼𝛿)𝛼𝛿𝑋22𝑥(𝑡𝛼𝛿)+𝑥𝑇(𝑡)𝑋𝑇13𝑡𝑡𝛼𝛿̇𝑥(𝑠)𝑑𝑠+𝑥𝑇(𝑡𝛼𝛿)𝑋𝑇23𝑡𝑡𝛼𝛿+̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑑𝑠𝑋13𝑥(𝑡)+𝑡𝑡𝛼𝛿̇𝑥𝑇(𝑠)𝑑𝑠𝑋23𝑥(𝑡𝛼𝛿)=𝑥𝑇(𝑡)𝛼𝛿𝑋11+𝑋𝑇13+𝑋13𝑥(𝑡)+𝑥𝑇(𝑡)𝛼𝛿𝑋12𝑋13+𝑋𝑇23𝑥(𝑡𝛼𝛿)+𝑥𝑇(𝑡𝛼𝛿)𝛿𝑋𝑇12+𝑋23𝑋𝑇13𝑥(𝑡)+𝑥𝑇(𝑡𝛼𝛿)𝛼𝛿𝑋22𝑋23𝑋𝑇23𝑥(𝑡𝛼𝛿).(2.16)
Similarly, we obtain 𝑡𝛼𝛿𝑡(𝑡)̇𝑥𝑇(𝑠)𝑌33̇𝑥(𝑠)𝑑𝑠𝑥𝑇(𝑡𝛼𝛿)2𝑌𝛼𝛿11+𝑌𝑇13+𝑌13𝑥(𝑡𝛼𝛿)+𝑥𝑇(𝑡𝛼𝛿)2𝑌𝛼𝛿12𝑌13+𝑌𝑇23𝑥(𝑡(𝑡))+𝑥𝑇(𝑡(𝑡))2𝑌𝛼𝛿𝑇12𝑌𝑇13+𝑌23𝑥(𝑡𝛼𝛿)+𝑥𝑇(𝑡(𝑡))2𝑌𝛼𝛿22𝑌23𝑌𝑇23𝑥(𝑡(𝑡)),(2.17)𝑡(𝑡)𝑡2̇𝑥𝑇(𝑠)𝑍33̇𝑥(𝑠)𝑑𝑠𝑥𝑇(𝑡(𝑡))2𝑍𝛼𝛿11+𝑍13+𝑍𝑇13𝑥(𝑡(𝑡))+𝑥𝑇(𝑡(𝑡))2𝑍𝛼𝛿12𝑍13+𝑍𝑇23𝑥𝑡2+𝑥𝑇𝑡22𝑍𝛼𝛿𝑇12𝑍𝑇13+𝑍23𝑥(𝑡(𝑡))+𝑥𝑇𝑡22𝑍𝛼𝛿22𝑍23𝑍𝑇23𝑥𝑡2.(2.18)
Substituting the above equations (2.14)–(2.18) into (2.13), we obtain ̇𝑉𝑥𝑡𝜉𝑇(𝑡)Ω𝜉(𝑡)𝑡𝑡𝛼𝛿̇𝑥𝑇𝑅(𝑠)1𝑋33̇𝑥(𝑠)𝑑𝑠𝑡𝛼𝛿𝑡(𝑡)̇𝑥𝑇𝑅(𝑠)2𝑌33̇𝑥(𝑠)𝑑𝑠𝑡(𝑡)𝑡2̇𝑥𝑇𝑅(𝑠)2𝑍33̇𝑥(𝑠)𝑑𝑠,(2.19) where 𝜉𝑇(𝑡)=[𝑥𝑇(𝑡)𝑥𝑇(𝑡(𝑡))𝑥𝑇(𝑡2)𝑥𝑇(𝑡𝛼𝛿)] and ΩΩ=11Ω120Ω14Ω𝑇12Ω22Ω23Ω240Ω𝑇23Ω330Ω𝑇14Ω𝑇240Ω44,(2.20) where Ω11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13+𝐴𝑇𝛼𝛿𝑅1+2𝑅𝛼𝛿2Ω𝐴,12=𝑃𝐴𝑑+𝐴𝑇𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑑,Ω14=𝛼𝛿𝑋12𝑋13+𝑋𝑇23,Ω22=1𝑑𝑄2+2𝑍𝛼𝛿11+𝑍13+𝑍𝑇13+2𝑌𝛼𝛿22𝑌23𝑌𝑇23+𝐴𝑇𝑑𝛼𝛿𝑅1+2𝑅𝛼𝛿2𝐴𝑑,Ω23=2𝑍𝛼𝛿12𝑍13+𝑍𝑇23,Ω24=2𝑌𝛼𝛿𝑇12𝑌𝑇13+𝑌23,Ω33=𝑄3+2𝑍𝛼𝛿22𝑍23𝑍𝑇23,Ω44=𝑄1+2𝑌𝛼𝛿11+𝑌13+𝑌𝑇13+𝛼𝛿𝑋22𝑋23𝑋𝑇23.(2.21)
If Ω<0,𝑅1𝑋330,𝑅2𝑌330, and 𝑅2𝑍330, then ̇𝑉(𝑥𝑡)<𝜀𝑥(𝑡)2 for a sufficiently small 𝜀>0. By the Schur complement of Lemma 2.2, Ω<0 is equivalent to the following inequality and is true: ΨΨ=11Ψ120Ψ14Ψ15Ψ16Ψ𝑇12Ψ22Ψ23Ψ24Ψ25Ψ260Ψ𝑇23Ψ33Ψ000𝑇14Ψ𝑇240Ψ44Ψ00𝑇15Ψ𝑇2500Ψ550Ψ𝑇16Ψ𝑇26000Ψ66<0,(2.22) where Ψ11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ψ12=𝑃𝐴𝑑,Ψ14=𝛼𝛿𝑋12𝑋13+𝑋𝑇23,Ψ15=𝛼𝛿𝐴𝑇𝑅1,Ψ16=2𝐴𝛼𝛿𝑇𝑅2,Ψ22=1𝑑𝑄2+2𝑌𝛼𝛿22𝑌23𝑌𝑇23+2𝑍𝛼𝛿11+𝑍13+𝑍𝑇13,Ψ23=2𝑍𝛼𝛿12𝑍13+𝑍𝑇23,Ψ24=2𝑌𝛼𝛿𝑇12𝑌𝑇13+𝑌23,Ψ25=𝛼𝛿𝐴𝑇𝑑𝑅1,Ψ26=2𝐴𝛼𝛿𝑇𝑑𝑅2,Ψ33=𝑄3+2𝑍𝛼𝛿22𝑍23𝑍𝑇23,Ψ44=𝑄1+2𝑌𝛼𝛿11+𝑌13+𝑌𝑇13+𝛼𝛿𝑋22𝑋23𝑋𝑇23,Ψ55=𝛼𝛿𝑅1,Ψ66=2𝑅𝛼𝛿2.(2.23)
That is to say, if Ψ<0,  𝑅1𝑋330,𝑅2𝑌330, and 𝑅2𝑍330, then ̇𝑉(𝑡)<𝜀𝑥(𝑡)2 for a sufficiently small 𝜀>0. Furthermore, (2.10a) implies 𝑟𝑖=1𝜃𝑖(𝑧(𝑡))Ω𝑖<0, which is equivalent to (2.22). Therefore, if LMIs (2.10a) are feasible, the system (2.4) is asymptotically stable.Case 2. When 1(𝑡)𝛼𝛿.
For this case, the Lyapunov-Krasovskii functional candidate is chosen 𝑉𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝛼𝛿𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡(𝑡)𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠+𝑡𝑡1𝑥𝑇(𝑠)𝑄3+𝑥(𝑠)𝑑𝑠0𝛼𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑅1̇𝑥(𝑠)𝑑𝑠𝑑𝜃+1𝛼𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑅2̇𝑥(𝑠)𝑑𝑠𝑑𝜃,(2.24) where 𝑃>0,𝑅1>0,𝑅2>0,𝑄𝑖>0(𝑖=1,2,3).
Choosing 𝜉𝑇(𝑡)=[𝑥𝑇(𝑡)𝑥𝑇(𝑡(𝑡))𝑥𝑇(𝑡1)𝑥𝑇(𝑡𝛼𝛿)] and then using a proof process similar to that for Case 1, we derive the same condition (2.10a), (2.10b), (2.10c), and (2.10d) as that for Case 1. This completes the proof.

When the information of the time derivative of delay is unknown, by eliminating 𝑄2we have the following result from Theorem 2.3.

Corollary 2.4. For given scalars 1, 2, and 𝛼(0<𝛼<1), the system (2.4) is asymptotically stable if there exist positive-definite matrices 𝑃=𝑃𝑇>0, 𝑄1=𝑄𝑇1>0, 𝑄3=𝑄𝑇3>0, 𝑅1=𝑅𝑇1>0, 𝑅2=𝑅𝑇2>0 and positive semidefinite matrices 𝑋𝑋=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33𝑌0,𝑌=11𝑌12𝑌13𝑌𝑇12𝑌22𝑌23𝑌𝑇13𝑌𝑇23𝑌33𝑍0,𝑍=11𝑍12𝑍13𝑍𝑇12𝑍22𝑍23𝑍𝑇13𝑍𝑇23𝑍330,(2.25) such that the following LMIs hold:Ξ𝑖=Ξ11Ξ120Ξ14Ξ15Ξ16Ξ𝑇12Ξ22Ξ23Ξ24Ξ25Ξ260Ξ𝑇23Ξ33Ξ000𝑇14Ξ𝑇240Ξ44Ξ00𝑇15Ξ𝑇2500Ξ550Ξ𝑇16Ξ𝑇26000Ξ66𝑅<0,(2.26a)1𝑋33𝑅0,(2.26b)2𝑌33𝑅0,(2.26c)2𝑍330,(2.26d)where Ξ𝑖𝑗(𝑖,𝑗=1,2,...,6) are defined in (8) and Ξ11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ξ22=2𝑌𝛼𝛿22𝑌23𝑌𝑇23+2𝑍𝛼𝛿11+𝑍13+𝑍𝑇13.(2.27)

Proof. If the matrix 𝑄2=0 is selected in (2.10a), (2.10b), (2.10c), and (2.10d), this proof can be completed in a similar formulation to Theorem 2.3.

When 1=0, Theorem 2.3 reduces to the following Corollary 2.5.

Corollary 2.5. For given scalars ,𝑑, and 𝛼(0<𝛼<1), the system (2.4) is asymptotically stable if there exist symmetry positive-definite matrices 𝑃=𝑃𝑇>0,  𝑄1=𝑄𝑇1>0,  𝑄2=𝑄𝑇2>0,𝑅=𝑅𝑇>0, and positive semidefinite matrices 𝑋𝑋=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33𝑌0,𝑌=11𝑌12𝑌13𝑌𝑇12𝑌22𝑌23𝑌𝑇13𝑌𝑇23𝑌330,(2.28) such that the following LMIs hold for 𝑖=1,2,...,𝑟, Σ𝑖=Σ11Σ120Σ14Σ𝑇12Σ22Σ23Σ240Σ𝑇23Σ330Σ𝑇14Σ𝑇240Σ44<0,𝑅𝑋330,𝑅𝑌330,(2.29) where Σ11=𝐴𝑇𝑖𝑃+𝑃𝐴𝑖+𝑄1+𝑄2+𝛼𝑋11+𝑋13+𝑋𝑇13,Σ12=𝑃𝐴𝑑𝑖+𝛼𝑋12𝑋13+𝑋𝑇23,Σ14=𝛼𝐴𝑇𝑖𝑅,Σ22=1𝑑𝑄2+𝛼𝑋22𝑋23𝑋𝑇23+𝑌11+𝑌13+𝑌𝑇13,Σ23=𝛼𝑌12𝑌13+𝑌𝑇23,Σ24=𝛼𝐴𝑇𝑑𝑖Σ𝑅,33=𝑄1+𝛼𝑌22𝑌23𝑌𝑇23,Ω44=𝛼𝑅.(2.30)

Proof. Choose the following fuzzy Lyapunov-Krasovskii functional candidate to be 𝑉(𝑡)=𝑉1(𝑡)+𝑉2(𝑡)+𝑉3(𝑡)+𝑉4(𝑡),(2.31) where 𝑉1(𝑡)=𝑥𝑇𝑉(𝑡)𝑃𝑥(𝑡)2(𝑡)=𝑡𝑡𝛼𝑥𝑇(𝑠)𝑄1𝑉𝑥(𝑠)𝑑𝑠3(𝑡)=𝑡𝑡(𝑡)𝑥𝑇(𝑠)𝑄2𝑥𝑉(𝑠)𝑑𝑠4(𝑡)=0𝛼𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑅̇𝑥(𝑠)𝑑𝑠𝑑𝜃.(2.32)
Then, taking the time derivative of 𝑉(𝑡) with respect to 𝑡 along the system (2.4) yield ̇̇𝑉𝑉(𝑡)=1̇𝑉(𝑡)+2̇𝑉(𝑡)+3̇𝑉(𝑡)+4(𝑡).(2.33)
Then the proof follows a linear similar to the proof of Theorem 2.3 and thus is omitted here.

Remark 2.6. In our Theorems 2.3, 𝑑 can be any value or unknown due to Ξ22=(1𝑑)𝑄2+(2𝛼𝛿)𝑌22𝑌23𝑌𝑇23+(2𝛼𝛿)𝑍11+𝑍13+𝑍𝑇13. Therefore, Theorem 2.3 is applicable to both cases of fast and slow time varying delay. We will show the other characteristic of Theorem 2.3. When the distance between 1 and 2 is sufficiently small, the upper bound 2 of delay for unknown 𝑑 will be very close to the upper bound for 𝑑=0. This characteristic is not included in previous Lyapunov functional based work where the upper bound of delay for 𝑑0 is always less than that for 𝑑=0.

Remark 2.7. It is seen from the proof of Theorem 2.3 and Corollary 2.4 that the main characteristics of the method developed in this paper can be generalized as the following two steps. (i) Construct a Lyapunov function to integrated both lower and upper delay bounds, for example, 𝛼𝛿=(2+1)/2 in (2.12) and (2.24). (ii) Employ Lemma 2.1 to deal with cross-product terms, for example, those in (2.15)–(2.18). It is also seen from the proof that neither model transformation nor free-weighting matrices have been employed to deal with the cross-product terms. Therefore, the stability criteria derived in this paper are expected to be less conservative. This will be demonstrated later through numerical examples. It is noted that although it has been observed that using 𝛼𝛿=(2+1)/2 in the constructed Lyapunov function can improve stability performance for many examples, theoretical evidence has not been found so far to explain the observations.
Based on that, a convex optimization problem is formulated to find the bound on the allowable delay time 01(𝑡)2 which maintains the delay-dependent stability of the time delay system (2.4).

Remark 2.8. It is interesting to note that 1,2 appears linearly in (2.10a) and (2.26a). Thus, a generalized eigenvalue problem (GEVP) as defined in Boyd et al. [28] can be formulated to solve the minimum acceptable 1/1(or1/2) and, therefore, the maximum 1(or2) to maintain robust stability as judged by these conditions.
In this way, our optimization problem becomes a standard generalized eigenvalue problem, which can be then solved using GEVP technique. From this discussion, we have the following Remark 2.9.

Remark 2.9. Theorem 2.3 provides delay-dependent asymptotic stability criteria for the T-S fuzzy systems with an interval time-varying delay (2.4) in terms of solvability of LMIs [28]. Based on them, we can obtain the maximum allowable delay bound (MADB) 01(𝑡)2 such that (2.4) is stable by solving the following convex optimization problem: Maximize2SubjecttoTheorem2.3(Corollary2.4)(2.34)
Inequality (2.34) is a convex optimization problem and can be obtained efficiently using the MATLAB LMI Toolbox.
About how to seek an appropriate 𝛼 satisfying 0<𝛼<1, such that the upper bound of delay (𝑡) subjecting to (2.29) is maximal, we give an algorithm as follows.

Algorithm 2.10 ((Maximizing >0)). Step 1: For given 𝑑, choose an upper bound on satisfying (2.29), and then select this upper bound as the initial value 0 of .
Step 2: Set appropriate step lengths, step and 𝛼step, for and 𝛼, respectively. Set 𝑘 as a counter, and choose 𝑘=1. Meanwhile, let =0+step and the initial value 𝛼0 of 𝛼 equals to 𝛼step.
Step 3: Let 𝛼=𝑘𝛼step, if inequality (2.29) is feasible, go to Step 4; otherwise, go to Step 5.
Step 4: Let 0=,𝛼0=𝛼,𝑘=1 and =0+step, go to step 3.
Step 5: Let 𝑘=𝑘+1. If 𝑘𝛼step<1, then go to step 3; otherwise, stop.

Remark 2.11. For Algorithm 2.10, the final 0 is the desired maximum of the upper bound of delay (𝑡) satisfying (2.29) and 𝛼0 is the corresponding value of 𝛼.

3. Illustrative Examples

To illustrate the usefulness of our results, this section will provide numerical examples. It will be shown that the proposed results can provide less conservative results that recent ones have given [3, 57, 9, 10, 12, 13, 15, 1721, 23, 27]. It is worth pointing out that our criteria carried out more efficiently for computation.

Example 3.1. Consider a time-delayed fuzzy system without controlling input. The T-S fuzzy model of this fuzzy system is of the following form.
Plant rules:Rule 1: If 𝑥1(𝑡) is 𝑀1, then ̇𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐴𝑑1𝑥(𝑡(𝑡)).(3.1)Rule 2: If 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐴𝑑2𝑥(𝑡(𝑡)),(3.2) and the membership functions for rule 1 and rule 2 are 𝑀1𝑥1=1(𝑡)11+𝑒5(𝑥1(𝑡)(𝜋/6))11+𝑒5(𝑥1(𝑡)(𝜋/6)),𝑀2𝑥1(𝑡)=1𝑀1𝑥1(𝑡),(3.3) where 𝐴1=2000.9,𝐴𝑑1=1011,𝐴2=1.5100.75,𝐴𝑑2=1010.85.(3.4)

Solution 3.2. For system (3.1) and (3.2), by taking the parameter 𝑑=0 and 𝛼=0.6, we get the Corollary 2.5 which remains feasible for any delay time 2.2459. In case of maximum allowable delay bound (MADB) =2.2459, solving Corollary 2.5 yields the following set of feasible solutions: 𝑃=33.23427.99927.999228.4879,𝑄1=0.00390.00070.00070.0029,𝑄2=,78.84372.54712.547122.8291𝑅=1.62804.50934.509313.7942,𝑋11=1.89331.58771.58778.4013,𝑋12=,𝑋0.16633.39813.43816.770013=1.20513.34693.345410.2327,𝑋22=2.02591.51081.51088.4337,𝑋23=,𝑋1.20303.34883.347310.231033=1.62464.50974.509713.7915,𝑌11=5.12990.41160.41165.9858,𝑌12=,𝑌0.11020.16240.15880.530213=0.19710.58720.59151.7945,𝑌22=0.29620.65220.65222.0458,𝑌23=,𝑌0.34560.80550.80582.509233=.0.80932.11412.11416.4977(3.5)
Applying the criteria in [9, 10, 12, 17] and in this paper, the maximum values of for the stability of system under considerations are listed in Table 1. It is easy to see that the stability criterion in this paper gives a much less conservative result than the ones in [9, 10, 12, 17].

tab1
Table 1: Computation MADB for varying 𝑑 in Example 3.1.

Furthermore, by taking the various 1(𝑑=0.5), and from Theorem 2.3, we obtain the maximum allowable delay bound (MADB) 2 as shown in Table 2. From the above results of Table 2, if the 1 increases, the delay time length increases.

tab2
Table 2: MADB 2(𝑑=0.5) for various 1 in Example 3.1.

Remark 3.3. Similar to Algorithm 2.10, an algorithm for seeking an appropriate 𝛼 such that the upper bound of delay 01(𝑡)2, subjecting to (2.10a), (2.10b), (2.10c), and (2.10d) is maximal can be easily obtained.

Example 3.4. Consider a T-S fuzzy system with time-varying delay. The T-S fuzzy model of this fuzzy system is of the following form:Rule 1: If 𝑥1(𝑡) is 𝑀1, then ̇𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐴𝑑1𝑥(𝑡(𝑡)).(3.6)Rule 2: If 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐴𝑑2𝑥(𝑡(𝑡)),(3.7) and the membership functions for rule 1 and rule 2 are 𝑀11(𝑧(𝑡))=1+exp2𝑥1(𝑡),𝑀2𝑥1(𝑡)=1𝑀1(𝑧(𝑡)),(3.8) where 𝐴1=3.20.602.1,𝐴𝑑1=10.902,𝐴2=1013,𝐴𝑑2=0.9011.6.(3.9)

Solution 3.5. By taking the parameter 𝑑=0, using Corollary 2.5, the maximum value of delay time for the System (3.6) and (3.7) to be asymptotically stable is 1.0245. By the criteria in [9, 16], the system (3.6) and (3.7) is asymptotically stable for 0.58 and 0.6148, respectively. Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared with the result of [9, 16]. Employing the LMIs in [1821] and those in Corollary 2.5 yields upper bounds on 2 that guarantee the stability of system (3.6) and (3.7) for various 𝑑, which are listed in Table 3, in which “–” means that the results are not applicable to the corresponding cases. It can be seen from Table 3 that Corollary 2.5 in this paper yields the least conservative stability test than other approaches, showing the advantage of the stability result in this paper.

tab3
Table 3: MADB for various 𝑑 in Example 3.4.

Example 3.6. Consider a T-S fuzzy system with time-varying delay is of the following form:
Plant rules.Rule 1: If 𝑥1(𝑡) is 𝑀1, then ̇𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐴𝑑1𝑥(𝑡(𝑡)).(3.10)Rule 2: If 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐴𝑑2𝑥(𝑡(𝑡)),(3.11) and the membership functions for rule 1 and rule 2 are 𝑀11(𝑧(𝑡))=1+exp2𝑥1(𝑡),𝑀2𝑥1(𝑡)=1𝑀1(𝑧(𝑡)),(3.12) where 𝐴1=0.1100.2,𝐴𝑑1=0.3200.5,𝐴2=0.1200.2,𝐴𝑑2=0.3420.010.4.(3.13)

Solution 3.7. By using Corollary 2.4, the maximum allowable delay bound (MADB) can be calculated as 2=2.3725. The results for stability conditions in different methods are compared in Table 4. It can be seen that the delay-dependent stability condition in this paper is less conservative than earlier reported ones in the literature [5, 7, 10, 15, 18, 27]. Compared with Guan and chen [7] who used 5 LMI variables, Yoneyama [27] employed 20 LMI variables to get better stability results. To obtain improved stability results than those in [15, 18, 27], we need 8 variables in Corollary 2.4 the same as [5, 10]. It is also seen from Table 4 that the larger r is the more unknown LMI variables are required in [18, 27]. However, the unknown number of LMI variables is independent of r in the results of this paper. It can be shown that the delay-dependent stability condition in this paper is the best performance.

tab4
Table 4: MADB 2 for unknown 𝑑(1=0) in Example 3.6 for different methods (Number of variables: 𝑁𝑣; Number of fuzzy rules: 𝑟).

Remark 3.8. Similar to Algorithm 2.10, we can also find an appropriate scalar 𝛼, such that the upper bound of delay 01(𝑡)2, subjecting to (2.26a), (2.26b), (2.26c), and (2.26d) reaches the maximum.

Example 3.9. Consider a nominal time-delay fuzzy system. The T-S fuzzy model of this fuzzy system is of the following form.Rule 1: If 𝑥1(𝑡) is 𝑀1, theṅ𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐴𝑑1𝑥(𝑡(𝑡)).(3.14a)Rule 2: If 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐴𝑑2𝑥(𝑡(𝑡)),(3.14b)and the membership function for rules 1 and 2 are 𝑀11(𝑧(𝑡))=1+exp2𝑥1(𝑡),𝑀2𝑥1(𝑡)=1𝑀1(𝑧(𝑡)),(3.15) where 𝐴1=2.10.10.20.9,𝐴𝑑1=1.10.10.80.9,𝐴2=1.901.11.1,𝐴𝑑2=0.901.11.2.(3.16)

Solution 3.10. The objective is to determine the maximum value of constant time-delay =2(1=0)for which the system is stable. Table 5 compares works based on common quadratic functionals [7, 1820, 23] with the fuzzy functional of Corollary 2.4. It is clear by inspecting Table 5 that Corollary 2.4 provides the largest time delays. For comparison, Table 6 also lists the maximum allowable delay bound (MADB) obtained from the criterion [3]. It is clear that Corollary 2.5 gives much better results than those obtained by [3]. It is illustrated that the proposed stability criteria are effective in comparison to earlier and newly published results existing in the literature.

tab5
Table 5: MADB =2(1=0) for different methods in Example 3.9.
tab6
Table 6: MADB for different 𝑑 for Example 3.9.

Example 3.11. Consider a nominal time-delay fuzzy system. The T-S fuzzy model of this fuzzy system is of the following form.Rule 1: If 𝑥1(𝑡) is 𝑀1, then ̇𝑥(𝑡)=𝐴1𝑥(𝑡)+𝐴𝑑1𝑥(𝑡(𝑡)).(3.17)Rule 2: If 𝑥1(𝑡) is 𝑀2, then ̇𝑥(𝑡)=𝐴2𝑥(𝑡)+𝐴𝑑2𝑥(𝑡(𝑡)),(3.18) and the membership functions for rule 1 and rule 2 are 𝑀1(𝑧(𝑡))=sin2𝑥1(𝑡),𝑀2𝑥1(𝑡)=cos2𝑥1,(𝑡)(3.19) where 𝐴1=0186,𝐴2=0180,𝐴𝑑1=𝐴𝑑2=0012.(3.20)

Solution 3.12. Considering a constant time delay, by using Corollary 2.4, the maximum allowable delay bound (MADB) can be calculated as 2=0.4130(1=0,𝛼=0.5). By the criteria in [13, 23], the systems (3.17) and (3.18) is asymptotically stable for any that satisfies 0.322 and 0.4060, respectively. Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared with the result of [13, 23].

4. Conclusion

In this paper, we have dealt with the stability problem for T-S fuzzy systems with interval time-varying delay. By constructing a Lyapunov-Krasovskii functional, the supplementary requirement that the time derivative of time-varying delays must be smaller than one is released in the proposed delay-range-dependent stability criterion. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-range-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Since the delay term is concerned more exactly, it is less conservative and more computationally efficient than those obtained from existing methods. Thus, the present method could largely reduce the computational burden in solving LMIs. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Numerical examples are given to illustrate the effectiveness of our theoretical results.

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