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 [3–21]. 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, 17–20], 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 β„Ž(𝑑) 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, 24–26] 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 [π‘‘βˆ’β„Ž,𝑑] is decomposed into [π‘‘βˆ’β„Ž,π‘‘βˆ’π›Όβ„Ž] and [π‘‘βˆ’π›Όβ„Ž,𝑑]. Since a tuning parameter 𝛼 is introduced, the information about π‘₯(π‘‘βˆ’π›Όβ„Ž) can be taken into full consideration; thus, the upper bound of βˆ«π‘‘π‘‘βˆ’β„ŽΜ‡π‘₯𝑇(𝑠)𝑅̇π‘₯(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, 5–7, 9, 10, 12, 13, 15, 17–21, 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 π‘–βˆΆ if 𝑧1(𝑑) is 𝑀𝑖1, and …, and 𝑧𝑝(𝑑) is 𝑀𝑖𝑝 thenΜ‡π‘₯(𝑑)=𝐴𝑖π‘₯(𝑑)+𝐴di[]π‘₯(π‘‘βˆ’β„Ž(𝑑)),(2.1a)π‘₯(𝑑)=πœ™(𝑑),π‘‘βˆˆβˆ’β„Ž,0,𝑖=1,2,...,π‘Ÿ,(2.1b)where 𝑧1(𝑑),𝑧2(𝑑),...,𝑧𝑝(𝑑) are the premise variables; 𝑀𝑖𝑗,𝑖=1,2,...,π‘Ÿ,𝑗=1,2,...,𝑝 are the fuzzy sets; π‘₯(𝑑)βˆˆπ‘…π‘› is the state; πœ™(𝑑) 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, β„Ž(𝑑), is a time-varying delay. We will consider the following two cases for the time-varying delay.

Case 1. β„Ž(𝑑) is a differentiable function satisfying 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2,||Μ‡||β„Ž(𝑑)β‰€β„Žπ‘‘,βˆ€π‘‘β‰₯0.(2.2)

Case 2. β„Ž(𝑑) is a differentiable function satisfying 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2,(2.3) where β„Ž1 and β„Ž2 are the lower and upper delay bounds, respectively; β„Ž1,β„Ž2, and β„Žπ‘‘ are constants. Here β„Ž1 the lower bound of delay may not be equal to 0, and when β„Žπ‘‘=0 we have β„Ž1=β„Ž2. 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 β„Žπ‘‘<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: βˆ‘Μ‡π‘₯(𝑑)=π‘Ÿπ‘–=1𝑀𝑖𝐴(𝑧(𝑑))𝑖π‘₯(𝑑)+𝐴diπ‘₯ξ€»(π‘‘βˆ’β„Ž(𝑑))βˆ‘π‘Ÿπ‘–=1𝑀𝑖=(𝑧(𝑑))π‘Ÿξ“π‘–=1πœƒπ‘–ξ€Ίπ΄(𝑧(𝑑))𝑖π‘₯(𝑑)+𝐴diξ€»π‘₯(π‘‘βˆ’β„Ž(𝑑))=𝐴π‘₯(𝑑)+𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))π‘₯(𝑑)=π‘Ÿξ“π‘–=1πœƒπ‘–(𝑧(𝑑))πœ™π‘–[],(𝑑),π‘‘βˆˆβˆ’β„Ž,0(2.4) where 𝑧=[𝑧1,𝑧2,...,𝑧𝑝];π‘€π‘–βˆΆπ‘…π‘β†’[0,1],𝑖=1,2,...,π‘Ÿ, is the membership function of the system with respect to the plant rule 𝑖; πœƒπ‘–(𝑧(𝑑))=π‘€π‘–βˆ‘(𝑧(𝑑))/π‘Ÿπ‘–=1𝑀𝑖(𝑧(𝑑)); βˆ‘π΄=π‘Ÿπ‘–=1πœƒπ‘–(𝑧(𝑑))𝐴𝑖,  𝐴𝑑=βˆ‘π‘Ÿπ‘–=1πœƒπ‘–(𝑧(𝑑))𝐴di, and It is assumed that 𝑀𝑖(𝑧(𝑑))β‰₯0,𝑖=1,2,...,π‘Ÿ,β€‰β€‰βˆ‘π‘Ÿπ‘–=1𝑀𝑖(𝑧(𝑑))β‰₯0forall𝑑, so we have πœƒπ‘–βˆ‘(𝑧(𝑑))β‰₯0,π‘Ÿπ‘–=1πœƒπ‘–(𝑧(𝑑))=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 βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘‹=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦β‰₯0.(2.5) Then, one obtains βˆ’ξ€œπ‘‘π‘‘βˆ’β„ŽΜ‡π‘₯𝑇(𝑠)𝑋33ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ β‰€π‘‘π‘‘βˆ’β„Žξ‚ƒπ‘₯𝑇(𝑑)π‘₯𝑇(π‘‘βˆ’β„Ž)Μ‡π‘₯𝑇(ξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘ )11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇230⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦π‘₯(𝑑)π‘₯(π‘‘βˆ’β„Ž)Μ‡π‘₯(𝑠)𝑑𝑠.(2.6)

Lemma 2.2 (see [28]). The following matrix inequality βŽ‘βŽ’βŽ’βŽ£π‘†π‘„(π‘₯)𝑆(π‘₯)𝑇(⎀βŽ₯βŽ₯⎦π‘₯)𝑅(π‘₯)<0,(2.7) where 𝑄(π‘₯)=𝑄𝑇(π‘₯),𝑅(π‘₯)=𝑅𝑇(π‘₯)and𝑆(π‘₯) depend on affine on π‘₯ is equivalent to 𝑄𝑅(π‘₯)<0,(π‘₯)<0,𝑄(π‘₯)βˆ’π‘†(π‘₯)π‘…βˆ’1(π‘₯)𝑆𝑇(π‘₯)<0.(2.8)

In this paper, a new Lyapunov functional is constructed, which contains the information of the lower bound of delay β„Ž1 and upper bound β„Ž2.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 β„Ž1,β„Ž2,β„Žd, 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 βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘‹=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Œβ‰₯0,π‘Œ=11π‘Œ12π‘Œ13π‘Œπ‘‡12π‘Œ22π‘Œ23π‘Œπ‘‡13π‘Œπ‘‡23π‘Œ33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘β‰₯0,𝑍=11𝑍12𝑍13𝑍𝑇12𝑍22𝑍23𝑍𝑇13𝑍𝑇23𝑍33⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦β‰₯0(2.9) such that the following LMIs hold:Ξi=⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣Ξ11Ξ120Ξ14Ξ15Ξ16ΞT12Ξ22Ξ23Ξ24Ξ25Ξ260ΞT23Ξ33Ξ000T14ΞT240Ξ44Ξ00T15ΞT2500Ξ550ΞT16ΞT26000Ξ66⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘…<0,(2.10a)1βˆ’π‘‹33𝑅β‰₯0,(2.10b)2βˆ’π‘Œ33𝑅β‰₯0,(2.10c)2βˆ’π‘33β‰₯0,(2.10d)where Ξ11=𝐴𝑇𝑖𝑃+𝑃𝐴𝑖+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ξ12=𝑃𝐴di,Ξ14=𝛼𝛿𝑋12βˆ’π‘‹13+𝑋𝑇23,Ξ15=𝛼𝛿𝐴𝑇𝑖𝑅1,Ξ16=ξ€·β„Ž2ξ€Έπ΄βˆ’π›Όπ›Ώπ‘‡π‘–π‘…2,Ξ22ξ€·=βˆ’1βˆ’β„Žπ‘‘ξ€Έπ‘„2+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ11+𝑍13+𝑍𝑇13,Ξ23=ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ12βˆ’π‘13+𝑍𝑇23,Ξ24=ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώπ‘‡12βˆ’π‘Œπ‘‡13+π‘Œ23,Ξ25=𝛼𝛿𝐴𝑇di𝑅1,Ξ26=ξ€·β„Ž2ξ€Έπ΄βˆ’π›Όπ›Ώπ‘‡di𝑅2,Ξ33=βˆ’π‘„3+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ22βˆ’π‘23βˆ’π‘π‘‡23,Ξ44=βˆ’π‘„1+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ11+π‘Œ13+π‘Œπ‘‡13+𝛼𝛿𝑋22βˆ’π‘‹23βˆ’π‘‹π‘‡23,Ξ55=βˆ’π›Ώπ‘…1,Ξ66ξ€·β„Ž=βˆ’2ξ€Έπ‘…βˆ’π›Όπ›Ώ2.(2.11)

Proof. If we can proof that Theorem 2.3 holds for two cases, that is, π›Όπ›Ώβ‰€β„Ž(𝑑)β‰€β„Ž2 and β„Ž1β‰€β„Ž(𝑑)≀𝛼𝛿, where 𝛼𝛿=(β„Ž2+β„Ž1)/2, then Theorem 2.3 is true.Case 1. When π›Όπ›Ώβ‰€β„Ž(𝑑)β‰€β„Ž2.
Construct a Lyapunov-Krasovskii functional candidate as 𝑉π‘₯𝑑=π‘₯𝑇(ξ€œπ‘‘)𝑃π‘₯(𝑑)+π‘‘π‘‘βˆ’π›Όπ›Ώπ‘₯𝑇(𝑠)𝑄1ξ€œπ‘₯(𝑠)𝑑𝑠+π‘‘π‘‘βˆ’β„Ž(𝑑)π‘₯𝑇(𝑠)𝑄2ξ€œπ‘₯(𝑠)𝑑𝑠+π‘‘π‘‘βˆ’β„Ž2π‘₯𝑇(𝑠)𝑄3+ξ€œπ‘₯(𝑠)𝑑𝑠0βˆ’π›Όπ›Ώξ€œπ‘‘π‘‘+πœƒΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘πœƒ+βˆ’π›Όπ›Ώβˆ’β„Ž2ξ€œπ‘‘π‘‘+πœƒΜ‡π‘₯𝑇(𝑠)𝑅2Μ‡π‘₯(𝑠)π‘‘π‘ π‘‘πœƒ.(2.12)
Calculating the derivative of (2.12) with respect to 𝑑>0 along the trajectories of (2.1a) and (2.1b) leads to ̇𝑉π‘₯𝑑=π‘₯𝑇(𝑑)𝑃𝐴+𝐴𝑇𝑃π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))𝐴𝑇𝑑𝑃π‘₯(𝑑)+π‘₯𝑇(𝑄𝑑)1+𝑄2+𝑄3ξ€Έπ‘₯(𝑑)βˆ’π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝑄1π‘₯(π‘‘βˆ’π›Όπ›Ώ)βˆ’π‘₯π‘‡ξ€·Μ‡β„Žξ€Έπ‘„(π‘‘βˆ’β„Ž(𝑑))1βˆ’(𝑑)2π‘₯(π‘‘βˆ’β„Ž(𝑑))βˆ’π‘₯π‘‡ξ€·π‘‘βˆ’β„Ž2𝑄3π‘₯ξ€·π‘‘βˆ’β„Ž2ξ€Έ+Μ‡π‘₯𝑇(𝑑)𝛼𝛿𝑅1Μ‡π‘₯(𝑑)+Μ‡π‘₯𝑇(ξ€·β„Žπ‘‘)2ξ€Έπ‘…βˆ’π›Όπ›Ώ2ξ€œΜ‡π‘₯(𝑑)βˆ’π‘‘π‘‘βˆ’π›ΏΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Ώπ‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑅2Μ‡π‘₯(𝑠)𝑑𝑠≀π‘₯𝑇(𝑑)𝑃𝐴+𝐴𝑇𝑃π‘₯(𝑑)+π‘₯𝑇(𝑑)𝑃𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))𝐴𝑇𝑑𝑃π‘₯(𝑑)+π‘₯𝑇𝑄(𝑑)1+𝑄2+𝑄3ξ€Έπ‘₯(𝑑)βˆ’π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝑄1π‘₯(π‘‘βˆ’π›Όπ›Ώ)βˆ’π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))1βˆ’β„Žπ‘‘ξ€Έπ‘„2π‘₯(π‘‘βˆ’β„Ž(𝑑))βˆ’π‘₯π‘‡ξ€·π‘‘βˆ’β„Ž2𝑄3π‘₯ξ€·π‘‘βˆ’β„Ž2ξ€Έ+Μ‡π‘₯𝑇(𝑑)𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2ξ€»ξ€œΜ‡π‘₯(𝑑)βˆ’π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑅2Μ‡π‘₯(𝑠)𝑑𝑠,(2.13) with the operator for the term Μ‡π‘₯𝑇(𝑑)[𝛼𝛿𝑅1+(β„Ž2βˆ’π›Όπ›Ώ)𝑅2]Μ‡π‘₯(𝑑) as follows: Μ‡π‘₯𝑇(𝑑)𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2ξ€»=ξ€ΊΜ‡π‘₯(𝑑)𝐴π‘₯(𝑑)+𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))𝑇𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴π‘₯(𝑑)+𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))=π‘₯𝑇(𝑑)𝐴𝑇𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴π‘₯(𝑑)+π‘₯𝑇(𝑑)𝐴𝑇𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))𝐴𝑇𝑑𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴π‘₯(𝑑)+π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))𝐴𝑇𝑑𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴𝑑π‘₯(π‘‘βˆ’β„Ž(𝑑)).(2.14)
Alternatively, the following equations are true: βˆ’ξ€œπ‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑅2ξ€œΜ‡π‘₯(𝑠)𝑑𝑠=βˆ’π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž(𝑑)Μ‡π‘₯𝑇(𝑠)𝑅2ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’β„Ž(𝑑)π‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑅2ξ€œΜ‡π‘₯(𝑠)𝑑𝑠=βˆ’π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇𝑅(𝑠)1βˆ’π‘‹33ξ€Έξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž(𝑑)Μ‡π‘₯𝑇𝑅(𝑠)2βˆ’π‘Œ33ξ€Έβˆ’ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘βˆ’β„Ž(𝑑)π‘‘βˆ’β„Ž2Μ‡π‘₯𝑇𝑅(𝑠)2βˆ’π‘33ξ€Έξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑋33βˆ’ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž(𝑑)Μ‡π‘₯𝑇(𝑠)π‘Œ33ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’β„Ž(𝑑)π‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑍33Μ‡π‘₯(𝑠)𝑑𝑠.(2.15)
By utilizing Lemma 2.1 and the Leibniz-Newton formula, we have βˆ’ξ€œπ‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑋33β‰€ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘π‘‘βˆ’π›Όπ›Ώξ‚ƒπ‘₯𝑇(𝑑)π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)Μ‡π‘₯𝑇(ξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘ )11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇230⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦π‘₯(𝑑)π‘₯(π‘‘βˆ’π›Όπ›Ώ)Μ‡π‘₯(𝑠)𝑑𝑠≀π‘₯𝑇(𝑑)𝛼𝛿𝑋11π‘₯(𝑑)+π‘₯𝑇(𝑑)𝛼𝛿𝑋12π‘₯(π‘‘βˆ’π›Όπ›Ώ)+π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝛼𝛿𝑋𝑇12π‘₯(𝑑)+π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝛼𝛿𝑋22π‘₯(π‘‘βˆ’π›Όπ›Ώ)+π‘₯𝑇(𝑑)𝑋𝑇13ξ€œπ‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯(𝑠)𝑑𝑠+π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝑋𝑇23ξ€œπ‘‘π‘‘βˆ’π›Όπ›Ώ+ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑑𝑠𝑋13ξ€œπ‘₯(𝑑)+π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇(𝑠)𝑑𝑠𝑋23π‘₯(π‘‘βˆ’π›Όπ›Ώ)=π‘₯𝑇(𝑑)𝛼𝛿𝑋11+𝑋𝑇13+𝑋13ξ€»π‘₯(𝑑)+π‘₯𝑇(𝑑)𝛼𝛿𝑋12βˆ’π‘‹13+𝑋𝑇23ξ€»π‘₯(π‘‘βˆ’π›Όπ›Ώ)+π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝛿𝑋𝑇12+𝑋23βˆ’π‘‹π‘‡13ξ€»π‘₯(𝑑)+π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)𝛼𝛿𝑋22βˆ’π‘‹23βˆ’π‘‹π‘‡23ξ€»π‘₯(π‘‘βˆ’π›Όπ›Ώ).(2.16)
Similarly, we obtain βˆ’ξ€œπ‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž(𝑑)Μ‡π‘₯𝑇(𝑠)π‘Œ33Μ‡π‘₯(𝑠)𝑑𝑠≀π‘₯π‘‡β„Ž(π‘‘βˆ’π›Όπ›Ώ)ξ€Ίξ€·2ξ€Έπ‘Œβˆ’π›Όπ›Ώ11+π‘Œπ‘‡13+π‘Œ13ξ€»π‘₯(π‘‘βˆ’π›Όπ›Ώ)+π‘₯π‘‡β„Ž(π‘‘βˆ’π›Όπ›Ώ)ξ€Ίξ€·2ξ€Έπ‘Œβˆ’π›Όπ›Ώ12βˆ’π‘Œ13+π‘Œπ‘‡23ξ€»π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯π‘‡β„Ž(π‘‘βˆ’β„Ž(𝑑))ξ€Ίξ€·2ξ€Έπ‘Œβˆ’π›Όπ›Ώπ‘‡12βˆ’π‘Œπ‘‡13+π‘Œ23ξ€»π‘₯(π‘‘βˆ’π›Όπ›Ώ)+π‘₯π‘‡β„Ž(π‘‘βˆ’β„Ž(𝑑))ξ€Ίξ€·2ξ€Έπ‘Œβˆ’π›Όπ›Ώ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23ξ€»βˆ’ξ€œπ‘₯(π‘‘βˆ’β„Ž(𝑑)),(2.17)π‘‘βˆ’β„Ž(𝑑)π‘‘βˆ’β„Ž2Μ‡π‘₯𝑇(𝑠)𝑍33Μ‡π‘₯(𝑠)𝑑𝑠≀π‘₯π‘‡β„Ž(π‘‘βˆ’β„Ž(𝑑))ξ€Ίξ€·2ξ€Έπ‘βˆ’π›Όπ›Ώ11+𝑍13+𝑍𝑇13ξ€»π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯𝑇(β„Žπ‘‘βˆ’β„Ž(𝑑))ξ€Ίξ€·2ξ€Έπ‘βˆ’π›Όπ›Ώ12βˆ’π‘13+𝑍𝑇23ξ€»π‘₯ξ€·π‘‘βˆ’β„Ž2ξ€Έ+π‘₯π‘‡ξ€·π‘‘βˆ’β„Ž2β„Žξ€Έξ€Ίξ€·2ξ€Έπ‘βˆ’π›Όπ›Ώπ‘‡12βˆ’π‘π‘‡13+𝑍23ξ€»π‘₯(π‘‘βˆ’β„Ž(𝑑))+π‘₯π‘‡ξ€·π‘‘βˆ’β„Ž2β„Žξ€Έξ€Ίξ€·2ξ€Έπ‘βˆ’π›Όπ›Ώ22βˆ’π‘23βˆ’π‘π‘‡23ξ€»π‘₯ξ€·π‘‘βˆ’β„Ž2ξ€Έ.(2.18)
Substituting the above equations (2.14)–(2.18) into (2.13), we obtain ̇𝑉π‘₯π‘‘ξ€Έβ‰€πœ‰π‘‡ξ€œ(𝑑)Ξ©πœ‰(𝑑)βˆ’π‘‘π‘‘βˆ’π›Όπ›ΏΜ‡π‘₯𝑇𝑅(𝑠)1βˆ’π‘‹33ξ€Έξ€œΜ‡π‘₯(𝑠)π‘‘π‘ βˆ’π‘‘βˆ’π›Όπ›Ώπ‘‘βˆ’β„Ž(𝑑)Μ‡π‘₯𝑇𝑅(𝑠)2βˆ’π‘Œ33ξ€Έβˆ’ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘βˆ’β„Ž(𝑑)π‘‘βˆ’β„Ž2Μ‡π‘₯𝑇𝑅(𝑠)2βˆ’π‘33ξ€ΈΜ‡π‘₯(𝑠)𝑑𝑠,(2.19) where πœ‰π‘‡(𝑑)=[π‘₯𝑇(𝑑)π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))π‘₯𝑇(π‘‘βˆ’β„Ž2)π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)] and ⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣ΩΩ=11Ξ©120Ξ©14Ω𝑇12Ξ©22Ξ©23Ξ©240Ω𝑇23Ξ©330Ω𝑇14Ω𝑇240Ξ©44⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(2.20) where Ξ©11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13+𝐴𝑇𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2Ω𝐴,12=𝑃𝐴𝑑+𝐴𝑇𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴𝑑,Ξ©14=𝛼𝛿𝑋12βˆ’π‘‹13+𝑋𝑇23,Ξ©22ξ€·=βˆ’1βˆ’β„Žπ‘‘ξ€Έπ‘„2+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ11+𝑍13+𝑍𝑇13+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23+𝐴𝑇𝑑𝛼𝛿𝑅1+ξ€·β„Ž2ξ€Έπ‘…βˆ’π›Όπ›Ώ2𝐴𝑑,Ξ©23=ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ12βˆ’π‘13+𝑍𝑇23,Ξ©24=ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώπ‘‡12βˆ’π‘Œπ‘‡13+π‘Œ23,Ξ©33=βˆ’π‘„3+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ22βˆ’π‘23βˆ’π‘π‘‡23,Ξ©44=βˆ’π‘„1+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ11+π‘Œ13+π‘Œπ‘‡13+𝛼𝛿𝑋22βˆ’π‘‹23βˆ’π‘‹π‘‡23.(2.21)
If Ξ©<0,𝑅1βˆ’π‘‹33β‰₯0,𝑅2βˆ’π‘Œ33β‰₯0, and 𝑅2βˆ’π‘33β‰₯0, then ̇𝑉(π‘₯𝑑)<βˆ’πœ€β€–π‘₯(𝑑)β€–2 for a sufficiently small πœ€>0. By the Schur complement of Lemma 2.2, Ξ©<0 is equivalent to the following inequality and is true: ⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣ΨΨ=11Ξ¨120Ξ¨14Ξ¨15Ξ¨16Ψ𝑇12Ξ¨22Ξ¨23Ξ¨24Ξ¨25Ξ¨260Ψ𝑇23Ξ¨33Ξ¨000𝑇14Ψ𝑇240Ξ¨44Ξ¨00𝑇15Ψ𝑇2500Ξ¨550Ψ𝑇16Ψ𝑇26000Ξ¨66⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<0,(2.22) where Ξ¨11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄2+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ξ¨12=𝑃𝐴𝑑,Ξ¨14=𝛼𝛿𝑋12βˆ’π‘‹13+𝑋𝑇23,Ξ¨15=𝛼𝛿𝐴𝑇𝑅1,Ξ¨16=ξ€·β„Ž2ξ€Έπ΄βˆ’π›Όπ›Ώπ‘‡π‘…2,Ξ¨22ξ€·=βˆ’1βˆ’β„Žπ‘‘ξ€Έπ‘„2+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ11+𝑍13+𝑍𝑇13,Ξ¨23=ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ12βˆ’π‘13+𝑍𝑇23,Ξ¨24=ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώπ‘‡12βˆ’π‘Œπ‘‡13+π‘Œ23,Ξ¨25=𝛼𝛿𝐴𝑇𝑑𝑅1,Ξ¨26=ξ€·β„Ž2ξ€Έπ΄βˆ’π›Όπ›Ώπ‘‡π‘‘π‘…2,Ξ¨33=βˆ’π‘„3+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ22βˆ’π‘23βˆ’π‘π‘‡23,Ξ¨44=βˆ’π‘„1+ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ11+π‘Œ13+π‘Œπ‘‡13+𝛼𝛿𝑋22βˆ’π‘‹23βˆ’π‘‹π‘‡23,Ξ¨55=βˆ’π›Όπ›Ώπ‘…1,Ξ¨66ξ€·β„Ž=βˆ’2ξ€Έπ‘…βˆ’π›Όπ›Ώ2.(2.23)
That is to say, if Ξ¨<0,  𝑅1βˆ’π‘‹33β‰₯0,𝑅2βˆ’π‘Œ33β‰₯0, and 𝑅2βˆ’π‘33β‰₯0, then ̇𝑉(𝑑)<βˆ’πœ€β€–π‘₯(𝑑)β€–2 for a sufficiently small πœ€>0. Furthermore, (2.10a) implies βˆ‘π‘Ÿπ‘–=1πœƒπ‘–(𝑧(𝑑))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 β„Ž1β‰€β„Ž(𝑑)≀𝛼𝛿.
For this case, the Lyapunov-Krasovskii functional candidate is chosen 𝑉π‘₯𝑑=π‘₯𝑇(ξ€œπ‘‘)𝑃π‘₯(𝑑)+π‘‘π‘‘βˆ’π›Όπ›Ώπ‘₯𝑇(𝑠)𝑄1ξ€œπ‘₯(𝑠)𝑑𝑠+π‘‘π‘‘βˆ’β„Ž(𝑑)π‘₯𝑇(𝑠)𝑄2ξ€œπ‘₯(𝑠)𝑑𝑠+π‘‘π‘‘βˆ’β„Ž1π‘₯𝑇(𝑠)𝑄3+ξ€œπ‘₯(𝑠)𝑑𝑠0βˆ’π›Όπ›Ώξ€œπ‘‘π‘‘+πœƒΜ‡π‘₯𝑇(𝑠)𝑅1ξ€œΜ‡π‘₯(𝑠)π‘‘π‘ π‘‘πœƒ+βˆ’β„Ž1βˆ’π›Όπ›Ώξ€œπ‘‘π‘‘+πœƒΜ‡π‘₯𝑇(𝑠)𝑅2Μ‡π‘₯(𝑠)π‘‘π‘ π‘‘πœƒ,(2.24) where 𝑃>0,𝑅1>0,𝑅2>0,𝑄𝑖>0(𝑖=1,2,3).
Choosing πœ‰π‘‡(𝑑)=[π‘₯𝑇(𝑑)π‘₯𝑇(π‘‘βˆ’β„Ž(𝑑))π‘₯𝑇(π‘‘βˆ’β„Ž1)π‘₯𝑇(π‘‘βˆ’π›Όπ›Ώ)] 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 β„Ž1, β„Ž2, 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 βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘‹=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Œβ‰₯0,π‘Œ=11π‘Œ12π‘Œ13π‘Œπ‘‡12π‘Œ22π‘Œ23π‘Œπ‘‡13π‘Œπ‘‡23π‘Œ33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘β‰₯0,𝑍=11𝑍12𝑍13𝑍𝑇12𝑍22𝑍23𝑍𝑇13𝑍𝑇23𝑍33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦β‰₯0,(2.25) such that the following LMIs hold:Ξžπ‘–=⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣Ξ11Ξ120Ξ14Ξ15Ξ16Ξžπ‘‡12Ξ22Ξ23Ξ24Ξ25Ξ260Ξžπ‘‡23Ξ33Ξ000𝑇14Ξžπ‘‡240Ξ44Ξ00𝑇15Ξžπ‘‡2500Ξ550Ξžπ‘‡16Ξžπ‘‡26000Ξ66⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘…<0,(2.26a)1βˆ’π‘‹33𝑅β‰₯0,(2.26b)2βˆ’π‘Œ33𝑅β‰₯0,(2.26c)2βˆ’π‘33β‰₯0,(2.26d)where Ξžπ‘–π‘—(𝑖,𝑗=1,2,...,6) are defined in (8) and Ξ11=𝐴𝑇𝑃+𝑃𝐴+𝑄1+𝑄3+𝛼𝛿𝑋11+𝑋13+𝑋𝑇13,Ξ22=ξ€·β„Ž2ξ€Έπ‘Œβˆ’π›Όπ›Ώ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23+ξ€·β„Ž2ξ€Έπ‘βˆ’π›Όπ›Ώ11+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 β„Ž1=0, Theorem 2.3 reduces to the following Corollary 2.5.

Corollary 2.5. For given scalars β„Ž,β„Žπ‘‘, 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 βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘‹π‘‹=11𝑋12𝑋13𝑋𝑇12𝑋22𝑋23𝑋𝑇13𝑋𝑇23𝑋33⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Œβ‰₯0,π‘Œ=11π‘Œ12π‘Œ13π‘Œπ‘‡12π‘Œ22π‘Œ23π‘Œπ‘‡13π‘Œπ‘‡23π‘Œ33⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦β‰₯0,(2.28) such that the following LMIs hold for 𝑖=1,2,...,π‘Ÿ, Σ𝑖=⎑⎒⎒⎒⎒⎒⎒⎣Σ11Ξ£120Ξ£14Σ𝑇12Ξ£22Ξ£23Ξ£240Σ𝑇23Ξ£330Σ𝑇14Σ𝑇240Ξ£44⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<0,π‘…βˆ’π‘‹33β‰₯0,π‘…βˆ’π‘Œ33β‰₯0,(2.29) where Ξ£11=𝐴𝑇𝑖𝑃+𝑃𝐴𝑖+𝑄1+𝑄2+π›Όβ„Žπ‘‹11+𝑋13+𝑋𝑇13,Ξ£12=𝑃𝐴𝑑𝑖+π›Όβ„Žπ‘‹12βˆ’π‘‹13+𝑋𝑇23,Ξ£14=π›Όβ„Žπ΄π‘‡π‘–π‘…,Ξ£22ξ€·=βˆ’1βˆ’β„Žπ‘‘ξ€Έπ‘„2+π›Όβ„Žπ‘‹22βˆ’π‘‹23βˆ’π‘‹π‘‡23+β„Žπ‘Œ11+π‘Œ13+π‘Œπ‘‡13,Ξ£23=π›Όβ„Žπ‘Œ12βˆ’π‘Œ13+π‘Œπ‘‡23,Ξ£24=π›Όβ„Žπ΄π‘‡π‘‘π‘–Ξ£π‘…,33=βˆ’π‘„1+π›Όβ„Žπ‘Œ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23,Ξ©44=βˆ’π›Όβ„Žπ‘….(2.30)

Proof. Choose the following fuzzy Lyapunov-Krasovskii functional candidate to be 𝑉(𝑑)=𝑉1(𝑑)+𝑉2(𝑑)+𝑉3(𝑑)+𝑉4(𝑑),(2.31) where 𝑉1(𝑑)=π‘₯𝑇𝑉(𝑑)𝑃π‘₯(𝑑)2ξ€œ(𝑑)=π‘‘π‘‘βˆ’π›Όβ„Žπ‘₯𝑇(𝑠)𝑄1𝑉π‘₯(𝑠)𝑑𝑠3ξ€œ(𝑑)=π‘‘π‘‘βˆ’β„Ž(𝑑)π‘₯𝑇(𝑠)𝑄2π‘₯𝑉(𝑠)𝑑𝑠4ξ€œ(𝑑)=0βˆ’π›Όβ„Žξ€œπ‘‘π‘‘+πœƒΜ‡π‘₯𝑇(𝑠)𝑅̇π‘₯(𝑠)π‘‘π‘ π‘‘πœƒ.(2.32)
Then, taking the time derivative of 𝑉(𝑑) with respect to 𝑑 along the system (2.4) yield ̇̇𝑉𝑉(𝑑)=1̇𝑉(𝑑)+2̇𝑉(𝑑)+3̇𝑉(𝑑)+4(𝑑).(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, β„Žπ‘‘ can be any value or unknown due to Ξ22=βˆ’(1βˆ’β„Žπ‘‘)𝑄2+(β„Ž2βˆ’π›Όπ›Ώ)π‘Œ22βˆ’π‘Œ23βˆ’π‘Œπ‘‡23+(β„Ž2βˆ’π›Όπ›Ώ)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 β„Ž1 and β„Ž2 is sufficiently small, the upper bound β„Ž2 of delay for unknown β„Žπ‘‘ will be very close to the upper bound for β„Žπ‘‘=0. This characteristic is not included in previous Lyapunov functional based work where the upper bound of delay for β„Žπ‘‘β‰ 0 is always less than that for β„Žπ‘‘=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=(β„Ž2+β„Ž1)/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=(β„Ž2+β„Ž1)/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 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2 which maintains the delay-dependent stability of the time delay system (2.4).

Remark 2.8. It is interesting to note that β„Ž1,β„Ž2 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/β„Ž1(or1/β„Ž2) and, therefore, the maximum β„Ž1(orβ„Ž2) 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) 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2 such that (2.4) is stable by solving the following convex optimization problem: Maximizeβ„Ž2SubjecttoTheorem2.3(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 β„Ž of delay β„Ž(𝑑) subjecting to (2.29) is maximal, we give an algorithm as follows.

Algorithm 2.10 ((Maximizing β„Ž>0)). Step 1: For given β„Žπ‘‘, choose an upper bound on β„Ž satisfying (2.29), and then select this upper bound as the initial value β„Ž0 of β„Ž.
Step 2: Set appropriate step lengths, β„Žstep and 𝛼step, for β„Ž and 𝛼, respectively. Set π‘˜ as a counter, and choose π‘˜=1. Meanwhile, let β„Ž=β„Ž0+β„Žstep and the initial value 𝛼0 of 𝛼 equals to 𝛼step.
Step 3: Let 𝛼=π‘˜π›Όstep, if inequality (2.29) is feasible, go to Step 4; otherwise, go to Step 5.
Step 4: Let β„Ž0=β„Ž,𝛼0=𝛼,π‘˜=1 and β„Ž=β„Ž0+β„Žstep, go to step 3.
Step 5: Let π‘˜=π‘˜+1. If π‘˜π›Όstep<1, then go to step 3; otherwise, stop.

Remark 2.11. For Algorithm 2.10, the final β„Ž0 is the desired maximum of the upper bound of delay β„Ž(𝑑) 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, 5–7, 9, 10, 12, 13, 15, 17–21, 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 π‘₯1(𝑑) is 𝑀1, then Μ‡π‘₯(𝑑)=𝐴1π‘₯(𝑑)+𝐴𝑑1π‘₯(π‘‘βˆ’β„Ž(𝑑)).(3.1)Rule 2: If π‘₯1(𝑑) is 𝑀2, then Μ‡π‘₯(𝑑)=𝐴2π‘₯(𝑑)+𝐴𝑑2π‘₯(π‘‘βˆ’β„Ž(𝑑)),(3.2) and the membership functions for rule 1 and rule 2 are 𝑀1ξ€·π‘₯1ξ€Έ=ξ‚΅1(𝑑)1βˆ’1+π‘’βˆ’5(π‘₯1(𝑑)βˆ’(πœ‹/6))ξ‚Ά11+π‘’βˆ’5(π‘₯1(𝑑)βˆ’(πœ‹/6)),𝑀2ξ€·π‘₯1ξ€Έ(𝑑)=1βˆ’π‘€1ξ€·π‘₯1ξ€Έ(𝑑),(3.3) where 𝐴1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’200βˆ’0.9,𝐴𝑑1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’10βˆ’1βˆ’1,𝐴2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’1.510βˆ’0.75,𝐴𝑑2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’101βˆ’0.85.(3.4)

Solution 3.2. For system (3.1) and (3.2), by taking the parameter β„Žπ‘‘=0 and 𝛼=0.6, we get the Corollary 2.5 which remains feasible for any delay time β„Žβ‰€2.2459. In case of maximum allowable delay bound (MADB) β„Ž=2.2459, solving Corollary 2.5 yields the following set of feasible solutions: ⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘ƒ=33.23427.99927.999228.4879,𝑄1=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.0039βˆ’0.0007βˆ’0.00070.0029,𝑄2=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦78.8437βˆ’2.5471βˆ’2.547122.8291𝑅=1.62804.50934.509313.7942,𝑋11=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1.89331.58771.58778.4013,𝑋12=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,𝑋0.1663βˆ’3.3981βˆ’3.4381βˆ’6.770013=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’1.2051βˆ’3.3469βˆ’3.3454βˆ’10.2327,𝑋22=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦2.02591.51081.51088.4337,𝑋23=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,𝑋1.20303.34883.347310.231033=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1.62464.50974.509713.7915,π‘Œ11=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦5.12990.41160.41165.9858,π‘Œ12=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Œβˆ’0.1102βˆ’0.1624βˆ’0.1588βˆ’0.530213=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.19710.58720.59151.7945,π‘Œ22=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.29620.65220.65222.0458,π‘Œ23=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Œ0.34560.80550.80582.509233=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦.0.80932.11412.11416.4977(3.5)
Applying the criteria in [9, 10, 12, 17] and in this paper, the maximum values of β„Ž 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].

Furthermore, by taking the various β„Ž1(β„Žπ‘‘=0.5), and from Theorem 2.3, we obtain the maximum allowable delay bound (MADB) β„Ž2 as shown in Table 2. From the above results of Table 2, if the β„Ž1 increases, the delay time length increases.

Remark 3.3. Similar to Algorithm 2.10, an algorithm for seeking an appropriate 𝛼 such that the upper bound of delay 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2, 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 π‘₯1(𝑑) is 𝑀1, then Μ‡π‘₯(𝑑)=𝐴1π‘₯(𝑑)+𝐴𝑑1π‘₯(π‘‘βˆ’β„Ž(𝑑)).(3.6)Rule 2: If π‘₯1(𝑑) is 𝑀2, then Μ‡π‘₯(𝑑)=𝐴2π‘₯(𝑑)+𝐴𝑑2π‘₯(π‘‘βˆ’β„Ž(𝑑)),(3.7) and the membership functions for rule 1 and rule 2 are 𝑀11(𝑧(𝑑))=ξ€·1+expβˆ’2π‘₯1ξ€Έ(𝑑),𝑀2ξ€·π‘₯1ξ€Έ(𝑑)=1βˆ’π‘€1(𝑧(𝑑)),(3.8) where 𝐴1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’3.20.60βˆ’2.1,𝐴𝑑1=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦10.902,𝐴2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’101βˆ’3,𝐴𝑑2=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.9011.6.(3.9)

Solution 3.5. By taking the parameter β„Žπ‘‘=0, using Corollary 2.5, the maximum value of delay time for the System (3.6) and (3.7) to be asymptotically stable is β„Žβ‰€1.0245. By the criteria in [9, 16], the system (3.6) and (3.7) is asymptotically stable for β„Žβ‰€0.58 and β„Žβ‰€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 [18–21] and those in Corollary 2.5 yields upper bounds on β„Ž2 that guarantee the stability of system (3.6) and (3.7) for various β„Žπ‘‘, which are listed in Table 3, in which β€œβ€“β€ 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.

Example 3.6. Consider a T-S fuzzy system with time-varying delay is of the following form:
Plant rules.Rule 1: If π‘₯1(𝑑) is 𝑀1, then Μ‡π‘₯(𝑑)=𝐴1π‘₯(𝑑)+𝐴𝑑1π‘₯(π‘‘βˆ’β„Ž(𝑑)).(3.10)Rule 2: If π‘₯1(𝑑) is 𝑀2, then Μ‡π‘₯(𝑑)=𝐴2π‘₯(𝑑)+𝐴𝑑2π‘₯(π‘‘βˆ’β„Ž(𝑑)),(3.11) and the membership functions for rule 1 and rule 2 are 𝑀11(𝑧(𝑑))=ξ€·1+expβˆ’2π‘₯1ξ€Έ(𝑑),𝑀2ξ€·π‘₯1ξ€Έ(𝑑)=1βˆ’π‘€1(𝑧(𝑑)),(3.12) where 𝐴1=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.1100.2,𝐴𝑑1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’0.320βˆ’0.5,𝐴2=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦0.120βˆ’0.2,𝐴𝑑2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’0.3420.01βˆ’0.4.(3.13)

Solution 3.7. By using Corollary 2.4, the maximum allowable delay bound (MADB) can be calculated as β„Ž2=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.

Remark 3.8. Similar to Algorithm 2.10, we can also find an appropriate scalar 𝛼, such that the upper bound of delay 0β‰€β„Ž1β‰€β„Ž(𝑑)β‰€β„Ž2, 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 π‘₯1(𝑑) is 𝑀1, thenΜ‡π‘₯(𝑑)=𝐴1π‘₯(𝑑)+𝐴𝑑1π‘₯(π‘‘βˆ’β„Ž(𝑑)).(3.14a)Rule 2: If π‘₯1(𝑑) is 𝑀2, then Μ‡π‘₯(𝑑)=𝐴2π‘₯(𝑑)+𝐴𝑑2π‘₯(π‘‘βˆ’β„Ž(𝑑)),(3.14b)and the membership function for rules 1 and 2 are 𝑀11(𝑧(𝑑))=ξ€·1+expβˆ’2π‘₯1ξ€Έ(𝑑),𝑀2ξ€·π‘₯1ξ€Έ(𝑑)=1βˆ’π‘€1(𝑧(𝑑)),(3.15) where 𝐴1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’2.10.1βˆ’0.2βˆ’0.9,𝐴𝑑1=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’1.10.1βˆ’0.8βˆ’0.9,𝐴2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’1.90βˆ’1.1βˆ’1.1,𝐴𝑑2=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’0.90βˆ’1.1βˆ’1.2.(3.16)

Solution 3.10. The objective is to determine the maximum value of constant time-delay β„Ž=β„Ž2(β„Ž1=0)for which the system is stable. Table 5 compares works based on common quadratic functionals [7, 18–20, 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) β„Ž 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.

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 π‘₯1(𝑑) is 𝑀1, then Μ‡π‘₯(𝑑)=𝐴1π‘₯(𝑑)+𝐴𝑑1π‘₯(π‘‘βˆ’β„Ž(𝑑)).(3.17)Rule 2: If π‘₯1(𝑑) is 𝑀2, then Μ‡π‘₯(𝑑)=𝐴2π‘₯(𝑑)+𝐴𝑑2π‘₯(π‘‘βˆ’β„Ž(𝑑)),(3.18) and the membership functions for rule 1 and rule 2 are 𝑀1(𝑧(𝑑))=sin2ξ€·π‘₯1ξ€Έ(𝑑),𝑀2ξ€·π‘₯1ξ€Έ(𝑑)=cos2ξ€·π‘₯1ξ€Έ,(𝑑)(3.19) where 𝐴1=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦01βˆ’8βˆ’6,𝐴2=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦01βˆ’80,𝐴𝑑1=𝐴𝑑2=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦001βˆ’2.(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 β„Ž2=0.4130(β„Ž1=0,𝛼=0.5). By the criteria in [13, 23], the systems (3.17) and (3.18) is asymptotically stable for any β„Ž that satisfies β„Žβ‰€0.322 and β„Žβ‰€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.