New relaxed stability and stabilization conditions for T-S fuzzy systems with time-varying delays

This paper investigates the stability analysis and stabilization of T-S fuzzy systems with time-varying delays. First, a new augmented Lyapunov–Krasovskii functional is constructed, delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) are obtained by combining them with the integral inequality technique and the reciprocally convex combination inequality. Based on the state space decomposition method, some piecewise membership functions are employed to approximate the membership functions. The piecewise membership functions can be locally represented in terms of the convex combinations of the supremum and inﬁmum of some local basis functions. The boundary information of the membership functions is adequately taken into consideration in stability analysis, and then some relaxed membership-function-dependent stability results are obtained. Second, state feedback controllers for fuzzy systems with time-varying delays are presented under the imperfect premise-matching technique, whose membership functions and the number of fuzzy rules are allowed to be designed freely, consequently, the ﬂexibility of controller design is improved. Finally, four numerical examples are given to demonstrate the effectiveness of the presented approaches.


INTRODUCTION
The T-S fuzzy model offers a universal framework to represent complex nonlinear systems, dynamic nonlinear systems can be represented as weighted sum of several local linear sub-systems via nonlinear fuzzy weights, and each linear sub-system can effectively describe the local characteristics of the nonlinear systems [1]. With the help of the favourable semi-linear property, perfect analysis and synthesis methods of linear systems can be introduced into the control problem of nonlinear systems via the T-S fuzzy model, which provides a powerful method for the control of nonlinear systems [2]. On the other hand, the phenomenon of time delay is inevitable in various practical control systems, the existence of time delay often affects the control performance and may even lead to the instability of the systems [3,4]. Consequently, the stability analysis and controller synthesis of time-delay systems are of great significance for practical engineering applications. In industrial production and practical control systems, time-delay systems are often highly nonlinear. Therefore, it is difficult to model and control the time-delay membership functions need to be bounded when the fuzzy Lyapunov-Krasovskii functional method is employed. Stability conditions of T-S fuzzy systems with time-varying delays were obtained by constructing novel Lyapunov-Krasovskii functionals dependent on the delay-fractioning technique in [13,14], where the conservatism could be reduced by increasing the segmentation step. However, while increasing the segmentation step, the delay-fractioning technique may introduce the larger decision variables and computational complexity. In [15,16], novel augmented Lyapunov-Krasovskii functionals were constructed, which had made full use of the time-delay information, and relaxed stability results of T-S fuzzy time-delay systems were gained. In order to further reduce the conservatism, delay-product-type augmented Lyapunov-Krasovskii functionals were constructed to introduce more time-delay information into stability analysis in [17,18]. Obviously, compared with the simple form of the Lyapunov-Krasovskii functional, the Lyapunov-Krasovskii functionals with augmented terms introduce several extra matrices, which provide more freedom for checking the feasibility of the LMIs conditions in the criteria [19]. The conservative of stability analysis can be effectively reduced by choosing an appropriate augmented Lyapunov-Krasovskii functional. However, introducing too many augmented vectors will inevitably increase computational complexity, how to balance computational complexity and conservatism is worth exploring. In addition, the utilization of powerful boundary techniques is another effective way to reduce the conservatism of stability and stabilization conditions for time-delay systems. Common methods include free-weighting matrix approach [20][21][22], the inequality technique [23] and convex combination technique [24]. The integral inequality technique is an effective approach to estimate the integral term of the derivative of Lyapunov-Krasovskii functional. In the analysis of time-varying delay systems via the integral inequality method, some other techniques often need to be combined to deal with the time-varying delay terms generated in the estimation results. A reciprocally convex combination lemma was presented in [25], which could be employed to evaluate the time-varying delays in the estimation terms directly. Moreover, the stability criterion could be obtained with fewer decision variables, and it will not increase the conservatism of stability analysis. Therefore, the reciprocally convex combination method has been widely used in the stability analysis of time-varying delay systems. By introducing relaxation matrix variables in [26], the reciprocally convex combination lemma in [25] was generalized, and the generalized reciprocally convex inequality was proved to be less conservative. Nevertheless, the introduction of relaxation matrix variables will increase the computational complexity of stability criteria. One of the motivations of this paper is to choose the more advanced reciprocal convex inequality and obtain some more relaxed delay-dependent stability criterion.
On the other hand, the results mentioned above and most of the controllers of T-S fuzzy systems are designed based on the parallel distributed compensation (PDC) technique. Fuzzy controllers based on the PDC strategy are required to share the same membership functions and the number of fuzzy rules with the fuzzy models, which facilitates the stability analysis of T-S fuzzy systems. However, the PDC strategy limits the flexibility of fuzzy controllers design and may complicate their construction [27]. A new PDC design approach for T-S fuzzy control systems with affine matched membership functions in the system and controller was presented in [45], and the conservativeness of stabilization condition for the T-S fuzzy system is significantly relaxed by utilizing the constraints condition of the controllers membership functions, which is determined from the difference of each transformed membership function. In recent years, the imperfect premise matching technique has been proposed to solve the above issue successfully. Different from the traditional PDC controllers, imperfect-premise-matching-based fuzzy controllers are not required to employ the same premise membership functions and the number of fuzzy rules as the fuzzy model. Consequently, the flexibility of controllers design is greatly increased by an imperfect premise-matching design scheme. Nevertheless, the mismatched premise membership functions and the number of fuzzy rules so that the stability analysis of T-S fuzzy systems is more challenging. Consequently, how to reduce the conservatism of stability analysis is the primary problem in the research of controller synthesis based on an imperfect premise-matching scheme. In recent years, the membership-function-dependent method has been proposed to reduce the conservativeness of the stability analysis of fuzzy systems. Based on the membership-function-dependent analysis method, the information of membership functions is taken into consideration to reduce the conservatism in stability analysis. Under the imperfect premise-matching scheme, a static output feedback controller was designed to guarantee the finite-time stability of closed-loop fuzzy control systems with time-varying delays in [28]. The information about membership functions was considered in the stabilization conditions to reduce conservativeness via the membership-functiondependent approach. The stability analysis and stabilization problems of T-S fuzzy time-delay systems were investigated in [29], an imperfect-premise-matching-based state feedback controller was designed, and the information of membership functions was integrated into the stability and stabilization conditions to reduce conservatism. However, the design of the fuzzy controllers under the imperfect premise matching scheme only considered the mismatch of the membership function, but not the number of fuzzy rules in [28]. Furthermore, the case of constant time delays was studied in [29], rather than time-varying delay.
Inspired by the above discussion, the stability and stabilization of T-S fuzzy systems with time-varying delays are investigated. Firstly, a new augmented Lyapunov-Krasovskii functional is constructed by introducing some double integral terms of delays into the augmented vectors. In addition, some piecewise membership functions are constructed to approximate the original membership functions, and less conservative membership-function-dependent stability conditions are obtained by combining with the integral inequality technique and the reciprocally convex combination inequality. Secondly, imperfect-premise-matching-based fuzzy state feedback controllers are presented by employing the Finsler lemma. Finally, four numerical examples are given to demonstrate the effectiveness of the presented approaches.
The main contributions of this paper are provided as follows: 1. A new augmented Lyapunov-Krasovskii functional is constructed by introducing some double delay integral terms into the augmented vectors, some relaxed stability criteria are obtained by combining with integral inequality technique and the reciprocally convex combination inequality; 2. In order to obtain further relaxed stability criteria, some piecewise membership functions are constructed to approximate the membership functions. Therefore, some less conservative membership-function-dependent stability results are obtained; 3. Under the imperfect premise matching technique, a fuzzy state feedback control design method is to be presented for T-S fuzzy systems with time-varying delays, the fuzzy controller is not required to employ the same premise membership functions and the number of fuzzy rules as the fuzzy model, and then the flexibility of the controller design is improved.
This paper is organized as follows. The problem description and some useful lemmas are given in Section 2. The main derivation process is presented in Section 3. Four numerical examples are provided to demonstrate the effectiveness of the proposed approaches in Section 4. The last section concludes this paper.
Notations. Throughout this study, the superscripts T and −1 denote the transpose and the inverse of a matrix, respectively. I and 0 denote the identity matrix and zero matrix with compatible dimensions, respectively. ℝ n and ℝ n×m stand for ndimensional Euclidean space and the set of all n × m real matrices, respectively. P > 0(≥ 0) means that P is a positive definite (positive semidefinite) matrix. diag{⋯} means a block diagonal matrix, and Sym{X } = X + X T . If not specially explained, matrices are supposed to be with suitable dimensions.

Fuzzy models
Consider a class of non-linear systems with time-varying delays, which can be described by the following T-S fuzzy model with p plant rules: where F i denote the premise variables with = 1, 2, … , , i = 1, 2, … , p; 1 (x(t )), … , (x(t )) are the fuzzy membership functions. (t ) is the vector-valued initial condition on [− M , 0]; A i , A di , B i are system matrices with appropriate dimensions, and (t ) is a time-varying delay satisfying where M , 1 and 2 are known constant scalars. Employing the singleton fuzzifier, product inference, and centre-average defuzzifier, the global dynamics of the delayed fuzzy model can be inferred as follows:

Fuzzy controllers
Based on imperfect premise matching technique, a fuzzy state feedback controller with c rules is presented as follows: Then, the overall controller is presented as: where K j ∈ ℝ m×n are control gains matrix to be determined. Consequently, the closed-loop delayed T-S fuzzy system can be described as: with compact forṁ where Remark 1. According to (3) and (5), supposing that p ≠ c or/and m(x(t )) ≠ h(x(t )), where p and c stand for the number of rules of the fuzzy model and the fuzzy controller, respectively, m(x(t )) and h(x(t )) mean membership functions of the fuzzy model and fuzzy controller, respectively. Imperfectpremise-matching-based fuzzy controllers do not employ the same membership functions and the number of rules as the fuzzy models, the ones can be selected flexibly, and then the flexibility of the controller design is increased. By selecting a smaller number of fuzzy rules and choosing some simpler membership functions to replace the complex ones in the fuzzy models, and the complexity and the implementation cost of the controllers are reduced. Therefore, the imperfect-premisematching-based fuzzy controllers of the T-S fuzzy time-delay system have more practical value and application prospect. In the case that p = c and m(x(t )) = h(x(t )), imperfect-premisematching-based fuzzy controllers are to degenerate into the ones based on PDC strategy, this shows that the imperfect premise matching method is the extension of the traditional PDC technique.
The main aim of this paper is to obtain the imperfectpremise-matching-based fuzzy controller design approach, so that the closed-loop system (6) is asymptotically stable.

Related lemmas
The following lemmas will be introduced, which are indispensable to derive our main results.

Membership-function-dependent stability analysis
In this subsection, a novel augmented Lyapunov-Krasovskii functional is constructed, and a relaxed delay-dependent stability criterion is obtained by employing the auxiliary function integral inequality together with the extended reciprocally convex matrix inequality. The boundary information of membership functions is considered by introducing piecewise membership functions, and further relaxed membership-function-dependent stability results are presented.
In order to consider the information of membership functions, we introduced the following results in [34].
Consider the state space being divided into k connected state subspaces l , l = 1, 2, … , k, such that = ∪ k l =1 l . Considering a stateX = [x 1 ⋯ x ] such that X ∈ l , define the minimum and maximum of x r as x ri r l andx ri r l , respectively, r = 1, 2, … , ,i r = 1, 2, such that x ri r l ≤ x r ≤x ri r l . Denoting the vertices of the subspaces l as where ri r l (x r ) is a function to be determined, which exhibits the properties as follows: An example is given below to better illustrate the definition of the piecewise membership function̂(x). Considering the state X = [ x 1 x 2 ], the state space Γ is divided into 12 subspaces l , l = 1, 2, … , 12. As shown in Figure 1, the black dots represent vertices, and each subspace l is represented by four vertices. The grades of membership corresponding to the four vertices of the l-th subspace are recorded aŝ(x 11l ),̂(x 12l ),̂(x 21l ),̂(x 22l ), respectively. The func- (7) are predefined with 11l (x 1 ) + 12l (x 1 ) = 1 and 21l (x 2 ) + 22l (x 2 ) = 1 in l-th subspace, and we have i jl (x i )=0 in other subspaces. Based on the definition of the piecewise membership function, the grades of membership in the l-th subspace can be expressed aŝ Remark 2. In order to obtain further relaxed stability criteria, instead of introducing global membership function and premise variable knowledge among the overall state space, additional information is to be introduced into the stability analysis, this includes: 1) the regional bounds of piecewise member-ship functions; 2) the property knowledge of interpolation membership function of piecewise membership functions and regional bounds of premise variables corresponding to each substate region; and 3) the regional approximation error between piecewise membership functions and original membership functions. Furthermore, some less conservative membership-function-dependent stability results are obtained.

Theorem 1. For given scalars
, and any matrices Y 1 ∈ ℝ 3n×3n , Y 2 ∈ ℝ 3n×3n , such that the following inequalities hold: where the piecewise membership functionsm i (x) are defined in (26), k is the number of divided state subspaces for the piecewise membership functions, x i 1 i 2 ⋯i n l represents the apexes of the l-th state subspace of . Δm i , Δm i are predefined constant scalars satisfying Δm i ≤ Δm i (x) ≤ Δm i . Concurrently, some other matrices are defined as follows: Proof. Construct an augmented Lyapunov-Krasovskii functional candidate as: where where P ∈ ℝ 7n×7n , Q 1 ∈ ℝ 6n×6n , Q 2 ∈ ℝ 6n×6n , R ∈ ℝ n×n are positive definite matrixes to be solved.
First, the derivatives of individual Lyapunov-Krasovskii functionals along the trajectory of (3) are computed as: where Then, by utilizing Lemma 1, the integral terms in (17) can be respectively estimated as: T 16 ] T , by applying Lemma 2 to deal with (18) and (19) together, the following inequality can be obtained: Based on (15), (16), (17) and (20), the derivative of V (t )can be calculated as follows: Therefore, if 1i (h(t ),̇h(t )) + 2 (h(t )) < 0 holds, the T-S fuzzy system with time-varying delays (3) under u(t ) = 0 is asymptotically stable. By Schur complement lemma, 1i (h(t ),̇h(t )) + 2 (h(t )) < 0 is equivalent to If conditions (t ) ∈ [0, M ] anḋ(t ) ∈ { 1 , 2 } hold, by Lemma 3,(22) is equivalent to To reduce the conservatism of stability analysis of T-S fuzzy system (3) with time-varying delays, piecewise membership functionsm i (x) are constructed based on (7) as: In the following parts, the piecewise membership functionŝ m i (x) are used to approximate the membership functions m i (x(t )) of the fuzzy model, and some less conservative membership-function-dependent stability criterion are obtained. Let , the maximum and minimum of Δm i (x) are denoted as Δm i and Δm i , respectively, such that Δm i ≤ Δm i (x) ≤ Δm i . From the properties of the piecewise membership functions and the definition of Δm i (x), we obtain Introducing a slack matrix E = E T ∈ ℝ 12n×12n , and the following equation is obtained: Let n = 1, 2, 3, combining (27) with (23)- (25), we arrive at: Furthermore, slack matrices 0 < F i ∈ ℝ 12n×12n , i = 1, 2, … , p are introduced into (28), and then we can obtain the following equation: (26) in (29), we obtain: Consequently, if LMIs in (8)-(13) are feasible, and it implies thatV (t ) < 0, which in turn guarantees the asymptotic stability of the T-S fuzzy system (3) under u(t ) = 0. Then, the proof is completed.
Remark 3. Considering the issue of balancing conservatism and computational complexity, we do not select the delayfractioning technique in [13] and [14] to construct Lyapunov-Krasovskii functionals. It is well known that the augmented Lyapunov-Krasovskii functionals can effectively reduce the conservatism of stability analysis. In Theorem 1, a novel augmented Lyapunov-Krasovskii functional is constructed, which has made full use of the information of time delays. Double integral terms of delays  Remark 4. In order to obtain further relaxed stability results based on Lyapunov-Krasovskii functionals method, an augmented Lyapunov-Krasovskii functional was constructed in [35], which contains single, double, triple and quadruple-integral terms. Meanwhile, multiple integral inequalities were introduced to estimate the integral terms of the derivative of Lyapunov-Krasovskii functional. However, simply introducing multiple integral terms to construct complex Lyapunov-Krasovskii functionals may increase computational complexity, and it is not conducive to controller synthesis. In this paper, a novel augmented Lyapunov-Krasovskii functional with the double integral terms is constructed, and we employ the auxiliary function-based integral inequalities to estimate the integral terms. The simulation results show that the stability results obtained in this paper are less conservative. This processing approach well balances the computational complexity and conservatism, and also facilitates the design of the fuzzy controllers.
In order to further illustrate the benefits of the membershipfunction-dependent method, Corollary 1 is given based on the membership-function-independent method. In this case, the information of membership functions is not considered in stability analysis, and the following corollary can be obtained.

Controller design
In this subsection, the stabilization problem for the T-S fuzzy closed-loop system (6) with time-varying delays is considered. On the basis of Theorem 1, novel state feedback controllers are presented under the imperfect premise matching design scheme. In order to reduce the conservatism of design conditions, some piecewise membership functions are employed to approximate the membership functions, and then some relaxed membershipfunction-dependent stabilization results are obtained by considering the boundary information of the membership functions.

Theorem 2.
For given scalars M > 0, ∈ { 1 , 2 } and tuning parameter , T-S fuzzy closed-loop system (6) with time-varying delays is asymptotically stable, if there exist matrices 0 <P ∈ ℝ 7n×7n , 0 <Q 1 ∈ ℝ 6n×6n , 0 <Q 2 ∈ ℝ 6n×6n , 0 <R ∈ ℝ n×n , 0 <F i j ∈ ℝ 13n×13n ,Ẽ =Ẽ T ∈ ℝ 13n×13n , and any matricesỸ 1 ∈ ℝ 3n×3n , Y 2 ∈ ℝ 3n×3n , X ∈ ℝ n×n , G j ∈ ℝ m×n , such that the following inequalities hold: 3i The corresponding state feedback control gain matrices are given by where the piecewise membership functionŝi j (x) are defined in (49), k is the number of divided state subspaces for the piecewise membership functions, x i 1 i 2 ⋯i n l represents the apexes of the l-th state subspace of . Δ̄i j , Δ i j are predefined constant scalars satisfying Δ i j ≤ Δ i j (x) ≤ Δ̄i j . Concurrently, some other matrices are defined as follows: Proof. For the convenience of presentation, the following matrices are defined as: x T (s) ds On the basis of Theorem 1, it follows from (21) and (40) thaṫ +̃T 7 ( (t )) Q 2̃8 (̇(t )) } − (1 −̇(t ))̃T 9 ( (t )) Q 1̃9 ( (t )) From (40) and (41), we obtain (t )̃(t ) = 0. By applying the statements of (1) and (4) of Lemma 4, the closed-loop T-S fuzzy system (6) is asymptotically stable if there exists L ∈ R 10n×10n such that , where X ∈ R n×n is any invertible matrix, and is a tuning parameter. By pre-and postmultiplying both sides of (43) with T and , respectively, we can obtain Here, if LMI in (44) holds, the closed-loop T-S fuzzy system (6) is asymptotically stable. By Schur complement lemma, (44) is equivalent to To reduce the conservatism of stabilization conditions of the closed-loop T-S fuzzy system (6), piecewise membership func-tionŝi j (x) are constructed based on (7) as: In the following parts, the piecewise membership func-tionŝi j (x) are used to approximate the product of membership functions i j (x) = m i (x)h j (x), and some less conservative membership-function-dependent stabilization conditions are obtained. Let , the maximum and minimum of Δ i j (x) are denoted as Δ̄i j and Δ i j , respectively, such that Expanding the piecewise membership functionŝi j (x(t )) as (49) in (50), we obtain:  Therefore, if LMIs in (34)-(39) are feasible, and it implies thaṫ V (t ) < 0, which in turn guarantees the asymptotic stability of the closed-loop T-S fuzzy system (6). This completes the proof of Theorem 2.

SIMULATION EXAMPLES
In this subsection, four numerical examples are presented to demonstrate the advantages and effectiveness of the proposed stability criteria and stabilization approach. For comparison, the first two numerical examples effectively demonstrate that the membership-function-dependent stability results presented in this paper is less conservative, and the last two simulation examples show the effectiveness of the fuzzy state feedback controllers presented under the imperfect premise matching technique. In Tables 2 and 3, "−" means results are not provided. Example 1. Consider the following non-linear system with time-varying delay given in [36]: which can be described by the following two-rule fuzzy model and the membership functions are defined by , As the membership functions m i (x 1 (t )), i = 1, 2 depend on the system state x 1 , it is logical to construct the piecewise membership functionsm i (x) in the form of (26) depending on is divided into k connected state subspaces, and the l-th subspace is described by ( where a and b are positive real numbers. In this example, we select 11l (x 1 ) = 1 − ). In addition, a = b = 10, k = 20 are employed, respectively. Therefore, based on the definition of Δm i (x) = m i (x) −m i (x), we can obtain the minimum and maximum of Δm i (x) as shown in Table 1.
The example is introduced for conservatism comparison, and an open-loop delayed fuzzy system is explored to demonstrate the improvements of the developed methods. This system is also considered in [1,11,17,[36][37][38][39][40][41][42]. The objective is to calculate the maximum allowable delay M which guarantees the asymptotic stability of the system, and the conservatism is evaluated by the calculated M . Under the same conditions, larger M usually means wider stability regions.
From Table 2, the maximum allowable delay M obtained from Theorem 1 with k = 20 is larger than the results given in [1,11,17,[36][37][38][39][40][41][42] at different values, this shows that the membership-function-dependent stability criteria given in Theorem 1 are less conservative than those proposed in the aforementioned literatures.
Furthermore, as shown in Table 2, the maximum allowable delay M obtained by Theorem 1 is much larger than that by Corollary 1, which demonstrates that the membership-functiondependent analysis method can effectively reduce conservatism. Compared with the membership-function-independent approach, the membership-function-dependent method can obtain further relaxed results.

Example 2.
Consider the T-S fuzzy time-delay system with u(t ) = 0 given in [11], which is in the form of (3) with two plant rules where ] , , and the membership functions are defined by According to Example 1, we select the number of divided state subspaces for the piecewise membership functions k = 20, moreover, the minimum and maximum of Δm i (x) are obtained as shown in Table 1. In this example, the maximum allowable delays M are obtained, which guarantee the asymptotic stability of the T-S fuzzy system with time-varying delays (53) by different literatures. As shown in Table 3, we choose = 0, 0.1 and 0.5, respectively, and the maximum allowable delays M is obtained via Theorem 1 as 5.6897, 4.9091 and 3.6397, respectively, which are larger than the results given in [10,11,43,44,46]. This indicates that the membership-function-dependent stability criteria given in Theorem 1 are less conservative than those proposed in the aforementioned literature.
Remark 5. Comparing with the existing results, the number of utilized decision variables of Theorem 1 is larger than the reported ones listed in Tables 2 and 3. The main reason for obtaining such a larger number is that Theorems 1 In this example, the following simple membership functions are selected for the imperfect-premise-matching-based fuzzy controllers: Utilizing the calculation method in Example 1, the minimum and maximum of Δ i j (x 1 ), i = 1, 2, j = 1, 2 with k= 20 are obtained as shown in Table 4.
The truck-trailer model is also studied in [29,37,38,44,46]. The objective is to calculate the maximum allowable delay M which guarantees the asymptotic stability of the closed-loop system, and the conservatism of stabilization conditions is evaluated by the calculated M . The maximum allowable delays M are shown in Table 5, which are obtained by the six different approaches.
As shown in Table 5, the maximum allowable delays M calculated in [29,37,38,44,46] are 4465, 3.4815, 246.0938, 8.1951 and 262.4908, respectively, which guarantee the asymptotic stability of the closed-loop system. However, the maximum allowable delay M calculated by Theorem 2 is 66,691, which is far bigger than the above ones. Consequently, the proposed membership-function-dependent stabilization conditions can obtain further relaxed results. Especially, assuming = 12, = 2 = − 1 = 0.2 and = 0.05, the imperfect-premise-matching-based fuzzy state feedback controllers are obtained based on Theorem 2: ). The state responses and control signals of the closed-loop system are shown in Figures 2 and 3, respectively. Figure 2 shows that all the state trajectories go to zero as time increases. Therefore, the closed-loop system is asymptotically stable under the above controllers, and the effectiveness of the imperfect-premisematching-based controller design approach in Theorem 2 is illustrated.
Remark 6. It can be seen from Example 3 that the approaches proposed in this paper can obtain more relaxed results. Moreover, the fuzzy controllers presented in [29,37,38,44,46] should employ the same membership functions with the fuzzy models, which is not required in Theorem 2. In Example 3, we select some simpler membership functions to replace the complex ones in the fuzzy models, and the complexity and the implementation cost of the controllers are reduced.

Example 4.
In order to further illustrate the advantages and effectiveness of the imperfect-premise-matching-based fuzzy state feedback controllers presented in Theorem 2, considering the following continuous stirred tank reactor (CSTR) system given in [47,48], the T-S fuzzy models with three rules for CSTR with time-varying delays can be represented as follows: Rule 2: IF x 2 (t ) is 2.7520 (temperature is middle), THEṄ The membership functions for this fuzzy system are as the same as those defined in [47,48]   Based on the imperfect premise matching method, the tworule fuzzy controllers are presented: Based on the imperfect premise matching design strategy, the following simpler membership functions are employed for the fuzzy state feedback controllers: , h 2 (x 1 ) = 1 − h 1 (x 1 ) Utilizing the calculation method in Example 1, the minimum and maximum of Δ i j (x 1 ), i = 1, 2, 3, j = 1, 2 with k= 20 are obtained as shown in Table 6.
Especially, assuming = 2, = 2 = − 1 = 0.2 and = 0.245, the imperfect-premise-matching-based fuzzy state feedback controllers are obtained based on Theorem 2:  Figure 4 that the considered CSTR system has been globally stabilized, which means that the proposed control method is effective.
Remark 7. For the CSTR system given in Example 4, the fuzzy controllers were presented under the traditional PDC technique in [47,48], which need to employ the same membership functions and the same number of rules as the fuzzy models. However, the imperfect-premise-matching-based fuzzy controllers presented in Theorem 2 are not required to share the same membership functions and number of rules with the fuzzy mod- els, the ones could be selected flexibly. In Example 4, the number of rules for T-S fuzzy models is 3, while the number of rules selected for the fuzzy controllers is 2, and the membership functions chosen for the fuzzy controllers are not the same as fuzzy models. By selecting a smaller number of fuzzy rules and choosing some simpler membership functions to replace the complex ones in the fuzzy models, the flexibility of the controller design is increased, and the complexity and the implementation cost of the controllers are reduced. Therefore, the imperfect-premisematching-based fuzzy controllers of the T-S fuzzy time-delay system have more practical value and application prospect.
Remark 8. From Examples 3 and 4, it can be proved that the imperfect premise matching method is the extension of the traditional PDC technique and has the ability to deal with more general T-S fuzzy time-delay systems.

CONCLUSIONS
In this paper, the membership-function-dependent stability and stabilization for T-S fuzzy systems with time-varying delays are investigated. In Theorem 1, a new augmented Lyapunov-Krasovskii functional with more time-delay information is constructed, and some relaxed stability criteria are obtained by combining with integral inequality technique and the reciprocally convex combination inequality. In order to obtain further relaxed stability criteria, some piecewise membership functions are constructed to approximate the membership functions, the boundary information of membership functions is taken into consideration adequately, meanwhile, some slack matrices are employed, and then some more relaxed membership-functiondependent stability results are obtained. The membershipfunction-independent stability conditions are presented in Corollary 1, which is to demonstrate the effectiveness of the augmented Lyapunov-Krasovskii functional constructed in this paper for reducing conservatism, and further illustrate the advantages of the membership-function-independent method. The imperfect-premise-matching-based fuzzy state feedback controllers are presented in Theorem 2. By constructing a smaller number of fuzzy rules and choosing some simpler membership functions to replace the complex ones in the fuzzy models, the flexibility of the controller design is increased, and the complexity and the implementation cost of the controllers are reduced. Finally, four numerical examples are given to demonstrate the effectiveness of the presented approaches.