Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach
This paper develops some improved stability and stabilization conditions of T-S fuzzy system with constant time-delay and interval time-varying delay with its derivative bounds available, respectively. These conditions are presented by linear matrix inequalities (LMIs) and derived by applying an augmented Lyapunov-Krasovskii functional (LKF) approach combined with a canonical Bessel-Legendre (B-L) inequality. Different from the existing LKFs, the proposed LKF involves more state variables in an augmented way resorting to the form of the B-L inequality. The B-L inequality is also applied in ensuring the positiveness of the constructed LKF and the negativeness of derivative of the LKF. By numerical examples, it is verified that the obtained stability conditions can ensure a larger upper bound of time-delay, the larger number of Legendre polynomials in the stability conditions can lead to less conservative results, and the stabilization condition is effective, respectively.
T-S fuzzy time-delay systems are well-recognized by the integration of time-delay systems and fuzzy systems which are often employed to model several nonlinear systems in practice . Such systems have two appealing advantages: T-S fuzzy system is capable of modeling the nonlinear systems  and the unavoidable time-delay phenomenon is explicitly incorporated in the system . When both nonlinearity and time-delay phenomena are considered, T-S fuzzy system with time-delay indeed offers a feasible system representation. For such a system, much work has been done in the past few decades to achieve stability and performance conditions, delay-dependent ones in particular [3–8]. The presence of time-delay in the fuzzy system can be either constant or time-varying delay, and systems with time-varying delay are more difficult to be handled than constant time-delay case. In the light of flourishing research on linear time-delay systems, more and more interesting stability results have been published recently for T-S fuzzy systems with time-varying delays [3–8]. Since these delay-dependent stability and performance conditions are only sufficient conditions, the admissible maximum upper bound of time-delay computed by the conditions is commonly treated as an essential index to evaluate the conservatism of the conditions. Thus, a primary purpose of delay-dependent stability conditions is to search for the admissible maximum upper bound of time-delay as large as possible while ensuring the system stability and performance.
Recalling the existing delay-dependent stability results on T-S fuzzy time-delay systems, one can see that a LKF approach is prevalent and well-studied. The basic idea is to derive the condition by estimating the time-derivative of the constructed LKF which satisfies the conditions of the LKF theorem in . Thus, the construction of the LKF and the estimation method of its derivative play a key role in developing less conservative stability conditions. In the derivative operation process of the LKF, various integral inequalities are established to produce an estimation as tight as possible, such as Jensen-type inequality [1, 3, 4], Wirtinger-type inequality [5, 10], B-L inequality [11–14], and reciprocally convex inequality with free weighting matrices [7, 15, 16]. Moreover, different types of LKFs such as fuzzy weighting-dependent LKFs [17, 18] and augmented LKFs [16, 19] are proposed for T-S fuzzy system. Recently, there are increasing works on exploring the connection or relationship between the process of integral inequality and the construction of the LKF , and they confirm that the augmented LKF approach can be considered as a competitive method  and the B-L inequality can cover Jensen-type inequality and Wirtinger-type inequality as special cases [11–13]. Accordingly, it is expected to develop some less conservative results for T-S fuzzy system with time-delay by updating the augmented LKF approach together with the B-L inequality, which mainly inspires the current research. More recently, for linear time-delay systems, the augmented LKF together with B-L inequalities are introduced to develop stability results [11–14]. However, this method in [11–14] is not presented for stabilization problem of T-S fuzzy systems with time-delay. More importantly, the augmented LKF in [13, 14, 16, 19] requires all Lyapunov matrix variables to be positive and the delay-induced convex combination in the augmented LKF is not sufficiently used, which leaves us much room for improvement.
In summary, the contributions of this paper are mainly in two aspects as follows.(i)Augmented LKFs based on the form of B-L inequality are proposed to achieve less conservative stability conditions of T-S fuzzy systems with constant or time-varying delay. In particular, for the time-varying delay case, free-weighting matrices are technically introduced in the process of the positive definiteness of the LKF and the negative definiteness of its derivative resorting to a reciprocally convex method.(ii)Both stability and stabilization conditions of T-S fuzzy systems with constant or time-varying delay are expressed by tractable LMIs. The advantages of the proposed stability conditions over than those in the existing literature and the design validity are shown in the examples.
Notation. is the -dimensional Euclidean space. (respectively, ) denotes a set of real symmetric (respectively, positive definite) matrices with dimensional. For a square matrix , means the sum of and , i.e., . For two positive integers , we define . We use to represent a column vector and to denote the block-diagonal matrix. The symmetric term in a symmetric matrix is denoted by “”. is a polytope generated by two vertices and .
2. System Description and Preliminaries
Consider the time-delay system with fuzzy rules as follows.
. IF is , is , and is , THENwhere , and , are the premise variables; , , , are the fuzzy sets; and represent the state vector and control input, respectively; , , and are known system matrices. is the time-delay that can be either constant (let ) or time-varying. For the time-varying case, it satisfieswhere is the upper bound of time-delay and and are the interval bounds of the derivative of . We define where as the initial condition of system (1).
According to the technique of fuzzy blending, system (1) can be inferred as the following overall system:where is the normalized membership function; is the membership function of the fuzzy set , where is the grade of membership of in and
Consider the following fuzzy controller for th rule.
. IF is , is , and is , THENwhere () is the fuzzy controller gain. The fuzzy controller is given by
To end this section, we give the following lemmas which will be used in deriving our main result.
Lemma 1 (see ). For a matrix , an integer , a matrix , two scalars and with , and a vector function such that the integrations below are well defined; thenwhere
Lemma 2 (see ). For given and , if one can find matrices and such thatsatisfied for and , then one has which holds for any .
3. Stability Conditions of T-S Fuzzy Time-Delay Systems
3.1. Constant Delay Case
Suppose that . Choose the following augmented LKF candidate:where
and then we can derive the following stability condition.
Theorem 3. For any given integer and a positive constant , if there exist , , and and of appropriate dimensions such that for , hold, then system (7) with constant delay is asymptotically stable, where with and being defined in Lemma 1, and .
Proof. The derivative of is given byDenote ThenApply Lemma 1 to deal with the integral term in (19), and we haveLet . (22) can be rewritten asFor any matrices () with appropriate dimensions, the following equation holds for system (7)Then combining (19) with (23) and (24) yieldsIf LMI (17) is satisfied, by Schur complement, one has , which implies that . The proof of Theorem 3 is completed.
3.2. Time-Varying Delay Case
For simplicity, we denote equations as follows: where .
Firstly, we deal with the positive definiteness of . Proposition 5 is proposed through which the positive definiteness of the LKF can be proved.
Remark 4. The augmented term of the proposed LKF is constructed based on the form of B-L inequality, which contributes to reducing the conservatism of the conditions. As proved in , (i) the augmented LKF approach is useful for conservatism-reducing, (ii) for the same nonaugmented LKF, the use of Wirtinger-based inequality and Jensen-based inequality does not make a difference in conservatism-reducing, and (iii) for an augmented LKF, the use of Wirtinger-based inequality is better than Jensen-based inequality to obtain less conservative results. As these inequalities are the special cases of B-L inequality, the proposed augmented LKF is significant and effective in deriving the main results. For time-varying delay case, delay-product-type terms of LKF are introduced in . Clearly, these terms in  are special cases of the terms in , which are proposed based on the form of B-L inequality.
Proposition 5. For a given scalar and any given integer , if there exist , , and such thatare satisfied, and then there exist and such that satisfieswhere and with , and represents the space of functions and with .
Proof. By using Jensen’s inequality to , one haswhere Then substituting (33) and (34) into (27), we obtain where If there exists a matrix such that (31) is satisfied, similar to , using Lemma 2 with and to , together with , and , then one has . If (30) and , and are satisfied, and there exists such that , which verifies the first inequality of (32). Similar to , the proof of the second inequality in (32) can be obtained, and the specific steps are omitted for brevity. This ends the proof.
Remark 6. Some extended reciprocally convex inequalities compared to Lemma 2 are proposed in [22, 23], and some improved results with less complexity  or less conservatism  are provided. It would be interesting to incorporate these inequalities with the proposed augmented LKF method to achieve some potential results in future work. The computing complexity of conditions and stochastic feature of systems  also could be studied.
Secondly, the negative definiteness of is shown by the following proposition.
Proposition 7. For given scalars , and and any given integer , there exists such that the functional satisfies if there exist , , , and , and of appropriate dimensions such that for , the following LMIs hold for :where and are defined in (10) and (11), respectively.
Proof. For in (27), we compute its derivative and obtainwhereDenote We can rewrite in (41) as follows:Using Lemma 1 to estimate the upper bounds of the above integral terms, respectively, we getSetting and . Then, according to (45) and (46), we yieldwhere andSimilar to the proof of Theorem 3, we introduce the following equation:Thus, according to (44), (45), (46), (47), (48), and (49) we haveIt can be clearly seen that the matrix is a convex combination in and . Similar to the two-dimensional convex combination method in , if LMIs (38) and (39) are satisfied, one can easily derive for . The proof of Proposition 7 is completed.
Theorem 8. For given scalars , and and any given integer , if there exist , , , and (), and of appropriate dimensions such that the LMIs (30), (31) and the LMIs as follows are satisfied for , then system (7) with time-varying delay subject to (2) is asymptotically stable.
For the case of time-varying delay, [12, 14] propose a refined delay set which has been proven to derive less conservative results than delay set . Taking the refined delay set into account and using Theorem 8, we obtain a stability condition as follows.
Theorem 9. For given scalars , and and any given integer , if there exist , , , and (), and of appropriate dimensions such that (30), (31) are satisfied and the following LMIs hold for , then system (7) is asymptotically stable for
4. Fuzzy Controller Design
In this section, controller design conditions will be given by two cases of the time delay.
4.1. Constant Delay Case
Theorem 10. For given scalars , , and a given integer , if there exist , , and (), and of appropriate dimensions such that the following LMIs hold for , then the closed-loop system (7) with constant delay is asymptotically stable where