Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach

Gao, Zifan; Yang, Jiaxiu; Zhu, Shuqian

doi:https://doi.org/10.1155/2019/8725610

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8725610 | https://doi.org/10.1155/2019/8725610

Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach

Zifan Gao,¹Jiaxiu Yang,¹and Shuqian Zhu²

Academic Editor: Libor Pekař

Received09 Mar 2019

Revised04 Jun 2019

Accepted13 Jun 2019

Published04 Jul 2019

Abstract

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.

1. Introduction

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 [1]. Such systems have two appealing advantages: T-S fuzzy system is capable of modeling the nonlinear systems [2] and the unavoidable time-delay phenomenon is explicitly incorporated in the system [1]. 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 [9]. 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 [20], and they confirm that the augmented LKF approach can be considered as a competitive method [19] 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

Substituting (6) into (3), we obtain the following closed-loop system:

To end this section, we give the following lemmas which will be used in deriving our main result.

Lemma 1 (see [14]). 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 [12]). 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

In this section, we establish some asymptotic stability conditions of system (6) with two cases of time delay: constant delay and time-varying delay satisfying (2).

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 .

In order to make full use of B-L inequality, we construct the following augmented LKF for system (3)with where , , , and and is defined in (16).

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 [21], (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 [21]. Clearly, these terms in [21] 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 [12], 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 [12], 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 [22] or less conservatism [23] 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 [24] 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 [16], if LMIs (38) and (39) are satisfied, one can easily derive for . The proof of Proposition 7 is completed.

Now, we establish the stability conditions of the T-S fuzzy system (7) based on Propositions 5 and 7 as follows.

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

Based on Theorem 3, the fuzzy control gains of system (7) with constant delay can be derived from the following theorem.

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 with , , , , and being defined in Theorem 3. Then, the fuzzy control gains can be calculated by .

Proof. Pre- and postmultiplying both sides of (17) with and its transpose, introducing , , and defining , , , , , and , we obtain the LMI (53). This completes the proof.

4.2. Time-Varying Delay Case

For the case of , the next theorem is given to obtain the fuzzy control gains.

Theorem 11. For a given integer , given parameters , and , and given scalars , if there exist , , , and (), , and of appropriate dimensions such that the following LMIs hold for , , then the closed-loop system (7) with time-varying delay is asymptotically stable where

The proof of Theorem 11 is similar to the one of Theorem 10 which is omitted here for brevity. Moreover, the fuzzy control gains are given by .

5. Numerical Examples

In this section, we give two numerical examples to verify the effectiveness of the proposed methods, where Example 1 is widely used for the comparison of the admissible delay upper bound computed by the delay-dependent stability conditions of this paper and some existing results, and Example 2 is employed to confirm the validity of the controller design conditions.

5.1. Example 1

Consider system (3) with , and system matrices as follows:

5.1.1. Constant Delay Case

When , using Theorem 3 for different and the results of some recent literature, the obtained admissible maximum time-delay upper bounds are listed in Table 1 for comparison. Meanwhile, the computing complexity of the methods is compared by listing the number of decision variables. We can clearly see that for an integer , Theorem 3 proposed in this paper can provide a larger upper bound than some existing literature, e.g., in [25] and in [26], which verifies that Theorem 3 is less conservative. But the computing complexity of Theorem 3 is larger than others.

5.1.2. Time-Varying Delay Case

Set the lower and upper bounds of the derivative of time delay to be and . Considering , one can compute the admissible upper bound by Theorem 8 with . Using the existing results in [27–30], the obtained maximum delay bounds are listed in Table 2 for comparison. The results for , and are also shown in Table 2. From Table 2, it is clear that Theorem 8 produces a larger delay upper bound than Theorem 2 in [27] and others listed in Table 2, which implies that Theorem 8 it is less conservative.

Assume that . Then we use Theorem 9 for different . The obtained maximum delay bounds are given in Table 3, from which one can see that Theorem 9 delivers a larger delay upper bound than Theorem 8. Tables 2 and 3 also show that both Theorems 8 and 9 ensure a larger with increasing .

Furthermore, to check that the system can tolerate the time-delay limited by the proposed results, we employ the simulation by Matlab. In simulation, let and . Then we choose and set the initial condition for . We depict the states and by Figure 1, which confirms asymptotically stable responses.

5.2. Example 2

Consider system (7) with s=2 and system matrices as follows:

Obviously, when and without any control input, both the subsystems of the above system are unstable, because and are not Hurwitz matrices; that is, it is an unstable system. Now we demonstrate that the designed controllers by Theorems 10 and 11 are effective, respectively.

5.2.1. Constant Delay Case

Using Theorem 10 with , , the admissible maximum upper bound of constant delay can be computed to be , and the corresponding control gains are as follows:According to the feasibility of Theorem 10, one can say that the controller (6) with the proposed gains (59) ensures the stability of the closed-loop system.

Similar to Example 1, we let and and set the initial condition for . The state responses of the closed-loop system under the controller (6) with gains (59) are demonstrated by Figure 2 which shows a positive result. From Figure 2, one can say that closed-loop system under the controller (6) with gains (59) is asymptotically stable. Then it can be concluded that Theorem 10 is effective.

5.2.2. Time-Varying Delay Case

Let . Then, according to Theorem 11 with , , and , one can deliver the admissible maximum bound while the control gains areFrom the feasibility of Theorem 11, it can be considered that the closed-loop system can be stabilized by the controller (6) with the proposed gains (60).

For further confirmation, the normalized membership functions are set as and . Besides, we introduce the initial condition for and choose the time-varying delay . The state responses in Figure 3 shows the positive effect of the controller (6) with gains (60) on the stabilization of the closed-loop system, which demonstrates the effectiveness of Theorem 11.

6. Conclusion

In this paper, we have proposed an augmented LKF approach to derive some improved stability and stabilization conditions of fuzzy system with constant delay and interval time-varying delay with its derivative bounds available, respectively. In particular, the proposed LKF have been constructed on the basis of the form of the B-L inequality. Two numerical examples have illustrated the improvements of the obtained conditions comparing with some existing recently results and the design validity of fuzzy controllers.

Data Availability

The data used to support the findings of this study are included within the article. These data include three tables in the manuscript, which are used to make comparisons between the proposed conditions with the existing results.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant 61773238, the Fundamental Research Funds of Shandong University, and the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi.

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Copyright © 2019 Zifan Gao et al. 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.

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