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Volume 2019, Article ID 7875305, 15 pages
Research Article

New Stabilization Results for Semi-Markov Chaotic Systems with Fuzzy Sampled-Data Control

1School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, China
2School of Mathematics, Southeast University, Nanjing 210096, China
3School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
4Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1

Correspondence should be addressed to Tao Wu; moc.621@631025uwoat and Lianglin Xiong; moc.621@8135_nilgnail

Received 18 July 2019; Accepted 26 September 2019; Published 3 November 2019

Academic Editor: Wenqin Wang

Copyright © 2019 Tao Wu 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.


This paper investigates the problem of stabilization for semi-Markov chaotic systems with fuzzy sampled-data controllers, in which the semi-Markov jump has generally uncertain transition rates. The exponential stability condition is firstly obtained by the following two main techniques: To make full use of the information about the actual sampling pattern, a novel augmented input-delay-dependent Lyapunov–Krasovskii functional (LKF) is firstly introduced. Meanwhile, a new zero-value equation is established to increase the combinations of component vectors of the resulting vector. The corresponding fuzzy sampled-data controllers are designed based on the stability condition. Finally, the validity and merits of the developed theories are shown by two numerical examples.

1. Introduction

As we all know, Takagi–Sugeno fuzzy model (T-SFM) [1] is an applicable and effective way for us to approximate complex nonlinear systems. It is often used for handling some practical nonlinear systems in many controlling engineering areas because the T-SFM could show topical linear input-output relations of a nonlinear system. For example, some famous nonlinear systems can be accurately expressed by the T-SFM, such as Chua’s system [2], Lorenz system [3], and Rösler system [4]. These systems are called chaotic systems. Consequently, T-S fuzzy control has a significant role in chaotic systems, and plenty of meaningful results on T-S fuzzy chaotic systems have been proposed based on various kinds of approaches [59].

Because of the fast development of microelectronics technology and communication networks, sampled-data control enjoys great popularity owing to its maintenance with low cost, better reliability, and efficiency [1016]. In general, in the realization of sampled-data control systems, only the sampling information is sent to the controller at discrete time instants, which would greatly decrease the amount of information while increasing the efficiency of bandwidth usage. Thus, to investigate the sampled-data control for fuzzy chaotic systems has considerable realistic significance. Up to now, three methods are mainly used, namely, impulsive control approach [13, 14], discrete-time technique [15], and input delay method [12, 17], to deal with the sampled-data control systems, in which the most favorite way is the input delay method. Combining the input delay method with the zero-order holder (ZOH), the sampled-data control system can be transformed as a continuous-time system with time-varying delayed control input. In recent years, according to the input delay method, lots of important stabilization results on chaotic systems with fuzzy sampled-data control were presented [1827]. In [18, 23, 24], by designing fuzzy sampled-data control, the stabilization conditions for chaotic systems have been obtained. In [19, 20], by constructing input-delay-dependent LKFs, less conservative criteria for chaotic system have been studied. Very recently, by utilizing the fuzzy input-delay-dependent LKF approach, fewer constraint conditions for chaotic systems have been established in [25, 26]. Most recently, by constructing a new input-delay-dependent LKF, improved results have been established for chaotic systems in [21, 27]. It is well known that the input-delay-dependent Lyapunov method is very effective for reducing conservatism because it can fully capture the information about the actual sampling pattern. However, in the most existing works, input-delay-dependent Lyapunov terms are mainly constructed based on linear function of input delay. Input-delay-dependent Lyapunov terms with high-order function of input delay are difficultly contained because their derivation may be nonlinear input delay. Then, the convex combination technique cannot be directly utilized. Thus, it is still a challenging issue to construct input-delay-dependent LKFs with high-order function of input delay to further improve the existing results for fuzzy sampled-data chaotic systems, which is the first motivation of this paper.

Moreover, it should be pointed out that the existing works [1827] considered how to obtain less conservative stabilization conditions, while they ignored calculation load. Actually, during the application of LMI-based criteria to large-scale physical systems, the computational complexity is a very significant problem [28]. An LMI-based condition with too much complexity may be limited in many practical applications. Hence, it is important to establish stabilization criteria discussing both the conservatism and calculation complexity for fuzzy sampled-data chaotic systems, which is the second motivation of this paper.

On the contrary, in the real world, the Markov jump usually exists in chaotic systems since the abrupt changes in their structure and parameters. So, the study on chaotic systems with Markov jump parameters is of great importance [2931]. It is noted that the transition rates in the aforementioned works are always supposed to be totally known. The transition rates are usually difficult to accurately appraise in general; thus, the Markov jump systems with generally uncertain transition rates were considered in [3234] and the references therein. Reference [27] studied the problem of stability and stabilization for fuzzy chaotic systems with fuzzy sampled data and Markov jump of generally uncertain transition rates. Some studied Markov jump has native limitations since the Markov chain is assumed to be exponentially distributed and the transition rates are considered to be constants. As a consequence, the obtained results are conservative to some extent. In order to avoid this situation, semi-Markov jump systems were discussed in [3539] by supposing the transition rates to be time varying instead of constant ones. Meanwhile, one can easily see that the obtained results for the Markov jump systems are special cases of those for semi-Markov jump systems. Clearly, semi-Markov jump systems are more general than Markov jump systems in applications. Therefore, it is necessary to consider semi-Markov chaotic systems with fuzzy sampled-data control. However, to the best of our knowledge, the issues have not been considered yet. On the basis of this fact, the previous stability and stabilization criteria will no longer be applicable. Hence, the second motivation of this study becomes more and more important.

Based on the above discussions, the aim of this paper is to further investigate the problem of stabilization for fuzzy chaotic systems with sampled-data control and semi-Markov jump of generally uncertain transition rates. The remainder of this article is given as follows: Many useful notations are given and the main problems are described in Section 2. The stability and stabilization conditions are provided in Section 3. In Section 4, two numerical examples are presented to show the feasibility and superiority of our results. Section 5 gives a conclusion. The advantages of this paper are summarized as follows:(i)This discussion for chaotic systems with both fuzzy sampled-data control and semi-Markov jump of generally uncertain transition rates is for the first time to be proposed. The obtained results in practical situation are more applicable than the previous results in [18, 2027].(ii)To make more cross terms of vector components in the resulting vector, a novel zero equality is established.(iii)To take full advantage of the available information with regard to the practical sampling mode, a new augmented time-dependent LKF is constructed, which is firstly presented for the fuzzy sampled-data semi-Markov chaotic systems.(iv)Compared with the existing stabilization conditions in [21, 2426], our results are not only less computationally complex but also less conservative.

2. Problem Statement and Preliminaries

To be clear, the following symbols are firstly explained in a simple way:T: transpose of a matrix or a vector: n-dimensional Euclidean space: set of all real matrices: matrix P is symmetric and positive definite: symmetric terms in a symmetric matrix: identity matrix: zero matrices: a block-diagonal matrix: (): the largest (smallest) eigenvalue of G: Euclidean norm for a given vector: complete probability space with a natural filtration satisfying the usual conditions (i.e., the natural filtration contains all -null sets and is right continuous): expectation function in regard to the given probability measure

Consider the semi-Markov chaotic system expressed by the following differential equation:where denotes the state vector, denotes the control input vector, and denotes a known nonlinear continuous function and satisfies . denotes the right-continuous semi-Markov process on the probability space and takes values in a finite state set with the transition rate matrix given bywhere and ; the transition rates are positive, if , while for each mode .

Actually, not all the transition rates are easy to accurately estimate in the jumping process; most of them always contain the following two assumptions: (1) is totally uncharted and (2) is not exactly known, but upper and lower bounded. Based on the second condition, we can further define , in which and are known real constants to represent the lower and upper bounds of , respectively. Furthermore, we can denote that , in which and with . Thus, the time-varying transition rate matrix with s jumping modes may be described aswhere is unknown time-varying transition rates. For convenience, , the set denotes with

Moreover, if and , we can obtain the following:

By using the following IF-THEN rules, system (1) can be developed into the following systems: Plant rule i: IF is and … and is , THENwhere , represent premise variables, represent the fuzzy sets, and represent the known constant matrices with suitable dimensions, and shows the number of fuzzy rules.

By using the fuzzy blending, the fuzzy semi-Markov jump system (6) can be rewritten as follows:where and is the membership function satisfyingin which stands for the grade of membership of in . For all , we have and .

Throughout this paper, the control signal is supposed to be generated by using a ZOH function with a series of holding moments . Accordingly, the fuzzy sampled-data controllers can be easily given as follows.

Controller rule j: IF is and … and is , THENwith , , and denotes the controller gain matrices to be computed. The sampling interval meets the condition , where τ is the upper bound of the sampling periods.

Hence, the entire fuzzy sampled-data controller can be represented as follows:

For the sake of convenience, is noted as hereinafter. Introducing (10) into (7), the fuzzy system (7) is further developed into

To clearly give the main results, we firstly provide the following definition and lemmas.

Definition 1. System (11) is exponentially stable if scalars and satisfywhere β and δ are defined as the decay rate and decay coefficient, respectively.

Lemma 1. The system (11) holds the following conclusion:with , , , and .

Proof. One can obtain for from (11) thatUsing the Cauchy–Schwarz inequality, we can get from (14) thatUtilizing the Gronwall–Bellman inequality, it is easy to see from (15) that (13) concludes. This proves Lemma 1.

Lemma 2. (Park et al. [40]). For a given matrix , any differentiable function satisfies the following inequality:with

3. Main Results

Two important results will be established in this section: First, the exponential stability condition of the fuzzy semi-Markov jump system (11) is given for known control gain matrices . Second, by designing the control gain matrices , the exponential stabilization criterion of the fuzzy semi-Markov jump system (11) is obtained. For clarity, we need to define the following matrices and vectors:

Theorem 1. Given scalars τ and β, the fuzzy system (11) with control gain matrices is exponentially stable; if there exist positive definite matrices , , , , and , any matrices , , , , , , and with appropriate dimensions, such that the following conditions hold for all i, , and .

Case 1. , , ,wherewith

Case 2. , , , ,where

Proof. Noticing the fact that , , and , for any matrix with suitable dimensions, the following zero equality holds:Consider an LKF candidate aswithLet be the weak infinitesimal operator for the semi-Markov process acting on as follows:where is a scalar. Then, similar to the computation in [35, 36], we haveSince , by Lemma 2 and the reciprocally convex approach in [41], we haveAccording to the fuzzy system (11), for any appropriate dimensional matrices , the following zero equality holds:with .
Merging (29) and (33)–(38), we have for thatwhere and .
On the contrary, since is generally uncertain transition rates in this paper, the proof needs further derivation by the following two cases (i.e., and ).
When , let . According to , we have . Then, the formula can be represented asIt is clear that and . Thus, for , one can obtainAccordingly, for , is equivalent toThen, we can find from the above inequality (42) thatFrom the inequality [42], for any , we haveCombining (40)–(44) and using Schur complement, it can be concluded (19)–(21) guarantee for all .
When , define . Because , we have . So, the expression can be rewritten asIt is obvious that and . Consequently, for , , we getTherefore, for , the equivalent form of can be described as follows:Since , the inequality (47) holds if we haveIn addition, similar to (43) and (44), for any , we obtainFrom (45)–(49), by Schur complement and inequalities (24)–(27), it is known that for all . In summary, based on the above discussions, one can obtainFollowing this, we illustrate is continuous at sampling times. By (30), we can obtainFrom (51) and (52), one can getHence, is continuous at sampling times. Besides, from (50)–(53), since , we have for thatwhich implies that at the sampling times.
Combining Dynkin’s formula with (50), we haveMoreover, it is followed from (30) thatwhereOn the contrary, on the basis of Lemma 1 and (55), for , one can derivewhich implies thatTherefore, we havewith .
Thus, the semi-Markov chaotic system (11) is exponentially stable with exponential convergence rate β according to Definition 1. This proves Theorem 1.

Remark 1. Noting that the aforementioned system has been studied in [27], the modeling and method in handling uncertain transition rates are completely dissimilar to those in this paper. Firstly, we investigated a more common subject on fuzzy sampled-data chaotic systems with semi-Markov jump. Secondly, reference [27] provided lots of conservative stipulations in modeling these uncertain transition rates, for example, the upper and lower bounds of uncertain transition rates are often assumed to be the same when the absolute values are taken but, however, different from the study in [27], where a more general approach takes the average of the upper and lower bounds of jumping time-dependent transition rates. Thirdly, from the perspective of dealing with uncertain transition rates which also are not alike for the proof, our approach is more universal than that in [27] and also can be used to deal with the case of uncertain transition rates with constant numbers. Nevertheless, the approach presented in [27] is very particular, and many assumptions in coping with uncertain transition rates are much conservative, which are not applicable in this paper.

Remark 2. Recently, the input-delay-dependent LKF approach has been widely used in some reported works (e.g., see [21, 24, 25]). However, we find that the constructed LKFs are simple and not easy to involve the high-order function of the sampling delay . Thus, in order to further solve this difficulty, we take , , and into the intermediate vector such that the cubic function of is successfully presented in the novel additional term . Meanwhile, the equality (29) is used to increase the combination among the vectors of , , , , , , and . It is also worth noting that in Theorem 1 is nonlinear matrix inequalities since it contains the quadratic function of the sampling input delay . So, it cannot be directly solved by using Matlab LMI Toolbox. Fortunately, by applying Lemma 7 in [43], the nonlinear matrix inequalities can be further transformed as general LMIs. Moreover, in comparison with the mentioned results in [21, 24, 25], our less conservative condition is built, which will be verified by examples in Numerical Simulation.

Theorem 2. Given scalars τ, β, and μ, the system (11) can be exponentially stabilized; if there exist positive definite matrices , , , , and , any matrices , , , , , , , and with appropriate dimensions, such that the following predonditions hold for all i, , .

Case 1. , , ,with

Case 2. , , , ,withMoreover, the controller gain matrices in (10) are designed as

Proof. Define , , , , , ,