$H_{\infty}$ Synchronization for Fuzzy Markov Jump Chaotic Systems with Piecewise-Constant Transition Probabilities Subject to PDT Switching Rule

This article investigates the nonfragile <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> synchronization issue for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems. With regard to the T–S fuzzy model, a novel processing method based on the matrix transformation is introduced to deal with the double summation inequality containing fuzzy weighting functions, which may be beneficial to obtain conditions with less conservatism. In view of the fact that the uncertainties may occur randomly in the execution of the actuator, a nonfragile controller design scheme is presented by virtue of the Bernoulli distributed white sequence. The main novelty of this article lies in that the transition probabilities of the Markov chain are considered to be piecewise time-varying, and whose variation characteristics are described by the persistent dwell-time switching regularity. Then, based on the Lyapunov stability theory, it is concluded that the resulting synchronization error system is mean-square exponentially stable with a prescribed <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> performance in the presence of actuator gain variations. Finally, an illustrative example about Lorenz chaotic systems is provided to show the effectiveness of the established results.


I. INTRODUCTION
O VER the past few decades, hybrid systems (HSs) involving a set of subsystems and the interconnection of logic have received much attention due to the powerful capability of modeling practical systems displaying switching features [1]- [5]. Generally, the switching signal that can reveal the logic association of HSs can be modeled either in stochastic or deterministic framework depending on whether the switching rule contains stochastic statistical properties [6]. An extensively investigated class of stochastic HSs are Markov jump systems (MJSs) whose jumps are determined by Markov chains. On account of the probabilistic properties, MJSs are quite suitable for describing the phenomena of random changes caused by unexpected events, and fruitful results focusing on the stability and stabilization analysis for MJSs have emerged accordingly [7]- [11].
For MJSs, the transition probabilities (TPs) which are usually calculated through a large amount of statistical data occupy an important position in determining the dynamic behavior of the whole system. Most of the current researches on MJSs are based on an implicit assumption that the TPs are time-invariant. Nevertheless, this assumption may not be consistent with the actual engineering applications since the data utilized for obtaining TPs may exhibit different features during different periods. Consequently, motivated by the increasing needs of more realistic stochastic models, abundant research efforts have been devoted to exploring suitable TPs that can better reveal the random property. Currently, an available strategy is the introduction of nonhomogeneous TPs, which can be usually described in two ways. One is to consider the TPs displaying memory property, i.e., semi-Markov chain. While the other is the consideration of piecewise-constant TPs, for which the TPs are time-varying but remain constant within a fixed interval, and the variation is governed by another highlevel switching signal [12], [13]. Generally, the deterministic switching signal is a preponderant option [14]. However, due perhaps to the complexity of two-level switching mechanism, relevant researches on this issue are far from maturity, and the mechanism that characterizes the switching of TPs for MJSs is mainly confined to dwell time (DT) or average dwell-time (ADT) switching [15], [16]. To the best of the authors' knowledge, there exists few available literature on MJSs with persistent dwell-time (PDT) switched TPs. As the PDT switching has been verified to be more flexible than DT and ADT switching mechanism [17]- [19], exploring an appropriate disposing method to deal with MJSs subject to PDT switching in the stochastic jumps is of great significance.
On another research forefront, nonlinearity is an intrinsic feature that widely exists in a majority of actual physical systems, which is also an essential element that cannot be neglected in the process of system modeling [20]- [23]. It is recognized that the T-S fuzzy model proposed in [24] is a prevailingly adopted tool for the approximation of nonlinear systems [25]- [29]. Described by a series of local linear sub-models, T-S fuzzy model can approximate systems with nonlinear term accurately [30]- [35]. As a consequence, considerable attention has been attached to the investigation of T-S fuzzy MJSs (TFMJSs) since the proposal of the magnificent scheme [36]- [38]. It should be mentioned that the majority of the works on TFMJSs involve the processing of the double summation inequality containing membership function terms, and at present, there are mainly three types of widely utilized disposal methods, as presented in [8], [18], [39]. Obviously, with respect to these methods, the conservatism of the results may increase with the accretion of the fuzzy rules. Therefore, seeking a more effective method to solve this issue deserves further investigation, which gives rise to our great interests.
Moreover, the synchronization of chaotic systems has captured attention and enthusiasm of many researchers during the past decades as the chaos phenomenon provides broader space for exploring the irregular and unpredictable behavior of natural nonlinear systems [39]. It is well known that a crucial character of chaotic systems is that they are particularly sensitive to initial conditions and system parameters, and a minor variation in initial state can lead to great changes in system response [40]. Therefore, exploring an effective control scheme to achieve precise synchronization of the chaotic master and slave systems is especially important, and an extraordinary volume of literature has been published on relevant issues.
To name a few, in allusion to a class of nonlinear chaotic systems, the fault-tolerant dissipative synchronization issue was investigated in [41], and the composite nonlinear feedback controller design issue was explored in [42]. What should be mentioned is that the actuator may work improperly during the operation of the systems when some unexpected errors occur, such as equipment failures or external interference. In response to this issue, we are interested in designing a resilient or non-fragile controller which possesses the advantage of effective implementation in the presence of gain variation in the controller parameters.
The aforementioned analyses motivate us to construct a nonfragile H ∞ controller for T-S fuzzy Markov jump chaotic systems (TFMJCSs) with PDT switched TPs to achieve the synchronization of the master and slave systems. The main contributions can be synthesized as the following three aspects: (i) As the first attempt, the discrete-time TFMJCSs with time-varying TPs is investigated, of which the variation of TPs are supposed to be governed by a deterministic switching signal. On account of the generality compared with the DT or ADT switching, the PDT switching regularity is introduced to describe the variation of TPs.
(ii) With regard to the double summation inequality containing membership function terms of T-S fuzzy model, a novel processing method based on [43] is presented. Compared with the disposing approach given in [8], [18] and [39], it may be conducive to deriving conditions with less conservatism.
(iii) For ensuring the effective implementation of the designed controller, we consider that the gain uncertainty may occur in the controller parameters, and the Bernoulli distribution is employed to describe the randomness of uncertainty. Then, a non-fragile controller which can guarantee the meansquare exponential stability and H ∞ performance of the resulting synchronization error system is constructed.
Notations: The notations employed in this work are standard [15]. Z ≥a : the set of integers no less than a; E{·}: the expectation operator; sym{A}: A + A T ; λ max {A}/λ min {A}: To illustrate the concept of PDT switching regularity, the following definition is provided.
Definition 1: [19] Given positive integers τ P (the persistent dwell-time) and T P (the period of persistent), λ(k) complies PDT switching regularity if the constraints presented in the following are satisfied: (a) For a series of nonadjacent intervals of length greater or equal to τ P , λ(k) is a constant in each of these intervals.
(b) The intervals mentioned above are separated by segments no longer than T P , λ(k) can take different values in these segments as long as the duration of each value is less than τ P .
Remark 1: To facilitate the subsequence analysis, Fig.1 presents the possible variation trends of the conceived Lyapunov function under the Markov jump sequence ϑ(k) and PDT switching sequence λ(k). With regard to the PDT switching signal, the value of the Lyapunov function can increase within a certain range at the switching instant while is required to decrease at the non-switching instant. To ensure the function as a whole tends to decay, the range that the function can rise at the switching instant should be confined. Therefore, we consider that the function value at the beginning of the current stage is less than that at the beginning of the previous stage. As the jumping of the investigated system is governed by the Markov chain, the mode jumping exhibits memoryless characteristics, and whether the system jumps or not is irrelevant with the previous mode. Furthermore, the sth stage of PDT switching signal λ(k) consists of two portions: the τ -portion (actual length τ (s) , τ (s) ≥ τ P ) and the T -portion (actual length T (s) = T (d) +T (g) +. . .+T (w) , T (s) ≤ T P ). k fs , k fs+1 , . . ., k fs+1−1 and k fs+1 represent the switching instants of the TPs.

B. System Description
We consider the following nonlinear Markov jump chaotic system: is a nonlinear function. Then, with the employing of the T-S fuzzy method, the ath rule of TFMJCS is expressed as: where z(k) ∈ R nz signifies the system output and I(k) ∈ R nx means the external input. Consider that the total number of IF-THEN rules is N , and a belongs to the set is the fuzzy sets. A aϑ(k) and B aϑ(k) are given constant matrices and their dimensions are suitable. For convenience, we denote A aϑ(k) A ai and B aϑ(k) B ai for ∀ϑ(k) = i ∈ M. Other relevant symbols are defined in a similar way. Subsequently, by fuzzy-blending the above given sub-models, the global model corresponding to the ith subsystem is presented as: where r a (η(k)) is the fuzzy weighting function satisfying For the purpose of simplification, in the following, we use r (k) a to denote r a (η(k)). System (3) is the master system, based on which the following controlled fuzzy slave system can be established by using the similar method as above are used to denote the state vector, the external disturbance belonging to l 2 [0, ∞), the control input vector, and the output vector of the slave system, respectively.
In the following, we denote e(k) y(k) − x(k) andz(k) z(k) − z(k) with e(k) being the synchronization error, and the resulting synchronization error system (Σ) can thereby be given as: One of the main objectives of this paper is to design a non-fragile controller which can guarantee the stability and prescribed performance of the synchronization error system. To achieve this purpose, a mode-dependent fuzzy controller is constructed as follows: where ς(k) ∈ {0, 1} is the Bernoulli white sequence which is independent of Markov chain and satisfies the following probability distribution law: Besides, K bi denotes the controller gain which needs to be determined. ∆K bi means the gain variation of the controller, whose form is as follows: and U bi , V bi are known constant matrices. By fuzzy-blending the individual linear controller (7), the overall controller can be given as: Synthesize the above, the synchronization error system (Σ) can finally be expressed as: Remark 2: For better illustrating the HSs with stochastic and deterministic switching properties, Fig. 2 is presented, where the HSs are composed of M stochastic modes, and the jumping of theses modes is governed by Markov chain. Meanwhile, the decision of the supervisor will have an effect on the TPs, which means the TPs are time-varying. Our purpose is to construct an appropriate control scheme, such that the stability and performance of the overall system can be ensured when the supervisor complies with PDT switching constraint. The advantage of this constraint lies in that the PDT switching rule can describe the switching with both fast and slow characteristics, and it is more flexible than DT or ADT switching.

C. Necessary Lemmas and Definitions
In order to investigate the stabilization and performance issue of the synchronization error system, the following lemmas, and definitions are provided.
Remark 3: For T-S fuzzy models, the generally utilized processing methods for ensuring the establishment of N a=1 N b=1 r (k) a r (k) b Θ ab < 0 are presented as the following three types: (I) Θ ab < 0, ∀a, b ∈ N [18]; (II) Θ aa < 0, ∀a ∈ N and Θ ab + Θ ba < 0, ∀a, b ∈ N , a < b [8]; (III) Θ aa < 0, ∀a ∈ N and 2 N −1 Θ aa + Θ ab + Θ ba < 0, ∀a, b ∈ N , a = b [39]. It can be noted that, if the value of N is large, these three methods may lead to relatively conservative results. Therefore, inspired by [43], Lemma 2 is introduced to deal with the double summation inequality containing fuzzy weighting function terms. Although the calculation burden may increase with regard to the form of (15), the method presented here may be beneficial for obtaining conditions with less conservatism.

III. MAIN RESULTS
In this section, the attention is focused on deriving sufficient conditions which can guarantee the mean-square exponential stability and H ∞ performance of the resulting synchronization error system (10). Then, the concrete form of the desired nonfragile fuzzy controller gains can be obtained based on the established criteria.

A. Stabilization and Performance Analysis
Before presenting further, some notations are given: Besides, we denote M (k) max{f s + n|k ≥ k fs+n }, which signifies the mark of the nearest switching from k.
Thus, from Definitions 2&3, one can conclude that system (10) is MSES with a prescribed H ∞ performance indexσ.

B. Controller Design
In the rest part of this section, by virtue of proper matrix processing methods, a non-fragile fuzzy controller is constructed based on Theorem 1.
Theorem 2: Given scalars α ∈ (0, 1), β ∈ (1, ∞), ς ∈ [0, 1] and σ > 0, if there exist symmetrical positive definite [1,6] and positive scalars bi , b ∈ N , i ∈ M, such that for ∀a, b ∈ N , i ∈ M, and θ, δ 1 , δ 2 ∈ H, (20) and the following inequalities hold where This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2020.3012761, IEEE Transactions on Fuzzy Systems Then, the synchronization error system (10) is MSES with a prescribed H ∞ performance indexσ. Furthermore, the expected controller gains of (9) can be given as

Proof 3: The Lyapunov function is constructed as
from which one can get which means condition (17) is satisfied. For convenience, we consider that ϑ(k) = i, λ(k M (k) ) = θ, i ∈ M and θ ∈ H.
For inequality (25), it can be inferred from Schur complement that Similarly, (26) is equivalent tõ Then, by virtue of Lemma 2, it can be elicited from (30) and which combining with Lemma 1 and (8) means i and Schur complement, one can get By virtue of (10) and (29), it can be calculated from (34) that Furthermore, it can be inferred from (27) that Therefore, for any i ∈ M, and δ 1 , δ 2 ∈ H, one has which implies (19) holds. In addition, it can be inferred from (26) Then, J i < 0 can be derived from Q θ i > 0. Thus, J i is a non-singular matrix and the expected controller gains can be calculated by (28). This completes the proof.
Remark 4: The conditions derived in Theorem 2 have certain conservatism. This mainly stems from the following aspects: i) The Lyapunov function constructed does not cover all characteristic information of the system, and some effective information may not be fully utilized. One of the possible improvements is the introduction of fuzzy Lyapunov function. ii) During the processing of gain uncertainty and the inverse of unknown matrices such as (Q θ i ) −1 , some inequalities have been utilized. iii) Inequality scaling is introduced to eliminate fuzzy weighting functions in the condition. As illustrated in Remark 3, the inequality scaling conditions employed in Lemma 2 may be conducive to reducing conservatism. However, the computational complexity, especially the dimension and number of matrices, will increase rapidly with the increase of N . Besides, it can be observed from (25)- (27) that the computation burden is also raising with the increase of jump modes and the PDT switching rules. Therefore, it is of great significance to find a suitable strategy that can effectively lower the computational complexity while reducing the conservatism of the obtained results.
Remark 5: In order to get conditions in terms of parameterindependent linear matrix inequality, the fuzzy weighting function in (32) should be disposed of. It can be noted from Remark 3 thatΘ iθ ab < 0 (∀a, b ∈ N ) utilized in [18] can also ensure the establishment of (32), which means the meansquare exponential stability and H ∞ performance of system (10) can be also obtained under (20), (27) and the following inequality Obviously, the computational complexity of (35) may be less than (25) and (26). However, this method will inevitably bring about great conservatism. To show the superiority of the utilized method of the paper in deriving conditions with less conservatism, some comparison results will be provided in the numerical example part.

IV. ILLUSTRATIVE EXAMPLE
In this section, a Lorenz chaotic system with jumping parameters is employed to demonstrate the effectiveness of the proposed synchronization control scheme. The considered chaotic system modified from [46] is described as follows: 2} is a jumping parameter governed by a Markov chain with piecewise constant TPs. The variation of TPs is considered to be subject to the PDT switching rule. It is well known that the value of b i has great influence on the chaos phenomenon of the system. Without loss of generality, we consider b 1 = 46 and b 2 = 32. Then, based on the T-S fuzzy theory (consider two fuzzy rules) and Euler's discretization approach (sampling timeT = 0.01s), together with the consideration of the output of the master system, one can obtain the following discrete-time fuzzy master system: with the fuzzy weighting function chosen as  with ν = 1 and the occurrence expectation being ς = 0.8.
For Markov chain, the TPM is given as: In addition, the parameters relevant to PDT switching regularity are provided as T P = 8, τ P = 6, α = 0.9, β = 1.1 Then, under the prescribed performance indexσ = 2.8930, the following controller gains can be calculated by virtue of Theorem 2. Given a set of PDT switching sequences, based on which the evolution of Markov chain can be derived with the aid of Algorithm 1, as presented in Fig. 3, and a set of Bernoulli sequences relevant to ς(k) are also given there (For additional clarity and  and the state responses of the synchronization error system are provided in Fig. 6. It can be observed that the synchronization error curves are divergent when the controller gains are set as zeros, while converges to zero when the controller works, which indicates the validity of the developed design method. Furthermore, under zero-initial conditions, the H ∞ performance is examined as follows: In what follows, the influence of the stochastic variable ς(k), Lyapunov variation rates α and β on the performance of the fuzzy chaotic system is investigated. The optimal performance index corresponding to different parameters is denoted as σ min . We set ν = 10 and other parameters are the same as above. Then, by adjusting the value of ς, α and β, the corresponding value ofσ min can be calculated, as listed in Table I and Fig. 7. Particularly, the optimal performance index obtained on account of Lemma 2 in this paper (corresponding to conditions presented in Theorem 2) andΘ iθ ab < 0 utilized in [18] (corresponding to conditions presented in Remark 5) are denoted as Case I and Case II, respectively. Then, it can be observed from Table I that the value ofσ min grows larger with the increase of ς, which means the occurrence of controller gain variation may have adverse effects on system performance. Besides, compared with the method of disposing fuzzy weighting function in [18], the processing method in this paper can get smallerσ min . Thus, the method employed in this paper may be beneficial in deriving conditions with less conservatism. Moreover, Fig. 7 indicates that the selection of different pair (α, β) will also affect the performance index of the synchronization error system. Thus, choosing proper Lyapunov function variation rate to ensure the prescribed performance index of the investigated system is significant.

V. CONCLUSION
The non-fragile H ∞ controller design issue for a class of discrete-time nonlinear hybrid chaotic systems has been addressed in this paper, where the random occurrence of gain variation has been described by a set of Bernoulli-distributed white sequences. Moreover, the stochastic jump of system parameters has been supposed to be described by a Markov chain with time-varying TPM subject to a deterministic switching mechanism, i.e., the PDT switching. The nonlinearity of the systems has been disposed by the T-S fuzzy model for which a novel disposing method for the double summation inequality containing fuzzy weighting function terms has been introduced to derive criteria with less conservatism. Furthermore, an applicable synchronization controller which can ensure the mean-square exponential stability and H ∞ performance of the fuzzy chaotic MJSs with PDT switched TPs has been constructed via Lyapunov stability theory. Finally, the Lorenz chaotic system with jumping parameters has been provided to illustrate the validity of the proposed control scheme. An interesting future work is to extend the proposed scheme to the adaptive tracking control of the interval type-2 T-S fuzzy switched model. Furthermore, constructing more suitable Lyapunov functions that are beneficial for deriving less conservative conditions is also a meaningful issue deserved to be further explored.
Jing Wang received received the Ph.D. degree in power system and automation from Hohai University, Nanjing, China, in 2019. She is currently an Associate Professor with the School of Electrical and Information Engineering, Anhui University of Technology, China. Her current research interests include Markov jump nonlinear systems, singularly perturbed systems, power systems, nonlinear control.