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

Authors: Tao Wu; Jinde Cao; Lianglin Xiong; Haiyang Zhang
Journal: Complexity
 (2019)

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## Abstract

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.


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## Body

## 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 [5–9].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 [10–16]. 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 [18–27]. 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 [18–27] 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 [29–31]. 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 [32–34] 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 [35–39] 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 Section2. 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, 20–27].(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 LKFV3xt,rt is constructed, which is firstly presented for the fuzzy sampled-data semi-Markov chaotic systems.(iv)
Compared with the existing stabilization conditions in [21, 24–26], 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 vectorℛn: n-dimensional Euclidean spaceℛn×m: set of all n×m real matricesP>0: matrix P is symmetric and positive definite∗: symmetric terms in a symmetric matrixIn: identity matrix0n×m: zero matricesdiag⋯: a block-diagonal matrixsymX: X+XTλmaxG (λminG): the largest (smallest) eigenvalue of G⋅: Euclidean norm for a given vectorΩ,ℱ,P: complete probability space with a natural filtration ℱtt≥0 satisfying the usual conditions (i.e., the natural filtration contains all P-null sets and is right continuous)E⋅: expectation function in regard to the given probability measure PConsider the semi-Markov chaotic system expressed by the following differential equation:(1)x˙t=gxt,ut,rt,rt,where xt∈ℛn denotes the state vector, ut,rt∈ℛm denotes the control input vector, and g⋅ denotes a known nonlinear continuous function and satisfies g0,0,rt=0. rt,t≥0 denotes the right-continuous semi-Markov process on the probability space Ω,F,P and takes values in a finite state set S=1,2,…,s with the transition rate matrix Πh=πpqhs×s given by(2)Prrt+h=qrt=p=πpqhh+oh,p≠q,1+πpphh+oh,p=q,where h>0 and limh⟶0oh/h=0; the transition rates πpqh are positive, if p≠q, while πpph=−∑q=1,q≠psπpqh for each mode p∈S.Actually, not all the transition ratesπpqh are easy to accurately estimate in the jumping process; most of them always contain the following two assumptions: (1) πpqh is totally uncharted and (2) πpqh is not exactly known, but upper and lower bounded. Based on the second condition, we can further define πpqh∈π¯pq,π¯pq, in which π¯pq and π¯pq are known real constants to represent the lower and upper bounds of πpqh, respectively. Furthermore, we can denote that πpqh≜πpq+Δπpqh, in which πpq=1/2π¯pq+π¯pq and Δπpqh≤λpq with λpq=1/2π¯pq−π¯pq. Thus, the time-varying transition rate matrix Πh with s jumping modes may be described as(3)Πh=π11+Δπ11h∇∇⋯π1s+Δπ1sh∇∇π23+Δπ23h⋯π2s+Δπ2sh⋮⋮⋮⋱⋮∇πs2+Δπs2h∇⋯∇,where ∇ is unknown time-varying transition rates. For convenience, ∀p∈S, the set Gp denotes Gp,k∪Gp,uk with(4)Gp,k≜q:πpqcan be determined forq∈S,Gp,uk≜q:πpqis unknown forq∈S.Moreover, ifGp,k≠∅ and Gp,uk≠∅, we can obtain the following:(5)Gp,k≜kp,1,kp,2,…,kp,v,1≤v<s.By using the following IF-THEN rules, system (1) can be developed into the following systems: Plant rule i: IF θ1t is α1i,rt and … and θlt is αli,rt, THEN(6)x˙t=Ai,rtxt+ℬi,rtut,rt,where i∈Θ=1,2,…,w, θ1t,…,θlt represent premise variables, α1i,rt,…,αli,rt represent the fuzzy sets, Ai,rt∈ℛn×n and ℬi,rt∈ℛn×m represent the known constant matrices with suitable dimensions, and w shows the number of fuzzy rules.By using the fuzzy blending, the fuzzy semi-Markov jump system (6) can be rewritten as follows:(7)x˙t=∑i=1wωi,rtθtAi,rtxt+ℬi,rtut,rt,where θt=θ1t,…,θltT and ωi,rtθt is the membership function satisfying(8)ωi,rtθt=∏j=1lαji,rtθjt∑n=1w∏j=1lαjn,rtθjt,in which αji,rtθjt stands for the grade of membership of θjt in αji,rt. For all t>0, we have ωi,rtθt≥0 and ∑i=1wωi,rtθt=1.Throughout this paper, the control signal is supposed to be generated by using a ZOH function with a series of holding moments0=t0<t1<⋯<tk<⋯. Accordingly, the fuzzy sampled-data controllers can be easily given as follows.Controller rulej: IF θ1t is α1i,rt and … and θlt is αli,rt, THEN(9)ut,rt=Kj,rtxt−τt,tk≤t<tk+1,with j=1,2,…,w, τt=t−tk, and Kj,rtj∈Θ denotes the controller gain matrices to be computed. The sampling interval τk meets the condition 0≤τt≤τk=tk+1−tk≤τ, where τ is the upper bound of the sampling periods.Hence, the entire fuzzy sampled-data controller can be represented as follows:(10)ut,rt=∑j=1wωj,rtθtkKj,rtxt−τt.For the sake of convenience,rt is noted as rt=pp∈S hereinafter. Introducing (10) into (7), the fuzzy system (7) is further developed into(11)x˙t=∑i=1w∑j=1wωi,pθtωj,pθtkAi,pxt+ℬi,pKj,pxt−τt,tk≤t<tk+1.To clearly give the main results, we firstly provide the following definition and lemmas.Definition 1.
System (11) is exponentially stable if scalars β>0 and δ>0 satisfy(12)Ext≤δe−βtsup−τ≤ψ≤0Exψ,x˙ψ,where β and δ are defined as the decay rate and decay coefficient, respectively.Lemma 1.
The system (11) holds the following conclusion:(13)xt2≤εxtk2,tk≤t<tk+1,with ε1=maxi∈Θ,p∈Sℬi,p, ε2=maxj∈Θ,p∈SKj,p, ε3=maxi∈Θ,p∈SAi,p, and ε=31+τ2ε1ε2e3τ2ε3.Proof.
One can obtain for∀t∈tk,tk+1 from (11) that(14)xt≤xtk+∫tkt∑i=1wωi,pθϵAi,pxϵdϵ+∫tkt∑i=1w∑j=1wωi,pθϵωj,pθtkℬi,pKj,pxtkdϵ.
Using the Cauchy–Schwarz inequality, we can get from (14) that(15)xt2≤3xtk2+3∫tkt∑i=1wωi,pθϵAi,pxϵdϵ2+3∫tkt∑i=1w∑j=1wωi,pθϵωj,pθtkℬi,pKj,pxtkdϵ2≤3xtk2+3τ∫tkt∑i=1wωi,pθϵAi,pxϵ2dϵ+3τ∫tkt∑i=1w∑j=1wωi,pθϵωj,pθtkℬi,pKj,pxtk2dϵ≤3xtk2+3τε3∫tktxε2dϵ+3τε1ε2∫tktxtk2dϵ≤31+τ2ε1ε2xtk2+3τε3∫tktxε2dϵ.
Utilizing 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 V>0, any differentiable function x:c,d⟶ℛn satisfies the following inequality:(16)∫cdx˙TsVx˙sds≥ϒ1TVϒ1+3ϒ2TVϒ2d−c,with(17)ϒ1=xd−xc,ϒ2=xd+xc−2d−c∫cdxsds.


## 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 Kj,p. Second, by designing the control gain matrices Kj,p, 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:(18)z1t=τtxt,z2t=τtx˙t,z3t=τt∫tktxsds,vt=1t−tk∫tktxsds,ξt=xTt,x˙Tt,xTtk,xTt−τ,vTt,1τ−t+tk∫t−τtkxTsds,z1Tt,z2Tt,z3TtT,ηt=xTt−xTtk,z1Tt,z3TtT,ei=0n×i−1nIn0n×9−in,Ψ=e1T,e2T,e3T,e4T,e5T,e6T,e7T,e8T,e9TT.Theorem 1.
Given scalarsτ and β, the fuzzy system (11) with control gain matrices Kj,p is exponentially stable; if there exist positive definite matrices Pp, Q, ℛ, Tpq, and Zpq, any matrices S11, S12, S22, ℋ, X, Y, and ℱmm=1,2 with appropriate dimensions, such that the following conditions hold for all i, j∈Θ, and p∈S.Case 1.
p
∈
G
p
,
k, ∀l∈Gp,uk, Gp,k≜kp,1,kp,2,…,kp,v1,(19)ℛ¯=ℛ0S11S12∗3ℛS12TS22∗∗ℛ0∗∗∗3ℛ≥0,(20)∑11Γ1∗−Λ1<0,(21)∑21Γ1∗−Λ1<0,where(22)∑11=Φτt=0+e1TTpqe1,∑21=Φτt=τ+e1TTpqe1,Γ1=e1TPkp,1−Pl,e1TPkp,2−Pl,…,e1TPkp,v1−Pl,Λ1=diagTp,kp,1,Tp,kp,2,…,Tp,kp,v1,Tpq=∑q∈Gp,kλpq24Tpq+πpqPq−Pl,with(23)Φτt=Φ0τt+Φ1τt,Φ0τt=symΨTXτte1−e7τte2−e8τ2te5−e9+syme1Tℱ1T+e2Tℱ2TAi,pe1+ℬi,pKj,pe3−e2,Φ1τt=syme1TPpe2+2βe1TPpe1+e1TQe1−e−2βτe4TQe4+τ2e2Tℛe2−e−2βτΠ1Tℛ¯Π1−symΠ2TℋΠ2+τ−τtsym2βΠ2TℋΠ2+τ−τtsyme2e1+e8τte5+e7TℋΠ2+2βτ−τtτte5TYe5+τ−2τte5TYe5+symτ−τte5TYe1−e5,Π1=e1−e3e1+e3−2e5e3−e4e3+e4−2e6,Π2=e1−e3e7e9.Case 2.
p
∈
G
p
,
u
k, ∀l∈Gp,uk, Gp,k≜kp,1,kp,2,…,kp,v2, l≠p,(24)ℛ¯=ℛ0S11S12∗3ℛS12TS22∗∗ℛ0∗∗∗3ℛ≥0,(25)Pp−Pl≥0,(26)∑12Γ2∗−Λ2<0,(27)∑22Γ2∗−Λ2<0,where(28)∑12=Φτt=0+e1TZpqe1,∑22=Φτt=τ+e1TZpqe1,Γ2=e1TPkp,1−Pl,e1TPkp,2−Pl,…,e1TPkp,v2−Pl,Λ2=diagZp,kp,1,Zp,kp,2,…,Zp,kp,v2,Zpq=∑q∈Gp,kλpq24Zpq+πpqPq−Pl.Proof.
Noticing the fact thatτtxt−z1t=0, τtx˙t−z2t=0, and τt∫tktxsds−z3t=0, for any matrix X with suitable dimensions, the following zero equality holds:(29)2e2βtτtxt−z1tτtx˙t−z2tτt∫tktxsds−z3tTXTξt=0.
Consider an LKF candidate as(30)Vxt,rt=∑m=14Vmxt,rt,t∈tk,tk+1,with(31)V1xt,rt=e2βtxTtPrtxt,V2xt,rt=∫t−τte2βsxTsQxsds+τ∫t−τt∫θte2βsx˙Tsℛx˙sdsdθ,V3xt,rt=2e2βtτ−τtηTtℋηt,V4xt,rt=e2βtτ−τtτtvTtYvt.
Letℒ be the weak infinitesimal operator for the semi-Markov process acting on Vxt,rt as follows:(32)ℒVxt,rt=limι⟶0EVxt+ι,rt+ιxt,rt−Vxt,rtι,where ι>0 is a scalar. Then, similar to the computation in [35, 36], we have(33)ℒV1xt,rt=e2βt2βxTtPpxt+2xTtPpx˙t+xTt∑q=1sπpqhPqxt,(34)ℒV2xt,rt≤e2βtxTtQxt−e−2βτxTt−τQxt−τ+τ2x˙Ttℛx˙t−τe−2βτ∫t−τtx˙Tsℛx˙sds,(35)ℒV3xt,rt=e2βt4βτ−τtηTtℋηt−2ηTtℋηt+2τ−τtx˙txt+z2t∫tktxsds+z1tTℋηt,(36)ℒV4xt,rt=e2βt2βτ−τtτtvTtYvt+2τ−τtxt−vtTYvt+τ−2τtvTtYvt.
Sinceℛ¯>0, by Lemma 2 and the reciprocally convex approach in [41], we have(37)−τe−2βτ∫t−τtx˙Tsℛx˙sds≤−e−2βτξTtΠ1Tℛ¯Π1ξt.
According to the fuzzy system (11), for any appropriate dimensional matrices ℱmm=1,2, the following zero equality holds:(38)0=2e2βt∑i=1w∑j=1wωi,pθtωj,pθtkϒt−x˙t+Ai,pxt+ℬi,pKj,pxt−τt,with ϒt=xTtℱ1T+x˙Ttℱ2T.
Merging (29) and (33)–(38), we have for tk≤t<tk+1 that(39)EℒVxt,rt≤Ee2βtWt,where Wt=∑i=1w∑j=1wωi,pθtωj,pθtkξTtΞτtξt and Ξτt=Φτt+e1T∑q=1sπpqhPqe1.
On the contrary, sinceπpqh is generally uncertain transition rates in this paper, the proof needs further derivation by the following two cases (i.e., p∈Gp,k and p∈Gp,uk).
Whenp∈Gp,k, let λp,k≜∑q∈Gp,kπpqh. According to Gp,uk≠Ø, we have λp,k<0. Then, the formula ∑q=1sπpqhPq can be represented as(40)∑q=1sπpqhPq=∑q∈Gp,kπpqhPq+∑q∈Gp,ukπpqhPq=∑q∈Gp,kπpqhPq−λp,k∑q∈Gp,ukπpqh−λp,kPq.
It is clear that0≤πpqh/−λp,k≤1q∈Gp,uk and ∑q∈Gp,ukπpqh/−λp,k=1. Thus, for ∀l∈Gp,uk, one can obtain(41)Ξτt=∑q∈Gp,ukπpqh−λp,kΦτt+e1T∑q∈Gp,kπpqhPq−Ple1.
Accordingly, for0≤πpqh≤−λp,k, Ξτt<0 is equivalent to(42)Φτt+e1T∑q∈Gp,kπpqhPq−Ple1<0.
Then, we can find from the above inequality (42) that(43)∑q∈Gp,kπpqhPq−Pl=∑q∈Gp,kπpqPq−Pl+∑q∈Gp,kΔπpqhPq−Pl.
From the inequalitycM+MT≤c2S+MS−1MT [42], for any Tpq>0, we have(44)∑q∈Gp,kΔπpqhPq−Pl=∑q∈Gp,k12ΔπpqhPq−Pl+Pq−Pl≤∑q∈Gp,kλpq22Tpq+Pq−PlTpq−1Pq−Pl.
Combining (40)–(44) and using Schur complement, it can be concluded (19)–(21) guarantee Ξτt<0 for all p∈Gp,k.
Whenp∈Gp,uk, define λp,k≜∑q∈Gp,kπpqh. Because Gp,k≠Ø, we have λp,k>0. So, the expression ∑q=1sπpqhPq can be rewritten as(45)∑q=1sπpqhPq=∑q∈Gp,kπpqhPq+πpphPp+∑q∈Gp,uk,q≠pπpqhPq=∑q∈Gp,kπpqhPq+πpphPp−πpph+λp,k∑q∈Gp,uk,q≠pπpqhPq−πpph−λp,k.
It is obvious that0≤πpqh/−πpph−λp,k≤1q∈Gp,uk and ∑q∈Gp,uk,q≠pπpqh/−πpph−λp,k=1. Consequently, for ∀l∈Gp,uk, l≠p, we get(46)Ξτt=∑q∈Gp,uk,q≠pπpqh−πpph−λp,kΦτt+e1TπpphPp−Ple1+e1T∑q∈Gp,kπpqhPq−Ple1.
Therefore, for0≤πpqh≤−πpph−λp,k, the equivalent form of Ξτt<0 can be described as follows:(47)Φτt+e1TπpphPp−Ple1+e1T∑q∈Gp,kπpqhPq−Ple1<0.
Sinceπpph<0, the inequality (47) holds if we have(48)Pp−Pl≥0,Φτt+e1T∑q∈Gp,kπpqhPq−Ple1<0.
In addition, similar to (43) and (44), for any Zpq>0, we obtain(49)∑q∈Gp,kπpqhPq−Pl≤∑q∈Gp,kπpqPq−Pl+∑q∈Gp,kλpq22Zpq+Pq−PlZpq−1Pq−Pl.
From (45)–(49), by Schur complement and inequalities (24)–(27), it is known that Ξτt<0 for all p∈Gp,uk. In summary, based on the above discussions, one can obtain(50)EℒVxt,rt≤0,t∈tk,tk+1.
Following this, we illustrateVxt,rt is continuous at sampling times. By (30), we can obtain(51)limt⟶tkVmxt,rt=Vmxtk,rtk≥0,m=1,2,(52)limt⟶tk−Vnxt,rt=limt⟶tk+Vnxt,rt=Vnxtk,rtk=0,n=3,4.
From (51) and (52), one can get(53)limt⟶tkVxt,rt=Vxtk,rtk≥0.
Hence,Vxt,rt is continuous at sampling times. Besides, from (50)–(53), since ξt≠0, we have for t∈tk,tk+1 that(54)Vxt,rt>Vxtk+1,rtk+1>0,which implies that Vxt,rt>0 at the sampling times.
Combining Dynkin’s formula with (50), we have(55)EVxt,rt≤EVx0,r0.
Moreover, it is followed from (30) that(56)EVx0,r0=∑m=14EVmx0,r0≤maxp∈SλmaxPpEx02+1−e−2βτ2βλmaxQsup−τ≤ψ≤0Exψ2+τe−2βτ+2βτ−14β2λmaxℛsup−τ≤ψ≤0Ex˙ψ2≤ℳEsup−τ≤ψ≤0xψ,x˙ψ2,where(57)ℳ=maxp∈SλmaxPp+1−e−2βτ2βλmaxQ+τe−2βτ+2βτ−14β2λmaxℛ.
On the contrary, on the basis of Lemma 1 and (55), for ∀t∈tk,tk+1, one can derive(58)Ext2≤εExtk2=εe2βtkminp∈SλminPpe2βtkminp∈SλminPpExtk2≤εminp∈SλminPpe2βtkEe2βtkxTtkPrtkxtk=εminp∈SλminPpe2βtkEV1xtk,rtk≤εminp∈SλminPpe2βtkEVxtk,rtk≤εminp∈SλminPpe−2βtkEVx0,r0=εminp∈SλminPpe−2βte2βt−tkEVx0,r0≤εe2βτminp∈SλminPpe−2βtEVx0,r0,which implies that(59)Ext2≤εe2βτℳe−2βtminp∈SλminPpEsup−τ≤ψ≤0xψ,x˙ψ2.
Therefore, we have(60)Ext≤Ue−βtsup−τ≤ψ≤0Exψ,x˙ψ,with U=εe2βτℳ/minp∈SλminPp.
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 τt. Thus, in order to further solve this difficulty, we take τtxt, τtx˙t, and τt∫tktxsds into the intermediate vector ξt such that the cubic function of τt is successfully presented in the novel additional term V3xt,rt. Meanwhile, the equality (29) is used to increase the combination among the vectors of xt, x˙t, xtk, 1/t−tk∫tktxsds, τtxt, τtx˙t, and τt∫tktxsds. It is also worth noting that Ξτt in Theorem 1 is nonlinear matrix inequalities since it contains the quadratic function of the sampling input delay τt. 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 scalarsτ, β, and μ, the system (11) can be exponentially stabilized; if there exist positive definite matrices P^p, Q^, ℛ^, T^pq, and Z^pq, any matrices S^11, S^12, S^22, ℋ^, X^, Y^, ℱ, and K^j,p with appropriate dimensions, such that the following predonditions hold for all i, j∈Θ, p∈S.Case 1.
p
∈
G
p
,
k, ∀l∈Gp,uk, Gp,k≜kp,1,kp,2,…,kp,v1,(61)ℛ¯^=diagℛ^,3ℛ^S^∗diagℛ^,3ℛ^≥0,(62)Σ^11Γ^1∗−Λ^1<0,(63)Σ^21Γ^1∗−Λ^1<0,with(64)S^=S^11S^12∗S^22,Σ^11=Φ^τt=0+e1TT^pqe1,Σ^21=Φ^τt=τ+e1TT^pqe1,Γ^1=e1TP^kp,1−P^l,e1TP^kp,2−P^l,…,e1TP^kp,v1−P^l,Λ^1=diagT^p,kp,1,T^p,kp,2,…,T^p,kp,v1,T^pq=∑q∈Gp,kλpq24T^pq+πpqP^q−P^l,Φ^τt=Φ^0τt+Φ^1τt,Φ^0τt=symΨTX^τte1−e7τte2−e8τ2te5−e9+symμe1T+e2TAi,pℱe1+ℬi,pK^j,pe3−ℱe2,Φ^1τt=syme1TP^pe2+2βe1TP^pe1+e1TQ^e1−e−2βτe4TQ^e4+τ2e2Tℛ^e2−e−2βτΠ1Tℛ¯^Π1+2βτ−τtτte5TY^e5+τ−2τte5TY^e5+symτ−τte5TY^e1−e5−symΠ2Tℋ^Π2+τ−τtsym2βΠ2Tℋ^Π2+τ−τtsyme2e1+e8τte5+e7Tℋ^Π2,Π1=e1−e3e1+e3−2e5e3−e4e3+e4−2e6,Π2=e1−e3e7e9.Case 2.
p
∈
G
p
,
u
k, ∀l∈Gp,uk, Gp,k≜kp,1,kp,2,…,kp,v2, l≠p,(65)ℛ¯^=diagℛ^,3ℛ^S^∗diagℛ^,3ℛ^≥0,(66)P^p−P^l≥0,(67)Σ^12Γ^2∗−Λ^2<0,(68)Σ^22Γ^2∗−Λ^2<0,with(69)Σ^12=Φ^τt=0+e1TZ^pqe1,Σ^22=Φ^τt=τ+e1TZ^pqe1,Γ^2=e1TP^kp,1−P^l,e1TP^kp,2−P^l,…,e1TP^kp,v2−P^l,Λ^2=diagZ^p,kp,1,Z^p,kp,2,…,Z^p,kp,v2,Z^pq=∑q∈Gp,kλpq24Z^pq+πpqP^q−P^l.
Moreover, the controller gain matrices in (10) are designed as(70)Kj,p=K^j,pℱ−1.Proof.
Defineℱ2=ℱ−1, ℱ1=μℱ−1, K^j,p=Kj,pℱ, Jn=diagℱ,ℱ,…,ℱ︸n, P^p=ℱTPpℱ, Q^=ℱTQℱ, ℛ^=ℱTℛℱ, T^pq=ℱTTpqℱ, Z^pq=ℱTZpqℱ, X^=J9TXJ3, ℋ^=J3TℋJ3, Y^=ℱTQℱ, S^11=ℱTS11ℱ, S^12=ℱTS12ℱ, and S^22=ℱTS22ℱ. By pre- and postmultiplying J4T and J4 with (19) and (24), the inequalities (61) and (65) are easily obtained. By pre- and postmultiplying Jv1+9T and Jv1+9 with (20) and (21), the inequalities (62) and (63) hold. By pre- and postmultiplying J1T and J1 with (25), the inequality (66) can be easily obtained. By pre- and postmultiplying Jv2+9T and Jv2+9 with (26) and (27), the inequalities (67) and (68) hold. This proves Theorem 2.
Specially, when there are no semi-Markov jump parameters considered, the semi-Markov processrt,t≥0 only takes account of S=1, and the fuzzy sampled-data controller (10) and the system (11) are rewritten as(71)ut=∑j=1wωjθtkKjxt−τt,t∈tk,tk+1,(72)x˙t=∑i=1w∑j=1wωiθtωjθtkAixt+ℬiKjxt−τt,t∈tk,tk+1,which have been extensively discussed in [18–27]. Choosing Prt=P in (30), one can obtain the following corollary from Theorem 2.Corollary 1.
For given scalarsτ, β, and μ, the chaotic system (72) is exponentially stabilized; if there exist P^>0, Q^>0, and ℛ^>0, any matrices S^11, S^12, S^22, ℋ^, X^, Y^, ℱ, and K^j with appropriate dimensions, such that the following conditions hold for all i, j∈Θ.(73)ℛ¯^=diagℛ^,3ℛ^S^∗diagℛ^,3ℛ^≥0,(74)Φ¯τt=0<0,(75)Φ¯τt=τ<0,where(76)Φ¯τt=Φ¯0τt+Φ^1τt,Φ¯0τt=symΨTX^τte1−e7τte2−e8τ2te5−e9+symμe1T+e2TAiℱe1+ℬiK^je3−ℱe2.
The controller gain matrices in (71) are provided as Kj=K^jℱ−1.Remark 3.
It should be pointed out that Theorem 2 and Corollary 1 involve the regulation parameterμ. Accordingly, to find a suitable value of this parameter is a very importance issue. Taking Theorem 2 as an example, in line with the grid search algorithm [22, 44], we can look for the optimal tuning parameter μ in Theorem 2. Similarly, this algorithm can also be used for Corollary 1 to find an optimal value of parameter μ.


## 4. Numerical Simulation

In this section, we will give two conventional models to show the merits and effectiveness of our results obtained in this paper.Example 1.
Inspect the dynamic model of the Lorenz system [21, 24, 27] as follows:(77)x˙1t=−a1x1t+a1x2t+u1t,x˙2t=a3x1t−x2t−x1tx3t,x˙3t=x1tx2t−a2x3t.
Assuming thatx1t∈−a4,a4, the system (77) can be expressed as the fuzzy system (11) with(78)A1,p=−a1a10a3−1−a40a4−a2,A2,p=−a1a10a3−1a40a4−a2,ℬ1,p=ℬ2,p=100T,and the membership functions are(79)ω1,px1t=a4+x1t2a4,ω2,px1t=1−ω1,px1t.
a
1, a2, a3, and a4 follow three jumping modes with the values provided in Table 1, and the generally uncertain transition rate matrix for unlike modes is given as follows:(80)Πh=−1.5+Δπ11h∇∇∇∇0.8+Δπ23h0.5+Δπ31h∇−1+Δπ33h,with Δπpqh≤λpq=⟵0.1∗πpq.
Now, we start to check the effectiveness and advantages of Theorem 2 from three aspects. At first, ifKj,p=0, the initial conditions are set as x0=12,15,−13T regarding mode 1, x0=2,−6,−2T regarding mode 2, and x0=−3,6,9T regarding mode 3, and the relevant chaotic actions of (77) are shown in Figures 1–3, respectively.
Then, ifKj,p≠0, choosing the adjustable parameters μ=90 and β=0 and resolving the LMIs (61)–(68) in Theorem 2 by Matlab LMI Control Toolbox, the maximum value of the sampling interval τ=0.0537 can be obtained. Simultaneously, we can obtain the following state feedback gain matrices:(81)K1,1=−29.5405−17.36958.2551,K2,1=−30.1036−17.2315−8.1225,K1,2=−28.5657−17.11058.1691,K2,2=−29.9845−17.1972−8.1155,K1,3=−29.5317−17.25598.2034,K2,3=−30.0220−17.1328−8.0798.
With the above controller gain matrices and the initial conditionsx0=−3,6,9T and r0=1, we have the relevant numerical simulation, as shown in Figures 4–6. The semi-Markov jump process is illustrated in Figure 4. Figure 5 demonstrates the state response of the fuzzy semi-Markov jump chaotic system (11). Figure 6 shows the fuzzy sampled-data control input. It is easily seen from Figure 5 that the system (11) can be stabilized by the presented fuzzy controller (10), which verifies the feasibility of Theorem 2.
Third, whenrt=1, taking a1=10, a2=8/3, a3=28, a4=25, μ=90, and β=0, and by utilizing Matlab LMI Control Toolbox to calculate the LMIs (73)–(75), we can get the largest value τ=0.0729. Actually, in [21, 24, 25, 27], the largest sampling intervals are 0.0347, 0.0412, 0.0438, and 0.0450, respectively. And those results are listed in Table 2. It is clearly seen that our proposed approach gives importantly better results than those obtained in [21, 24, 25, 27]. Additionally, in comparison with the number of variables in [21, 24], the number of variables in our presented approach in Corollary 1 is much smaller. Hence, it is easy to see that our proposed method gives a less conservative and less computationally complex result than those obtained in [21, 24], which checks the availability of Remark 2 discussed.
When the largest sampling periodτ reaches 0.0729, the relevant controller gains are computed as(82)K1=−21.8910−17.58234.7085,K2=−21.8910−17.5823−4.7085.
Based on the above fuzzy sampled-data control gain matrices, the simulations are provided in Figures7 and 8 with the initial condition x0=−1.5,−1,2T. Figure 7 depicts the state trajectory of the system (72). Figure 8 shows the fuzzy sampled-data control input variables. From Figure 7, it is clearly seen that system (72) is exponentially stabilized, which shows the usefulness of our method provided in this paper.Table 1
Values ofa1, a2, a3, and a4 for different modes in example 1.
Modep
a
1
a
2
a
3
a
4
1
10
8/3
28
25
2
10.1
2.7
28.1
25.2
3
9.9
2.6
27.8
24.8Figure 1
Behavior of (77) with x0=12,15,−13T for mode 1.Figure 2
Curve of (77) with x0=2,−6,−2T for mode 2.Figure 3
Trajectory of (77) with x0=−3,6,9T for mode 3.Figure 4
Semi-Markov jump signal.Figure 5
State response of the fuzzy semi-Markov chaotic system (11) with control input.Figure 6
Fuzzy sampled-data control inputut.Table 2
Largest sampling intervalτ with different approaches.
Approach
τ
Number of variables
[25]
0.0347
24n2 + 3n + 2mn
[24]
0.0412
513.5n2 + 111.5n + 2mn
[21]
0.0438
48.5n2 + 8.5n + 2mn
[27]
0.0450
19.5n2 + 1.5n + 2mn
Corollary 1
0.0729
40.5n2 + 1.5n + 2mnFigure 7
State trajectory of the fuzzy chaotic system (72) with control input.Figure 8
Fuzzy sampled-data controllerut.Example 2.
Consider the system (72) with the following parameters [24–26]:(83)A1=−1010028−1−25025−83,A2=−1010028−125025−83,ℬ1=ℬ2=001001,and the fuzzy membership functions are(84)ω1x1t=121+x1t25,ω2x1t=1−ω1x1t.
In this example, similar to that in [24–26], we choose the same tuning parameter μ=1 and take β=0; the comparisons on the allowable maximum sampling period by different methods are shown in Table 3. From Table 3, we can observe that a larger sampling period has been obtained by our approach. Moreover, to verify the effectiveness of Remark 3 discussed, by specifying the range μ∈0,10 and increments Δμ=1 for μ and Δτ=0.0001 for τ and then by applying a similar algorithm in [26], we can find the optimal tuning parameter μ=8, and the corresponding allowable sampling period τ is 0.0689, which improves the former one 20.5%. As a result, finding an optimal tuning parameter is very important for us to obtain the maximum value of the sampling period τ. On the contrary, choosing τ=0.0572, and by utilizing Matlab LMI Control Toolbox to solve the conditions (73)–(75), the related controller gain matrices are obtained as follows:(85)K1=−17.7941−9.390224.1495−0.9946−23.2460−3.5656,K2=−17.7941−9.3902−24.14950.994623.2460−3.5656.
Under the controller (68) with above gain matrices, taking the initial condition x0=7,−4,−5T, the state path of system (72) is shown in Figure 9. The simulation result shows that the designed fuzzy sampled-data controller achieves exponential stabilization successfully with a larger sampling period than that in [24–26].Table 3
Maximum value of the sampling periodτ.
Method
τ
Degree of improvement (%)
Corollary 1
0.0572
[25]
0.0016
3475
[24]
0.0022
2500
[26]
0.0253
126.1Figure 9
State path of the fuzzy chaotic system (72) with control input.


## 5. Conclusion

The issue of stabilization for semi-Markov jump chaotic systems based on fuzzy sampled-data control has been considered in this paper. The semi-Markov jump has generally uncertain transition rates. A new augmented LKF with the cubic function of the sampling input delayτt has been constructed in which the available information with regard to the practical sampling mode is fully used. At the same time, we have also constructed a novel zero equality to make more cross terms of vector components in the resulting vector. In view of this, less conservative criteria with reduced computational complexity have been established to guarantee the stability and stabilization of the fuzzy sampled-data semi-Markov chaotic systems. Furthermore, according to the results obtained, the desired fuzzy sampled-data controller gain matrices have been given during larger sampling periods compared to the existing results. Finally, the validity and practicality of the derived results have been demonstrated by two interesting numerical examples. In the future, the employed results and methods can be extended to several cases including those studied extensively in the literature studies such as event-triggered sampling control [45, 46], quantized sampled-data control [47, 48], and asynchronous control [49, 50].



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