# Aperiodic Sampled-Data Control for Chaotic System Based on Takagi–Sugeno Fuzzy Model

Authors: Minjie Zheng; Shenhua Yang; Lina Li
Journal: Complexity
 (2021)

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## Abstract

This paper investigates the aperiodic sampled-data control for a chaotic system. Firstly, Takagi–Sugeno (T-S) fuzzy models for the chaotic systems are established. The lower and upper bounds of the sampling period are taken into consideration. Then, the criteria for mean square exponential stability analysis and aperiodic sampled-data controller synthesis are provided by means of linear matrix inequalities. And the real sampling patterns can be fully captured by constructing suitable Lyapunov functions. Finally, an illustrative example shows that the proposed method is effective to guarantee that the system’s states are stable with aperiodic sampled data.


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## Body

## 1. Introduction

Recently, the chaotic system has gradually become one of the hot topics in the field of nonlinear system. The chaotic system has wide application in several areas such as information processing, chemical reaction, power conversion, secure communication biological system, and other aspects. Thus, the control problem for the chaotic system has attracted considerable attention (see [1, 2]). Among these references, fuzzy control is an effective method to deal with chaotic systems, especially when the plant’s knowledge is incomplete or the action cannot be accurately controlled. So far, many scholars have focused on the fuzzy chaotic systems (see [3–8]), and many methods have been developed, such as robust control [9], adaptive control [10], pulse control [11], and sliding mode control [12].On the other hand, the chaotic system can be described with T-S fuzzy models (TSFM) [13–15], which can be analyzed by using mature linear system theory. Recently, considerable references have been reported about chaotic system with TSFM. In [16], the issue about asymptotic stability for the chaotic system with TSFM is studied, and an impulsive controller is designed. In [17], an adaptive synchronization method for the chaotic system with TSFM is proposed for solving the issue of parameter mismatch. In [18], the predictive control and synchronization for the chaotic system with TSFM is discussed. And the identical satellite systems are synchronized by the predictive control technique.In the past few years, the sampled-data system has become an important topic because modern control systems widely used the digital computers to control continuous-time systems (see [19–21]). Compared with the analog controller, the digital controller has better reliability, lower installation, and easier maintenance, which are the advantages of the systems. The outstanding feature of the systems is the coexistence of continuous signal and discrete signal, which is difficult to be analyzed and designed. Until now, the system has attracted much attention of scholars, and considerable results have been reported for that system (see [22–25]). Besides, several sampled-data control approaches such as lifting technology and input delay approach have been used in the systems including many real systems such as the near-space hypersonic vehicles, autonomous airships, unmanned marine vehicles, and so on. Recently, the fuzzy sampled-data chaotic system has attracted considerable attention [26]. In [27], by adaptive event-triggered scheme, the issue about fault-tolerant synchronization for chaotic system is discussed. In [28], a new Lyapunov–Krasovskii function (LKF) method is introduced for the chaotic systems with TSFM, and a sampled-data controller is produced to obtain a long sampling period. In [29], the exponential stabilization problem of the fuzzy semi-Markov chaotic system is discussed by establishing a new zero-value equation. In [30], the stochastic sampled-data controllers for the chaotic system are designed, and an improved LKF is constructed to fully exploit the sampling characteristics.It is noted that the sampling periods of the existing references for the chaotic system are ideally assumed to be constant. However, the sampling period is aperiodic due to the aging of sensors and the interference of noise environment. Hence, to consider nonperiodicity sampling for designing sampled-data controller is practically significant. Besides, in these papers, the lower bound of the sampling period is often considered to be 0 which will lead to considerable conservatism because the value of the variable period may change in a range. Therefore, the lower bound and upper bound of the sampling period should be both considered. Finally, LKF has room for improving to fully capture the real sampling patterns.Motivated by the above, in this paper, the issue about aperiodic fuzzy sampled-data control of the chaotic system is discussed. Firstly, TSFM is represented for chaotic systems. Then, both lower and upper bounds of the variable period are taken into consideration. In terms of LMI approach, Lyapunov theorem is involved for the stability analysis which can fully capture the sampling patterns. Then, the designed method of fuzzy sampled-data controller is introduced. Finally, a simulation of a chaotic system is conducted to verify the effectiveness of the given strategy.Notations: SymM represents M+MT. “∗” represents the symmetric term of a matrix.


## 2. Problem Formulation

Consider a chaotic system as follows:(1)x˙t=fxt,ut,where xt∈Rn denotes the state vector, ut∈Rm denotes the input vector, and fxt is a nonlinear function which satisfies f0,0=0. Based on the TSFM, system (1) can be described as follows.Mode Rule i: IFz1tis χi1, and ⋯znt is χin, THEN(2)x˙t=Aixt+Biut,i=1,2,…,r,where χi1,χi2,…,χin denote the fuzzy sets, r denotes the number of rules, z1t,z2t,…,znt represent premise variables, and Ai,Bi denote the appropriated dimensioned matrices. Similar to [9], fuzzy system (2) is given by(3)x˙t=∑i=1rμiztAixt+Biut,where(4)μizt=υizt∑i=1rυizt≥0,υizt=∏j=1nχijzjt,∑i=1rμizt=1,zt=z1t,z2t,…,znt,with χijzjt representing the membership grade of zjt.Assume that the state variables of chaotic systems are measured in0=t0<t1<t2<⋯<tk<⋯limk⟶∞tk=+∞, where tk,k=1,2,… is sampling instant. Moreover, the sampling period is aperiodic and satisfies(5)d1≤tk+1−tk=dk≤d2,∀k≥0,d1≥0,d2>0.The framework of the sampled-data chaotic system is given in Figure1.Figure 1
The framework of the sampled-data chaotic system.Then, based on parallel distributed compensation, the fuzzy sampled-data controller is designed.Controller Rule i: IFz1t is χi1, and …znt is χin, THEN(6)ut=Kixtk,tk≤t<tk+1,where K is a controller matrix. Then, the overall fuzzy model is represented as follows:(7)ut=∑j=1rμjztKjxtk,tk≤t<tk+1,k=0,1,2,….Substituting (7) into equation (3), we obtain(8)x˙t=∑i=1r∑j=1rμiztμjztAixt+BiKjxtk.The paper’s purpose is designing a fuzzy sampled-data controller (FSDC) to satisfy that(1)
System (8) is mean square exponentially stable.(2)
A longer sampling period is achieved.


## 3. Main Results

In this section, the sufficient stability criteria for system (8) are exhibited by establishing LKF firstly. Then, the FSDC will be provided to analyze the stability criteria.Theorem 1.
For scalesε1,ε2 and d2>d1≥0, system (8) is mean square exponentially stable, if there exist matrices Y11,Y22,M,P>0,Q11Q12∗Q22>0, such that(9)Ψ˜1ijd=Ψ1ij+de−2λd2I3Q22I3T+dϒ1<0,(10)Ψ˜2ijd=Ψ1ij−de−2λd2Ψ3∗−de−2λd2Q11<0,d∈d1,d2,(11)Θ=P+d2Y11+Y11T−d2Y11+Y22∗d2Y22+Y22T>0,where(12)Ψ1ij=SymI1PI2T+λI1PI1T−I1Y11I1T+I1Y11I3T+I1Y22I3T−ε1I1MAiI1T−I1A¯iTMTI2T−ε1I1MB¯iθKjI3T+I2MI2T−I3Y22I3T+ε1I1MI2T−ε1I1A¯iTMTI3T−I2MB¯iθKjI3T+ε2I2MTI3T−ε1I3MB¯iθKjI3T+I−de−2λd2I3Q22I3T,ϒ1=SymI1Y11+Y11TI2T−I2Y11+Y22I3T+I2Q11I2T+I2Q12I3T+λI1Y11+Y11TI1T−λI1Y11+Y22I3T+I3Q22I3T+λI3Y22+Y22TI3T,Ψ3=00Q12T.Proof.
The novel LKF is proposed:(13)Vt=∑i=13Vit,t∈tk,tk+1,V1t=xtTPxt,V2t=h2t∫tkte2λs−tx˙sxtkTQ11Q12∗Q22x˙sxtkds,V3t=h2txtxtkTY11+Y11T−Y11−Y22∗Y22+Y22Txtxtk.
It is noted thatlimt⟶tk−Vit=limt⟶tk+Vit=Vit=0,i=2,3, and hence Vtis continuous on 0,∞. To reduce the conservatism, the matrix in V3t is free; then, according to condition (11), it can be obtained that(14)V1t+V3t=d2−h2td2xTtPxt+h2td2xTtPxt+h2td2d2xtxtkTY11+Y11T−Y11−Y22∗Y22+Y22Tx˙txtk=d2−h2td2xTtPxt+h2td2xtxtkTΘx˙txtk≥0.
Taking the derivative ofVt, the following can be obtained:(15)V˙1t+2λV1t=2xTtPx˙t+2λxTtPxt,V˙2t+2λV2t=h2tx˙txtkTQ11Q12∗Q22x˙txtk−e−2λd2∫tktx˙sxtkTQ11Q12∗Q22x˙sxtkds,V˙3t+2λV3t=2h2txtxtkTY11+Y11T−Y11−Y22∗Y22+Y22Tx˙t0−xtxtkTY11+Y11T−Y11−Y22∗Y22+Y22Txtxtk+2λh2txtxtkTY11+Y11T−Y11−Y22∗Y22+Y22Txtxtk.
For any free matrixM and scalars ε1,ε2, we can obtain the equation as follows:(16)2∑i=13∑j=13μiztμjztε1xTtM+x˙TtM+ε2xTtkM×x˙t−A¯ixt+B¯iKjxtk=0.
Then, combining (15) and (16), we have(17)V˙t+2λVt=2∑i=13∑j=13μiztμjzth2tdkζTtΨ˜1ijdkζt1dk∫tktζTt,sΨ˜2ijdkζt,sds,where(18)ζt=xTtx˙TtxTtkT,ζt,s=xTtx˙TtxTtkx˙TsT.
From (9) and (10), the following can be obtained:(19)Ψ˜1ijdk=dk−d1d2−d1Ψ˜1ijd2+d2−dkd2−d1Ψ˜1ijd1<0,Ψ˜2ijdk=dk−d1d2−d1Ψ˜2ijd2+d2−dkd2−d1Ψ˜2ijd1<0,which implies that(20)V˙t+2λVt<0,and then we have(21)λminPxt2≤Vt<e−2λt−tkVtk<e−2λt−tk−1Vtk−1<⋯<e−2λtV0≤λmaxPe−2λtxt02.
Then(22)xt≤λmaxPλminPe−λtxt0,which implies that system (8) is mean square exponentially stable. This completed the proof.Remark 1.
The mean square exponential stability criteria for system (8) are introduced in Theorem 1. To fully characterize the sampling pattern, a suitable time-dependent LKF (11) is constructed and a novel quadratic functionV3t is added in LKF (11).Remark 2.
For the purpose of reducing the conservatism, more relaxed constraint matrices are introduced in LKF (13). The matrices are not required to be positive, which can effectively reduce conservatism.
Furthermore, the sampled-data controller (7) will be designed for stabilizing system (8) based on the following theorems.Theorem 2.
For scalesε1,ε2 and d2>d1≥0, system (8) is mean square exponentially stable, if there exist matrices Y¯11,Y¯22,M¯,P¯>0,Q¯11Q¯12∗Q¯22>0, satisfying(23)Ψ˜¯1ijd=Ψ¯1ij+de−2λd2I3Q¯22I3T+dϒ¯1Ψ¯4∗−I<0,(24)Ψ˜¯2ijd=Ψ¯1ij−de−2λd2Ψ¯3Ψ¯4∗−de−2λd2Q¯110∗∗−I<0,d∈d1,d2,(25)Θ¯=P¯+d2Y¯11+Y¯11T−d2Y¯11+Y¯22∗d2Y¯22+Y¯22T>0,where(26)Ψ¯1ij=SymI1P¯I2T+λI1P¯I1T−I1Y¯11I1T+I1Y¯11I3T+I1Y¯22I3T−ε1I1A¯iM¯I1T−I1M¯A¯iTI2T−ε1I1B¯iθK¯jI3T+I2M¯I2T−I3Y¯22I3T+ε1I1M¯I2T−ε1I1M¯A¯iTI3T−I2B¯iθK¯jI3T+ε2I2M¯TI3T−ε1I3B¯iθK¯jI3T−de−2λd2I3Q¯22I3T,ϒ¯1=SymI1Y¯11+Y¯11TI2T−I2Y¯11+Y¯22I3T+I2Q¯11I2T+I2Q¯12I3T+λI1Y¯11+Y¯11TI1T−λI1Y¯11+Y¯22I3T+I3Q¯22I3T+λI3Y¯22+Y¯22TI3T,Ψ¯3=00Q¯12T,Ψ¯4=M¯T00T.
Then, the controller gain matrixKcan be obtained as follows:(27)K=K¯M¯−T.Proof.
Let(28)χ1=M¯,χ2=diagM¯,M¯,χ3=diagχ2,M¯,I.
Define(29)M¯=M−1,P¯=M¯PM¯T,Q11=M¯Q11M¯T,Q12=M¯Q12M¯T,Q22=M¯Q22M¯T,Y11=M¯Y11M¯T,Y22=M¯Y22M¯T,K¯=KM¯T.
Pre- and post-multiply (9)–(11) by χ1,χ2, and χ3respectively; then, according to Schur complement, (23)–(25) can be obtained. This completed the proof.Remark 3.
Note that the lower and upper bounds of the variable period are both considered in Theorem1. If d1=d2=d, the periodic sampled-data issue for chaotic system can be solved, and the sampling period will reduce to be constant like [31], which have strict limitations on the lower bound of the delay and it will lead to considerable conservatism. So, the result in Theorem 1 covers [31] as a special case and has more significance than [31].


## 4. Numerical Examples

In the section, a simulation example for a chaotic system will be used to verify the effectiveness of given methods.Consider a chaotic system as follows [15]:(30)x˙1t=−ax1t+ax2t,x˙2t=cx1t−x2t−x1tx3t+u1t,x˙3t=x1tx2t−bx3t+u2t,where(31)a=10,b=83,c=28.Assume thatx1t∈m1,m2,x˙1t∈−5,5. According to [15], system (30) is represented with TSFM as follows.Mode rule 1: IFx1t is ω1, THEN(32)x˙t=A1xt+B1ut.Mode rule 2: IFx1t is ω2, THEN(33)x˙t=A2xt+B2ut,where(34)A1=−aa0c−1m10−m1−b,A2=−aa0c−1m20−m2−b,B1=B2=001001.Then, the membership functions are(35)μ1xt=x1t−m1m2−m1,μ2x1t=−x1t−m2m2−m1,and the trajectories of system (30) are exhibited in Figure2.Figure 2
Trajectory of the chaotic system.Letm1=−25,m2=25. Firstly, we compare the results in Theorem 2 with the references which use different sampled-data control methods for chaotic systems. We suppose that τm=τM and choose the same parameters; then, the maximum sampling period τM obtained by different references is shown in Table 1. It is obvious that Theorem 2 has longer sampling period than that in [15, 28, 31], which improves about 131.2%, 68.1%, and 19.35%, respectively. Hence, the result in the paper can reduce the conservatism to some extent. Besides, the lower and upper bounds of the sampling period are both considered in the paper, which has more significance than the previous work.Table 1
The maximum upper bound forτM.
Method[28][31][15]Theorem2τM0.00160.00220.00310.0037Then, we verify the effectiveness of the given method. The initial state isxt=−536. Assuming the sampling interval τm=0.6and ε1=1,ε2=0.01, then the maximum upper bound of sampling period is obtained as τM=1.33 by Theorem 2. Then, the related matrices by solving conditions (23)–(25) are obtained as(36)K¯1=10−9×0.00280.23690.0174,K¯2=10−9×0.00280.23690.0174,M¯=10−9×0.39040.0219−0.0005−0.61320.12330.00230.0083−0.18640.0007.Then, from (27), we can obtain the controller gain matrix.(37)K1=K¯1M−T=13.50978.67786.0571,K2=K¯2M−T=13.50938.67766.0569.Under controller (37), the responses of the state are shown in Figures 3–5 , which indicate that the states are stable in a short time. It is easy to know that the designed FSDC achieves system’s stabilization successfully, and a longer sampling period is obtained.Figure 3
Responses ofx1(t).Figure 4
Responses ofx2(t).Figure 5
Responses ofx3(t).


## 5. Conclusions

This paper discusses the aperiodic sampled-data control problem for a chaotic system with TSFM. And both lower and upper bounds of the sampling period are considered in the paper. Then, the criteria of mean square exponential stability are given. By constructing an appropriate LKF, the sampling patterns are fully captured and less conservative result is obtained. The simulation result is used to verify that the proposed fuzzy aperiodic sampled-data control strategy is effective.



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