# New Results on Fuzzy Synchronization for a Kind of Disturbed Memristive Chaotic System

Authors: Bo Wang; L. L. Chen
Journal: Complexity
 (2018)

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## Abstract

This paper concerns the problem on the fuzzy synchronization for a kind of disturbed memristive chaotic system. First, based on fuzzy theory, the fuzzy model for a memristive chaotic system is presented; next, based on H-infinity technique, a multidimensional fuzzy controller and a single-dimensional fuzzy controller are designed to realize the synchronization of master-slave chaotic systems with disturbances. Finally, some typical examples are included to illuminate the correctness of the given control method.


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## Body

## 1. Introduction

Since May 2008, by using nanotechnology physical techniques, HP laboratory research team successfully obtain the resistance with memory characteristic [1], which confirmed the concept of memristor proposed by Chua [2, 3]. As the fourth basic passive device, memristor establishes the relationship between the magnetic flux and the charge. It has been reported that memristor can be applied in the field of computer science [4], biological engineering [5], and electronic engineering [6]. Especially, memristor can be used to construct the chaotic circuits.For chaotic circuits, the nonlinear device is the key component. In 2008, Itoh and Chua built the first memristor-based chaotic system by replacing the diode with a piecewise linear magnetron Chua’s memristor [7]. Compared with the conventional nonlinear-device-based chaotic circuits, the memristor-based circuit has two main characteristics: first, the memristor-based circuit can produce the complicated dynamical behavior, which is different from the general chaotic dynamical behavior; secondly, the memristor-based circuit is more suitable to generate the high-frequency chaotic signal and have potential applications in chaotic secure communication, signal generator, and image process [8–12]. Hence, up to now, a number of memristor-based chaotic circuit with different structures are proposed. For example, the chaotic circuit with one memristor is studied in [13, 14], the chaotic circuit with two memristor is concerned in [15, 16], the integer-order chaotic memristor circuit is investigated in [17, 18], and the fractional chaotic memristor circuit are researched in [19, 20].Chaos synchronization is a common phenomenon and can be found in biological systems, chemical reactions, power converters, secure communication system, and so on. Fuzzy technique is a powerful tool [21–27] and especially suitable for the chaos synchronization in the case that disturbances exist. For general fuzzy control, the control input is multidimensional and requires all system state information. However, in practical engineering, it is not easy to get all system state information. The multidimensional control can not only increase the control cost but also result in disturbance input problem. Hence, it is meaningful to design a single-dimensional fuzzy controller which is just based on one system state variable. In addition, disturbance inputs exist in actual system widely, which should be considered in synchronization control. All these motivate our research.The paper is schemed as follows: the fuzzy model for a memristor-based chaotic circuit is constructed and the preliminary knowledge will be given in Section2; a multidimensional fuzzy controller and a single-dimensional fuzzy controller will be designed to achieve the chaos synchronization of the master-slave systems in Section 3; the typical simulation example will be included to validate the correctness of the scheme in Section 4; and finally, the paper will be concluded in Section 5.Notations used in this paper are fairly standard.diag… represents a block diagonal matrix, Rn is the n-dimensional Euclidean space, Rn×m denotes the set of n×m real matrix, the superscript T stands for matrix transposition, ⋅2 refers to the Euclidean vector norm or the induced matrix 2-norm, and λmin⋅ represents the maximum eigenvalue.


## 2. System Description and Preliminaries

First, consider a memristor-based circuit as Figure1.Figure 1
The memristor-based chaotic circuit.One can get the equivalent dynamic system as(1)C1dVC1tdt=iL1t−WφtVC1t,L1diL1tdt=VC2t−VC1t+R1iL1t,C2dVC2tdt=1RVC3t−VC2t−iL1t,C3dVC3tdt=1RVC2t−VC3t−iL2t,L2diL2tdt=VC3t−R2iL2,dφtdt=VC1t.Define the state variable as(2)x1t=VC1t,x2t=RiL1t,x3t=VC2t,x4t=VC3t,x5t=RiL2t,x6t=φt.One can obtain the equivalent dynamical equation as(3)ẋ1t=r1x2t−W¯x6tx1t,ẋ2t=r2x3t−x1t+r7x2t,ẋ3t=r3x4t−x3t−x2t,ẋ4t=r4x3t−x4t−x5t,ẋ5t=r5x4t−r6x5t,ẋ6t=x1t,with
(4)W¯x6t=RWφt=a,x6t≤1,b,x6t>1,where xi,i=1,2,…,7 is the state variable of the system and ri>0,i=1,2,…,7 is the system parameter. The memristive system will possess the chaotic dynamical behavior when the system parameters are r1=5, r2=2, r3=2, r4=4, r5=3, r6=0.1, r7=0.8, a=0.1, and b=6.Next, consider the fuzzy modeling of the memristive chaotic system.Forẋ1t=r1x2t−W¯x6x1t.Rule 1.
Ifx1t is H11, then
(5)ẋ1t=r1x2t−ax1t,where H11 means x6t≤1, and define
(6)M11=1,x6t≤1,0,x6t>1.Rule 2.
Ifx1t is H12, then
(7)ẋ1t=r1x2t−bx1t,where H12 means x6t>1, and define
(8)M12=0,x6t≤1,1,x6t>1.Hence, the fuzzy model of the memristive chaotic system is defined as(9)ẋ1tẋ2tẋ3tẋ4tẋ5tẋ6t=M11000000M11000000M11000000M11000000M11000000M11r1x2t−ax1tr2x3t−x1t+r7x2tr3x4t−x3t−x2tr4x3t−x4t−x5tr5x4t−r6x5tx1t+M12000000M12000000M12000000M12000000M12000000M12r1x2t−bx1tr2x3t−x1t+r7x2tr3x4t−x3t−x2tr4x3t−x4t−x5tr5x4t−r6x5tx1t.Above model can be rewritten as(10)ẋt=∑i=12ΘiAixt,where
(11)Θi=diagM1i,M1i,M1i,M1i,M1i,M1i,A1=−ar1r10000−r2r2r7r20000−r3−r3r30000r4−r4−r40000r5−r5r60100000,A2=−br1r10000−r2r2r7r20000−r3−r3r30000r4−r4−r40000r5−r5r60100000.System (3) is supposed as the master system, and the slave system is constructed as
(12)ẏ1t=r1y2t−W¯y6ty1t+u1t+w1t,ẏ2t=r2y3t−y1t+r7y2t+u2t+w2t,ẏ3t=r3y3t−y4t−y2t+u3t+w3t,ẏ3t=r4y4t−y3t−y5t+u4t+w4t,ẏ5t=r5y4t−r6y5t+u5t+w5t,ẏ6t=y1t+u6t+w6t,where yt=y1t,y2t,y3t,y4t,y5t,y6tT is the state variable vector of the slave system and wt=w1t,w2t,w3t,w4t,w5t,w6tT is the disturbance input of the slave system.Hence, the fuzzy model of the slave system can be represented as(13)ẏt=∑i=12ΘiAiyt+wt+ut,where ut=u1t,u2t,u3t,u4t,u5t,u6tT is the synchronization fuzzy controller.Define the synchronization error vector of the master-slave systems as(14)Et=yt−xt,where Et=e1t,e2t,e3t,e4t,e5t,e6tT.One can get the error dynamic system as(15)Ėt=ẏt−ẋt=∑i=12ΘiAiyt−∑i=12ΘiAixt+wt+ut=∑i=12ΘiAiEt+wt+ut.In this paper, the following lemmas are concerned:Lemma 1
(see [28]). If ft∈L∞∩L2 and ḟt∈L∞, one can get
(16)limt→+∞ft=0.Definition 1.
For nonzerowt∈L2t0,∞ and under the assumption of zero initial condition, if there exists a positive scalar γ such that
(17)Et2≤γwt2.Then, the slave system will synchronize to the master system with H∞ norm bound γ.


## 3. Main Results

Based on fuzzy theory and Lyapunov theory, a controller is presented as follows.Theorem 1.
If there exist scalarKji>0,i=1,2,j=1,…,6, design the multidimensional fuzzy controller with following control regulation
(18)ut=−∑i=12ΘikiEt,with
(19)k11=K11−r1a+2,k21=K21+r1−r224+r2r7+1,k31=K31+r2−r324−r3+1,k41=K41+−r3−r424−r4+1,k51=K51+r5−r424−r5r6,k61=K61+14,k12=K12−r1b+2,k22=K22+r1−r224+r2r7+1,k32=K32+r2−r324−r3+1,k42=K42+−r3−r424−r4+1,k52=K52+r5−r424−r5r6,k62=K62+14,(20)I−∑i=12ΘiKiI2∗−γ2I<0,(21)ki=diagk1i,k2i,k3i,k4i,k5i,k6i,(22)Ki=diagK1i,K2i,K3i,K4i,K5i,K6i.Then, the slave system (12) can synchronize to the master system (3) with H∞ norm bound γ.Proof 1.
With (18), the error dynamic system can be transformed as
(23)ė1tė2tė3tė4tė5tė6t=M11000000M11000000M11000000M11000000M11000000M11r1e2t−ae1t−k11e1t+w1tr2e3t−e1t+r7e2t−k21e2t+w2tr3e4t−e3t−e2t−k31e3t+w3tr4e3t−e4t−e5t−k41e4t+w4tr5e4t−r6e5t−k51e5t+w5te1t−k61e6t+w6t+M12000000M12000000M12000000M12000000M12000000M12r1e2t−be1t−k12e1t+w1tr2e3t−e1t+r7e2t−k22e2t+w2tr3e4t−e3t−e2t−k32e3t+w3tr4e3t−e4t−e5t−k42e4t+w4tr5e4t−r6e5t−k52e5t+w5te1t−k62e6t+w6tChoose the Lyapunov function candidate as(24)Vt=12e12t+e22t+e32t+e42t+e52t+e62tOne can get the time derivative ofVt as
(25)V̇t=e1tM11r1e2t−ae1t−k11e1t+w1t+M12r1e2t−be1t−k12e1t+w1t+e2tM11r2e3t−e1t+r7e2t−k21e2t+w2t+M12r2e3t−e1t+r7e2t−k22e2t+w2t+e3tM11r3e4t−e3t−e2t−k31e3t+w3t+M12r3e4t−e3t−e2t−k32e3t+w3t+e4tM11r4e3t−e4t−e5t−k41e4t+w4t+M12r4e3t−e4t−e5t−k42e4t+w4t+e5tM11r5e4t−r6e5t−k51e5t+w5t+M12r5e4t−r6e5t−k52e5t+w5t+e6tM11e1t−k61e6t+w6t+M12e1t−k62e6t+w6t=−r1M11a+M12be12t+r1−r2e1te2t+r2r7e22t+r2−r3e2te3t−r3e32t+r3+r4e3te4t−r4e42t+r5−r4e4te5t−r5r6e52t+e1te6t−∑j=16M11kj1ej2t+M12kj2ej2t+wTtEt≤−r1M11a+M12b−2+M11k11+M12k12e12t−e1t−r1−r2e2t22−−r1−r224−1−r2r7+M11k21+M12k22e22t−e2t−r2−r3e3t22−−r2−r324−1+r3+M11k31+M12k32e32t−e3t−r3+r4e4t22−−r3+r424−1+r4+M11k41+M12k42e42t−e4t−r5−r4e5t22−−r5−r424+r5r6+M11k51+M12k52e52t−e1t−e6t22−−14+M11k61+M12k62e62t+wTtEt.With (19), one can conclude that
(26)V̇t≤−ETt∑i=12ΘiKiEt+wTtEt.Consider theH∞ performance index as
(27)J=∫t0tTETtEt−γ2wTtwtdt=∫t0tTETtEt−γ2wTtwt+V̇tdt+Vt0−VtT.ForVt0=0 and VtT≥0,
(28)J≤∫t0tTETtEt−γ2wTtwt+V̇tdt=∫t0tTηTtΩηtdt,where
(29)ηt=ETt,wTtT,Ω=I−∑i=12ΘiKiI2∗−γ2I.Consider (20), it can be concluded that J≤0. Based on Definition 1, slave system (12) can synchronize to master system (3) with H∞ norm bound γ.Next, consider the design for the single-dimensional fuzzy synchronization controller.Construct the slave system as(30)ẏ1t=r1y2t−W¯y6ty1t,ẏ2t=r2y3t−y1t+r7y2t+wt+ut,ẏ3t=r3y4t−y3t−y2t,ẏ3t=r4y3t−y4t−y5t,ẏ5t=r5y4t−r6y5t,ẏ6t=y1t,where ut is the single-dimensional synchronization fuzzy controller.Theorem 2.
If there exist scalarki>0,i=1,2, design the single-dimensional fuzzy controller with following control regulation
(31)ut=−∑i=12M1ikie2t,where
(32)ki=ki+r7r2−M11a+M12b00001−M11k1+M12k2012r200−r60012r20−γ2≤0.Then, slave system (30) with any initial conditions can synchronize to master system (3) with H∞ norm bound γ.Proof 2.
With (31), the error dynamic system can be transformed as
(33)ė1tė2tė3tė4tė5tė6t=M11000000M11000000M11000000M11000000M11000000M11r1e2t−ae1tr2e3t−e1t+r7e2t+wt−k1e2tr3e4t−e3t−e2tr4e3t−e4t−e5tr5e4t−r6e5te1t+M12000000M12000000M12000000M12000000M12000000M12r1e2t−be1tr2e3t−e1t+r7e2t+wt−k2e2tr3e4t−e3t−e2tr4e3t−e4t−e5tr5e4t−r6e5te1t.Choose the Lyapunov function candidate as(34)V0t=12r1e12t+12r2e22t+12r3e32t+12r4e42t+12r5e52t.One can get the time derivative ofV0t as
(35)V̇0t=1r1e1tė1t+1r2e2tė2t+1r3e3tė3t+1r4e4tė4t+1r5e5tė5t=e1tM11e2t−ae1t+M12e2t−be1t+e2tM11e3t−e1t+r7e2t+M12e3t−e1t+r7e2t+wt/r2+ut/r2+e3tM11e4t−e3t−e2t+M12e4t−e3t−e2t+e4tM11e3t−e4t−e5t+M12e3t−e4t−e5t+e5tM11e4t−r6e5t+M12e4t−r6e5t=e1tM11e2t−ae1t+M12e2t−be1t+e2tM11e3t−e1t−k1e2t+M12e3t−e1t−k2e2t+e3tM11e4t−e3t−e2t+M12e4t−e3t−e2t+e4tM11e3t−e4t−e5t+M12e3t−e4t−e5t+e5tM11e4t−r6e5t+M12e4t−r6e5t+et2wt/r2=−M11a+M12be12t−M11k1+M12k2e22t−e3t−e4t2−r6e52t+e2twt/r2.With (31), one can conclude that
(36)V̇0t≤−E¯Tt∑i=12Θ¯iK¯iE¯t+1r2wte2t,where(37)K¯1=diaga,k1,r6≥0,K¯2=diagb,k2,r6≥0,E¯=e1t,e2t,e5tT,Θ¯i=diagM1i,M1i,M1i.Consider theH∞ performance index as
(38)J=∫t0tTe22t−γ2wTtwtdt=∫t0tTe22t−γ2wTtwt+V̇0tdt+Vt0−VtT.ForVt0=0 and VtT≥0,
(39)J≤∫t0tTe22t−γ2wTtwt+V̇0tdt=∫t0tTηTtΩηtdt,where
(40)ηt=E¯Tt,wTtT,Ω=−M11a+M12b00001−M11k1+M12k2012r200−r60012r20−γ2.Consider (32), it can be concluded that J≤0. Based on Definition 1, slave system (30) can synchronize to master system (3) with H∞ norm bound γ.


## 4. Example and Simulation

First, consider the dynamics of the memristive chaotic system, and the simulation result is shown in Figure2.Figure 2
Attractor of the memristive chaotic system.Next, we study the synchronization control of the master-slave systems. In the simulation, the system initial values arey0=1,−1,0.5,−1,2,−1T and x0=0.001×1,1,1,1,1,1T. The disturbance input is wt=wit=2sin2tsinet/t+1,t≥20s. Let γ=0.4, and based on Theorem 1, the control parameters for the multidimensional fuzzy controller are k1=diag4.06,7.41,1.56,8.56,2.21,2.81 and k2=diag−35.43,7.41,1.56,8.56,2.21,2.81; the simulation result is shown in Figure 3. Then, based on Theorem 2, the control parameters for the single-dimensional fuzzy controller are k1=9.73 and k2=7.38; the simulation result is shown in Figure 4.Remark 2.
Figure3 depicts the time response of the synchronization error variables of the memristive master-slave systems with the multidimensional fuzzy controller. Figure 4 depicts the time response of the synchronization error variables of the master-slave systems with the single-dimensional fuzzy controller. It can be seen that although there exists just one disturbance wt for single-dimensional fuzzy control, the disturbance has impact on all error synchronization variables. In addition, it can be seen that both controllers can be able to realize the synchronization of the master-slave systems; the multidimensional fuzzy controller has the better control performance and realizes the chaos synchronization during 2.0 seconds. However, the possession of the good control performance is at the cost of the acquirement of all system state information. In addition, multidimensional control may introduce more disturbance input. The single-dimensional synchronization controller has the general control performance but requires just one system state information, which can decrease the control cost and the disturbance input. Hence, two kinds of controllers are useful and recommended for the different applied cases.Remark 3.
For the nonlinear disturbed chaotic system, the fuzzy modeling technique is adopted to realize the exact linearization control, which can eliminate the constraint on the system nonlinear term, compared with the general nonlinear control method; in addition, H-infinity approach is introduced to deal with the case that disturbances exist.Figure 3
Time response of synchronization error variables with multidimensional fuzzy controller.Figure 4
Time response of synchronization error variables with single-dimensional fuzzy controller.


## 5. Conclusion

This paper focuses on the fuzzy synchronization for a new memristive chaotic system with disturbances. Based on fuzzy theory and Lyapunov stability theory, we have built the fuzzy model for the memristive chaotic system. Then, by using H-infinity technique, we have presented two kinds of fuzzy controllers for the possible application in chaos synchronization of slave-master systems. Finally, we have included some example to demonstrate the effectiveness of the given fuzzy controllers. In addition, the proposed results can be extended to the memristive chaotic control system with daelay or event trigger, which is our future work.



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