# Observer-Based Fuzzy Control for Memristive Circuit Systems

Authors: Qian Ye; Xuyang Lou
Journal: Complexity
 (2018)

---

## Abstract

This paper proposes an observer-based fuzzy control scheme for a class of memristive chaotic circuit systems. First, the Takagi-Sugeno fuzzy model is adopted to reconstruct the nonlinear chaotic circuit system. Next, based on the proposed fuzzy model, an observer-based fuzzy controller is developed to estimate the states and stabilize the origin. Third, the results are extended to explore theL∞-gain observer-based fuzzy control for the chaotic system with disturbances. Finally, simulation results are also addressed to show the effectiveness of the proposed control scheme.


---

## Body

## 1. Introduction

In 1971, Chua postulated the existence of a fourth circuit element [1], called memristor, which was realized by Williams’s group of HP Labs only 37 years later [2]. In recent years, the memristor has attracted much attention due to its potential application in associative memory [3], image processing [4], filter [5], programmable analog circuits [6], and so on. In particular, Pershin and Ventra experimentally demonstrated the formation of associative memory in a simple neural network, which consists of three electronic neurons using memristor-emulator synapses [3]. A new image encryption algorithm was presented in [4] based on chaos with the piecewise-linear memristor in Chua’s circuit. The authors in [5] experimentally demonstrated an adaptive filter by introducing a memristor and using the memristive properties of vanadium dioxide. In [6], a memristor was designed for a pulse-programmable midband differential gain amplifier with fine resolution.The HP memristor is described by a nonlinear constitutive relation as introduced in [7]
(1)v=Mqi,ori=Wφv,between the device terminal voltage v and terminal current i, where φ=∫vdt, Mq=dφq/dq, and Wφ=dqφ/dφ, where Mq and Wφ are the memristance and memductance. Memristor-based systems may exhibit complex behaviors, such as chaotic and hyperchaotic dynamics. Recently, chaos control, hybrid control, and synchronization of memristor-based or memristive chaotic systems have received intensive investigation [8–14]. However, the “piecewise-linear” nonlinearity characterization by introducing a memristor may lead to challenges in dealing with the chaotic systems. Fuzzy modeling approaches result in a way that the original systems can be decomposed into a number of linear subsystems. In the context of the Takagi-Sugeno fuzzy models, Zhong et al. [8] addressed fuzzy modeling and impulsive control of the memristor-based Chua chaotic system. Cafagna and Grassi presented a novel fractional-order memristor-based chaotic system and carried out the theoretical analysis of the system dynamics [9]. The Takagi-Sugeno fuzzy method emerged as a promising approach for approximating nonlinear systems [8, 15, 16]. More recently, a new fuzzy model of the memristor-based Lorenz circuit, which was employed to synchronize with the memristor-based Chua circuit, was explored in [14].Despite the rich achievements, most of the above results mainly focused on stability or synchronization of memristive chaotic circuit systems rather than state estimation [17, 18]. However, in real chaotic circuits, it is often the case that only partial information about the states (for instance, voltage) is available in the system outputs. Therefore, in order to utilize the memristive chaotic circuit systems, one often needs to estimate the system state through available measurement, and then use the estimated system to achieve synchronization, optimal control [19], or tracking performances. In addition, general results on state estimation and observer-based control for such memristive systems do not seem to have received much attention so far. To the best of our knowledge, there is a lack of effort in the observer-based control and synchronization of memristive systems [18].Inspired by [16], this paper aims at investigating the observer-based fuzzy control for the stabilization of the memristive Chua circuit systems with or without external disturbances. An observer-based fuzzy control scheme based on the Takagi-Sugeno fuzzy model of the Chua systems is proposed. The controller design based on linear matrix inequality (LMI) conditions is developed. The results are extended to explore the L∞-gain control problem for the chaotic system with disturbances using the observer-based fuzzy control approach. In addition, the nonlinear L∞-gain control problem is transformed into a suboptimal control problem, that is, to minimize the upper bound of the L∞-gain of the closed-loop system subject to LMI constraints.Notations.
ℝn denotes the n-dimensional Euclidian space. Given a vector x∈ℝn, xT denotes its transpose. ∣x∣=∑i=1nxi21/2 denotes the Euclidean norm. For a function s∈0,+∞→ℝn, st∞≜suptst. AT and A−1 represent the transpose of matrix A and the inverse of matrix A, respectively. We use A>0A<0 to denote a positive- (negative-) definite matrix A, and I (resp., 0) denotes the identity matrix (resp., zero matrix) of appropriate dimension. diag· denotes a block diagonal matrix. The symbol “⋆” within a matrix represents the symmetric term of the matrix. λmaxP and λminP represent the maximum and minimum eigenvalue of the real symmetric matrix P, respectively. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Modeling and Control of Memristive Chua Systems

### 2.1. Memristive System without Disturbances

Consider the memristive Chua circuit system [7]:
(2)ẋ1=σ1x2−Wx4x1,ẋ2=x3−x1,ẋ3=−σ2x2+σ3x3,ẋ4=x1,with the output y=x4, and
(3)Wx4=dqx4dx4=a,x4<1,b,x4>1,qx4=bx4+0.5a−bx4+1−x4−1.When taking the parametersσ1=4, σ2=1, σ3=0.65, a=0.2, and b=10, (2) exhibits the chaotic behavior in [8] as shown in Figure 1.Figure 1
Memristive Chua system.
(a)
x1−x2−x3
(b)
x2−x4
(c)
x3−x4Our aim is to estimate the states of (2) and stabilize the origin of the system. To do this, by imposing a controller into the system, one gets
(4)ẋt=fxt+But,yt=Cxt,where xt=x1t,…,x4tT∈ℝ4 denotes the vector of the states, ut=u1t,…,umtT∈ℝm denotes the vector of the control inputs, C=0001, B∈ℝ4×m, and
(5)fx=σ1x2−Wx4x1x3−x1−σ2x2+σ3x3x1.According to [19], we construct the Takagi-Sugeno fuzzy model for (2) as follows:Rule 1.
Ifyt is M1, then
(6)ẋt=A1x+Bu.Rule 2.
Ifyt is M2, then
(7)ẋt=A2x+Bu,where M1 is ∣yt∣<1, and M2 is ∣yt∣>1,
(8)A1=−aσ1σ100−10100−σ2σ301000,A2=−bσ1σ100−10100−σ2σ301000.Through a center-average defuzzifier, the overall fuzzy system is represented as(9)ẋ=∑i=12hitAixt+But,where the membership functions are
(10)h1t=1,yt<1,0,yt>1,h2t=0,yt<1,1,yt>1.Suppose that the state variables are not fully measurable and only the flux of the capacitor (i.e.,x4) is measurable (when choosing other partial states, the results are also applicable), we propose the following fuzzy state observer for fuzzy model (9).Observer rulei: if yt is Mi, i∈1,2, then
(11)x^̇=Aix^t+But+Liyt−Cx^t,where x^ is an estimate of x and Li is the observer gain for the ith observer rule.Note thatyt is applied for the observer rule as it is available from the system (2). Similarly, the overall fuzzy observer is represented as follows:
(12)x^̇=∑i=12hitAix^t+But+Lyt−Cx^t.To stabilize the memristive chaotic system (2), the observer-based fuzzy controller is given by
(13)ut=∑j=12hjtKjx^t,where j∈1,2 is the observer rule index, Kj is the control gain for the jth controller rule.The proposed observer-based fuzzy control scheme is depicted in Figure2. The fuzzy observer estimates the states of the memristive chaotic systems based on the measurable output. Then, the estimated states are utilized in the observer-based fuzzy controller and the output of the controller will be imposed on the chaotic systems.Figure 2
Observer-based fuzzy control structure diagram.We denote the estimation error aset=xt−x^. Hence, (9) is equivalent to
(14)ẋ=∑i=12∑j=12hithjtAixt+BKjx^t=∑i=12∑j=12hithjtAi+BKjxt−BKjet.By differentiatinget, one can readily obtain
(15)ė=∑i=12hitAi−LiCet=∑i=12hit∑j=12hjtAi−LiCet,where we used the fact ∑j=12hjt=1.Combining with (14)-(15) yields
(16)ẋtėt=∑i=12∑j=12hithjtAijxtet,where Aij=Ai+BKj−BKj0Ai−LiC,i,j∈1,2.Let us define(17)η=xtet,then the augmented system in (16) can be rewritten as
(18)η̇=∑i=12∑j=12hithjtAijη.Proposition 1.
For the augmented system (16), if there exists a symmetric positive-definite matrix P>0 and a positive scalar α>0 such that the following matrix inequality
(19)PAij+Aji2T+PAij+Aji2+αP<0,holds for all i≤j i,j∈1,2, then the augmented system (16) is asymptotically stable.Proof.
Consider the Lyapunov function candidate(20)Vt=ηTtPηt,where η=xTteTtT, P>0 is symmetric. After calculating the derivative of V along the trajectories of (18), we have
(21)V̇t=∑i=12∑j=12hithjtηTtAijTP+PAijηt=∑i=12hi2tηTtPAii+Aii2T+PAii+Aii2+αPηt+∑i=12hit∑i<j2hjtηTtPAij+Aji2T+PAij+Aji2+αPη−αηTPηt,where α>0 is an arbitrary scalar. Using (19), we get
(22)V̇≤−αηTPηt.It is obvious thatVt is positive definite and V̇t<0, ∀x≠0, and ∀e≠0. According to the Lyapunov stability theorem [20], Theorem 5.16, the origin of (16) is globally asymptotically stable. This completes the proof.Remark 1.
It is worth pointing out that this matrix inequality (19) is hardly tractable numerically. Moreover, linearizing the matrix inequality is also a very difficult task due to the presence of the coupling term in PAij.For the convenience of design, letP=P100P2.Hence, the matrix inequalities in (19) are equivalent to the following matrix inequalities:
(23)Φ11+αP1−Φ12⋆Φ2+αP2<0,where
(24)Φ11=Ω11+Ω11T,Ω11=P1Ai+BKj+P1Aj+BKi2,Φ12=P1BKj+P1BKi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2,Yi=P2Li,for all i≤j, i,j∈1,2.Since there are no effective algorithms for solvingP1,Ki,P2,andYi simultaneously, we use the separation method to solve the problem. Note that (23) can be decoupled as follows:
(25)Φ11+αP1−Φ12⋆Φ2+αP2=Φ11+αP1−Φ12⋆−βP1+00⋆Φ22+αP2+βP1,where β is a positive scalar. It is obvious that if
(26)Φ11+αP1−Φ12⋆−βP1<0,and
(27)00⋆Φ22+αP2+βP1<0,then (23) holds.Note that (26) is related to the controller part (the parameters are P1 and Ki) and (27) is related to the observer part (the parameters are P1,P2 and Li), respectively. Now we can determine P1,Ki,P2,andYi simultaneously by the following arrangement.It follows by pre- and postmultiplying the inequality (26) by the matrix diagP1−1,P1−1
(28)Ξ11+αX1−Ξ12⋆−βX1<0,where
(29)X1=P1−1,Ξ11=Θ11+Θ11T,Θ11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Wi=KiX1.By the well-known Schur complement [21], the inequality (28) is equivalent to
(30)Φ22+αP2I⋆−β−1X1<0.Based on the above analysis, we can obtain the following observer-based control theorem.Theorem 1.
System (2) is asymptotically stabilizable by (13) if for fixed scalars α>0 and β>0, there exist two positive-definite matrices X1∈ℝ4×4 and P2∈ℝ4×4 and two matrices Wi and Yi, so that the following two LMI conditions are feasible:
(31)Ξ11+αX1−Ξ12⋆−βX1<0,(32)Φ22+αP2I⋆−β−1X1<0,for all i≤j,i,j∈1,2, where
(33)Ξ11=Ω11+Ω11T,Ω11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2.Moreover, the stabilizing observer-based control gains are given byKi=WiX1−1, Li=P2−1Yi.According to the analysis above, the design procedure is summarized as follows:Design procedure:Step 1.
Construct the fuzzy plant rules in (9) and the observer rules in (12).Step 2.
Take a set of positive scalarsα>0 and β>0 iteratively.Step 3.
Solve the following linear matrix inequality problem(34)Ξ11+αX1−Ξ12⋆−βX1<0,Φ22+αP2I⋆−β−1X1<0,to obtain X1,P2,Wi,andYi (thus Ki=WiX1−1, Li=P2−1Yi can also be derived).Step 4.
IfX1>0 and P2>0 cannot be found, try another set of α>0,β>0,andc>0 iteratively and repeat Steps 3-4.Step 5.
Construct the fuzzy observer (12).Step 6.
Construct the fuzzy controller (13).


### 2.2. Memristive System with Disturbances

In this subsection, we will extend the above results to deal with theL∞-gain control problem for the memristive chaotic system with external disturbances.Consider the following the memristive Chua system with external disturbances(35)ẋ=∑i=12hitAixt+But+Diwt,where Di∈ℝ4×p and wt=w1t,…,wptT∈ℝp denotes the vector of the bounded external disturbances.Applying the same fuzzy observer (12) and the observer-based fuzzy control law (13), we have
(36)ẋtėt=∑i=12∑j=12hithjtAijxtet+Di00Diwtwt,where Aij=Ai+BKj−BKj0Ai−LiC.Denote(37)η=xtet,w¯t=wtwt,D¯i=Di00Di.Then (36) can be expressed as
(38)η̇=∑i=12∑j=12hithjtAijηt+D¯iw¯t.The objective is to determine a fuzzy controller for the closed-loop system (38) with the following L∞-gain performance as small as possible. Given a disturbance attenuation ρ, we need to achieve
(39)∥ηt∥∞≤γ∣η0∣+ρ∥w¯t∥∞,where γ and ρ are some positive scalars.Following [16], to extend the results in the previous subsection, the L∞-gain control performance in (39) is guaranteed for an attenuated level ρ=c/αλminP, and β=λmaxP/λminP.TheL∞ control problem can be transformed to solve the following minimization problem:
(40)mincαλminPs.t.X1=X1T>0,P2=P2T>0,Wi,Yi,β>0,α>0,c>0,Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,for all i≤j, i,j∈1,2, where
(41)Ξ11=Ω11+Ω11T,Ω11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2,Ξ13=Di+Dj2,Φ23=P2Di+P2Dj2.Note thatX1 instead of P1 is formulated in the constraint conditions, thus, the term c/αλminP should be modified as follows to solve the minimization problem. It is observed that c/αλminP<ε2 if c/αI<λminPε2I<ε2P, that is,
(42)cαI00cαI<ε2P100ε2P2,which is equivalent to
(43)X1<ε2αcI,and
(44)cε2αI<P2.Therefore, we can summarize the theorem for theL∞-gain performance as follows.Theorem 2.
The closed-loop system (38) has the L∞-gain performance
(45)∥ηt∥∞<γ∣η0∣+ε∥w¯t∥∞,if for fixed scalars α>0 and β>0, there exist two positive-definite matrices X1∈ℝ4×4andP2∈ℝ4×4, and two matrices Wi,Yi and i∈1,2, the following minimization problem is solved,
(46)minX1,P2,Wi,Yi,β,α,cε2s.t.X1=X1T>0,P2=P2T>0,Wi,Yi,β>0α>0,c>0,Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,X1<ε2αcI,cε2aI<P2.Design procedure:Step 1.
Construct the fuzzy plant rules in (9) and the observer rules in (12).Step 2.
Take a set of positive scalarsα>0,β>0,andc>0 iteratively.Step 3.
Given an initialε2.Step 4.
Solve the following linear matrix inequality problem(47)Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,X1<ε2αcI,cε2αI<P2,
to obtainX1,P2,Wi,andYi (thus Ki=WiX1−1 and Li=P2−1Yi can also be derived).Step 5.
Decreaseε2 and repeat Steps 4-5 until X1>0 and P2>0 cannot be found. If X1>0 and P2>0 cannot be found for all possible ε2, try another set of α>0,β>0,andc>0 iteratively and repeat Steps 3–5.Step 6.
Construct the fuzzy observer (12).Step 7.
Construct the fuzzy controller (13).


## 2.1. Memristive System without Disturbances

Consider the memristive Chua circuit system [7]:
(2)ẋ1=σ1x2−Wx4x1,ẋ2=x3−x1,ẋ3=−σ2x2+σ3x3,ẋ4=x1,with the output y=x4, and
(3)Wx4=dqx4dx4=a,x4<1,b,x4>1,qx4=bx4+0.5a−bx4+1−x4−1.When taking the parametersσ1=4, σ2=1, σ3=0.65, a=0.2, and b=10, (2) exhibits the chaotic behavior in [8] as shown in Figure 1.Figure 1
Memristive Chua system.
(a)
x1−x2−x3
(b)
x2−x4
(c)
x3−x4Our aim is to estimate the states of (2) and stabilize the origin of the system. To do this, by imposing a controller into the system, one gets
(4)ẋt=fxt+But,yt=Cxt,where xt=x1t,…,x4tT∈ℝ4 denotes the vector of the states, ut=u1t,…,umtT∈ℝm denotes the vector of the control inputs, C=0001, B∈ℝ4×m, and
(5)fx=σ1x2−Wx4x1x3−x1−σ2x2+σ3x3x1.According to [19], we construct the Takagi-Sugeno fuzzy model for (2) as follows:Rule 1.
Ifyt is M1, then
(6)ẋt=A1x+Bu.Rule 2.
Ifyt is M2, then
(7)ẋt=A2x+Bu,where M1 is ∣yt∣<1, and M2 is ∣yt∣>1,
(8)A1=−aσ1σ100−10100−σ2σ301000,A2=−bσ1σ100−10100−σ2σ301000.Through a center-average defuzzifier, the overall fuzzy system is represented as(9)ẋ=∑i=12hitAixt+But,where the membership functions are
(10)h1t=1,yt<1,0,yt>1,h2t=0,yt<1,1,yt>1.Suppose that the state variables are not fully measurable and only the flux of the capacitor (i.e.,x4) is measurable (when choosing other partial states, the results are also applicable), we propose the following fuzzy state observer for fuzzy model (9).Observer rulei: if yt is Mi, i∈1,2, then
(11)x^̇=Aix^t+But+Liyt−Cx^t,where x^ is an estimate of x and Li is the observer gain for the ith observer rule.Note thatyt is applied for the observer rule as it is available from the system (2). Similarly, the overall fuzzy observer is represented as follows:
(12)x^̇=∑i=12hitAix^t+But+Lyt−Cx^t.To stabilize the memristive chaotic system (2), the observer-based fuzzy controller is given by
(13)ut=∑j=12hjtKjx^t,where j∈1,2 is the observer rule index, Kj is the control gain for the jth controller rule.The proposed observer-based fuzzy control scheme is depicted in Figure2. The fuzzy observer estimates the states of the memristive chaotic systems based on the measurable output. Then, the estimated states are utilized in the observer-based fuzzy controller and the output of the controller will be imposed on the chaotic systems.Figure 2
Observer-based fuzzy control structure diagram.We denote the estimation error aset=xt−x^. Hence, (9) is equivalent to
(14)ẋ=∑i=12∑j=12hithjtAixt+BKjx^t=∑i=12∑j=12hithjtAi+BKjxt−BKjet.By differentiatinget, one can readily obtain
(15)ė=∑i=12hitAi−LiCet=∑i=12hit∑j=12hjtAi−LiCet,where we used the fact ∑j=12hjt=1.Combining with (14)-(15) yields
(16)ẋtėt=∑i=12∑j=12hithjtAijxtet,where Aij=Ai+BKj−BKj0Ai−LiC,i,j∈1,2.Let us define(17)η=xtet,then the augmented system in (16) can be rewritten as
(18)η̇=∑i=12∑j=12hithjtAijη.Proposition 1.
For the augmented system (16), if there exists a symmetric positive-definite matrix P>0 and a positive scalar α>0 such that the following matrix inequality
(19)PAij+Aji2T+PAij+Aji2+αP<0,holds for all i≤j i,j∈1,2, then the augmented system (16) is asymptotically stable.Proof.
Consider the Lyapunov function candidate(20)Vt=ηTtPηt,where η=xTteTtT, P>0 is symmetric. After calculating the derivative of V along the trajectories of (18), we have
(21)V̇t=∑i=12∑j=12hithjtηTtAijTP+PAijηt=∑i=12hi2tηTtPAii+Aii2T+PAii+Aii2+αPηt+∑i=12hit∑i<j2hjtηTtPAij+Aji2T+PAij+Aji2+αPη−αηTPηt,where α>0 is an arbitrary scalar. Using (19), we get
(22)V̇≤−αηTPηt.It is obvious thatVt is positive definite and V̇t<0, ∀x≠0, and ∀e≠0. According to the Lyapunov stability theorem [20], Theorem 5.16, the origin of (16) is globally asymptotically stable. This completes the proof.Remark 1.
It is worth pointing out that this matrix inequality (19) is hardly tractable numerically. Moreover, linearizing the matrix inequality is also a very difficult task due to the presence of the coupling term in PAij.For the convenience of design, letP=P100P2.Hence, the matrix inequalities in (19) are equivalent to the following matrix inequalities:
(23)Φ11+αP1−Φ12⋆Φ2+αP2<0,where
(24)Φ11=Ω11+Ω11T,Ω11=P1Ai+BKj+P1Aj+BKi2,Φ12=P1BKj+P1BKi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2,Yi=P2Li,for all i≤j, i,j∈1,2.Since there are no effective algorithms for solvingP1,Ki,P2,andYi simultaneously, we use the separation method to solve the problem. Note that (23) can be decoupled as follows:
(25)Φ11+αP1−Φ12⋆Φ2+αP2=Φ11+αP1−Φ12⋆−βP1+00⋆Φ22+αP2+βP1,where β is a positive scalar. It is obvious that if
(26)Φ11+αP1−Φ12⋆−βP1<0,and
(27)00⋆Φ22+αP2+βP1<0,then (23) holds.Note that (26) is related to the controller part (the parameters are P1 and Ki) and (27) is related to the observer part (the parameters are P1,P2 and Li), respectively. Now we can determine P1,Ki,P2,andYi simultaneously by the following arrangement.It follows by pre- and postmultiplying the inequality (26) by the matrix diagP1−1,P1−1
(28)Ξ11+αX1−Ξ12⋆−βX1<0,where
(29)X1=P1−1,Ξ11=Θ11+Θ11T,Θ11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Wi=KiX1.By the well-known Schur complement [21], the inequality (28) is equivalent to
(30)Φ22+αP2I⋆−β−1X1<0.Based on the above analysis, we can obtain the following observer-based control theorem.Theorem 1.
System (2) is asymptotically stabilizable by (13) if for fixed scalars α>0 and β>0, there exist two positive-definite matrices X1∈ℝ4×4 and P2∈ℝ4×4 and two matrices Wi and Yi, so that the following two LMI conditions are feasible:
(31)Ξ11+αX1−Ξ12⋆−βX1<0,(32)Φ22+αP2I⋆−β−1X1<0,for all i≤j,i,j∈1,2, where
(33)Ξ11=Ω11+Ω11T,Ω11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2.Moreover, the stabilizing observer-based control gains are given byKi=WiX1−1, Li=P2−1Yi.According to the analysis above, the design procedure is summarized as follows:Design procedure:Step 1.
Construct the fuzzy plant rules in (9) and the observer rules in (12).Step 2.
Take a set of positive scalarsα>0 and β>0 iteratively.Step 3.
Solve the following linear matrix inequality problem(34)Ξ11+αX1−Ξ12⋆−βX1<0,Φ22+αP2I⋆−β−1X1<0,to obtain X1,P2,Wi,andYi (thus Ki=WiX1−1, Li=P2−1Yi can also be derived).Step 4.
IfX1>0 and P2>0 cannot be found, try another set of α>0,β>0,andc>0 iteratively and repeat Steps 3-4.Step 5.
Construct the fuzzy observer (12).Step 6.
Construct the fuzzy controller (13).


## 2.2. Memristive System with Disturbances

In this subsection, we will extend the above results to deal with theL∞-gain control problem for the memristive chaotic system with external disturbances.Consider the following the memristive Chua system with external disturbances(35)ẋ=∑i=12hitAixt+But+Diwt,where Di∈ℝ4×p and wt=w1t,…,wptT∈ℝp denotes the vector of the bounded external disturbances.Applying the same fuzzy observer (12) and the observer-based fuzzy control law (13), we have
(36)ẋtėt=∑i=12∑j=12hithjtAijxtet+Di00Diwtwt,where Aij=Ai+BKj−BKj0Ai−LiC.Denote(37)η=xtet,w¯t=wtwt,D¯i=Di00Di.Then (36) can be expressed as
(38)η̇=∑i=12∑j=12hithjtAijηt+D¯iw¯t.The objective is to determine a fuzzy controller for the closed-loop system (38) with the following L∞-gain performance as small as possible. Given a disturbance attenuation ρ, we need to achieve
(39)∥ηt∥∞≤γ∣η0∣+ρ∥w¯t∥∞,where γ and ρ are some positive scalars.Following [16], to extend the results in the previous subsection, the L∞-gain control performance in (39) is guaranteed for an attenuated level ρ=c/αλminP, and β=λmaxP/λminP.TheL∞ control problem can be transformed to solve the following minimization problem:
(40)mincαλminPs.t.X1=X1T>0,P2=P2T>0,Wi,Yi,β>0,α>0,c>0,Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,for all i≤j, i,j∈1,2, where
(41)Ξ11=Ω11+Ω11T,Ω11=AiX1+BWj+AjX1+BWi2,Ξ12=BWj+BWi2,Φ22=Ω22+Ω22T,Ω22=P2Ai−YiC+P2Aj−YjC2,Ξ13=Di+Dj2,Φ23=P2Di+P2Dj2.Note thatX1 instead of P1 is formulated in the constraint conditions, thus, the term c/αλminP should be modified as follows to solve the minimization problem. It is observed that c/αλminP<ε2 if c/αI<λminPε2I<ε2P, that is,
(42)cαI00cαI<ε2P100ε2P2,which is equivalent to
(43)X1<ε2αcI,and
(44)cε2αI<P2.Therefore, we can summarize the theorem for theL∞-gain performance as follows.Theorem 2.
The closed-loop system (38) has the L∞-gain performance
(45)∥ηt∥∞<γ∣η0∣+ε∥w¯t∥∞,if for fixed scalars α>0 and β>0, there exist two positive-definite matrices X1∈ℝ4×4andP2∈ℝ4×4, and two matrices Wi,Yi and i∈1,2, the following minimization problem is solved,
(46)minX1,P2,Wi,Yi,β,α,cε2s.t.X1=X1T>0,P2=P2T>0,Wi,Yi,β>0α>0,c>0,Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,X1<ε2αcI,cε2aI<P2.Design procedure:Step 1.
Construct the fuzzy plant rules in (9) and the observer rules in (12).Step 2.
Take a set of positive scalarsα>0,β>0,andc>0 iteratively.Step 3.
Given an initialε2.Step 4.
Solve the following linear matrix inequality problem(47)Ξ11+αX1−Ξ12Ξ13⋆−βX10⋆⋆−c2I<0,Φ22+αP2IΦ23⋆−β−1X10⋆⋆−c2I<0,X1<ε2αcI,cε2αI<P2,
to obtainX1,P2,Wi,andYi (thus Ki=WiX1−1 and Li=P2−1Yi can also be derived).Step 5.
Decreaseε2 and repeat Steps 4-5 until X1>0 and P2>0 cannot be found. If X1>0 and P2>0 cannot be found for all possible ε2, try another set of α>0,β>0,andc>0 iteratively and repeat Steps 3–5.Step 6.
Construct the fuzzy observer (12).Step 7.
Construct the fuzzy controller (13).


## 3. Numerical Simulations

The numerical simulations are carried out using the fourth-order Rune-Kutta method (via ode45 in MATLAB). Consider the memristive Chua circuit system (2) with the parameters σ1=4,σ2=1, σ3=0.65, a=0.2 and b=10. Then, the overall fuzzy system to be stabilized can be represented as (9) and the corresponding fuzzy observer is given by (12). By Theorem 1, we need to verify (31) and (32). Take α=0.0003>0 and β=11.48>0. Using MATLAB to solve the LMIs (31) and (32), we can obtain a feasible solution as follows
(48)X1=136.959255.57891.1645−19.913255.578974.503438.5401−46.35941.164538.540144.8794−62.5622−19.9132−46.3594−62.5622155.0652>0,and
(49)P2=13.9891−3.41070.2092−26.6232−3.410711.5482−5.6816−58.00150.2092−5.68166.35450.0344−26.6232−58.00150.0344950.4500>0.Consequently, we can obtain the gains(50)K1=W1X1−1=−2.65363.1573−8.4596−2.6001,K2=W2X1−1=36.54643.1573−8.4596−2.6001,L1=P2−1Y1=150.7265197.9643162.827716.4382T,L2=P2−1Y2=350.5547528.3090451.475642.1846T.Therefore, all conditions of Theorem1 are satisfied, which means that the closed-loop system (16) asymptotically converges to the origin. By observation, the initial states of the original Chua system and the observer are set as 0.1,0.2,0.3,0.4T and 0,0,0,0T. Figure 3 shows the time evolution of the control input ut. Figure 4 shows the time evolutions between the state xit and its estimation x^it, i=1,2,3,4. It is revealed from Figure 4 that the states xit (i=1,2,3,4) are estimated, and converge to the origin, which implies that the memristive Chua system is stabilized.Figure 3
Time response of the control input under observer-based fuzzy control.Figure 4
Time responses of the state variables under observer-based fuzzy control.
(a)
x1 and x^1
(b)
x2 and x^2
(c)
x3 and x^3
(d)
x4 and x^4Next, we consider the memristive Chua system (35) with external disturbances with wt as the bounded external disturbances with zero mean and variance 0.52. The disturbance matrices are taken as D1=D2=0.110.120.130.14T and apply the same fuzzy observer in (12). Set α=10>0, β=13>0, and c=1. Then, using MATLAB to solve the LMIs in Theorem 2, we can solve the minimization problem in Theorem 2 and obtain the minimum value ε=7.99. Consequently, we can obtain the gains
(51)K1=W1X1−1=−12.80457.2009−8.1336−1.5516,K2=W2X1−1=−12.058334.1065−27.8910−2.8660,L1=P2−1Y1=103×0.75513.22924.64850.0185T,L2=P2−1Y2=103×0.93164.05385.83660.0200T.Therefore, all conditions of Theorem2 are satisfied, which means the closed-loop system (38) with external disturbances asymptotically converges to the origin. To see the simulation easily, the initial states of the original Chua system and the observer-based control are set as −1.0,0.8,2.0,0.4T and 0,0,0,0. Figure 5 shows the time evolution of the control input ut. Figure 6 shows the time evolutions between the state xit and its estimation x^it, i=1,2,3,4. Initial time evolutions are shown in the subfigures. It is revealed from Figure 6 that the states xit (i=1,2,3,4) are estimated, and converge to the origin, which implies that the memristive Chua system with external disturbances is stabilized.Figure 5
Time response of the control input for (35).Figure 6
Time responses of the state variables in (35) under observer-based fuzzy control.
(a)
x1 and x^1
(b)
x2 and x^2
(c)
x3 and x^3
(d)
x4 and x^4


## 4. Conclusion

In summary, we provide the fuzzy model for the memristive Chua system and an observer-based fuzzy control scheme to stabilize the system. Sufficient conditions that are easily verified based on LMI have been derived, with which the observer-based fuzzy controllers can be designed conveniently. TheL∞-gain observer-based fuzzy control design for the chaotic system with disturbances has also been discussed. Finally, numerical simulations have been carried out to show the effectiveness of the proposed method for one system with chaotic behavior used as an example.



---

*Source: 8731048-2018-09-24.xml*