# Adaptive Neural Networks Synchronization of a Four-Dimensional Energy Resource Stochastic System

Authors: Duo Meng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

An adaptive neural networks chaos synchronization control method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assuming the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on stochastic Lyapunov stability and robust adaptive theories, an adaptive neural networks synchronization control method is developed. In the design process, an auxiliary design system is employed to address input constraints. Simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results.


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## Body

## 1. Introduction

Energy resource system is a kind of complex nonlinear system. Over the last two decades, much attention has been paid to the chaos synchronization in this class system. Reference [1] established a three-dimensional energy resource demand-supply system based on the real energy resources demand-supply system in the East and the West of China. By adding a new variable to consider the renewable resources, a four-dimensional energy resource system was proposed in [2]. The dynamics behaviors of the four-dimensional energy resource system have been analyzed by means of the Lyapunov exponents and bifurcation diagrams. Also the same as the above-mentioned power systems, this four-dimensional energy resource system is with rich chaos behaviors. The problem of chaotic control for the energy resource system was considered in [3]. Feedback control and adaptive control methods were used to suppress chaos to unstable equilibrium or unstable periodic orbits, where only three of the system’s parameters were supposed to be unknown. Reference [4] investigated the robust chaos synchronization problem for the four-dimensional energy resource systems based on the sliding mode control technique. The control of energy resource chaotic system was investigated by time-delayed feedback control method in [5]. Four linear control schemes are proposed to a four-dimensional energy resource system in [6]. Based on stability criterion of linear system and Lyapunov stability theory, respectively, the chaos synchronization problems for energy resource demand-supply system were discussed using two novel different control methods in [7].In many practical dynamic systems (including the energy resource demand-supply system), physical input saturation on hardware dictates that the magnitude of the control signal is always constrained. Saturation is a potential problem for actuators of control systems. It often severely limits system performance, giving rise to undesirable inaccuracy or leading instability [8, 9]. The development of control schemes for systems with input saturation has been a task of major practical interest as well as theoretical significance. The proposed approaches in [1–7] assume that all the components of the considered energy resource demand-supply systems are in good operating conditions and do not consider the problem of saturation. Reference [10] proposed two different chaos synchronization methods for a class of energy resource demand-supply systems with input saturation, but the response system in [10] did not contain unknown uncertain nonlinearities and unknown stochastic disturbances. It is well known that stochastic disturbances also often exist in many practical systems. Their existence is a source of instability of the control systems; thus, the investigations on stochastic control systems have received considerable attention in recent years [11–22]. Since the emergence of the stochastic stabilization theory in the 1960s, the progress has been constructed by a fundamental technical Itô lemma, and the control design for stochastic systems is more difficult compared with deterministic systems.Motivated by the above observations, an adaptive neural networks chaos synchronization method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assume that the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances. In the design, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on Lyapunov stability, an adaptive synchronization method is developed in order to make the states of two chaotic systems asymptotically synchronized. The new auxiliary design system is employed to address input constraints. Numerical simulations are provided to illustrate the effectiveness of the proposed approach.Compared with the existing results, the main contributions of the proposed method are as follows: (i) the controlled response system of this paper contains unknown nonlinearities, and the proposed method can solve the unknown nonlinearity problem by neural networks, but the methods of [1–8, 10] cannot solve this problem; (ii) the controlled response system of this paper contains stochastic disturbances, and the proposed method can solve the stochastic disturbances problem based on Itô’s lemma and stochastic LaSalle’s theorem, but the methods of [1–8, 10] cannot solve this problem; (iii) an auxiliary design system is employed to address input constraints problem, and the methods of [1–8] can solve this problem.


## 2. Energy Resource Chaotic System

The four-dimensional energy resource system can be expressed as follows (see [2, 4, 6]):
(1)x˙=a1x(1-xM)-a2(y+z)-d3w,y˙=-b1y-b2z+b3x[N-(x-z)],z˙=c1z(c2x-c3),w˙=d1x-d2w,
where x(t) is the energy resource shortage in A region, y(t) is the energy resource supply increment in B region, and z(t) and w(t) are energy resource import in A region and renewable energy resource in A region, respectively; M, N, ai, bj, cj, and dj(i=1,2,j=1,2,3) are parameters that are all positive real. The dynamics of this system has been extensively studied in [2, 4, 6].When the system parameters are taken as the following values, this system exhibits chaotic behavior:M=1.8, N=1, a1=0.1, a2=0.15, b1=0.06, b2=0.082, b3=0.07, c1=0.2, c2=0.5, c3=0.4, d1=0.1, d2=0.06, and d3=0.07. Without the particular statement, these values are adopted in this whole paper. Figures 1, 2, and 3 show the phase portraits with initial conditions of x(0)=0.82, y(0)=0.29, z(0)=0.48, and w(0)=0.1.Figure 1
Three-dimensional viewx-y-z.Figure 2
Three-dimensional viewy-x-w.Figure 3
Three-dimensional vieww-z-x.


## 3. Synchronization of the Energy Resource System

In this section, a controller will be designed in order to make the response system track the drive system. The drive system with subscript 1 is written as(2)x˙1=a1x1(1-x1M)-a2(y1+z1)-d3w1,y˙1=-b1y1-b2z1+b3x1[N-(x1-z1)],z˙1=c1z1(c2x1-c3),w˙1=d1x1-d2w1.
Assume that the controlled response system with subscript 2 contained uncertain nonlinearities (unknown smooth nonlinear functions) and unknown external stochastic disturbance, and it can be expressed as the following dynamics:
(3)dx2=[a1x2(1-x2M)-a2(y2+z2)-d3w2ff+f1(x2)+u1(v1(t))(1-x2M)]dt+p1(e1)dW,dy2=[-b1y2-b2z2+b3x2[N-(x2-z2)]f0f+f2(y2)+u2(v2(t))]dt+p2(e2)dW,dz2=[c1z2(c2x2-c3)+f3(z2)+u3(v3(t))]dtff+p3(e3)dW,dw2=[d1x2-d2w2+f4(w2)+u4(v4(t))]dtff+p4(e4)dW,
where vi is the actual controller to be designed and ui(vi(t))(i=1,2,3,4) is the plant input subject to saturation type nonlinearly. pi(ei), i=1,2,3,4, are uncertain functions, and W∈Rn is an independent standard Brownian motion defined on a complete probability space, with the incremental covariance E{dW·dWT}=σ(t)σ(t)Tdt.Remark 1.
If no input saturation, uncertain nonlinearities, and unknown external stochastic disturbance (i.e.,ui(vi(t))=vi(t)), pi(t)=0, and fi(·)=0(i=1,2,3,4)) are included in (3), then (3) becomes the chaotic systems studied widely, see [8, 10], where ui(vi(t)) can be described as(4)ui(vi(t))=sat⁡(vi(t))={sign⁡(vi(t))uiM,|vi(t)|≥uiM,vi(t),|vi(t)|<uiM,
where uiM is a known bound of ui(vi(t)).To design an adaptive controller, the following basic assumption is made for the system (3).Assumption 2.
The disturbance covariancePTσσTP≤∑i=14eiσ-i(ei), where P=[p1(e1)p2(e2)p3(e3)p4(e4)]T and σ-i(ei) is a known function.To establish stochastic stability as a preliminary, consider a stochastic nonlinear system:(5)dx=f(x,t)dt+g(x,t)dW,
where x∈Rn, W is an independent r-dimensional Wiener process, defined on the probability space (Ω,F,P), with the incremental covariance E{dW,dWT=σ(t)σ(t)Tde}, and f:Rn×R→Rn and f:Rn×R→Rn×r are locally Lipschitz continuous in x, uniformly in t∈R, with f(0,t)=0 and g(0,t)=0, for all t≥0.Lemma 3 (see [16, 17] (stochastic LaSalle’s theorem)).
Consider (5) and suppose that there exists a twice continuously differentiable function V(x,t), which is positive definite, decrescent, and radially unbounded, and another nonnegative continuous function Q(x)≥0 such that the infinitesimal generator V(x,t) along (5) satisfies
(6)ℓV:=∂V∂t+∂V∂xf(x)+12Tr⁡{σTgT∂2V∂x2gσ}≤-Q(x),000000000000000000000000000000000∀x∈Rn,t≥0,
where Tr⁡ denotes the matrix trace. Then, the equilibrium x=0 is globally stable in probability and
(7)P{lim⁡t→∞Q(x(t))=0}=1,∀x(0)∈Rn.In order to solve the unknown nonlinearfi(·)(i=1,2,3,4), the following radial basis function neural networks (RBFNNs) [23] are used to identify them similar to fuzzy logic systems [24–27].An RBFNN can approximate a continuous functionh(X): Rq→R,
(8)hnn(X)=WTφ(X),
where the input vector X∈Ω⊂Rq, weight vector W=[W1,…,Wm]T∈Rm, the NN node number m>1, and φ(X)=[φ1(X)⋯φm(X)]T, with φi(X) being Gaussian functions, which have the form
(9)φi(X)=exp⁡[-(X-μi)T(X-μi)η2],i=1,2,…,m,
where μi=[μi1,…,μiq]T is the center of the receptive field and η is the width of the Gaussian function.According to the literatures [23], the neural network (8) can approximate any continuous function h(X) over a compact set D⊂Rq to arbitrary any accuracy as
(10)h(X)=W*Tφ(X)+ε(X),∀X∈D,
where W* is an ideal constant weight, ε(X) is the bounded approximation error, and W* is defined as
(11)W*=argmin⁡W∈Ω{sup⁡X∈D|h(X)-WTφ(X)|}.


## 4. Adaptive Synchronization of the Energy Resource System

For different initial conditions of systems (2) and (3), the two coupled systems can achieve synchronization by designing an appropriate control input ui(t). First, we define the synchronization error vector between systems (2) and (3) as
(12)e1=x2-x1-h1,e2=y2-y1-h2,e3=z2-z1-h3,e4=w2-w1-h4,
where hi(i=1,2,3,4) is filter signal and will be given later.From (2), (3), and (12), the error dynamical system can be written as
(13)de1=[-h˙1+a1e1-a2(e2+e3)-a1x22M+a1x12M-d3e4f+a1h1-a2(h2+h3)-d3h4+f1(x2)f+u1(v1(t))a1x22M]dt+p1(e1)dW,de2=[-h˙2-b1e2-b2e3+b3Ne1-b3x22+b3x12+b3x2z2d-b3x1z1-b1h2-b2h3+b3Nh1f+f2(y2)+u2(v2(t))-h˙2]dt+p2(e2)dW,de3=[-h˙3-c1c3e3+c1c2x2z2-c1c2x1z1-c1c3h3f+f3(z2)+u3(v3(t))-h˙3]dt+p3(e3)dW,de4=[-h˙4+d1e1-d2e4+d1h1-d2h4+f4(w2)d+u4(v4(t))-h˙4]dt+p4(e4)dW.
RBFNNs are used to identify fi(·)(i=1,2,3,4), and (13) can be rewritten as
(14)de1=[-h˙1+a1e1-a2(e2+e3)-a1x22M+a1x12M-d3e4d+a1h1-a2(h2+h3)-d3h4+W1*Tφ1(x2)+ε1(x2)d+u1(v1(t))a1x22M]dt+p1(e1)dW,de2=[-h˙2-b1e2-b2e3+b3Ne1-b3x22+b3x12f+b3x2z2-b3x1z1-b1h2-b2h3+b3Nh1f+W2*Tφ2(y2)+ε2(y2)+u2(v2(t))]dt+p2(e2)dW,de3=[-h˙3-c1c3e3+c1c2x2z2-c1c2x1z1-c1c3h3f+W3*Tφ3(z2)+ε3(z2)+u3(v3(t))]dt+p3(e3)dW,de4=[-h˙4+d1e1-d2e4+d1h1-d2h4+W4*Tφ4(w2)f+ε4(w2)+u4(v4(t))-h˙4]dt+p4(e4)dW,
where |εi|≤εi*(i=1,2,3,4) and εi* is a positive constant.In this section, we assume that all the parameters of the energy resource system are unknown. For convenience, similar to [7], we define a3=a1/M, b4=b3N, q1=c1c2, and q2=c1c3; the system (14) can be rewritten as
(15)de1=[-h˙1+a1e1-a2(e2+e3)-a3x22+a3x12-d3e4f+a1h1-a2(h2+h3)-d3h4+W1*Tφ1(x2)+ε1(x2)h+u1(v1(t))-h˙1]dt+p1(e1)dW,de2=[-h˙2-b1e2-b2e3+b4e1-b3x22d+b3x12+b3x2z2-b3x1z1-b1h2d-b2h3+b4h1+W2*Tφ2(y2)+ε2(y2)d+u2(v2(t))-h˙1]dt+p2(e2)dW,de3=[-h˙3-q2e3+q1x2z2-q1x1z1-q2h3+W3*Tφ3(z2)d+ε3(z2)+u3(v3(t))-h˙1]dt+p3(e3)dW,de4=[-h˙4+d1e1-d2e4+d1h1-d2h4+W4*Tφ4(w2)d+ε4(w2)+u4(v4(t))-h˙1]dt+p4(e3)dW.
Define the dynamic system as
(16)h˙i=-hi+(ui-vi),i=1,2,3,4.
Choose the following Lyapunov function candidate V as
(17)V=12(e12+e22+e32+e42)+W~1TW~1+W~2TW~2+W~3TW~3+W~4TW~4+a~12+a~22+a~32+b~12+b~22+b~32+b~42+d~12+d~22+d~32+q~12+q~22,
where a~i=ai-a^i, d~i=di-d^i(i=1,2,3), W~j=Wj*-W^j, b~j=bj-b^j(j=1,2,3,4), and q~k=qk-q^k(k=1,2).Similar to [16, 17], the ℓ infinitesimal generator of V along with the solutions of (15) is
(18)ℓV=e1[h1+v1+a1e^1-a^2(e2+e3)-a^3x22+a^3x12dd-d^3e4+a^1h1-a^2(h2+h3)-d^3h4+p1(t)dd+W^1Tφ1(x2)+ε1(x2)]+e2[h2+v2-b^1e2-b^2e3+b^4e1-b^3x22dddd+b^3x12+b^3x2z2-b^3x1z1-b^1h2-b^2h3dddd+b^4h1+p2(t)+W^2Tφ2(y2)+ε2(y2)]+e3[W^3Tφ3(z2)h3+v3-q^2e3+q^1x2z2-q^1x1z1-q^2h3ffff+p3(t)+W^3Tφ3(z2)+ε3(z2)]+e4[W^4Tφ4(w2)h4+v4+d^1e1-d^2e4+d^1h1-d^2h4ffff+p4(t)+W^4Tφ4(w2)+ε4(w2)]+W~1T[e1φ1(x2)-W^˙1]+W~2T[e2φ2(y2)-W^˙2]+W~3T[e3φ3(z2)-W^˙3]+W~4T[e4φ4(w2)-W^˙4]+a~1[e12+e1h1-a^˙1]+a~2[e1(e2+e3)-e1(h2+h3)-a^˙2]+a~3[-e1x22+e1x12-a^˙3]+b~1[-e22-e2h2-b^˙1]+b~2[-e2e3-e2h3-b^˙2]+b~3[-e2x22+e2x12+e2x2z2-e2x1z1-b^˙3]+b~4[e2e1+e2h1-b^˙4]+d~1[e4e1+e4h4+e4h1-d^˙1]+d~2[-e42-d^˙2]+d~3[-e1(e4+h4)-d^˙3]+q~1[e3x2z2-e3x1z1-q^˙1]+q~2[-e32-e3h3-q^˙2]+∑i=14eiσ-i(ei).
Design the actual controllers vj and parameters update laws of W^j, a^i, d^i(i=1,2,3), b^j(j=1,2,3,4), and q^k(k=1,2) as follows:
(19)v1=-l1e1-σ-1(e1)-h1-a^1e1+a^2(e2+e3)+a^3x22-a^3x12+d^3e4-a^1h1+a^2(h2+h3)+d^3h4-W^1Tφ1(x2)-sgn⁡(e1)(ε1*+α1),(20)v2=-l2e2-σ-2(e2)-h2+b^1e2+b^2e3-b^4e1+b^3x22-b^3x12-b^3x2z2+b^3x1z1+b^1h2+b^2h3-b^4h1-W^2Tφ2(y2)-sgn⁡(e2)(ε2*+α2),(21)v3=-l-3e3-σ-3(e3)-h3+q^2e3-q^1x2z2+q^1x1z1+q^2h3-W^3Tφ3(z2)-sgn⁡(e3)(ε3*+α3),(22)v4=-l4e4-σ-4(e4)-h4-d^1e1+d^2e4-d^1h1+d^2h4-W^4Tφ4(w2)-sgn⁡(e4)(ε4*+α4),
where li(i=1,2,3,4) are positive design parameters. Consider the following:
(23)W^˙1=e1φ1(x2),(24)W^˙2=e2φ2(y2),(25)W^˙3=e3φ3(z2),(26)W^˙4=e4φ4(w2),(27)a^˙1=e12+e1h1,(28)a^˙2=e1(e2+e3)-e1(h2+h3),(29)a^˙3=-e1x22+e1x12,(30)b^˙1=-e22-e2h2,(31)b^˙2=-e2e3-e2h3,(32)b^˙3=-e2x22+e2x12+e2x2z2-e2x1z1,(33)b^˙4=e2e1+e2h1,(34)d^˙1=e4e1+e4h4+e4h1,(35)d^˙2=-e42,(36)d^˙3=-e1(e4+h4),(37)q^˙1=e3x2z2-e3x1z1,(38)q^˙2=-e32-e3h3.
Substituting (19)–(38) into (18) results in
(39)ℓV≤-l1e12-l2e22-l3e32-l4e42.From (39) and Lemma 3, we can conclude that the states x2, y2, z2, and w2 of response system (2) and the states x1, y1, z1, and w1 of drive system (3) are ultimately synchronized asymptotically in probability.


## 5. Simulation Results

In this section, external perturbationspi(ei)=ei; uncertain nonlinear f1(x2)=0.1x22, f2(y2)=0.1y22, f3(z2)=z2, and f4(w2)=w2. Consider α1=α2=α2=α2=0.1 and ε1*=ε2*=ε3*=ε4*=0.1. RBFNNs, W^iTφi(·), contain 25 nodes, with centers evenly spaced in [-4,4] and width is 2. W˙(t) is assumed to be Gaussian white noise with zero mean and variance 1.0.The initial values are chosen asx1(0)=0.1, y1(0)=-0.8, z1(0)=0.2, w1(0)=0.1, x2(0)=0.4, y2(0)=0.1, z2(0)=0.6, and w2(0)=-0.3, and the other initial values are chosen as zeros. The saturation values are u2M=5, u3M=2, and u4M=2. Design parameters in controllers are l1=20, l2=20, l3=20, and l4=20. The simulation results are shown in Figures 4, 5, 6, 7, 8, 9, 10, and 11.Figure 4
The trajectory ofe1.Figure 5
The trajectory ofe2.Figure 6
The trajectory ofe3.Figure 7
The trajectory ofe4.Figure 8
The trajectories ofv1 (solid line) and u1 (dash-dotted line).Figure 9
The trajectories ofv2 (solid line) and u2 (dash-dotted line).Figure 10
The trajectories ofv3 (solid line) and u3 (dash-dotted line).Figure 11
The trajectories ofv4 (solid line) and u4 (dash-dotted line).Remark 4.
It is worth pointing out that the method of [10] cannot be used to control the systems of this paper. There exist three reasons: (i) the system of this paper is four dimensional and the system in [10] is three dimensional; (ii) the system of this paper contains stochastic disturbances, and the system in [10] does not contain them; (iii) the controlled response system of this paper contains unknown nonlinearities, and [10] does not contain them.


## 6. Conclusions

This paper has solved the synchronization problems of a class of unknown parameters four-dimensional energy resource system. The main features of the proposed algorithm are that (i) the problems of the input constraint have been solved by employing a new auxiliary system; (ii) the unknown nonlinearities and stochastic disturbances that existed in the response system have been overcome by the neural networks and some special robust terms, respectively; (iii) the stability of the energy resource demand-supply system has been guaranteed based on stochastic Lyapunov theory.



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