# A Novel Fast Convergence Control Scheme for a Class of 3D Chaotic Systems with Uncertain Parameters and External Disturbances

Authors: Haipeng Su; Runzi Luo; Ling Xu; Meichun Huang; Jiaojiao Fu
Journal: Complexity
 (2020)

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## Abstract

This paper studies the control of a class of 3D chaotic systems with uncertain parameters and external disturbances. A new method which is referred as the analytical solution approach is firstly proposed for constructing Lyapunov function. Then, for suppressing the trajectories of the 3D chaotic system to its equilibrium point00,0,0, a novel fast convergence controller containing parameter λ which determines the convergence rate of the system is presented. By using the designed Lyapunov function, the stability of the closed-loop system is proved via the Lyapunov stability theorem. Computer simulations are employed to a new chaotic system to illustrate the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Chaos is a very fascinating phenomenon which often appears in some nonlinear systems of physics, mathematics, psychology, engineering, and so on. A system is called as the chaotic system, provided that it exhibits chaos phenomenon. Due to its powerful potential applications in secure communications, biological systems, information processing, etc., the control of chaotic system has attracted extensive attention of scholars in recent 20 years, and many efficient approaches have been presented for controlling chaos, such as OGY control [1], impulsive control [2], fuzzy control [3, 4], sliding mode control [5, 6], adaptive control [7, 8], composite learning control [9, 10], and so on. Nowadays, many papers about controlling chaotic systems have been published. For instance, in [11], an adaptive control scheme is presented to realize the control and synchronization of the uncertain Liu chaotic dynamical system. In papers [12, 13], the control of the unified chaotic system is investigated by using the output feedback control strategy and the passivity-based control method, respectively. The authors in paper [14] propose an adaptive control approach for controlling the chaotic power systems via the passivity-based control method and finite-time stability theory. In paper [15], a conventional adaptive controller is presented for stabilizing the uncertain chaotic Zhang system via a Lyapunov-like approach. Paper [16] discusses the control of fractional-order nonlinear systems by the composite learning adaptive method. A fractional dynamic surface is introduced to avoid the explosion of complexity of the designed controller. The adaptive fuzzy backstepping control for a class of uncertain fractional-order nonlinear systems with unknown external disturbances is investigated in [17], where the stability is analyzed by using the Lyapunov function.It is easy to see that these control schemes presented in papers [11–15] are only valid for one kind of chaotic system, which limits their range of application. In addition, the proposed methods do not consider the effect of uncertain parameters and external disturbances which exist widely in practical systems. Moreover, the controllers proposed in papers [11–17] do not contain the parameters that can change the convergence rate of the controlled system which means once the controller is given, the convergence rate of the controlled system is fixed. From the practical viewpoint of application, it is desired that the chaos control or chaos synchronization process can be accomplished quickly in a short time. Thus, how to design a controller such that the process of chaos control or synchronization can be completed quickly in a short time is an important issue.Motivated by the above discussions, in this paper we investigate the control of a class of 3D chaotic systems with uncertain parameters and external disturbances. A novel fast convergence control scheme for controlling 3D chaotic system to its equilibrium point00,0,0 is presented. Numerical simulations verify the effectiveness of the presented method.The main contributions of this paper can be summarized into three aspects. First, a new method which is referred as the analytical solution approach is proposed. This technique has two advantages: (a) the rate of the convergence can be known by the designer; (b) a Lyapunov function for the controlled system can be easily constructed. Second, a new 3D chaotic model with fractional power terms of statexi is proposed. Third, a novel controller for controlling 3D chaotic system to its equilibrium point 00,0,0 is presented. The proposed controller contains parameter λ which determines the convergence rate of the system. In general, the larger the value of λ, the faster the rate of convergence.The rest of the paper is organized as follows. In the next section, the problem formulation and assumption are introduced. The main theoretical results are presented in Section3. In Section 4, the control of a novel chaotic attractor and its simulation results are given. Concluding remarks are finally given Section 5.


## 2. The Problem Formulation

Consider the following chaotic systems with three inputs described by(1)x˙1=f1x+α1Tg1x+d1+u1,x˙2=f2x+α2Tg2x+d2+u2,x˙3=f3x+α3Tg3x+d3+u3,where f1x=a11x1m11+a12x2m12,f2x=a21x1m21+a22x2m22+a23x1m23x2m24+a24x3m25, f3x is a continuous function. α1=α11,α12,…,α1m1T, α2=α21,α22,…,α2m2T, and α3=α31,α32,…,α3m3T are unknown parameter vectors. g1x=g11x,g11x,…,g1m1xT, g2x=g21x,g22x,…,g2m2xT, and g3x=g31x,g32x,…,g3m3xT are continuous functions, m1,m2,m3 are positive integers. di is external disturbance, ui is the controller, i=1,2,3. In addition, a11,a12≠0,a21,a22,a23,a24≠0 are parameters. m11,m12=p1/q1,m21,m22,m23,m24,m25=p2/q2 are positive constants. p1,p2,q1,q2 are positive odd numbers.The control aim of this paper is to design some adaptive control laws, in the presence of the uncertain parameters and unknown disturbances, such that the states of system (1) are asymptotically stable for any given initial conditions, i.e., limt⟶∞x1=limt⟶∞x2=limt⟶∞x3=0.Before proceeding to the main results, we make the following Assumption1.Assumption 1.
There exists a positive numberM>0 such that di≤M, i=1,2,3.


## 3. The Main Results

In order to obtain some adaptive controllers, we first consider the control problem of the following system:(2)x˙1=f1x,x˙2=f2x,x˙3=u0,where u0 is the controller to be designed such that the equilibrium point 0,0,0 of system (2) is asymptotically stable.The purpose of considering system (2) firstly is that through designing controller u0, one can get the analytical solution of system (2). By using the obtained analytical solution, we can easily construct the Lyapunov function which is helpful in proving Theorem 2.Theorem 1.
The equilibrium point0,0,0 of system (2) is asymptotically stable if u0 is chosen as(3)u0=−λ3w1/2−λ3+3λ2−3λ−1w2/2−λ3−3λ2−3λ+1w3/2a12m12x2m12−1a24m25x1m25−1−a12m12m12−1x2m12−2f2x2+a11m11m11−1f1x2a12m12x2m12−1a24m25x3m25−1−a11m11x1m11−1a11m11x1m11−1f1x+a12m12x2m12−1f2xa24m25x3m25−1−a21m21x1m21−1+a23m23x1m23−1x2m24f1xa24m25x3m25−1−a22m22x2m22−1+a23m24x1m23x2m24−1f2xa24m25x3m25−1,where λ<0 and(4)w1=x111a11x1m11+a12x2m12λ+1λ−1ρλ2+2λ−1λ2−2λ−1,w2=1x11λa11x1m11+a12x2m12λ−1λ2ρλ2−2λ−1,w3=11x1λλ+1a11x1m11+a12x2m12λ2λ2+2λ−1ρ,ρ=a11m11x1m11−1f1x+a12m12x2m12−1f2x.Proof.
Theorem1 will be proved by constructing the exact analytical solutions of system (2).
Let(5)x1=eλt1+cost+sint,be a solution of system (2), where λ<0. Obviously, we have limt⟶∞x2=0.
Taking the derivative on both sides of equation (5), we have(6)x˙1=λeλt1+cost+sint+eλtcost−sint=eλtλ+λ+1cost+λ−1sint.
Sincex˙1=f1x, we get(7)eλtλ+λ+1cost+λ−1sint=a11x1m11+a12x2m12.
Then, we obtain(8)a12x2m12=eλtλ+λ+1cost+λ−1sint−a11x1m11.
Taking the derivative on both sides of the above equation, it yields(9)a12m12x2m12−1x˙2=eλtλ2+λ2+2λ−1cost+λ2−2λ−1sint−a11m11x1m11−1a11x1m11+a12x2m12.i.e.,(10)eλtλ2+λ2+2λ−1cost+λ2−2λ−1sint=a11m11x1m11−1a11x1m11+a12x2m12+a12m12x2m12−1a21x1m21+a22x2m22+a23x1m23x2m24+a24x3m25.
According to (5), (7), and (10), we have(11)eλt1+cost+sint=x1,eλtλ+λ+1cost+λ−1sint=a11x1m11+a12x2m12,eλtλ2+λ2+2λ−1cost+λ2−2λ−1sint=ρ.
By using Cramer’s rule, the solution of system (11) is easily obtained as(12)eλt=−w12,costeλt=−w22,sinteλt=−w32.
Calculating the derivative on both sides of equation (10), we derive(13)eλtλ3+λ3+3λ2−3λ−1cost+λ3−3λ2−3λ+1sint=a11m11m11−1x1m11−2f1x2+a11m11x1m11−1a11m11x1m11−1f1x+a12m12x2m12−1f2x+a12m12m12−1x2m12−2f2x2+a12m12x2m12−1a21m21x1m21−1+a23m23x1m23−1x2m24f1x+a22m22x2m22−1+a23m24x1m23x2m24−1f2x+a24m25x3m25−1x˙3.
Based on equation (13), we get(14)x˙3=u0,where(15)u0=eλtλ3+λ3+3λ2−3λ−1cost+λ3−3λ2−3λ+1sinta12m12x2m12−1a24m25x3m25−1−a12m12m12−1x2m12−2f2x2+a11m11m11−1x1m11−2f1x2a12m12x2m12−1a24m25x3m25−1−a11m11x1m11−1a11m11x1m11−1f1x+a12m12x2m12−1f2xa24m25x3m25−1−a21m21x1m21−1+a23m23x1m23−1x2m24f1xa24m25x3m25−1−a22m22x2m22−1+a23m24x1m23x2m24−1f2xa24m25x3m25−1=−λ3w1/2−λ3+3λ2−3λ−1w2/2−λ3−3λ2−3λ+1w3/2a12m12x2m12−1a24m25x1m25−1−a12m12m12−1x2m12−2f2x2+a11m11m11−1x1m11−2f1x2a12m12x2m12−1a24m25x3m25−1−a11m11x1m11−1a11m11x1m11−1f1x+a12m12x2m12−1f2xa24m25x3m25−1−a21m21x1m21−1+a23m23x1m23−1x2m24f1xa24m25x3m25−1−a22m22x2m22−1+a23m24x1m23x2m24−1f2xa24m25x3m25−1.
According to (14), we know that if u0 is taken as (15), then equation (11) is the solution of system (2). From (11), it is obvious that the equilibrium point 0,0,0 of system (2) is asymptotically stable.Remark 1.
Theorem1 is proved without using the Lyapunov stability theory. In fact, Theorem 1 can also be proved by using the Lyapunov stability theory. To show this, we choose the following function as the Lyapunov candidate:(16)V0=w12λ+w224+w329.
This is becauseV0 can be rewritten as(17)V0=4e2λtλ+costeλt22+sinteλt32=4e2λt1λ+cos2t4+sin2t9.
The derivative ofV0 along system (2) is(18)V˙0=8λe2λt1λ+cos2t4+sin2t9+4e2λt−sin2t4+sin2t9=4e2λt−2+2λcos2t4+sin2t9−sin2t4+sin2t9.
Sinceλ<0, we have V˙0<0. According the Lyapunov stability theory, we derive limt⟶∞w1=limt⟶∞w2=limt⟶∞w3=0 which implies that the equilibrium point 0,0,0 of system (2) is asymptotically stable.Remark 2.
It is easy to see that the Lyapunov functionV0 is constructed according to the analytical solutions. So, in this paper, the new method of obtaining Lyapunov function is referred as the analytical solution approach. This new technique can be applied to other chaotic systems for designing Lyapunov function.Remark 3.
By (11), one can see that system (2) is globally exponentially stable at the origin and the rate of the convergence is determined by λ. The larger the value of λ, the faster the speed of convergence.
With the help ofV0 described by (17), we are now in a position to discuss the control of system (1).Theorem 2.
Suppose that Assumption1 holds. If controllers u1,u2,u3 are taken as(19)u1=−α¯1Tg1x−Msign∂V0∂x1,u2=−α¯2Tg2x−Msign∂V0∂x2,u3=−f3x+u0−α¯3Tg3x−Msign∂V0∂x3,with the update laws:(20)α¯˙1=∂V0∂x1g1x,α¯˙2=∂V0∂x2g2x,α¯˙3=∂V0∂x3g3x,Then, the equilibrium point 0,0,0 of system (1) is asymptotically stable, where α¯1,α¯2,α¯3 are the estimated values of α1,α2,α3, respectively.Proof.
Plugging (19) into system (1) yields(21)x˙1=f1x+α1−α¯1Tg1x+d1−Msign∂V0∂x1,x˙2=f2x+α2−α¯2Tg2x+d2−Msign∂V0∂x2,x˙3=u0+α3−α¯3Tg3x+d3−Msign∂V0∂x3.
Now, consider the following Lyapunov candidate:(22)V=V0+12α1−α¯1Tα1−α¯1+α2−α¯2Tα2−α¯2+α3−α¯3Tα3−α¯3.
The derivative of (22) along system (21) is(23)V˙=∂V0∂x1x˙1+∂V0∂x2x˙2+∂V0∂x3x˙3−α1−α¯1Tα¯˙1−α2−α¯2Tα¯˙2−α3−α¯3Tα¯˙3=∂V0∂x1f1x+α1−α¯1Tg1x+d1−Msign∂V0∂x1+∂V0∂x2f2x+α2−α¯2Tg2x+d2−Msign∂V0∂x2+∂V0∂x3u0+α3−α¯3Tg3x+d3−Msign∂V0∂x3−α1−α¯1Tα¯˙1−α2−α¯2Tα¯˙2−α3−α¯3Tα¯˙3=∂V0∂x1f1x+∂V0∂x2f2x+∂V0∂x3u0+α1−α¯1T∂V0∂x1g1x−α¯˙1+α2−α¯2T∂V0∂x2g2x−α¯˙2+α3−α¯3T∂V0∂x3g3x−α¯˙3+∂V0∂x1d1−Msign∂V0∂x1+∂V0∂x2d2−Msign∂V0∂x2+∂V0∂x3d3−Msign∂V0∂x3.
By Assumption1, we know that di≤M. Thus, we have(24)∂V0∂x1d1−Msign∂V0∂x1+∂V0∂x2d2−Msign∂V0∂x2+∂V0∂x3d3−Msign∂V0∂x3≤0.
According to update laws (20), we get(25)α1−α¯1T∂V0∂x1g1x−α¯˙1+α2−α¯2T∂V0∂x2g2x−α¯˙2+α3−α¯3T∂V0∂x3g3x−α¯˙3=0.
In view of Remark1, we obtain(26)∂V0∂x1f1x+∂V0∂x2f2x+∂V0∂x3u0<0.
Substituting (24), (25), and (26) into (23), one derives that(27)V˙<0.
Based on the Lyapunov stability theory, we concludelimt⟶∞w1=limt⟶∞w2=limt⟶∞w3=0, which implies that the equilibrium point 0,0,0 of system (1) is asymptotically stable.Remark 4.
Note thatλ determines the speed of the convergence of system (2) and system (2) is the main part of system (1); therefore, λ determines the speed of the convergence of system (1). The simulation results show that the smaller the value of λ, the faster the speed of convergence.


## 4. The Control of an Novel Chaotic Attractor and its Simulation Results

Consider the following new three-dimensional chaotic system [18]:(28)x˙1=x2−ax1+bx2x3,x˙2=cx2−x1x3+x3,x˙3=dx1x2−hx3,where x1,x2,x3T∈R3 is the state vector, and a,b,c,d and h are positive real constants. It is reported that this chaotic system has many interesting complex dynamical behaviors [18]. The new two-wing chaotic attractor of system (28) with a=3,b=2.7,c=4.7,d=2 and h=9 is shown in Figure 1. In the following numerical process, we assume a=3,b=2.7,c=4.7,d=2, and h=9. For simplicity, we suppose that b and d are unknown parameter.Figure 1
The chaos attractor of system (28) with x10=5,x20=0,x30=−4.If system (28) is affected by external disturbances, then system (28) with three inputs can be rewritten as(29)x˙1=x2−ax1+bx2x3+sinx2x1+u1,x˙2=cx2+θx1x3+x3+cosx2x3+u2,x˙3=dx1x2−hx3+2sinx1cosx3+u3,where θ=−1 and u1,u2,u3 are three controllers.Compared with system (1), we know that m11,m12,m21,m22,m23,m24,m25=1,1,1,1,1,1,1.a11=−a,a12=1,a21=0,a22=c,a23=0,a24=1. α11=b,α21=θ,α31=d.g11x=x2x3,g21x=x1x3,g31xx1x2,f3x=−hx3. The external disturbances are d1=sinx2x1,d2=cosx2x3,d3=2sinx1cosx3.The reason why we introduceθ is that in f2x of system (1) there is no term x2x3. So, in order to use the results obtained in the above section, we assume θ is the unknown parameter.Based on system (2), we derive the following auxiliary system:(30)x˙1=x2−3x1,x˙2=4.7x2+x3,x˙3=u0,Now, for convenience of comparison, we consider two cases:λ=−1 and λ=−10.Case 1.
λ
=
−
1.
According to Theorem2, in this case, one can obtain(31)w1=−10x1−37x25−2x3,w2=6x1+2.7x2+x3,w3=2x1+4.7x2+x3,u0=−122609x250−20x1+47x35,V01=10x1+37x25+2x32+146x1+2.7x2+x32+192x1+4.7x2+x32.
Therefore, the controllers can be taken as(32)u1=−b¯x2x3−2sign∂V01∂x1,u2=−θ¯x1x3−2sign∂V01∂x2,u3=9x3+u0−d¯x1x2−2sign∂V01∂x3,with the update laws:(33)b¯˙=∂V01∂x1x2x3,θ¯˙=∂V01∂x2x1x3,d¯˙=∂V01∂x3x2x1,where b¯,θ¯,d¯ are the estimated vales of b,θ,d, respectively.Case 2.
λ
=
−
10.
Based on Theorem2, in this case, we have(34)w1=−100x1−217x25−2x3,w2=42x1+20.7x2+x3,w3=56x1+22.7x2+x3,u0=−350x1−368.99x2−31.7x3,V02=−100x1−217x2/5−2x3210+42x1+20.7x2+x324+56x1+22.7x2+x329.
Therefore, the controllers can be taken as(35)u1=−b¯x2x3−2sign∂V02∂x1,u2=−θ¯x1x3−2sign∂V02∂x2,u3=9x3+u0−d¯x1x2−2sign∂V02∂x3,with the update laws:(36)b¯˙=∂V02∂x1x2x3,θ¯˙=∂V02∂x2x1x3,d¯˙=∂V02∂x3x2x1,where b¯,θ¯,d¯ are the estimated vales of b,θ,d, respectively.
According to Theorem2, we know that in the two cases, the equilibrium point 0,0,0 of system (29) is asymptotically stable. The Figures 2–7 show the simulation results with λ=−1 and λ=−10. The initial values are taken as x10,x20,x30=5,0,−4, b¯0,θ¯0,d¯0=1,1,1.
From Figures2–4, it can be seen that when λ=−1, the states x1,x2,x3 need about 2.5 s to reach the equilibrium point. However, for λ=−10, the states x1,x2,x3 only need about 0.6 s to reach the equilibrium point, which means that the convergence time is greatly shortened compared with that of λ=−1. Therefore, one can conclude that the larger the value of λ, the faster the rate of convergence. In addition, the vibration amplitude of λ=−10 is smaller than that of λ=−1 which implies that the vibration amplitude is decreased with the increase of the absolute value of λ.Figure 2
The time response of statex1 of system (29) with x10=5,x20=0,x30=−4.Figure 3
The time response of statex2 of system (29) with x10=5,x20=0,x30=−4.Figure 4
The time response of statex3 of system (29) with x10=5,x20=0,x30=−4.Figure 5
The time response of stateb¯ of system (29) with x10=5,x20=0,x30=−4.Figure 6
The time response of stateθ¯ of system (29) with x10=5,x20=0,x30=−4.Figure 7
The time response of stated¯ of system (29) with x10=5,x20=0,x30=−4.


## 5. Conclusions

The control of a class of 3D chaotic systems with uncertain parameters and external disturbances via an adaptive control method has been systematically investigated in this paper. Through designing the analytical solution of the chaotic system, a proper Lyapunov function can be easily obtained. This method of constructing Lyapunov function by using the analytical solution can be applied to other chaotic systems. By using the designed Lyapunov function, a novel fast convergence controller containing parameterλ which determines the convergence rate of the system is presented. In addition, as shown in Section 4, since the terms that are not included in functions f1x,f2x,f3x can be considered to contain uncertain parameters, the control scheme presented in this paper can be employed to any 3D continuous chaotic systems. Finally, the effectiveness of the theoretical results is illustrated by some numerical simulations.In this paper, only the integer-order chaotic system, which is a special case of fractional-order system, has been considered. It is well known that compared with integer-order derivatives, the fractional derivatives has many distinguishing features such as the long-term memory and self-similarity which can be used for the description of memory and hereditary properties of various materials and processes. Thus, the control and synchronization of fractional-order chaotic systems via the analytical solution approach is an important issue. This issue will be our research focus in the future.



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