Adaptive Control of Chaos in Chua's Circuit

Abstract

A feedback control method and an adaptive feedback control method are proposed for Chua's circuit chaos system, which is a simple 3D autonomous system. The asymptotical stability is proven with Lyapunov theory for both of the proposed methods, and the system can be dragged to one of its three unstable equilibrium points respectively. Simulation results show that the proposed methods are valid, and control performance is improved through introducing adaptive technology.

1. Introduction

In the last several decades, much effort has been devoted to the study of nonlinear chaotic systems. As more and more knowledge is gained about the nature of chaos, recent interests are now focused on controlling a chaotic system, that is, bringing the chaotic state to an equilibrium point or a small limit cycle.

After the pioneering work on controlling chaos of Ott et al. [1], there have been many other attempts to control chaotic systems. These attempts can be classified into two main streams: the first is parameters’ perturbation that is introduced in [2] and the references therein. The second is feedback control on an original chaotic system [3–8].

Chua’s circuit has been studied extensively as a prototypical electronic system [7–11]. Chen and Dong [4] applied the linear feedback control for guiding the chaotic trajectory of the circuit system to a limit cycle. Hwang et al. [5] proposed a feedback control on a modified Chua’s circuit to drag the chaotic trajectory to its fixed points. He et al. [10] proposed an adaptive tracking control for a class of Chua's chaotic systems. At the same time, the adaptive control technology for chaos systems has undergone rapid developments (see [6, 7, 10] and the references therein) in the past decade.

The aim of this paper is to introduce a simple, smooth, and adaptive controller for resolving the control problems of Chua’s circuit system. It is assumed that one state variable is availablefor implementing the feedback controller. In Section 2, Chua’s circuit system model is built and its three equilibrium points are analyzed. In Section 3, a feedback control approach and an adaptive feedback control approach are proposed. In Section 4, the numerical simulations are presented for two proposed control approaches. Section 5 is the conclusion.

2. Chua’s Circuit Modeling

Chua’s circuit is a well-known electronic system, which displays very rich and typical bifurcation and chaos phenomena such as double scroll, dual double scroll, and double hook, and so forth. The Chua’s circuit is illustrated Figure 1. In the circuit, there are one inductor (𝐿,𝑟 is its inner resistor), two capacitors (𝐶₁ and 𝐶₂), one linear resistor (𝑅), and one piece-linear resistor (𝑅_𝑛), which has the following volt-ampere characteristic: 𝑖𝐺=𝑢=𝑚₁,|𝑢|𝑚<𝐸,₀,|𝑢|>𝐸, (2.1) where, 𝑢 and 𝑖, respectively, are the voltage across 𝑅_𝑛 and the current through 𝑅_𝑛 and 𝐸 is a positive constant. The characteristic of 𝑅_𝑛 is illustrated in Figure 2.

Figure 1 

Chua’s Circuit Model.

Figure 2 

Nonlinear representation of 𝑅𝑛.

According to the circuit theory, the dynamics of Chua’s circuit systems can be obtained:𝐶₁𝑑𝑣𝑑𝑡^𝐶1=𝑣_𝐶2−𝑣_𝐶1𝑅−𝑖𝑣_𝐶1,𝐶₂𝑑𝑣𝑑𝑡^𝐶2=𝑣_𝐶1−𝑣_𝐶2𝑅+𝑖_𝐿,𝐿𝑑𝑖_𝐿𝑑𝑡=−𝑣_𝐶2, (2.2) where 𝑣_𝐶1 and 𝑣_𝐶2 are the voltages across 𝐶₁ and 𝐶₂, respectively, 𝑖_𝐿 is the current through the inductor 𝐿, and𝑖𝑣_𝐶1=𝑚₀𝑣_𝐶1+𝐸𝑚₁−𝑚₀,𝑣_𝐶1𝑚>𝐸,₁𝑣_𝐶1,𝑣_𝐶1𝑚≤𝐸,₀𝑣_𝐶1−𝐸𝑚₁−𝑚₀,𝑣_𝐶1<−𝐸. (2.3)

For Chua’s circuit system described by (2.2) and (2.3), let 𝑥₁=𝑣_𝐶1, 𝑥₂=𝑣_𝐶2, 𝑥₃=𝑅𝑖_𝐿, 𝜏=𝑡/𝑅𝐶₂, 𝑎=𝑚₁𝑅, 𝑏=𝑚₀𝑅, 𝑝=𝐶₂/𝐶₁, and 𝑞=𝑅²𝐶₂/𝐿, where 𝑥₁, 𝑥₂, and 𝑥₃ are system states. We can obtain the system model of Chua’s circuit: 𝑑𝑥₁𝑥𝑑𝜏=𝑝₂−𝑥₁−𝑓𝑥₁,𝑑𝑥₂𝑑𝜏=𝑥₁−𝑥₂+𝑥₃,𝑑𝑥₃𝑑𝜏=−𝑞𝑥₂, (2.4) where the differential is with respect to variable 𝜏 and 𝑓(𝑥₁) is a piece-linear function as: 𝑓𝑥₁=𝑏𝑥₁+𝐸(𝑎−𝑏),𝑥₁>𝐸,𝑎𝑥₁,𝑥₁≤𝐸,𝑏𝑥₁−𝐸(𝑎−𝑏),𝑥₁<−𝐸. (2.5)

Normally due to the piece-linear function, the system described by (2.4) has three equilibrium points, which are denoted by 𝐸(𝑥_1𝑟,𝑥_2𝑟,𝑥_3𝑟)(𝑟=1,2,3). For Chua’s circuit system described by (2.4), the following conclusion holds.

Theorem 2.1 (see [9]). For Chua’s circuit system described by (2.4) and (2.5), its first Lyapunov exponent is positive real number, that is, the system trajectory has some chaotic behaviors.

3. Chaos Control in Chua’s Circuit

When the feedback control is added to the system (2.4), the controlled closed-loop Chua’s circuit system can be written as 𝑥̇₁=𝑝𝑥₂−𝑥₁−𝑓𝑥₁−𝑢₁,𝑥̇₂=𝑥₁−𝑥₂+𝑥₃−𝑢₂,𝑥̇₃=−𝑞𝑥₂−𝑢₃, (3.1) where 𝑢₁, 𝑢₂, and 𝑢₃ are external control inputs that calculated according to system states. It is desired that the control inputs can drag the chaotic trajectory of Chua’s circuit system (2.4) to one of its three unstable equilibrium points. That is to say, the inputs can change three unstable equilibrium points of the open-loop system (2.4) to stable equilibrium points of the closed-loop Chua’s circuit system (3.1).

3.1. Feedback Control

Let the control law take the following form:𝑢₁=𝑘₁𝑥₁−𝑥_1𝑟,𝑢₂=𝑢₃=0, (3.2) where 𝑘₁ is a positive feedback gain and 𝑥_1𝑟 is the goal of the available state 𝑥₁.

Thus, the closed-loop Chua’s circuit model can be written as𝑥̇₁=𝑝𝑥₂−𝑥₁−𝑓𝑥₁−𝑘₁𝑥₁−𝑥_1𝑟,𝑥̇₂=𝑥₁−𝑥₂+𝑥₃,𝑥̇₃=−𝑞𝑥₂. (3.3)

For closed-loop system (3.3), the following conclusion can be drawn.

Theorem 3.1. When the feedback gain 𝑘₁ satisfies 𝑘₁>𝑘_1𝑟=−𝑝𝑐, (3.4) where both 𝑝 and 𝑐 are the system parameters and 𝑘_1𝑟 is the minimal stable feedback gain. The equilibrium points 𝐸(𝑥_1𝑟,𝑥_2𝑟,𝑥_3𝑟) of closed-loop Chua’s circuit system (3.3) are asymptotically stable.

Proof. In the neighbourhood of the equilibrium points 𝐸(𝑥_1𝑟,𝑥_2𝑟,𝑥_3𝑟), the jacobian matrix of feedback Chua’s circuit system is ()𝐽=−𝑝1+𝑐−𝑘₁𝑝01−110−𝑞0, (3.5) where 𝑐=𝑏or𝑎.
Let us define the state errors𝑒₁=𝑥₁−𝑥_1𝑟,𝑒₂=𝑥₂−𝑥_2𝑟,𝑒₃=𝑥₃−𝑥_3𝑟. (3.6)
The error linearized equation of controlled system in the neighborhood of the equilibrium points is𝑒₁̇=𝑝𝑒₂−(1+𝑐)𝑒₁−𝑘₁𝑒₁,𝑒₂̇=𝑒₁−𝑒₂+𝑒₃,𝑒₃̇=−𝑞𝑒₂. (3.7)
The Lyapunov function is defined as1𝑉=2𝑞𝑝𝑒²₁+𝑞𝑒²₂+𝑒²₃. (3.8)
The time derivative of 𝑉 in the neighborhood of the equilibrium point is𝑉̇=𝑞𝑝𝑒₁𝑒₁̇+𝑞𝑒₂𝑒₂̇+𝑒₃𝑒₃̇=𝑞𝑝𝑒₁𝑝𝑒₂−(1+𝑐)𝑒₁−𝑘₁𝑒₁+𝑞𝑒₂𝑒₁−𝑒₂+𝑒₃−𝑞𝑒₃𝑒₂=−𝑞𝑒₁−𝑒₂²𝑘−𝑞𝑐+¹𝑝𝑒²₁. (3.9)
It is clear that for the system parameters 𝑝, 𝑞, and 𝑐, if we choose𝑘₁>𝑘_1𝑟=−𝑝𝑐, (3.10) then 𝑉̇ is negative definite and the Lyapunov function 𝑉 is positive definite. From Lyapunov stability theorem it follows that the equilibrium point of the system (3.3) is asymptotically stable.

3.2. Adaptive Control

In the feedback control of Section 3.1, 𝑢₁=𝑘₁𝑥₁−𝑥_1𝑟,𝑢₂=𝑢₃=0 (3.11) the feedback gain 𝑘₁ is a constant. In this section, 𝑘₁ will be adjusted by an adaptive algorithm with respect to system state error 𝑥₁−𝑥_1𝑟, that is to say, the feedback gain 𝑘₁ automatically changes according to the system state 𝑥₁. For closed-loop system (3.3), the following adaptive algorithm is designed:𝑘̇₁=𝛾𝑥₁−𝑥_1𝑟², (3.12) where 𝛾 is the adaptive gain. For Chua’s circuit adaptive control system, the following conclusion holds.

Theorem 3.2. For 𝛾>0 and 𝑘̇₁=𝛾(𝑥₁−𝑥_1𝑟)², the equilibrium point 𝐸(𝑥_1𝑟,𝑥_2𝑟,𝑥_3𝑟) of the closed-loop Chua’s circuit adaptive control system is asymptotically stable.

Proof. In the neighborhood of the equilibrium points 𝐸(𝑥_1𝑟,𝑥_2𝑟,𝑥_3𝑟), the jacobian matrix of adaptive feedback Chua’s circuit system is ()𝐽=−𝑝1+𝑐−𝑘₁𝑝01−110−𝑞0, (3.13) where, 𝑐=𝑏or𝑎.
Let us define the state errors𝜂₁=𝑥₁−𝑥_1𝑟,𝜂₂=𝑥₂−𝑥_2𝑟,𝜂₃=𝑥₃−𝑥_3𝑟. (3.14)
The error linearized equation of controlled system in the neighbourhood of the equilibrium points is𝜂̇₁=𝑝𝜂₂−(1+𝑐)𝜂₁−𝑘₁𝜂₁,𝜂̇₂=𝜂₁−𝜂₂+𝜂₃,𝜂̇₃=−𝑞𝜂₂. (3.15)
The Lyapunov function is defined for the closed-loop Chua’s circuit adaptive control system as1𝑉=2𝑞𝑝𝑥₁−𝑥_1𝑟²+𝑞𝑥₂−𝑥_2𝑟²+𝑥₃−𝑥_3𝑟²+𝑞𝑘𝑝𝛾¹−𝑘_1𝑟². (3.16)
The time derivative of V in the neighbourhood of the equilibrium point is𝑉̇=𝑞𝑝𝑥₁−𝑥_1𝑟𝑥̇₁+𝑞𝑥₂−𝑥_2𝑟𝑥̇₂+𝑥₃−𝑥_3𝑟𝑥̇₃+𝑞𝑘𝑝𝛾¹−𝑘_1𝑟𝑘̇₁=𝑞𝑝𝜂₁𝑥̇₁+𝑞𝜂₂𝑥̇₂+𝜂₃𝑥̇₃+𝑞𝑘𝑝𝛾¹−𝑘_1𝑟𝑘̇₁=𝑞𝜂₁𝜂₂−𝑞(1+𝑐)𝜂²₁−𝑞𝑝𝑘₁𝜂²₁+𝑞𝜂₁𝜂₂−𝑞𝜂²₂+𝑞𝜂₂𝜂₃−𝑞𝜂₂𝜂₃+𝑞𝑝𝑘₁𝜂²₁−𝑞𝑝𝑘_1𝑟𝜂²₁=−𝑞𝜂₁²+2𝑞𝜂₁𝜂₂−𝑞𝜂²₂−𝑞𝑐𝜂₁²−𝑞𝑝𝑘_1𝑟𝜂²₁=−𝑞𝜂₁−𝜂₂²𝑘−𝑞𝑐+^1𝑟𝑝𝜂²₁. (3.17)
As we know, 𝑘_1𝑟=−𝑝𝑐, 𝑐=𝑏or𝑎, and the system parameter 𝑞>0. From Lyapunov stability theorem, 𝑉>0 and 𝑉̇<0, it follows that the equilibrium point of Chua’s circuit adaptive control system is asymptotically stable.

4. Simulation Studies

In the simulations, the following system parameters are used: 𝑝=10, 𝑞=100/7, 𝑎=−8/7, and 𝑏=−5/7. The fourth-order Runge-Kutta algorithm is applied to calculate the number integral. The system initial state is 𝑥₁₀=0.15264, 𝑥₂₀=−0.02281, and 𝑥₃₀=0.38127. In order to illustrate the effectiveness of the proposed control methods, the control is added at the 40th second. According to the given system parameters, the minimal stable feedback gains for different equilibrium points can be calculated: 𝑘_1𝑟=50/7 for equilibrium point 𝐸(1.5,0−1.5), 𝑘_1𝑟=80/7 for equilibrium point 𝐸(0,0,0), and 𝑘_1𝑟=50/7 for equilibrium point 𝐸(−1.5,0,1.5).

4.1. Feedback Control

The simulation results for the proposed feedback control of Chua’s circuit system are illustrated in Figures 2–7, where Figures 3 and 4, respectively, are the simulation results to drag Chua’s circuit system to one equilibrium point 𝐸(1.5,0−1.5) with the feedback gain 𝑘=15. Figure 3 illustrates the time response of system state 𝑥₁ and Figure 4 is the phase plane portrait of system states 𝑥₁ and 𝑥₂. Figures 5 and 6, respectively, are the simulation results to drag the Chua’s circuit system to one equilibrium point 𝐸(0,0,0) with the feedback gain 𝑘=15. Figure 5 illustrates the time response of system state 𝑥₁ and Figure 6 is the phase plane portrait of system states 𝑥₁ and 𝑥₂. Figures 7 and 8, respectively, are the simulation results to drag the Chua’s circuit system to one equilibrium point 𝐸(−1.5,0,1.5) with the feedback gain 𝑘=15. Figure 7 illustrates the time response of system state 𝑥₁ and, Figure 8 is the phase plane portrait of system states 𝑥₁ and 𝑥₂.

Figure 3 

𝑥1 time response of 𝐸(1.5,0,−1.5)  with feedback control.

Figure 4 

𝑥1-𝑥2 phase plane of 𝐸(1.5,0,−1.5)  with feedback control.

Figure 5 

𝑥1 time response of 𝐸(0,0,0) with feedback control.

Figure 6 

𝑥1-𝑥2 phase plane of 𝐸(0,0,0) with feedback control.

Figure 7 

𝑥1 time response of 𝐸(1.5,0,−1.5) with feedback control.

Figure 8 

𝑥1-𝑥2 phase plane of 𝐸(1.5,0,−1.5) with feedback control.

The simulation results show that feedback control with suitable feedback gain can drag Chua’s circuit system to one of its three equilibrium points. Moreover, the system initial state, feedback gain, and the position of system equilibrium point have some effects on the time responses of system states.

4.2. Adaptive Control

Next is the simulation of adaptive control with adaptive gain 𝛾=15 to drag the Chua’s circuit system to one of its three equilibrium points. The simulation results are illustrated in Figures 8–13, where Figures 9, 10, 11, 12, 13, and 14 are the time responses and phase plane portrait corresponding with Figures 3, 4, 5, 6, 7, and 8.

Figure 9 

𝑥1 time response of 𝐸(1.5,0,−1.5) with adaptive control.

Figure 10 

𝑥1-𝑥2 phase plane of 𝐸(1.5,0,−1.5) with adaptive control.

Figure 11 

𝑥1 time response of 𝐸(0,0,0) with adaptive control.

Figure 12 

𝑥1-𝑥2 phase plane of 𝐸(0,0,0) with adaptive control.

Figure 13 

𝑥1 time response of 𝐸(1.5,0,−1.5) with adaptive control.

Figure 14 

𝑥1-𝑥2 phase plane of 𝐸(1.5,0,−1.5) with adaptive control.

The simulation results show that adaptive control with suitable adaptive gain also can drag the Chua’s circuit system to one of its three equilibrium points. Moreover the adaptive control has the following advantages over feedback control: shorter time to drag the system state to its equilibrium point, more satisfactory dynamic response and wider application scope.

5. Conclusions

The advantage of using feedback control is that one can bring the system state away from chaotic motion and into any desired equilibrium point. In this paper, we have discussed a feedback control approach and an adaptive control approach for controlling the chaos in Chua’s circuit. The control schemes of the two proposed approaches are given and their stabilities are proven in detail. Simulation results show that, in the closed-loop system, system state asymptotically converges to the desired equilibrium point and the adaptive feedback control approach has some advantages over feedback control approach.

Acknowledgments

This work is supported by a Project of Shandong Province Higher Educational Science and Technology Program (under Grant no. J08LJ16) and in part by Research Award Fund for Outstanding Middle-Aged and Young Scientist of Shandong Province (under Grant no. BS2009DX021).