Chaos control of the modified Chua’s circuit system
Tzuyin Wu
∗, Min-Shin Chen
Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan Received 22 December 2000; accepted 12 January 2002
Communicated by M. Sano
Abstract
In this paper, a nonlinear controller called the backstepping controller is applied to suppress the chaotic motion of a modified Chua’s circuit system. The new controller can drive the system to the exact reference state at any prescribed speed. Most importantly, the controller achieves global exponential stability in the sense that the attraction basin for the reference state is the entire state space. Previous controllers for the Chua’s circuit can achieve only local stability in the sense that the attraction basin is a subset of the state space. © 2002 Elsevier Science B.V. All rights reserved.
PACS: 05.45.+b
Keywords: Chaos control; Chua’s circuit; Nonlinear control; Backstepping control
1. Introduction
In the last decade, 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 directed to controlling a chaotic system; that is, bringing the chaotic state to a fixed point or a limit cycle. In the literature, there are two basic approaches to the control of a chaotic system: non-feedback control, and feedback control. For the non-feedback control, it is demonstrated that a small periodic parametric perturbation can tame the chaos [1,2]. For the feedback control, there are two different methods: one is the data-based control, and the other model-based control. For the data-based feedback control, knowledge of the system model is not required. A time-varying small parametric perturbation is created based on a discrete feedback of the delay-coordinate vector of a state variable [3,4]. Such a control can bring the chaotic system state to an originally unstable periodic orbit which is embedded in the chaotic attractor. For the model-based feedback control, a linear or nonlinear feedback controller is designed based on the full knowledge of the system model. This kind of control has the capability to reshape the whole state space trajectory, and to create new stable fixed points or limit cycles [5,6].
In this paper, we consider the feedback control of a modified Chua’s circuit. The Chua’s circuit is an extensively studied and well-understood chaotic system [7]. The original Chua’s circuit contains a piecewise linear resistor. Hartley [8] suggested that the piecewise linear function can be replaced by a cubic polynomial with no major
∗Corresponding author. Tel.:+886-2-236-30231x2413; fax: +886-2-236-31755. E-mail address: [email protected] (T. Wu).
0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 3 6 0 - 3
change in the system’s dynamic characteristics. For the (modified) Chua’s circuit, there have been several controllers proposed to suppress the chaos. Hartley and Mossayebi [9] used the LQ optimal state feedback control and a classical lead compensator to drive the state to a desired fixed point on the equilibrium manifold. Hwang et al. [10] proposed a different linear state feedback control to regulate the system state, claiming faster settling time. Chen and Dong [11] used proportional control to drag the system state out of the strange attractor, and into an unstable limit cycle of the Chua’s circuit.
However, since the chaotic system is nonlinear, the above “linear” controls [3,4,9–11] do not guarantee stability or performance in the global sense. If the initial state of the system lies outside the attraction basin, the state trajectory will not converge to the target. Therefore, one has to resort to nonlinear control in order to obtain global results. In this paper, a nonlinear backstepping controller [12] is adopted to control the modified Chua’s circuit. Even though this new controller has a more complicated structure than previous ones, it has the advantage that the attraction basin of the desired fixed point is the whole state space. In other words, the controlled system is globally stable. Furthermore, it is always possible to tune the parameters in the backstepping controller such that the convergence to the target is as fast as desired.
2. Backstepping control
Consider the modified Chua’s circuit system proposed in [8]:
˙x = p(y − f (x)), f (x) = 2x3− x
7 , (1)
˙y = x − y + z, (2)
˙z = −qy + u, (3)
wherep = 10, q = 100/7 are positive system parameters, the control input u represents a voltage source in series with the inductor,x and y are voltages across two capacitors, and z is the current through the inductor.
Without the controlu, the modified Chua’s circuit is known to generate a chaotic phenomenon when the system parameters are set at values specified above. In this section, a backstepping control will be introduced to suppress this chaotic motion. Furthermore, the proposed control will bring the system state, sayx, to any reference state xr
at any prescribed speed. The proposed control is as follows:
u = a(x)x + b(x)y + c(x)z + d(x)y2+ ef(x) + gf(x)f(x) + p2f (x)f(x)2 + p2f2(x)f(x) + 1 8pkxkykzxr, (4) where a(x) = 1 −1 2(kx+ ky+ kz) − 1 8pkxkykz+ pf (x), b(x) = q − p − 1 +1 2(kx+ ky+ kz) − 1 4(kxky+ kykz+ kzkx) − 1−1 2(kx+ ky+ kz) pf(x) − p2f(x)2− 2p2f (x)f(x), c(x) = 1 −1 2(kx+ ky+ kz) + pf (x), d(x) = p2f(x), e = p +1 4(kxky+ kykz+ kzkx), g = − p 2(kx+ ky+ kz),
andkx,kyandkzare three positive control gains, which determine the convergence rate of the controlled system. The backstepping controller is somewhat more complicated than previous controllers; however, the added complexity allows one to claim global stability of the controlled system, and to have exact control over how fast the system state converges.
3. Stability analysis
The design philosophy of the proposed controller (4) is based on the backstepping principle [12]. The backstepping controller design, in our case, consists of three steps:
Step I: In thex-state Eq. (1), one treats the second state variable y as a control input, and finds out the desired y,
sayy = yd(x, xr) that will force x to converge to the reference state xrat a prescribed speed.
Step II: In they-state Eq. (2), one treats the third state variable z as a control input, and finds out the desired z,
sayz = zd(x, y, xr) that will force y to approach the previously designed function yd at a prescribed
speed.
Step III: Finally, in thez-state Eq. (3), one chooses the true control input u to drive z to approach the previously
designed functionzdat a prescribed speed.
The detailed design procedures and stability analysis are presented below. Firstly, in the design procedure Step I, the desiredydfunction is chosen as
yd(x, xr) = f (x) −
kx
2p(x − xr). (5)
Lemma 1. If the second state variable y in Eq. (1) is equal toyd, the first state variable x will approach xr
exponentially fast.
Proof. Define a Lyapunov function candidateVx = 12(x − xr)2≥ 0. From Eqs. (1) and (5), the time derivative of
Vxis found to be ˙Vx= −kxVx. Consequently,x(t) approaches xrexponentially fast in the following way: |x(t) − xr| = |x(0) − xr|e−(kx/2)t.
Secondly, in the design procedure Step II, the desiredzdfunction is chosen as
zd(x, y, xr) = − 1+kxky 4p x + 1−ky+ kx 2 y + ky+ kx 2 f (x) + kxky 4p xr+ pf (x)(y − f (x)). (6)
Lemma 2. If the third state variable z in Eq. (2) is equal tozd, the second state variable y will approach yd
exponentially fast.
Proof. Define a second Lyapunov function candidateVy= 12(y − yd)2≥ 0. From Eqs. (2) and (6), one can verify
that the time derivative ofVyis given by ˙Vy= −kyVy. Again, one has exponential convergence ofy(t) to yd(t): |y(t) − yd(t)| = |y(0) − yd(0)|e−(ky/2)t.
Finally, in the design procedure Step III, the control inputu is chosen to be that in the control law (4).
Lemma 3. If the control input u is chosen to be that in Eq. (4), the third state variable z will approachzdin Eq. (6)
Proof. Define a third Lyapunov function candidateVz = 12(z − zd)2 ≥ 0. From Eqs. (3) and (4), one can verify,
although tediously, that the time derivative ofVzis given by ˙Vz= −kzVz; hence,
|z(t) − zd(t)| = |z(0) − zd(0)|e−(kz/2)t.
The following theorem summarizes the above three Lemmas, which states that the control law (4) globally stabilizes the system to the reference state
(xr, yr, zr) = (xr, f (xr), f (xr) − xr). (7)
Theorem. Consider the modified Chua’s circuit system (1)–(3) and the feedback control (4). Given any initial
conditionx(0), y(0), and z(0), the system state converges to the reference state in Eq. (7) exponentially fast.
Proof. The stability of the controlled system is a direct consequence of Lemmas 1–3. Since in proving
Lemmas 1–3, one makes no constraints on the initial conditionx(0), y(0), and z(0), this stability result is global. Note that in Eq. (7) the reference stateyr is obtained by substitutingx = xr in Eq. (5), andzris obtained by
substitutingx = xrandy = f (xr) in Eq. (6).
4. Numerical simulations
In this Section, we use computer simulation to verify the performance of the proposed controller, and the results are compared with those in [10].
Consider the system (1) with the initial condition(x(0), y(0), z(0)) = (0.65, 0, 0). When the reference state is set at the origin; i.e.,(xr, yr, zr) = (0, 0, 0), the controller in [10] cannot stabilize the system. Fig. 1 shows the
controlled state response (with a control gainkp = 5), which remains oscillatory after the control is turned on at
Fig. 2. State response with backstepping control.
t = 40 s. In contrast, the backstepping control (4) proposed in this paper with kx = ky = kz= 2 can successfully
bring the state to the origin in about 7 s. The results are shown in Fig. 2, where similarly the backstepping control is turned on att = 40 s.
To further verify the effectiveness of the proposed control design, the backstepping control (4) is simulated with different control gainski = 1, 1.3, 2; i = x, y, z. Fig. 3 shows the state norm || X(t)|| =x2(t) + y2(t) + z2(t)
versus the time t. It is observed that, as predicted by the analysis in Lemmas 1–3, the state convergence rate is determined by the control gains ki. The larger the control gains are, the faster the convergence rate is.
5. Conclusions
The control of chaos may be achieved by open-loop control (non-feedback control) or by feedback control. The advantage of using feedback control is that one can bring the system state away from chaotic motion and into any desired fixed point. Furthermore, by employing the newly developed nonlinear feedback controls, the fixed point in the reshaped state space can possess global stability, which is impossible to attain by previously suggested linear controls. This paper demonstrates the effectiveness of nonlinear control by applying the nonlinear backstepping control to suppress the chaotic motion of the modified Chua circuit system, and reshape the entire state space into one with a globally stable fixed point.
References
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![Fig. 1. State response with linear control [10].](https://thumb-ap.123doks.com/thumbv2/9libinfo/8868731.247159/4.1263.414.865.1073.1503/fig-state-response-with-linear-control.webp)
