ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) c World Scientific Publishing Company ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT KEVIN BARONE and SAHJENDRA N. SINGH Department of Electrical and Computer Engineering, University of Nevada Las Vegas, Las Vegas, NV , USA sahaj@ee.unlv.edu Received December 15, 2000; Revised August 3, 2001 In this Letter a feedback linearizing adaptive control system for the control of Chua s circuits is presented. It is assumed that all the parameters of the system are unknown. Using a backstepping design procedure, an adaptive control system is designed which accomplishes trajectory control of the chosen node voltage by an independent voltage source. Simulation results are presented which show precise trajectory tracking and regulation of the state vector to the desired terminal state. Keywords: Chaos control; adaptive control; nonlinear system; Chua s circuit; adaptive feedback linearization. 1. Introduction Recently considerable effort has been made to control chaotic systems [Chen & Dong, 1998; Fradkov & Pogromsky, 1998; and the references therein]. In the literature, several design techniques have been applied for the control and synchronization of a variety of chaotic systems. Generalized synchronization of chaos via linear transformation has been considered [Yang & Chua, 1999]. Using Lyapunov theory, adaptive control and synchronization of chaotic systems have been considered [Wu et al., 1996; Bernardo, 1996; Yang et al., 1998]. Based on backstepping design techniques [Krstic et al., 1995], nonadaptive and adaptive control systems for several types of chaotic systems including the Chua circuits, and Lorenz and Rossler systems have been designed [Mascolo & Grassi, 1999; Barone & Singh, 1999; Wang & Ge, 2001a, 2001b; Ge & Wang, 2000; Ge et al., 2000]. A partially linearizable Lorenz system with nontrivial zero dynamics has been considered for adaptive control [Zeng & Singh, 1997]. In this Letter, for the control of Chua s circuits, an adaptive control system based on a backstepping design technique is presented. This Chua s diode has a cubic polynomial nonlinearity [Yang et al., 1998]. A feedback linearizing adaptive control law for the trajectory control of a node voltage using an independent voltage source is designed. In the closed-loop system, asymptotic trajectory tracking of the reference node voltage trajectory is accomplished, and the state vector converges to the equilibrium state. Simulation results are presented which show trajectory control in spite of the uncertainties in the circuit parameters. It is pointed out that the control problem considered here differs from that of [Ge & Wang, 2000]. In [Ge & Wang, 2000], the trajectory control of the inductor current has been considered, but here the node voltage is controlled. Thus the control input appears at a different branch in the circuit. Moreover, the synthesis of the controller for the node voltage control is relatively complicated, because the nonlinearity appears in the differential equation of the output which needs to be differentiated twice to design the control law. 1599

2 1600 K. Barone & S. N. Singh Fig. 1. The Chua s circuit. 3. Adaptive Control Law Define the state vector as x =(V c1,v c2,i L ) T.Then (1) can be written as ẋ 1 = b 1 x 2 + φ T 1 (x 1 )θ ẋ 2 = b 2 x 3 + φ T 2 (x 1,x 2 )θ (3) ẋ 3 = b 3 u + φ T 3 (x)θ where b 1 =(1/RC 1 ), b 2 =(1/C 2 ), b 3 =(1/L), and θ =( a 0 C 1 1, C 1 1 (a 1 + R 1 ), a 2 C 1 1, φ T 1 φ T 2 a 3 C 1 1, (1/C 2R), L 1, R 0 L 1 ) T R 7 1 x 1 x 2 1 x = x 1 x R Chua s Circuit and Control Problem Figure 1 shows Chua s circuit with the independent voltage source as a control input. The state equations are given by V c1 = C 1 1 [R 1 (V c2 V c1 ) f(v c1 )] V c2 = C 1 2 [ R 1 (V c2 V c1 )+I L ] (1) I L = L 1 [V c2 R 0 I L + u] where the current f(v c1 ) flowing through the Chua s diode is a nonlinear function of V c1 given by f(v c1 )=a 0 + a 1 V c1 + a 2 V 2 c1 + a 3V 3 c1 (2) Here a polynomial nonlinearity for f is chosen, but the design is applicable if any other linearly parameter-dependent function of the node voltage V c1 is used for f in (2). For the purpose of design it is assumed that the parameters C i (i =1, 2), R 0, R, L, and the diode parameters a k (k =0, 1,...,3) are not known. It is noted that [Ge & Wang, 2000] have introduced control input in the first equation for V c1 in which the nonlinearity appears, and designed the controller for the trajectory control of the inductor current I L instead. Suppose a smooth reference trajectory y r is given. We are interested in designing an adaptive control system so that the node voltage V c1 asymptotically tracks y r and that for y r (t) =y,aconstant, the state vector converges to the equilibrium point. φ T x 2 x 3 The system (3) is in the strict feedback form, and the parameters b i and θ are unknown. Following [Krstic et al., 1995], one designs the adaptive law for the trajectory control of the node voltage x 1 using a backstepping design procedure. Define the tracking error z 1 = x 1 y r, ρ i =(1/b i )(i =1, 2, 3), the parameter errors ρ i = ρ i ˆρ i, b i = b i ˆb i, β =(θ T,b 1,b 2 ) T R 9, β = β ˆβ, and α i =ˆρ i α i, i =1, 2, 3 (4) z i+1 = x i+1 α i, i =1, 2 where an overhat denotes the parameter estimate, and α (i =1, 2, 3) are yet to be determined. The design is completed in three steps. At each step a Lyapunov function is used to obtain the stabilization signals α i, and the adaptation law is obtained in the final step. The readers can find the details of derivation in [Barone & Singh, 1999; Ge & Wang, 2000], and, therefore, these are not repeated here. Following the derivation and the notation of the conference paper of [Barone & Singh, 1999], the stabilization signals (α k ) and the tuning functions (τ k )aregivenby α 1 (x 1,y r, ẏ r, ˆθ) = c 1 z 1 φ T 1 ˆθ +ẏ r τ 1 (x 1,y r )=z 1 [φ T 1, 0, 0]T R 9 ψ 2 =[φ T 2 ( α 1/ x 1 )φ T 1, ( α 1 / x 1 )x 2 + z 1, 0] T R 9 τ 2 (x 1,x 2, ˆρ 1,y r, ẏ r, ˆθ) =τ 1 + ψ 2 z 2 R 9 w 2 =[ α 1 / ˆθ, 0, 0]Γτ 2 R

3 Adaptive Feedback Linearizing Control of Chua s Circuit 1601 α 2 = φ T 2 ˆθ +( α 1 / x 1 )(ˆb 1 x 2 + φ T 1 ˆθ) (5) 1 +( α 1 / ˆρ 1 ) ˆρ + ( α 1 / y r (i) )y r (i+1) i=0 c 2 z 2 ˆb 1 z 1 + w 2 ψ 3 =[φ T 3 ( α 2/ x 2 )φ T 2 ( α 2/ x 1 )φ T 1, ( α 2 / x 1 )x 2, ( α 2 / x 2 )x 3 + z 2 ] T R 9 τ 3 (x, ˆρ 1, ˆρ 2,y r, ẏ r, ÿ r, ˆθ) =τ 2 + ψ 3 z 3 w 3 =[( α 1 / ˆθ)z 2, 0, 0]ψ 3 +( α 2 / ˆβ) ˆβ α 3 = φ T 3 ˆθ +( α 2 / x 1 )(ˆb 1 x 2 + φ T 1 ˆθ) +( α 2 / x 2 )(ˆb 2 x 3 + φ T ˆθ) ( α 2 / ˆρ j ) ˆρ j + ( α 2 / y (k) j=1 c 3 z 3 ˆb 2 z 2 + w 3 k=0 r )y (k+1) r where y r (k) = d k y r /dt k,γisa9 9diagonal positive definite matrix, and c i > 0 are the feedback gains in the stabilization signals. Then the control law is given by and the adaptation laws are u =ˆρ 3 α 3 (6) ˆρ i = γ i sgn(b i )α i z i, i =1, 2, 3 (7) ˆβ =Γτ 3 (8) Taking the derivative of W and using (5) (8), one can show that 3 Ẇ = c i zi 2. (10) i=1 Now using LaSalle Yoshikawa theorem [Krstic et al., 1995], it follows that z i 0 as t. Furthermore, for y r (t) = y, x(t) converges to the largest invariant set Ω E, where E = {(z T, β T, ρ T ) T R 15 : z =(z 1,z 2,z 3 ) T =0}, where ρ =( ρ 1, ρ 2, ρ 3 ) T. But in E, z 1 0 implies that x 1 = y and ẋ 1 = 0. Then using (3), one finds that in Ω, x 2 = b 1 1 φt 1 (y )θ = x 2, and because ẋ 2 =0, one has x 3 = b 1 2 φt 2 (y,x 2 )θ = x Simulation Results This section presents the simulation results. The circuit parameters are C 1 = 0.1 F, C 2 = 1 F, R 1 = 1, R 0 = 0 and L = 0.07 H. The Chua s diode coefficients are (a 0,a 1,a 2,a 3 ) = ( 0.01, (8/7), 0.01, (2/7)). The initial conditions are V c1 (0) = , V c2 (0) = and I L (0) = The initial values of the parameter estimates are arbitrarily set to ˆθ(0) = 0, ˆρ i (0) = 0 (i = 1, 2, 3) and ˆb i (0) = 0 (i = 1, 2). This is rather a worse choice of the initial parameter estimates. However, these have been chosen to demonstrate the adaptation capability of the control system. The open-loop responses are shown in where γ i (i =1, 2, 3) are positive numbers. The derivatives of the parameter estimates are substituted in α i for the synthesis of the controller. Theorem 1. Consider the closed-loop system (1), (6) (8). Suppose that y r (t) and its derivatives are smooth and bounded. Then along the trajectory of the closed-loop system, z i (t) 0 as t. Moreover, for y r (t) = y, a constant, the state vector converges to the equilibrium point x = [y, b 1 1 φt 1 (y )θ, b 1 2 φt 2 (y,x 2 )θ]t. Proof. Consider a positive definite Lyapunov function ( 3 )/ 3 W = zi 2 + γ i b i ρ 2 i + β T Γ 1 β 2 (9) i=1 i=1 Fig. 2. Open-loop chaotic response.

4 1602 K. Barone & S. N. Singh (a) (d) (b) (e) (c) Fig. 3. Regulation to origin. (a) V c1,v c2,i L,(b)3-Dplot of V c1,v c2,i L, (c) Parameter estimate ˆθ 1, (d) Parameter estimate ˆb 1, (e) Parameter estimate ˆρ 1.

5 Adaptive Feedback Linearizing Control of Chua s Circuit 1603 (a) (b) Fig. 4. Nonzero set point control. (a) V c1,v c2,i L,(b)3-DplotofV c1,v c2,i L. Fig. 2. The chaotic nature of the Chua s circuit is observed for the chosen initial condition. To examine the trajectory tracking ability of the controller, the complete closed-loop system is simulated. Reference trajectory y r is generated by a third-order filter (s +1) 3 (y r y )=0 with repeated poles at s = 1. The initial conditions chosen are y r (0) = 0.5, y r (k) (0) = 0(k =1, 2). For the regulation of the state vector to the origin, y is set to zero. The control parameters are set to c i = γ i = 1 and Γ = I, the identity matrix, for simplicity. Smooth responses are obtained and the state vector converges to the origin (Fig. 3). Only selected parameter estimates are shown in order to save space, but it has been found that each parameter estimate converges to certain constant value which differs from its true value. This is well known that unless there is persistent excitation, parameters cannot converge to their true values. For the nonzero set point control, y r is generated by setting y =1. Selected responses are given in Fig. 4. We observe convergence of the state vector to the desired equilibrium point. Responses are also obtained to follow sinusoidal trajectory y r =0.4 cos(6πt/100). For the choice of the feedback parameters c 1 =0.25, c 2 =2.5, c 3 =1, γ 1 = γ 3 = 1, γ 2 = 0.75 and Γ = 1.5I, selected Fig. 5. Sinusoidal trajectory control: V c1,v c2,i L. responses are shown in Fig. 5. Again we observe that the node voltage V c1 asymptotically follows the time-varying trajectory. Furthermore, V c2 and I L also asymptotically tend to sinusoidal functions, which satisfy the first two equations given in (1). Extensive simulation has been performed also to examine the effect of feedback parameters c i and Γ. Larger values of c i and Γ usually give faster tracking of y r. These results can be found in [Barone, 2000].

6 1604 K. Barone & S. N. Singh 5. Conclusions In this Letter, an adaptive feedback linearizing control system was designed for the control of Chua s circuit. All the circuit parameters were assumed to be unknown. Using a backstepping design technique, an adaptive control law was designed. In the closed-loop system, the controlled node voltage can asymptotically track any smooth reference trajectory, and regulation to the origin or to any nonzero terminal state can be accomplished. The control system is capable of following sinusoidal reference trajectories in spite of the uncertainties in the circuit parameters. References Barone, K. & Singh. S. N. [1999] Adaptive control and synchronization of chaos in Chua s circuit, Proc. 13th Int. Conf. System Engineering, Las Vegas, NV, EE1 EE6. Barone, K. [2000] Adaptive control and synchronization of chaotic systems with unknown parameters, M. S. thesis, ECE Dept, University of Nevada, Las Vegas, NV. Bernardo, M. [1996] An adaptive approach to the control and synchronization of chaotic systems, Int. J. Bifurcation and Chaos 6(3), Chen, G. & Dong, X. [1998] From Chaos To Order (World Scientific, Singapore). Fradkov, A. L. & Pogromsky, A. Yu. [1996] Introduction to Control of Oscillations and Chaos (World Scientific, Singapore). Ge, S. S., Wang, C. & Lee, T. H. [2000] Adaptive backstepping control of a class of chaotic systems, Int. J. Bifurcation and Chaos 10(5), Ge, S. S. & Wang, C. [2000] Adaptive control of uncertain Chua s circuit, IEEE Trans. Circuits Syst. I: Fundamental Th. Appl. 47(9), Krstic, M., Kanellakopoulos, I. & Kokotovic, P. [1995] Nonlinear and Adaptive Control Design (John Wiely, NY). Mascolo, S. & Grassi, G. [1999] Controlling chaotic dynamics using backstepping design with application to the Lorenz system and Chua s circuit, Int. J. Bifurcation and Chaos 9(7), Wang, C. & Ge, S. S. [2001a] Adaptive backstepping control of uncertain Lorenz system, Int. J. Bifurcation and Chaos 11(4), Wang, C. & Ge, S. S. [2001b] Synchronization of uncertain chaotic systems via adaptive backstepping, Int. J. Bifurcation and Chaos 11(6), Wu, C. W., Yang, T. & Chua, L. O. [1996] On adaptive synchronization and control of nonlinear dynamical systems, Int. J. Bifurcation and Chaos 6(3), Yang, T., Yang, Ch.-M. & Yang, L.-B. [1998] Detailed study of adaptive control of chaotic systems with unknown parameters, Dyn. Contr. 8(3), Yang, T. & Chua, L. O. [1999] Generalized synchronization of chaos via linear transformations, Int. J. Bifurcation and Chaos 9(1), Zeng, Y. & Singh, S. N. [1997] Adaptive control of chaos in Lorenz system, Dyn. Contr. 7,