adaptive feedback linearizing control of chua's circuit - Semantic Scholar
International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1599–1604 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 89154-4026, USA ∗ [email protected] 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 backstep-
ping 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.
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3. Adaptive Control Law Define the state vector as x = (Vc1 , Vc2 , IL )T . Then (1) can be written as x˙ 1 = b1 x2 + φT1 (x1 )θ x˙ 2 = b2 x3 + φT2 (x1 , x2 )θ x˙ 3 = b3 u +
(3)
φT3 (x)θ
where b1 = (1/RC1 ), b2 = (1/C2 ), b3 = (1/L), and θ = (−a0 C1−1 , −C1−1 (a1 + R−1 ), −a2 C1−1 , − a3 C1−1 , (1/C2 R), L−1 , −R0 L−1 )T ∈ R7 Fig. 1.
The Chua’s circuit.
φT1
1
T φ2 = 0 0 φT3
2. 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 = C1−1 [R−1 (Vc2 − Vc1 ) − f (Vc1 )] V˙ c2 = C2−1 [−R−1 (Vc2 − Vc1 ) + IL ]
(1)
x21
x31
0
0
0
0
0
x1 − x2
0
0
0
0
0
x2
(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 Vc1 is used for f in (2). For the purpose of design it is assumed that the parameters Ci (i = 1, 2), R0 , R, L, and the diode parameters ak (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 IL instead. Suppose a smooth reference trajectory yr is given. We are interested in designing an adaptive control system so that the node voltage Vc1 asymptotically tracks yr and that for yr (t) = y ∗ , a constant, the state vector converges to the equilibrium point.
3×7 0 ∈R x3
i = 1, 2, 3
zi+1 = xi+1 − αi ,
where the current f (Vc1 ) flowing through the Chua’s diode is a nonlinear function of Vc1 given by
0
The system (3) is in the strict feedback form, and the parameters bi and θ are unknown. Following [Krstic et al., 1995], one designs the adaptive law for the trajectory control of the node voltage x1 using a backstepping design procedure. Define the tracking error z1 = x1 − yr , ρi = (1/bi )(i = 1, 2, 3), the parameter errors ρ˜i = ρi − ρˆi , ˜bi = bi − ˆbi , ˆ and β = (θ T , b1 , b2 )T ∈ R9 , β˜ = β − β, αi = ρˆi αi ,
I˙L = L−1 [Vc2 − R0 IL + u]
2 3 f (Vc1 ) = a0 + a1 Vc1 + a2 Vc1 + a3 Vc1
x1
i = 1, 2
(4)
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 αi , 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 (αk ) and the tuning functions (τk ) are given by ˆ = −c1 z1 − φT θˆ + y˙ r α1 (x1 , yr , y˙ r , θ) 1 τ1 (x1 , yr ) = z1 [φT1 , 0, 0]T ∈ R9 ψ2 = [φT2 − (∂α1 /∂x1 )φT1 , −(∂α1 /∂x1 )x2 + z1 , 0]T ∈ R9 ˆ = τ1 + ψ2 z2 ∈ R9 τ2 (x1 , x2 , ρˆ1 , yr , y˙ r , θ) ˆ 0, 0]Γτ2 ∈ R w2 = [∂α1 /∂ θ,
Adaptive Feedback Linearizing Control of Chua’s Circuit 1601
ˆ α2 = −φT2 θˆ + (∂α1 /∂x1 )(ˆb1 x2 + φT1 θ) + (∂α1 /∂ ρˆ1 )ρˆ˙ +
1 X
(5)
Taking the derivative of W and using (5)–(8), one can show that
(∂α1 /∂yr(i) )yr(i+1)
i=0
˙ =− W
− c2 z2 − ˆb1 z1 + w2
3 X
ci zi2 .
(10)
i=1
ψ3 = [φT3 − (∂α2 /∂x2 )φT2 − (∂α2 /∂x1 )φT1 , − (∂α2 /∂x1 )x2 , −(∂α2 /∂x2 )x3 + z2 ]T ∈ R9 ˆ = τ2 + ψ3 z3 τ3 (x, ρˆ1 , ρˆ2 , yr , y˙ r , y¨r , θ) ˆ 2 , 0, 0]ψ3 + (∂α2 /∂ β) ˆ βˆ˙ w3 = [(∂α1 /∂ θ)z ˆ α3 = −φT3 θˆ + (∂α2 /∂x1 )(ˆb1 x2 + φT1 θ) ˆ + (∂α2 /∂x2 )(ˆb2 x3 + φT2 θ) +
2 X
(∂α2 /∂ ρˆj )ρˆ˙ j +
j=1
2 X
(∂α2 /∂yr(k) )yr(k+1)
k=0
− c3 z3 − ˆb2 z2 + w3 (k)
where yr = dk yr /dtk , Γ is a 9 × 9 diagonal positive definite matrix, and ci > 0 are the feedback gains in the stabilization signals. Then the control law is given by u = ρˆ3 α3
(6)
and the adaptation laws are ρˆ˙ i = −γi sgn(bi )αi zi ,
i = 1, 2, 3
(7)
˙ βˆ = Γτ3
(8)
Now using LaSalle–Yoshikawa theorem [Krstic et al., 1995], it follows that zi → 0 as t → ∞. Furthermore, for yr (t) = y ∗ , x(t) converges to the largest invariant set Ω ⊂ E, where E = {(z T , β˜T , ρ˜T )T ∈ R15 : z = (z1 , z2 , z3 )T = 0}, where ρ˜ = (˜ ρ1 , ρ˜2 , ρ˜3 )T . But in E, z1 ≡ 0 implies that x1 = y ∗ and x˙ 1 = 0. Then using (3), one finds that T ∗ ∗ in Ω, x2 = −b−1 ˙ 2 = 0, 1 φ1 (y )θ = x2 , and because x −1 T ∗ ∗ one has x3 = −b2 φ2 (y , x2 )θ = x∗3 .
4. Simulation Results This section presents the simulation results. The circuit parameters are C1 = 0.1 F, C2 = 1 F, R1 = 1, R0 = 0 and L = 0.07 H. The Chua’s diode coefficients are (a0 , a1 , a2 , a3 ) = (−0.01, −(8/7), −0.01, (2/7)). The initial conditions are Vc1 (0) = 0.442006, Vc2 (0) = −0.213984 and IL (0) = −0.90913. The initial values of the parameter estiˆ mates are arbitrarily set to θ(0) = 0, ρˆi (0) = 0 ˆ (i = 1, 2, 3) and bi (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 γi (i = 1, 2, 3) are positive numbers. The derivatives of the parameter estimates are substituted in αi for the synthesis of the controller. Consider the closed-loop system (1), (6)–(8). Suppose that yr (t) and its derivatives are smooth and bounded. Then along the trajectory of the closed-loop system, zi (t) → 0 as t → ∞. Moreover, for yr (t) = y ∗ , a constant, the state vector converges to the equilibrium point x∗ = −1 T ∗ ∗ T ∗ T [y ∗ , −b−1 1 φ1 (y )θ, −b2 φ2 (y , x2 )θ] .
Theorem 1.
Consider a positive definite Lyapunov function
Proof.
W =
3 X i=1
zi2
+
3 X i=1
!,
γi | bi | ρ˜2i
+ β˜T Γ−1 β˜
2
(9) Fig. 2.
Open-loop chaotic response.
1602 K. Barone & S. N. Singh
(a) (d)
(b)
(e)
(c)
Fig. 3. Regulation to origin. (a) Vc1 , Vc2 , IL , (b) 3-D plot of Vc1 , Vc2 , IL , (c) Parameter estimate θˆ1 , (d) Parameter estimate bˆ1 , (e) Parameter estimate ρˆ1 .
Adaptive Feedback Linearizing Control of Chua’s Circuit 1603
(a) Fig. 4.
(b) Nonzero set point control. (a) Vc1 , Vc2 , IL , (b) 3-D plot of Vc1 , Vc2 , IL .
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 yr is generated by a third-order filter (s + 1)3 (yr − y ∗ ) = 0 with repeated poles at s = −1. The initial condi(k) tions chosen are yr (0) = 0.5, yr (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 ci = γi = 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, yr 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 yr = 0.4 cos(6πt/100). For the choice of the feedback parameters c1 = 0.25, c2 = 2.5, c3 = 1, γ1 = γ3 = 1, γ2 = 0.75 and Γ = 1.5I, selected
Fig. 5.
Sinusoidal trajectory control: Vc1 , Vc2 , IL .
responses are shown in Fig. 5. Again we observe that the node voltage Vc1 asymptotically follows the time-varying trajectory. Furthermore, Vc2 and IL 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 ci and Γ. Larger values of ci and Γ usually give faster tracking of yr . These results can be found in [Barone, 2000].
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), 557–568. 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), 1149–1156. Ge, S. S. & Wang, C. [2000] “Adaptive control of uncertain Chua’s circuit,” IEEE Trans. Circuits Syst. I: Fundamental Th. Appl. 47(9), 1397–1402. 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), 1425–1434. Wang, C. & Ge, S. S. [2001a] “Adaptive backstepping control of uncertain Lorenz system,” Int. J. Bifurcation and Chaos 11(4), 1115–1119. Wang, C. & Ge, S. S. [2001b] “Synchronization of uncertain chaotic systems via adaptive backstepping,” Int. J. Bifurcation and Chaos 11(6), 1743–1751. 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), 455–472. Yang, T., Yang, Ch.-M. & Yang, L.-B. [1998] “Detailed study of adaptive control of chaotic systems with unknown parameters,” Dyn. Contr. 8(3), 255–267. Yang, T. & Chua, L. O. [1999] “Generalized synchronization of chaos via linear transformations,” Int. J. Bifurcation and Chaos 9(1), 215–219. Zeng, Y. & Singh, S. N. [1997] “Adaptive control of chaos in Lorenz system,” Dyn. Contr. 7, 143–154.
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 89154-4026, USA ∗ [email protected] 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 backstep-
ping 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.
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1600 K. Barone & S. N. Singh
3. Adaptive Control Law Define the state vector as x = (Vc1 , Vc2 , IL )T . Then (1) can be written as x˙ 1 = b1 x2 + φT1 (x1 )θ x˙ 2 = b2 x3 + φT2 (x1 , x2 )θ x˙ 3 = b3 u +
(3)
φT3 (x)θ
where b1 = (1/RC1 ), b2 = (1/C2 ), b3 = (1/L), and θ = (−a0 C1−1 , −C1−1 (a1 + R−1 ), −a2 C1−1 , − a3 C1−1 , (1/C2 R), L−1 , −R0 L−1 )T ∈ R7 Fig. 1.
The Chua’s circuit.
φT1
1
T φ2 = 0 0 φT3
2. 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 = C1−1 [R−1 (Vc2 − Vc1 ) − f (Vc1 )] V˙ c2 = C2−1 [−R−1 (Vc2 − Vc1 ) + IL ]
(1)
x21
x31
0
0
0
0
0
x1 − x2
0
0
0
0
0
x2
(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 Vc1 is used for f in (2). For the purpose of design it is assumed that the parameters Ci (i = 1, 2), R0 , R, L, and the diode parameters ak (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 IL instead. Suppose a smooth reference trajectory yr is given. We are interested in designing an adaptive control system so that the node voltage Vc1 asymptotically tracks yr and that for yr (t) = y ∗ , a constant, the state vector converges to the equilibrium point.
3×7 0 ∈R x3
i = 1, 2, 3
zi+1 = xi+1 − αi ,
where the current f (Vc1 ) flowing through the Chua’s diode is a nonlinear function of Vc1 given by
0
The system (3) is in the strict feedback form, and the parameters bi and θ are unknown. Following [Krstic et al., 1995], one designs the adaptive law for the trajectory control of the node voltage x1 using a backstepping design procedure. Define the tracking error z1 = x1 − yr , ρi = (1/bi )(i = 1, 2, 3), the parameter errors ρ˜i = ρi − ρˆi , ˜bi = bi − ˆbi , ˆ and β = (θ T , b1 , b2 )T ∈ R9 , β˜ = β − β, αi = ρˆi αi ,
I˙L = L−1 [Vc2 − R0 IL + u]
2 3 f (Vc1 ) = a0 + a1 Vc1 + a2 Vc1 + a3 Vc1
x1
i = 1, 2
(4)
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 αi , 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 (αk ) and the tuning functions (τk ) are given by ˆ = −c1 z1 − φT θˆ + y˙ r α1 (x1 , yr , y˙ r , θ) 1 τ1 (x1 , yr ) = z1 [φT1 , 0, 0]T ∈ R9 ψ2 = [φT2 − (∂α1 /∂x1 )φT1 , −(∂α1 /∂x1 )x2 + z1 , 0]T ∈ R9 ˆ = τ1 + ψ2 z2 ∈ R9 τ2 (x1 , x2 , ρˆ1 , yr , y˙ r , θ) ˆ 0, 0]Γτ2 ∈ R w2 = [∂α1 /∂ θ,
Adaptive Feedback Linearizing Control of Chua’s Circuit 1601
ˆ α2 = −φT2 θˆ + (∂α1 /∂x1 )(ˆb1 x2 + φT1 θ) + (∂α1 /∂ ρˆ1 )ρˆ˙ +
1 X
(5)
Taking the derivative of W and using (5)–(8), one can show that
(∂α1 /∂yr(i) )yr(i+1)
i=0
˙ =− W
− c2 z2 − ˆb1 z1 + w2
3 X
ci zi2 .
(10)
i=1
ψ3 = [φT3 − (∂α2 /∂x2 )φT2 − (∂α2 /∂x1 )φT1 , − (∂α2 /∂x1 )x2 , −(∂α2 /∂x2 )x3 + z2 ]T ∈ R9 ˆ = τ2 + ψ3 z3 τ3 (x, ρˆ1 , ρˆ2 , yr , y˙ r , y¨r , θ) ˆ 2 , 0, 0]ψ3 + (∂α2 /∂ β) ˆ βˆ˙ w3 = [(∂α1 /∂ θ)z ˆ α3 = −φT3 θˆ + (∂α2 /∂x1 )(ˆb1 x2 + φT1 θ) ˆ + (∂α2 /∂x2 )(ˆb2 x3 + φT2 θ) +
2 X
(∂α2 /∂ ρˆj )ρˆ˙ j +
j=1
2 X
(∂α2 /∂yr(k) )yr(k+1)
k=0
− c3 z3 − ˆb2 z2 + w3 (k)
where yr = dk yr /dtk , Γ is a 9 × 9 diagonal positive definite matrix, and ci > 0 are the feedback gains in the stabilization signals. Then the control law is given by u = ρˆ3 α3
(6)
and the adaptation laws are ρˆ˙ i = −γi sgn(bi )αi zi ,
i = 1, 2, 3
(7)
˙ βˆ = Γτ3
(8)
Now using LaSalle–Yoshikawa theorem [Krstic et al., 1995], it follows that zi → 0 as t → ∞. Furthermore, for yr (t) = y ∗ , x(t) converges to the largest invariant set Ω ⊂ E, where E = {(z T , β˜T , ρ˜T )T ∈ R15 : z = (z1 , z2 , z3 )T = 0}, where ρ˜ = (˜ ρ1 , ρ˜2 , ρ˜3 )T . But in E, z1 ≡ 0 implies that x1 = y ∗ and x˙ 1 = 0. Then using (3), one finds that T ∗ ∗ in Ω, x2 = −b−1 ˙ 2 = 0, 1 φ1 (y )θ = x2 , and because x −1 T ∗ ∗ one has x3 = −b2 φ2 (y , x2 )θ = x∗3 .
4. Simulation Results This section presents the simulation results. The circuit parameters are C1 = 0.1 F, C2 = 1 F, R1 = 1, R0 = 0 and L = 0.07 H. The Chua’s diode coefficients are (a0 , a1 , a2 , a3 ) = (−0.01, −(8/7), −0.01, (2/7)). The initial conditions are Vc1 (0) = 0.442006, Vc2 (0) = −0.213984 and IL (0) = −0.90913. The initial values of the parameter estiˆ mates are arbitrarily set to θ(0) = 0, ρˆi (0) = 0 ˆ (i = 1, 2, 3) and bi (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 γi (i = 1, 2, 3) are positive numbers. The derivatives of the parameter estimates are substituted in αi for the synthesis of the controller. Consider the closed-loop system (1), (6)–(8). Suppose that yr (t) and its derivatives are smooth and bounded. Then along the trajectory of the closed-loop system, zi (t) → 0 as t → ∞. Moreover, for yr (t) = y ∗ , a constant, the state vector converges to the equilibrium point x∗ = −1 T ∗ ∗ T ∗ T [y ∗ , −b−1 1 φ1 (y )θ, −b2 φ2 (y , x2 )θ] .
Theorem 1.
Consider a positive definite Lyapunov function
Proof.
W =
3 X i=1
zi2
+
3 X i=1
!,
γi | bi | ρ˜2i
+ β˜T Γ−1 β˜
2
(9) Fig. 2.
Open-loop chaotic response.
1602 K. Barone & S. N. Singh
(a) (d)
(b)
(e)
(c)
Fig. 3. Regulation to origin. (a) Vc1 , Vc2 , IL , (b) 3-D plot of Vc1 , Vc2 , IL , (c) Parameter estimate θˆ1 , (d) Parameter estimate bˆ1 , (e) Parameter estimate ρˆ1 .
Adaptive Feedback Linearizing Control of Chua’s Circuit 1603
(a) Fig. 4.
(b) Nonzero set point control. (a) Vc1 , Vc2 , IL , (b) 3-D plot of Vc1 , Vc2 , IL .
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 yr is generated by a third-order filter (s + 1)3 (yr − y ∗ ) = 0 with repeated poles at s = −1. The initial condi(k) tions chosen are yr (0) = 0.5, yr (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 ci = γi = 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, yr 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 yr = 0.4 cos(6πt/100). For the choice of the feedback parameters c1 = 0.25, c2 = 2.5, c3 = 1, γ1 = γ3 = 1, γ2 = 0.75 and Γ = 1.5I, selected
Fig. 5.
Sinusoidal trajectory control: Vc1 , Vc2 , IL .
responses are shown in Fig. 5. Again we observe that the node voltage Vc1 asymptotically follows the time-varying trajectory. Furthermore, Vc2 and IL 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 ci and Γ. Larger values of ci and Γ usually give faster tracking of yr . These results can be found in [Barone, 2000].
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.
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