ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR Dr. SR Technical University Avadi, Chennai-600 062, Tamil Nadu, INDIA sundarvtu@gmail.com ABSTRACT The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one o the important models o ourdimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization o hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization o identical hyperchaotic Qi systems with unknown parameters. Numerical simulations are shown to demonstrate the eectiveness o the proposed adaptive stabilization and synchronization schemes. KEYWORDS Adaptive Control, Stabilization, Chaos Synchronization, Hyperchaos, Hyperchaotic Qi System. 1. INTRODUCTION Chaotic systems are dynamical systems that are highly sensitive to initial conditions. The sensitive nature o chaotic systems is commonly called as the butterly eect [1]. Since chaos phenomenon in weather models was irst observed by Lorenz in 1961, a large number o chaos phenomena and chaos behaviour have been discovered in physical, social, economical, biological and electrical systems. The control o chaotic systems is to design state eedback control laws that stabilizes the chaotic systems around the unstable equilibrium points. Active control technique is used when the system parameters are known and adaptive control technique is used when the system parameters are unknown [2-4]. Synchronization o chaotic systems is a phenomenon that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator. Because o the butterly eect, which causes the exponential divergence o the trajectories o two identical chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a very challenging problem in the chaos literature [5-16]. In 1990, Pecora and Carroll [5] introduced a method to synchronize two identical chaotic systems and showed that it was possible or some chaotic systems to be completely synchronized. From then on, chaos synchronization has been widely explored in a variety o ields including physical systems [6], chemical systems [7], ecological systems [8], secure communications [9-10], etc. DOI : 10.5121/cseij.2011.1202 14
In most o the chaos synchronization approaches, the master-slave or drive-response ormalism has been used. I a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea o synchronization is to use the output o the master system to control the slave system so that the output o the slave system tracks the output o the master system asymptotically. Since the seminal work by Pecora and Carroll [5], a variety o impressive approaches have been proposed or the synchronization o chaotic systems such as the sampled-data eedback synchronization method [11], OGY method [12], time-delay eedback method [13], backstepping method [14], adaptive design method [15], sliding mode control method [16], etc. This paper is organized as ollows. In Section 2, we derive results or the adaptive stabilization o hyperchaotic Qi system (Chen, Yang, Qi and Yuan, [17], 2007) with unknown parameters. In Section 3, we derive results or the adaptive synchronization o hyperchaotic Qi systems with unknown parameters. In Section 4, we summarize the main results obtained in this paper. 2. ADAPTIVE STABILIZATION OF HYPERCHAOTIC QI SYSTEM 2.1 Theoretical Results The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, [17], 2007) is one o the important models o our-dimensional hyperchaotic systems. It is a novel hyperchaotic system with only one equilibrium at the origin. The hyperchaotic Qi dynamics is described by = a( x x ) + ε x x 1 2 1 2 3 = c x d x x + x + x 2 1 1 3 2 4 = x x b x 3 1 2 3 = x 4 2 (1) where xi ( i = 1, 2,3, 4) are the state variables and a, b, c, d, ε, are positive constants. Figure 1. State Orbits o the Hyperchaotic Qi System 15
The system (1) is hyperchaotic when the parameter values are taken as a = 35, b = 4.9, c = 25, d = 5, ε = 35 and = 22. (2) The hyperchaotic state portrait o the system (1) is described in Figure 1. When the parameter values are taken as in (2), the system (1) is hyperchaotic and the system linearization matrix at the equilibrium point E 0 = (0,0,0,0) is given by 35 35 0 0 25 1 0 1 A = 0 0 4.9 0 0 22 0 0 which has the eigenvalues λ = 0.8989, λ = 16.6257, λ = 4.9 and λ 4 = 51.5245 1 2 3 Since λ1, λ2 are eigenvalues with positive real part, it is immediate rom Lyapunov stability theory [18] that the system (1) is unstable at the equilibrium point E 0 = (0,0,0,0). In this section, we design adaptive control law or globally stabilizing the hyperchaotic system (1) when the parameter values are unknown. Thus, we consider the controlled hyperchaotic Qi system as ollows. = a( x x ) + ε x x + u 1 2 1 2 3 1 = c x d x x + x + x + u 2 1 1 3 2 4 2 = x x b x + u 3 1 2 3 3 = x + u 4 2 4 (3) where u1, u2, u3 and u4 are eedback controllers to be designed using the states and estimates o the unknown parameters o the system. In order to ensure that the controlled system (3) globally converges to the origin asymptotically, we consider the ollowing adaptive control unctions u = a ( x x ) ε x x k x 1 2 1 2 3 1 1 u = c x + d x x x x k x 2 1 1 3 2 4 2 2 u = x x + b x k x 3 1 2 3 3 3 u = x k x 4 2 4 4 (4) where a, b, c, d, ε and are estimates o the parameters a, b, c, d, ε and, respectively, and ki,( i = 1, 2,3, 4) are positive constants. 16
Substituting the control law (4) into the hyperchaotic Qi dynamics (1), we obtain = ( a a ) ( x x ) + ( ε ε ) x x k x 1 2 1 2 3 1 1 = ( c c ) x ( d d ) x x k x 2 1 1 3 2 2 = ( b b ) x k x 3 3 3 3 = ( ) x k x 4 2 4 4 (5) Let us now deine the parameter errors as e = a a, e = b b, e = c c a b c e,, d = d d eε = ε ε e = (6) Using (6), the closed-loop dynamics (5) can be written compactly as = e ( x x ) + e x x k x 1 a 2 1 ε 2 3 1 1 = e x e x x k x 2 c 1 d 1 3 2 2 = e x k x 3 b 3 3 3 = e x k x 4 2 4 4 (7) For the derivation o the update law or adjusting the parameter estimates a, b, c, d, ε,, the Lyapunov approach is used. Consider the quadratic Lyapunov unction 1 2 2 2 2 2 2 2 2 2 2 V = ( x1 + x2 + x3 + x4 + ea + eb + ec + ed + eε + e ) (8) 2 which is a positive deinite unction on Note also that R 10.,, a = a b = b c = c e d =, e = ε, e = d ε (9) Dierentiating V along the trajectories o (7) and using (9), we obtain 2 2 2 2 2 V = k1 x1 k2 x2 k3 x3 k4 x 4 + e a x1 ( x2 x1 ) a e b x3 b + + e c x1x2 c e + d x1x2 x3 d + e ε x1x2 x3 ε e + x2x4 (10) 17
In view o Eq. (10), the estimated parameters are updated by the ollowing law: a = x1 ( x2 x1 ) + k5e 2 b = x3 + k6eb c = x1x2 + k7ec d = x1 x2x3 + k8ed ε = x1 x2x3 + k9eε = x x + k e 2 4 10 a (11) where ki,( i = 5, K,10) are positive constants. Substituting (11) into (10), we get V = k x k x k x k x k e k e k e k e k e k e (12) 2 2 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4 5 a 6 b 7 c 8 d 9 ε 10 which is a negative deinite unction on R 10. Thus, by Lyapunov stability theory [18], we obtain the ollowing result. Theorem 1. The hyperchaotic Qi system (1) with unknown parameters is globally and 4 exponentially stabilized or all initial conditions x(0) R by the adaptive control law (4), where the update law or the parameters is given by (11) and ki, ( i = 1, K,10) are positive constants. 2.2 Numerical Results For the numerical simulations, the ourth order Runge-Kutta method is used to solve the hyperchaotic system (1) with the adaptive control law (4) and the parameter update law (11). The parameters o the hyperchaotic Qi system (1) are selected as a = 35, b = 4.9, c = 25, d = 5, ε = 35 and = 22. For the adaptive and update laws, we take k = 2, ( i = 1, 2, K,10). Suppose that the initial values o the estimated parameters are a (0) = 4, b (0) = 5, c (0) = 2, d (0) = 7, ε (0) = 5 and (0) = 8. The initial values o the hyperchaotic Qi system (1) are taken as x (0) = (10,4,8,12). i When the adaptive control law (4) and the parameter update law (11) are used, the controlled hyperchaotic Qi system converges to the equilibrium E 0 = (0, 0, 0, 0) exponentially as shown in Figure 2. The parameter estimates are shown in Figure 3. 18
Figure 2. Time Responses o the Controlled Hyperchaotic Qi System Figure 3. Parameter Estimates a ( t), b ( t), c ( t), d ( t), ε ( t), ( t) 19
3. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC QI SYSTEMS 3.1 Theoretical Results In this section, we discuss the adaptive synchronization o identical hyperchaotic Qi systems (Chen, Yang, Qi and Yuan, [17], 2007) with unknown parameters. As the master system, we consider the hyperchaotic Qi dynamics described by = a( x x ) + ε x x 1 2 1 2 3 = c x d x x + x + x 2 1 1 3 2 4 = x x b x 3 1 2 3 = x 4 2 where xi, ( i = 1,2,3,4) are the state variables and a, b, c, d, ε, parameters. (13) are unknown system The system (13) is hyperchaotic when the parameter values are taken as a = 35, b = 4.9, c = 25, d = 5, ε = 35 and = 22. As the slave system, we consider the controlled hyperchaotic Qi dynamics described by y = a( y y ) + ε y y + u 1 2 1 2 3 1 y = c y d y y + y + y + u 2 1 1 3 2 4 2 y = y y b y + u 3 1 2 3 3 y = y + u 4 2 4 (14) where yi, ( i = 1, 2,3,4) are the state variables and ui, ( i = 1, 2,3,4) are the nonlinear controllers to be designed. The synchronization error is deined by e = y x, ( i = 1,2,3,4) (15) i i i Then the error dynamics is obtained as = a ( e e ) + ε ( y y x x ) + u 1 2 1 2 3 2 3 1 = c e d ( y y x x ) + e + e + u 2 1 1 3 1 3 2 4 2 = y y x x be + u 3 1 2 1 2 3 3 = e + u 4 2 4 (16) 20
Let us now deine the adaptive control unctions u1( t), u2( t), u3( t), u4( t) as u = a ( e e ) ε ( y y x x ) k e 1 2 1 2 3 2 3 1 1 u = c e + d ( y y x x ) e e k e 2 1 1 3 1 3 2 4 2 2 u = y y + x x + b e k e 3 1 2 1 2 3 3 3 u = e k e 4 2 4 4 (17) where a, b, c, d, ε and are estimates o the parameters a, b, c, d, ε and respectively, and ki,( i = 1,2,3,4) are positive constants. Substituting the control law (17) into (16), we obtain the error dynamics as = ( a a ) ( e e ) + ( ε ε ) ( y y x x ) k e 1 2 1 2 3 2 3 1 1 = ( c c ) e ( d d )( y y x x ) k e 2 1 1 3 1 3 2 2 = ( b b ) e k e 3 3 3 3 = ( ) e k e 4 2 4 4 (18) Let us now deine the parameter errors as e,,,, a = a a eb = b b ec = c c ed = d d eε = ε ε, e = (19) Substituting (19) into (18), the error dynamics simpliies to = e ( e e ) + e ( y y x x ) k e 1 a 2 1 ε 2 3 2 3 1 1 = e e e ( y y x x ) k e 2 c 1 d 1 3 1 3 2 2 = e e k e 3 b 3 3 3 = e e k e 4 2 4 4 (20) For the derivation o the update law or adjusting the estimates o the parameters, the Lyapunov approach is used. Consider the quadratic Lyapunov unction 1 2 2 2 2 2 2 2 2 2 2 V = ( e1 + e2 + e3 + e4 + ea + eb + ec + ed + eε + e ) (21) 2 which is a positive deinite unction on Note also that R 10.,, a = a b = b c = c e d =, e = ε, e = d ε (22) 21
Dierentiating V along the trajectories o (20) and using (22), we obtain 2 2 2 2 2 V = k1 e1 k2 e2 k3 e3 k4 e4 + e a e1 ( e2 e1 ) a e + b e3 b + e c e1e 2 c e + d e2 ( y1 y3 x1x3 ) d + e ε e1 ( y2 y3 x2x3 ) ε e + e2e4 (23) In view o Eq. (23), the estimated parameters are updated by the ollowing law: a = e1 ( e2 e1 ) + k5ea 2 b = e3 + k6eb c = e1e 2 + k7ec d = e2 ( y1 y3 x1x3 ) + k8ed ε = e1 ( y2 y3 x2x3 ) + k9eε = e e + k e 2 4 10 (24) where ki,( i = 5, K,10) are positive constants. Substituting (24) into (23), we get V = k e k e k e k e k e k e k e k e k e k e (25) 2 2 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4 5 a 6 b 7 c 8 d 9 ε 10 which is a negative deinite unction on R 10. Thus, by Lyapunov stability theory [18], it is immediate that the synchronization error and the parameter error decay to zero exponentially with time or all initial conditions. Hence, we have proved the ollowing result. Theorem 2. The identical hyperchaotic Qi systems (13) and (14) with unknown parameters are globally and exponentially synchronized or all initial conditions by the adaptive control law (17), where the update law or parameters is given by (24) and ki,( i = 1, K,10) are positive constants. 3.2 Numerical Results For the numerical simulations, the ourth order Runge-Kutta method is used to solve the two systems o dierential equations (13) and (14) with the adaptive control law (17) and the parameter update law (24). For the adaptive synchronization o the hyperchaotic Qi systems with parameter values a = 35, b = 4.9, c = 25, d = 5, ε = 35 and = 22, we apply the adaptive control law (17) and the parameter update law (24). 22
We take the positive constants ki, ( i = 1, K,10) as k i = 2 or i = 1, 2, K,10. Suppose that the initial values o the estimated parameters are a (0) = 5, b (0) = 4, c (0) = 3, d (0) = 6, ε (0) = 4 and (0) = 2. We take the initial values o the master system (13) as x (0) = (5,24,18, 20). We take the initial values o the slave system (14) as y (0) = (17, 20,8,12). Figure 4 shows the adaptive chaos synchronization o the identical hyperchaotic Qi systems. Figure 5 shows that the estimated values o the parameters a, b, c, d, ε, converge to the system parameters a = 35, b = 4.9, c = 25, d = 5, ε = 35 and = 22. Figure 4. Adaptive Synchronization o the Identical Hyperchaotic Qi Systems 23
4. CONCLUSIONS Figure 5. Parameter Estimates a ( t), b ( t), c ( t), d ( t), ε ( t), ( t) In this paper, we applied adaptive control theory or the stabilization and synchronization o the hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) with unknown system parameters. First, we designed adaptive control laws to stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then we derived adaptive synchronization scheme and update law or the estimation o system parameters or identical hyperchaotic Qi systems with unknown parameters. Our synchronization schemes were established using Lyapunov stability theory. Since the Lyapunov exponents are not required or these calculations, the proposed adaptive control method is very eective and convenient to achieve chaos control and synchronization o the hyperchaotic Qi system. Numerical simulations are shown to demonstrate the eectiveness o the proposed adaptive stabilization and synchronization schemes. REFERENCES [1] Alligood, K.T., Sauer, T. Yorke, J.A. (1997) Chaos: An Introduction to Dynamical Systems, Springer, New York. [2] Ge, S.S., Wang, C. Lee, T.H. (2000) Adaptive backstepping control o a class o chaotic systems, Internat. J. Biur. Chaos, Vol. 10, pp 1149-1156. [3] Wang, X., Tian, L. Yu, L. (2006) Adaptive control and slow maniold analysis o a new chaotic system, Internat. J. Nonlinear Science, Vol. 21, pp 43-49. 24
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