ANTI-SYNCHRONIZING BACKSTEPPING CONTROL DESIGN FOR ARNEODO CHAOTIC SYSTEM

International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 DOI : 10.5121/ijbb.2013.3103 21 ANTI-SYNCHRONIZING BACKSTEPPING CONTROL DESIGN FOR ARNEODO CHAOTIC SYSTEM Sundarapandian Vaidyanathan1 1Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600 062, Tamil Nadu, INDIA [email protected] ABSTRACT In this paper, we derive new results for backstepping controller design for the anti-synchronization of Arneodo chaotic system (1980). Backstepping control is a recursive procedure that combines the choice of a Lyapunov function with the design of a feedback controller. In anti-synchronization of chaotic systems, the states of the synchronized systems have the same absolute values, but opposite signs. First, we derive an active backstepping controller for the anti-synchronization of identical Arneodo chaotic systems. Next, we derive an adaptive backstepping controller for the anti-synchronization of identical Arneodo chaotic system, when the system parameters are unknown. The anti-synchronization results for Arneodo chaotic systems have been proved using Lyapunov stability theory. Numerical simulations have been shown to illustrate the backstepping controllers derived in this paper for Arneodo chaotic system. KEYWORDS Backstepping Control; Chaos; Anti-Synchronization; Arneodo System. 1. INTRODUCTION Chaos theory deals with the behaviour of nonlinear dynamical systems that are highly sensitive to initial conditions, an effect which is popularly known as the butterfly effect [1]. Small differences in initial conditions result in widely diverging outcomes for chaotic systems, rending long-term prediction impossible in general. The chaos phenomenon was first observed in weather models by the American scientist, Lorenz ([2], 1963). Since then, chaos theory has found applications in a variety of fields in science and engineering [3-9]. The problem of controlling a chaotic system was first introduced by Ott et al. ([10], 1990). The problem of chaos synchronization occurs when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator ([11], 1990). The idea of chaos antisynchronization is to use the output of the master system to control the output of the slave system so that the states of the master and slave systems have the same absolute values, but opposite signs, i.e. the sum of the output signals of the master and slave systems can converge to zero asymptotically. Since the pioneering work by Pecora and Carroll [11], various methods have been developed in the chaos literature for the synchronization of chaotic systems such as active control method [1215], adaptive control method [16-20], time-delay feedback control method [21], sampled-data control method [22-23], sliding mode control method [24-30], backstepping control method [3133], etc. In this paper, we deploy backstepping control method for the anti-synchronization of identical Arneodo chaotic systems ([34], 1980). Backstepping control method is a recursive procedure that combines the choice of a Lyapunov function with the design of a feedback controller. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 22 The organization of this research paper is as follows. In Section 2, we design an active backstepping controller for the anti-synchronization of identical Arneodo systems when the system parameters are known. In Section 3, we design an adaptive backstepping controller for the anti-synchronization of identical Arneodo systems when the system parameters are unknown. Section 4 contains the conclusions of this work. 2. ACTIVE BACKSTEPPING CONTROLLER DESIGN FOR THE ANTISYNCHRONIZATION OF ARNEODO SYSTEMS 2.1 Theoretical Results Arneodo system ([34], 1980) is one of the classical 3-D chaotic systems as it captures many features of chaotic systems. In this section, we investigate the problem of active backstepping controller design for the anti-synchronization of identical Arneodo chaotic systems, when the system parameters are known. As the master system, we consider the 3-D Arneodo dynamics 1 2 2 3 2 3 1 2 3 1 , , , x x x x x ax bx x x = = = − − −    (1) where 12 3 , ,x x x are the states and ,a b are positive, known parameters of the system. Figure 1. Strange Chaotic Attractor of the Arneodo System International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 23 The Arneodo system (1) undergoes chaotic behaviour when the system parameter values are chosen as 7.5a= and 3.8.b= The strange chaotic attractor of the Arneodo system (1) is shown in Figure 1. As the slave system, we consider the controlled 3-D Arneodo dynamics 1 2 2 3 2 3 1 2 3 1 , , , y y y y y ay by y y u = = = − − − +    (2) where 1 2 3 , ,y y y are the states and u is the active control to be designed. The anti-synchronization error between the master system (1) and the slave system (2) is defined as 1 1 1 2 2 2 3 3 3 ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ). e t y t x t e t y t x t e t y t x t = + = + = + (3) The design problem is to find a control ( )u t so that the error converges to zero asymptotically, i.e. ( ) 0 i e t → as t→ ∞ for 1,2,3.i= The error dynamics is easily derived as 1 2 2 3 2 2 3 1 2 3 1 1 , , . e e e e e ae be e y x u = = = − − − − +    (4) In this section, we apply the active backstepping control method to design a controller ( ).u t Theorem 1. The identical Arneodo chaotic systems (1) and (2) are globally and exponentially anti-synchronized for all initial conditions by the active backstepping controller 2 2 1 2 3 1 1 ( ) (3 ) (5 ) 2 .u t a e b e e y x= − + − − − + + (5) Proof. First, we define a Lyapunov function 2 1 1 1, 2 V z= (6) where 1 1.z e= (7) Its time derivative along the solutions of systems (1) and (2) is obtained as 2 1 1 1 1 1 1 2 1 1 1 2 ( ).V z z e e e e z z e e= = = = − + +   (8) International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 24 Next, we define 2 1 2.z e e= + (9) From (9), it follows that 2 1 1 1 2.V z z z= − +  (10) Secondly, we define the Lyapunov function ( ) 2 2 2 2 1 2 1 2 1 1 . 2 2 V V z z z= + = + (11) The time derivative of 2 V is given by 2 2 2 1 2 2 1 2 3 (2 2 ).V z z z e e e= − − + + +  (12) Next, we define 3 1 2 3 2 2 .z e e e= + + (13) From (13), it follows that 2 2 2 1 2 2 3.V z z z z= − − +  (14) Finally, we define the Lyapunov function ( ) 2 2 2 2 2 3 1 2 3 1 1 . 2 2 V V z z z z= + = + + (15) Clearly, V is a positive definite function on 3.R The time derivative of V is obtained as ( ) 2 2 2 2 1 2 2 3 3 2 3 1 2 3 1 1 2 2V z z z z z e e ae be e y x u= − − + + + + − − − − +  (16) A simple calculation gives 2 2 2 2 2 1 2 3 3 1 2 3 1 1 (3 ) (5 ) 2 .V z z z z a e b e e y x u   = − − − + + + − + − − +    (17) Substituting the backstepping controller u defined by (5) in (17), we get 222 1 2 3 .V z z z= − − −  (18) Clearly, V  is a negative definite function on 3.R Hence, by Lyapunov stability theory [35], the error dynamics (4) is globally exponentially stable. This completes the proof.  International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 25 2.2 Numerical Results For numerical simulations using MATLAB, the fourth order Runge-Kutta method with initial step 8 10h− = is used to solve the Arneodo systems (1) and (2) with the backstepping controller u defined by (5). The parameters of the Arneodo chaotic systems are selected as 7.5a= and 3.8.b= The initial values of the master system (1) are chosen as 1 2 3 (0) 14, (0) 5, (0) 6x x x= = − = The initial values of the slave system (2) are chosen as 1 2 3 (0) 18, (0) 12, (0) 16y y y= = = − Figure 2 depicts the anti-synchronization of Arneodo chaotic systems (1) and (2). Figure 3 depicts the time-history of the anti-synchronization errors 1 2 3 , , .e e e Figure 2. Anti-Synchronization of Arneodo Chaotic Systems International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 26 Figure 3. Time-History of the Anti-Synchronizing Errors 1 2 3 , ,e e e 3. REGULATING ACTIVE BACKSTEPPING CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF ARNEODO SYSTEMS 3.1 Theoretical Results In this section, we derive new results for the adaptive backstepping controller design for antisynchronization of Arneodo systems when the parameters a and b are unknown. As the master system, we consider the 3-D Arneodo dynamics 1 2 2 3 2 3 1 2 3 1 , , , x x x x x ax bx x x = = = − − −    (19) where 12 3 , ,x x x are the states and ,a b are unknown parameters of the system. As the slave system, we consider the controlled 3-D Arneodo dynamics 1 2 2 3 2 3 1 2 3 1 , , , y y y y y ay by y y u = = = − − − +    (20) where 1 2 3 , ,y y y are the states and u is the adaptive control to be designed. International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 27 The anti-synchronization error between the master system (19) and the slave system (20) is defined as 1 1 1 2 2 2 3 3 3 ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ). e t y t x t e t y t x t e t y t x t = + = + = + (21) The design problem is to find a control ( )u t so that the error converges to zero asymptotically, i.e. ( ) 0 i e t → as t→ ∞ for 1,2,3.i= The error dynamics is easily derived as 1 2 2 3 2 2 3 1 2 3 1 1 , , . e e e e e ae be e y x u = = = − − − − +    (22) In this section, we apply the adaptive backstepping control method to design a controller ( ).u t Inspired by the control law defined by Eq. (5) in the active backstepping controller design, we may consider the adaptive backstepping controller design law given by 2 2 1 2 3 1 1 ˆ ˆ ( ) (3 ) (5 ) 2 ,u t a e b e e y x= − + − − − + + (23) where ˆ( )a t and ˆ( )b t are estimates of the unknown parameters a and ,b respectively. We define the parameter estimation errors as ˆ ( ) ( ) a e t a a t= − and ˆ ( ) ( ) b e t b b t= − (24) Note that ˆ ( ) ( ) a e t a t= −  and ˆ ( ) ( ) b e t b t= −   (25) Next, we shall state and prove the second main result of this paper. Theorem 2. The identical Arneodo chaotic systems (19) and (20) with unknown parameters a and b are globally and exponentially anti-synchronized for all initial conditions by the adaptive backstepping controller 2 2 1 2 3 1 1 ˆ ˆ ( ) (3 ) (5 ) 2 ,u t a e b e e y x= − + − − − + + (26) where ˆ( )a t and ˆ( )b t are estimates of a and ,b respectively, and the parameter update law is given by 1 2 3 1 1 2 3 2 ˆ( ) (2 2 ) , ˆ( ) (2 2 ) , a a b b a t e e e e k e b t e e e e k e = + + + = − + + +   (27) International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 28 with positive control gains a k and . b k Proof. First, we define the Lyapunov function 2 1 1 1, 2 V z= (28) where 1 1.z e= (29) The time derivative of 1 V is given by 2 1 1 1 1 1 1 2 1 1 1 2 ( ).V z z e e e e z z e e= = = = − + +   (30) Next, we define 2 1 2.z e e= + (31) From (30), it follows that 2 1 1 1 2.V z z z= − +  (32) Secondly, we define the Lyapunov function ( ) 2 2 2 2 1 2 1 2 1 1 . 2 2 V V z z z= + = + (33) The time derivative of 2 V is given by 2 2 2 1 2 2 1 2 3 (2 2 ).V z z z e e e= − − + + +  (34) Next, we define 3 1 2 3 2 2 .z e e e= + + (35) From (34), it follows that 2 2 2 1 2 2 3.V z z z z= − − +  (35) Finally, we define the Lyapunov function ( ) ( ) 2 2 2 2 2 2 2 2 2 3 1 2 3 1 1 1 . 2 2 2 a b a b V V z e e z z z e e= + + + = + + + + (36) The time derivative of V is obtained as 2 2 2 2 2 1 2 3 3 1 2 3 1 1 ˆ ˆ (3 ) (5 ) 2 . a b V z z z z a e b e e y x u e a e b   = − − − + + + − + − − + − −      (37) Substituting the backstepping controller u defined by (26) in (37), we get ( ) ( ) 222 1 2 3 1 3 2 3 ˆ ˆ. a b V z z z e e z a e e z b= − − − + − + − −    (38) International Journal on Bioinformatics & Biosciences (IJBB) Vol.3, No.1, March 2013 29 Substituting the parameter law (27) in (38) and noting that 3 1 2 3 2 2 ,z e e e= + + we get 2 2 2 2 2 1 2 3 , a a b b V z z z k e k e= − − − − −  (39) which is a negative definite function on 5.R Thus, by Lyapunov stability theory [35], the proof is complete. 3.2 Numerical Results For numerical simulations with MATLAB, the fourth-order Runge-Kutta method with initial step 8 10h− = is used to solve the Arneodo systems (19) and (20) with the backstepping controller u defined by (26) and the parameter update law defined by (27). The parameters of the Arneodo chaotic systems are chosen as 7.5a= and 3.8.b= The initial values of the parameter estimates are chosen as ˆ(0) 16a= and ˆ(0) 9.b= The control gains are chosen as 6 a k= and 6. b k= The initial values of the master system (19) are chosen as 1 2 3 (0) 4, (0) 5, (0) 8x x x= = = − The initial values of the slave system (20) are chosen as 1 2 3 (0) 2, (0) 6, (0) 5y y y= = = − Figure 4 depicts the anti-synchronization of Arneodo chaotic systems. Figure 5 depicts the timehistory of the anti-synchronization errors 1 2 3 , , .e e e Figure 6 depicts the time-history of the parameter estimation errors , . a b e e