An Approach of Tracking Control for Chaotic Systems

SinceOtt et al. firstly proposed themethod of chaos control in 1990 [1], chaos control has attracted great attention in recent years and lots of successful methods have been reported [2–22], such as feedback control [7, 8], impulsive control [9, 10], backstepping method [11], adaptive control [12–15], adaptive fuzzy backstepping technique [16], adaptive sliding mode control [17, 18], neural network technique [19], andH ∞

In 1998, Lin et al. applied the idea of Chen's method [23] to discrete chaotic systems [24], but the method needed calculating an error feedback matrix to assure that a certain matrix is negative semidefinite.When the dimension of these matrixes is large, the calculation becomes rather complex.
Chen established the open-plus-closed-loop control law for discrete dynamical systems [25], but it required that the initial points were in the basin of entrainment.References [26][27][28] realized tracking control only for Henon chaotic system.References [29,30] presented tracking control schemes for continuous chaotic systems and did not refer to the application to discrete systems.Zheng et al. gave a rapid synchronization algorithm [31], but it involved the calculation of high order derivative matrix.Rehan et al. discussed stabilization and tracking control using linear matrix inequalities for a class of continuous systems satisfying global Lipschitz condition [32].
In practical engineering, we need to eliminate the chaos or transform them into some useful signals.Therefore, tracking control, which transforms the chaos signal into desired bounded signal, is significant in practice.Moreover, generalized projective synchronization (GPS) can transform into the problem of tracking control.
Based on the above discussions, we present a general tracking control scheme based on the Jacobian matrix and ergodicity of chaos.It is simple and does not refer to high order derivative matrix or other requirements such as global Lipschitz condition.The rest of the paper is organized as follows.In Section 2, the tracking control schemes for continuous and discrete systems and their mathematical proofs are given, respectively.In Section 3, we have applied this method to three chaotic systems and make simulations.Finally, the conclusions are drawn in Section 4.
Set the range of chaotic system as  and the range of reference signal as .As the range of chaotic system  is certain, we choose the reference signal  satisfying ∩ ̸ = Φ.It is easy to implement in practice.
Theorem 1.For systems (1) and ( 2), if we set  = − and add the controller where Df|  , Dg|  are the Jacobian matrix of (, ) and (, ) at , respectively,  = diag( 1 ,  2 , . . .,   ) is a constant feedback diagonal matrix, and  is the switch-off controller, which is depicted as where ‖ ⋅ ‖ denotes the Euclidean norm and  is a constant, then for the initial values (0) which make (1) chaotic, That is, system (1) tracks system (2) asymptotically.
Proof.The Taylor series of (, ) at  is where (, ) is a polynomial vector which contains quadratic term and finite higher order terms of  and (0, ) = 0. Similarly, where (, ) is a polynomial vector like (, ) and (0, ) = 0.
When ‖‖ ≥ , the controller does not work.As there exist  and  satisfying  ∩  ̸ = Φ, we let  =  ∩ .According to the ergodicity of chaos, there always exists a certain time  0 satisfying  = (,  0 ) − (,  0 ) = 0 in the set , and system (1) always comes into the domain of ‖‖ <  in a limited time; thus, it asymptotically tracks the reference signal with controller (3).The proof is completed.
Remark 2. The value of  is related to   ( = 1, 2, . . ., ) according to (10).The larger the   is, the quicker the convergence is.However, it is difficult to calculate specific value of .In practice, the value of  can be increased from zero for an appropriate convergence speed.
Theorem 4. For systems (11) and ( 12 where  is a constant less than 1, then for the initial values  1 which make (11) chaotic we have That is, system (11) tracks system (12) asymptotically.
Proof.The Taylor series of (,   ) at   is where (,   ) is a polynomial vector which contains quadratic term and finite higher order terms of   and (, 0) = 0. Similarly, where V(,   ) is a polynomial vector like (,   ) and V(, 0) = 0.
Remark 9.The method can be extended to GPS.
To display the robustness of the proposed method, we add a uniformly distributed random noise to ().Figure 2 indicates that () eventually tracks the reference signal and ultimately slightly fluctuates around it.
According to (3), we have   The tracking trajectory and the controller are shown in Figure 4, where the dotted line denotes the reference signals.The controlled Duffing system quickly tracks the reference signal whose range is larger than that of Duffing system.
Given the goal orbit [25] which is a period-4 orbit.Applying control law (14), we have The results with different  1 are shown in Figures 5-7.It can be seen that the system arrives at the desired goal in   a short time.It is quicker than [25] and does not have any requirement of initial point.
However, the speed of convergence is related to the value of  and  1 .We should choose appropriate  1 (choosing 0 as initial value) to get a desired speed of convergence, not the larger the better.

Conclusions
We design a general tracking controller for chaotic systems combining the ergodicity of chaos and Jacobian matrix.It is suitable for continuous and discrete chaotic systems.For continuous systems, the element of feedback matrix   < 0 ( = 1, 2, . . ., ) and the norm of error  > 0. For discrete system, the element of feedback matrix |  | <  ( = 1, 2, . . ., ) and the norm of error 0 <  < 1.
The control scheme has the ability to track a bounded reference signal satisfying  ∩  ̸ = Φ.Moreover, it can be generalized into GPS.The simulations demonstrate its good performance in terms of simplicity, feasibility, and robustness, which indicate it has better practical significance for real world applications.The number of steps  The number of steps ), we have ė = ẋ − ṙ =  (, ) +  +  (, ) −  (, ) − (Df|  + Dg     )  −  (, ) =  (, ) +  (, ) + .

Figure 3 :
Figure 3: The chaotic behavior of Duffing system.