New Results on the Control for a Kind of Uncertain Chaotic Systems Based on Fuzzy Logic

In this paper, the problem of the control for an uncertain nonlinear chaotic system has been studied; based on fuzzy logic, a kind of single-dimensional controller is constructed for the control of the chaotic systems in the situation that uncertainties and unknowns exist; at last some typical numerical simulations are carried out, and corresponding results illuminate the effectiveness of the controller.


Introduction
Nonlinear systems exist in real engineering widely.Since the pioneering work from Lurie in 1944, the research on nonlinear system control has become the challenging issue, and many techniques, such as differential geometry technique [1,2], sliding mode technique [3][4][5][6] and so on, have been proposed to deal with this problem.It can be noted that these approaches are based on multidimensional control.However, in some cases, the single-dimensional controller is more cherished for its simpler structure and more convenient application in practice.
As an important branch of nonlinear systems, chaotic system and its control received many attentions, and a lot of related results have been reported so far [7][8][9][10][11][12][13][14].For instance, in [7], based on output feedback control strategy, a method was presented to realize the control for unified chaotic systems; in [8], the synchronization control for Lü systems with unknown parameters was investigated; in [9], the adaptive control for the synchronization of hyperchaotic systems was studied; in [10], the fuzzy control for Arneodo chaotic system is discussed.However most of these researches focused on just one typical chaotic system.In addition, it is well known that there exist many kinds of uncertainties in practical control system, and the following chaotic system model is studied.ẋ  = (  + Δ  )  +1 +   (  ) , 1 ≤  ≤  − 1 ẋ  = ℎ  () + Δℎ  () +   () +     =  1 (1) where are the known system parameters and satisfy 0 <   ≤ |  | ≤   , where   and   are the positive scalars,   () and   (  ) are the unknown terms, Δ  and Δℎ  are the uncertainties, ℎ  is the known term,   is the control parameter,  is the system output, and  is the single-dimensional control input.A lot of chaotic systems can be transformed into the system with the form (1) through topological mapping.
As an important technique, fuzzy techniques are very suitable for the research of nonlinear and complex systems (see [15][16][17][18][19][20][21][22][23] and references therein), and they will be introduced to design the single-dimensional controller for system (1) in this paper.Some simulations will be included to illuminate the effectiveness of the constructed controller.

Model Description and Preliminaries
It is well known that fuzzy logic system can approximate the nonlinear function.Let () denote the smooth function and () denote the fuzzy logic system.There exists the optimal parameter  * = arg min ∈Ω 0 [sup ∈Ω |()−()|] for the least approximation error, where Ω 0 and Ω are bounded sets of  and x.
Define fuzzy rules as Define the following fuzzy logic system [16] where where In the paper, the following lemmas are concerned.
Step .Define the tracking error  1 =  −  d ;  d is the desired trajectory.
For the first subsystem of system (1), the virtual variable  1 is introduced, such that where  2 =  2 −  1 .
Step .For the second subsystem of system (1), the virtual variable  2 is introduced, such that where  3 =  3 −  2 .
Step k ( < ).For k-th subsystem of system (1), the virtual variable   is introduced, such that where  +1 =  +1 − Step n.For the n-th subsystem of system (1), one can get where   =   −  −1 Then, the following tracking error dynamic system can be derived where  0 =  d The object of this paper is to design a controller, such that lim Choose the first Lyapunov function as where Let Choose the second Lyapunov function as used to approximate the nonlinear function  2 , then where T (  ) −1 is used to approximate the nonlinear function Choose the k-th Lyapunov function ( < ) as Hence where Let   = −    − −1 −  ,   > 0, where   =   T ( +1 )  is used to approximate the nonlinear function   , then It is consistent with our notation that Choose the n-th Lyapunov function as Hence where Suppose that   =   T ()  approximate the nonlinear function   and that is based on Lyapunov theory, then the following theoretical result can be obtained.
Theorem 2. For   > 0,   > 0 and   > 0, based on the controller and the adaptive law then the output of chaotic system ( ) can track the desired trajectory.
Integrating both sides of inequality (48) from 0 to T, one can get which means () ∈  2 .From error dynamic system (10), it can be concluded that ė () ∈  ∞ .Accordingly based on Lemma 1, one can get lim →+∞ () = 0, which means the achievement of the track control.The proof of Theorem 2 is thus completed.
Remark .Figure 1 displays the chaotic attractor of Arneodo system.Figure 2 displays the state response of x 1 of Arneodo system.From Figures 1 and 2, it can be seen that Arneodo system has the complicated dynamical behavior.Figure 3 displays the state response of variable x 1 of uncertain Arneodo system.It can be seen that the existence of unknowns and uncertainties makes Arneodo system unstable.
Remark .Figure 4 displays the fuzzy membership function.Figure 5 displays the state response of control input.Figure 6 displays the state response of y d and y. Figure 7 displays the state response of position tracking error.From Figures 4-7, it can be seen that for uncertain Arneodo system, the position tracking can be achieved during 0.5 second based on the designed controller.

Conclusion
In this paper, based on fuzzy logic, a single-dimensional controller has been constructed for the control of a kind of uncertain chaotic systems.Some typical examples have been employed and corresponding simulation results have illuminated the effectiveness of proposed controller.

Figure 2 :
Figure 2: State response of x 1 of Arneodo system.