Lyapunov-Based Controller for a Class of Stochastic Chaotic Systems

This study presents a general control law based on Lyapunov’s direct method for a group of well-known stochastic chaotic systems. Since real chaotic systems have undesired random-like behaviors which have also been deteriorated by environmental noise, chaotic systems aremodeled by exciting a deterministic chaotic systemwith awhite noise obtained fromderivative ofWiener process which eventually generates an Ito differential equation. Proposed controller not only can asymptotically stabilize these systems in meansquare sense against their undesired intrinsic properties, but also exhibits good transient response. Simulation results highlight effectiveness and feasibility of proposed controller in outperforming stochastic chaotic systems.


Introduction
In last two decades, the problem of control of chaotic systems has been widely investigated by many researchers due to the existence of chaos in real practical systems [1][2][3][4][5][6][7][8][9][10].A chaotic system has some inherent characteristics, such as excessive sensitivity to initial conditions, fractal properties of the motion in phase space, and board spectrums of the frequency response; hence, it is usually difficult to accurately predict the future behavior of the chaotic system, which can end up in performance degradation and restriction on the operating range of dynamic systems.
Considering aforesaid, developing strategies for controlling chaos phenomenon based on the features of chaotic motion is highly important; therefore, many nonlinear techniques for chaos control were proposed, such as feedback control [1,2] and sliding mode control [3][4][5].To exploit their advantages, these approaches are integrated as a complex control algorithm such as adaptive sliding mode [6,7], adaptive fuzzy sliding mode control [8], and predictive feedback control [9].However, in practice, real systems are usually affected by external perturbations which, in many cases, are of great uncertainty and hence may be treated as random; therefore, controlling chaos in such concrete systems needs to be regarded by stochastic concepts.Stochastic chaotic systems appear in many fields of science and engineering such as mechanical engineering [10], biology systems [11,12], chemistry [13], physics and laser science [14], and financial systems [15].For modeling stochastic chaotic systems, an Ito stochastic differential form is utilized by using the derivative of a Wiener process which creates a white Gaussian noise [16].
In [17], the sliding mode control is used for controlling stochastic chaos toward desired unstable periodic orbits of the deterministic chaotic system but the sliding mode suffers from the deficiency (chatter), which is caused by the sign function switch term in the control input.As a result, the convergence of the stochastic states to the desired equilibrium point cannot be completely achieved and state variance converges to a bound around equilibrium point.In other words, they cannot satisfy asymptotical stability condition for stochastic chaotic systems in mean-square sense.
According to the above, the main contribution of this paper is design of a very simple controller for a group of stochastic chaotic systems on the basis of Lyapunov's direct method which guarantees the asymptotically stabilizing of these systems in mean-square sense.Therefore, by adding stochastic terms to the generalized form of a well-known group of chaotic systems, control problem of these systems is investigated.The Lyapunov direct method which is a useful tool for designing globally stabilizing control schemes is utilized in this work.One of the important advantages of this method is stabilizing systems globally without any linearization.Numerical simulations show that the proposed method can easily eliminate undesired characteristics of chaos phenomenon and stabilize the system.This paper is organized as follows: general description of a group of chaotic systems is presented in Section 2. In Section 3, Lyapunov-based control law is obtained and stability of the proposed scheme is analyzed.In Section 4, numerical simulation results are shown.Finally, conclusion is addressed in Section 5.
Remark 1.Note that, in [6], half of recent published chaotic systems are organized by (1).Adding financial system, Table 1 illustrates these chaotic models.

Stochastic Chaos Control
In this section, a nonlinear control law is proposed for a group of stochastic chaotic systems (3) based on Lyapunov's direct method.Lyapunov's direct method design is a systematic Lyapunov-based control technique, which can be applied to strict feedback systems, pure-feedback systems, and block strict-feedback systems.Prior to the move into design control law, two important definitions should be stated.
Definition 4 (Itô formula [18]).Consider an -dimensional stochastic vector process {(, )} with stochastic differential (, ) = (, ) ⋅  + (, ) ⋅   () on  ≥ 0, where   () is an -dimensional Brownian motion defined on where () is given by  where   ,   , and   denote the partial derivative on , the gradient of (), and the Hessian matrix of (), respectively, and tr{⋅} is the sum of the main diagonal entries.) . ( Proof.For designing control law in Lyapunov's direct approach (without linearization), at first, Lyapunov function is regarded as Notice that {()} ≥ 0. Using Itô differential formula given in Definition 4, the stochastic differential of () along the trajectories of the system (3) turns out to be or can be stated as follows: ) . ( Since {()} = {()}, in order to show that the stochastic chaotic system (3) is asymptotically stable in the mean-square sense, it is enough to have {()} < 0. Now, let control law be introduced as (7).By substituting proposed control law into (13), it can be obtained that So, by applying input vector (7), stochastic chaotic system (3) is asymptotically stable in mean-square sense.

Proposed Controller for Stochastic Chen System.
In this case, the efficiency of the proposed Lyapunov's direct method scheme is verified by an example of control of the stochastic Chen system.Here, the stochastic Chen system can be rewritten in the form of ( 3 ( In the absence of the controller, the stochastic Chen system (17) exhibits a chaotic behavior as shown in Figure 1.
For this case, it is easy to see that According to (7), input signal is acquired as The simulation results are illustrated with the initial condition [  (0)   (0)   (0)]  = [5 4 7]  in Figure 2. The control law ( 20) is applied to the system (17) after five seconds.The left column of Figure 2 illustrates the trajectories of (), (), () which shows that the obtained theoretic results are feasible and efficient for the controlling stochastic Chen system.

Proposed Controller for Stochastic Financial System.
In this subsection, we will compare our proposed controller with the sliding based controller which is proposed in [17] on the financial system (16).Adding stochastic terms and input vector to the financial system, system ( 16) can be reformed as The stochastic system (21) exhibits a chaotic behavior without the controller vector (), which is shown in Figure 3.
Finally, according to (7), our proposed controller is obtained as follows: On the other hand, based on sliding mode scheme proposed in [17], input vector is obtained as follows: It is worth to mention that, in order to design a sliding mode controller, we have two steps: first constructing a sliding surface and designing an equivalent control law to have a desired system dynamics, second developing a switching control law such that a sliding mode exists on every point of the sliding surface, and thus any states outside the surface are driven to reach the surface in a finite time.
The state () of the system ( 21) is shown in the left column of Figure 4, in which the first row shows the effect of our proposed control law and the second row is related to sliding based controller.The states () and () are not shown here due to space limitations.In the right column of Figure 4, the input signals are presented in which the first row is related to Lyapunov's direct method and the second row is related to sliding mode strategy.It should be pointed that, in the last subfigure of Figure 4, the effect of equivalent control law is from 25th second to 32nd second, and next seconds are related to switching control law that causes chattering phenomenon.
It can be seen by using proposed method that not only would the state () converge to zero and the transient response be fast, but also, in comparison with the sliding based input, the input signal is smooth and chattering-free.

Conclusions
In this paper, the control problem of a group of stochastic chaotic systems is studied.Based on Lyapunov's direct method and Itô formula, the control law has been designed to control chaotic systems.It has been shown that, by applying proposed controller, systems are asymptotically stable in mean square sense and the transient responses are fast.Finally, in order to demonstrate the feasibility and effectiveness, the proposed control scheme is applied to two wellknown chaotic systems.

Financial
and  are replaced with each other, just to adopt the chaotic systems with the general Model (1).

Figure 1 :
Figure 1: The chaotic trajectories of the stochastic Chen system.

Figure 2 :
Figure 2: State trajectories of the stabilized stochastic Chen system (a) and input signals (b).

Figure 3 :
Figure 3: Financial system state space trajectories.

Figure 4 :
Figure 4: State trajectories of the stabilized stochastic financial system (a) and input signals (b).

Table 1 :
List of published chaotic models, which belong to the general class.