Adaptive Neural Gradient Descent Control for a Class of Nonlinear Dynamic Systems with Chaotic Phenomenon

A neural network controller design is studied for a class of nonlinear chaotic systems with uncertain parameters. Because the chaos phenomena are often in this class of systems, it is indispensable to control this class of systems. At the same time, due to the presence of uncertainties in the chaotic systems, it results in the difficulties of the controller design. The neural networks are employed to estimate the uncertainties of the systems and a controller is designed to overcome the chaos phenomena.The main contribution of this paper is that the adaptation law can be determined via the gradient descent algorithm to minimize a cost function of error. It can prove the stability of the closed-loop system. The numerical simulation is specified to pinpoint the validation of the approach.


Introduction
The control problem for complex and ill-defined systems is a very significant work in practice.The uncertainties, unknown parameters, and unmodeled dynamics, and so forth, belong to ill-defined systems.There are many practical plants that are ill-defined, such as robot, chemical systems, underactuated systems, and chaotic systems, and their stability problems are hard to be solved.A good way for solving ill-defined systems is to use the neural network control scheme and this scheme has been used to perform important task in practice.Simultaneously, a lot of achievements have been gained for some different classes of ill-defined systems.In [1][2][3][4][5][6][7][8][9], the controller tracking designs were solved for ill-defined continuous-time systems which contain uncertainties and nonlinear property.For ill-defined discrete-time systems, several design strategies were given in [10][11][12][13].
Recently, the stability of chaotic systems, which belong to a class of ill-defined systems, has been an active research direction.In [14], the control problem for chaotic systems was studied.The functions of chaotic systems are nonlinear and do not require satisfying Lipschitz condition.A synchronization sliding mode control algorithm was proposed in [15] for two classes of chaotic systems based on RBF neural networks.A combined disturbance observer was proposed and the update law of uncertain parameters was shown to monitor the combined disturbance.A function projective synchronization for fractional order chaotic systems with unknown parameters was investigated in [16] by using adaptive control modified method.In [17], a novel synchronization modified function projective scheme for identical and nonidentical hyperchaotic complex chaotic systems with uncertain parameters was developed.An adaptive chatterfree control via sliding mode design was pointed out in [18] for uncertain chaotic system without the knowledge of the bounds of the uncertain term representing the model uncertainty and unknown disturbance.In [19], the time stability for a general chaotic system was analyzed by utilizing adaptive feedback terminal sliding mode control.However, the adaptation laws of these approaches are obtained without using gradient descent method.A main advantage of gradient descent method is that it can minimize a cost function of error.This paper will use gradient descent method to control a class of chaotic systems.
Based on the above presentations, an adaptive control algorithm via the neural networks is investigated for a class of chaotic systems.By using the neural network approximation, the unknown function is modeled and chaos phenomena can 2 Mathematical Problems in Engineering be solved by using a stable controller.Adaptive law design is proposed via the gradient descent algorithm to minimize a cost function of error.The Lyapunov analysis is employed to prove the stability of the closed-loop system and a simulation is to validate the effectiveness of the approach.

Problem and Formulation
Consider the chaotic systems as the following form: where   ,  = 1, . . ., 4, are unknown parameters.The system (1) is also called Arneodo chaotic system.When there does not exist a control input, the chaotic system is unstable.
Let () =    where  = [ 1 ,  2 ,  3 ,  4 ] and  = [ 1 ,  2 ,  3 ,  3  1 ]  .Then, one has where  = [ 1 ,  2 ,  3 ]  ∈  3 is the state vector of the system and  ∈  is the scalar control input.In this study, the control objective is to design an adaptive neural control algorithm so that the system is stable; that is, all the signals in the closed-loop system are bounded.
The error vector is defined as where Then, we know that It can be written as follows: where We select  = [ 0 ,  1 ,  2 ]  to ensure that  =  0 −   is stable.Therefore, there exists a matrix  such that where  and  are positive definite symmetric matrices.
Define  as where  > 0 and  > 0 are the constants.By adding and subtracting the term  to (6), we know that Based on the implicit function theorem, there exists an ideal controller  * (, c) so that If  is selected as  =  * , ( 10) is simplified as Consider the following Lyapunov function: Combining with ( 12) and ( 8), the time derivative of ( 13) can be written as Because    tanh(  /) > 0, V < 0.Then, the system is stable.Nevertheless,  * (, ) is not effective owing to unknown parameters.

Gradient Descent Controller Design
In Section 2, we assume that there is  * (, ) can realize the objective.We can determine the presence of the ideal controller with the aid of implicit function theorem, but we cannot get the way to build it.In this section, we use adaptive neural system to determine the controller. * (, ) is approximated as follows: where  = [  , ]  and () is the fuzzy approximation error,  * is an ideal parameter vector, and () is a basic function vector.In this study, the approximation error is bounded; that is, |()| ≤  with  > 0 being a constant. * is unknown; then, choose  as an estimate of  * .Define the control law as  () =   () .(16) We define the error between  and  * as follows: By using ( 15) and ( 16), ( 17) can be written as where θ =  * − .
According to (18), () +  can be expressed as Substituting ( 19) into (10) yields Substituting ( 11) into (20), Then, we have Then, consider a quadratic cost function, which is used to measure the discrepancy between the ideal and the neural controller.And the function can be defined as Here, we will minimize (23) based on the gradient descent algorithm.We obtain the following first-order differential equation where () is a positive parameter.By employing (23), it has Hence, the gradient descent algorithm can be represented as Because   is unavailable, we need to make (26) calculable; we select () =  0 where  0 > 0.Then, (26) can be redefined as By substituting ( 22) into (27), we know that θ =  0  () { (3) In order to improve the robustness of ( 27), we modify it as follows: where  > 0 is a small constant.Then, we consider the following function: Considering ( 22) and ( 29), the time derivative of ( 29) is From ( 18), we get By substituting (32) into (31), (31) becomes Mathematical Problems in Engineering Equation (33) can be rewritten as In addition, the parameter  * is constant, and () is presumed to be bounded, so it will define  as Then, (35) can be written as where  =  0 .

Theorem 1. The adaptive law (31) can guarantee that
(1) θ is bounded and converges to the compact set (2) the estimation error   is bounded.
On the base of (37), we can establish that This means that θ converges to the compact set Along with   ∈  ∞ and |()| ≤ , and integrating (35), then we can get with  0 = max(1, ‖ * ‖ 2 ),  1 = 2sup  (  () −   ( + )), and (41) implies that   is bounded.In order to analyze the convergence of the errors, let us consider the following Lyapunov function: Substituting ( 8) and ( 20) into (42) and then differentiating it with respect to time, Considering ( 18), (43) becomes Considering (39), we know that Using (45) and the fact that () and () are bounded, we can know that where  0 > 0 and  1 > 0 are some constants.Therefore, (44) can be written as By supporting that  is chosen large enough so that  ≥ where  min () denotes the minimum eigenvalue of the matrix  and satisfies the condition that  min () > 0.5.Equation (50) becomes where   = [ min () − 0.5]/ max () and  max () is the maximum eigenvalue of the matrix .
Theorem 2. Consider (2).Suppose that the approximation error in (18) is bounded.Then the control law can guarantee the boundedness of the signals  and  and the convergence of the error to the compact set Proof.Considering (51) we can notice that, for V  < 0 and   ≥ (2|  | 0  − + 2 1 )/  , the error vector  is bounded, and the state vector  is bounded.Moreover, since the term 2|  | 0  − in (51), we can also conclude that the function   will be asymptotically bounded as Thus, the error will converge asymptotically to the compact set Based on Theorem 1, we can know that θ ∈  ∞ ; we have that  ∈  ∞ , together with () ∈  ∞ , implies that  ∈  ∞ .

Simulation Results
In this section, in order to demonstrate the effectiveness of the proposed adaptive neural controller, a system is given in (2) where  1 = 5.5,  2 = 3,  3 = −1, and  4 = −1.
The purpose is to design an adaptive controller according to (16) which can make all the signals in the system (2) keep bounded.Based on Theorems 1 and 2, we can develop a controller for system (1) satisfying the purpose.
The simulation results are obtained in Figures 1, 2, 3, 4, and 5. From Figures 1-3, we get the trajectories of the states  1 ,  2 ,  3 .It means that the states are bounded.Figure 4 illustrates the trajectory of the adaptation law and the action of control input is given in Figure 5. Hence, we can see that the proposed controller makes the closed-loop systems stable.

Conclusion
In this study, an adaptive control approach for a class of chaotic systems has been developed.The main advantage is to design an adaptive law based on the gradient descent algorithm.This way can minimize a cost function of error.Finally, the theorems and the effectiveness can be validated by Lyapunov analysis and a simulation example.The future research works are to extend the approach to a class of multiinput multioutput systems.