Abstract
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.
1. 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 [1ā9], 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 [10ā13].
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 chatter-free 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 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.
2. Problem and Formulation
Consider the chaotic systems as the following form: where , , 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 and . Then, one has where 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 to ensure that is stable. Therefore, there exists a matrix such that where and are positive definite symmetric matrices.
Define as where and 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 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 , . Then, the system is stable. Nevertheless, is not effective owing to unknown parameters.
3. 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 being a constant.
is unknown; then, choose as an estimate of . Define the control law as We define the error between and as follows: By using (15) and (16), (17) can be written as where .
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 where . Then, (26) can be redefined as By substituting (22) into (27), we know that In order to improve the robustness of (27), we modify it as follows: where 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 Consider the following inequalities: Equationā(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 .
Theorem 1. The adaptive law (31) can guarantee that(1) is bounded and converges to the compact set (2)the estimation error is bounded.
Proof. Considering (37) we can notice that , and ; therefore, and are bounded, as well as and , resulting in .
On the base of (37), we can establish that
This means that converges to the compact set
Along with and , and integrating (35), then we can get
with , , 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 and are some constants.
Therefore, (44) can be written as
By supporting that is chosen large enough so that , and using , with , (47) becomes
Consider the inequality
Equationā(48) becomes
where denotes the minimum eigenvalue of the matrix and satisfies the condition that . Equationā(50) becomes
where and 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 and , the error vector is bounded, and the state vector is bounded. Moreover, since the term 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 ; we have that , together with , implies that .
4. Simulation Results
In this section, in order to demonstrate the effectiveness of the proposed adaptive neural controller, a system is given in (2) where , , , and .
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 designed parameters of the proposed control approach are chosen as , , , , and . The initial conditions for the system states are supposed as , , , and .
The simulation results are obtained in Figures 1, 2, 3, 4, and 5. From Figures 1ā3, we get the trajectories of the states . 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.
5. 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 multi-input multioutput systems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by Natural Science Foundation of Liaoning University of Technology (X201221).