An adaptive neural networks chaos synchronization control method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assuming the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on stochastic Lyapunov stability and robust adaptive theories, an adaptive neural networks synchronization control method is developed. In the design process, an auxiliary design system is employed to address input constraints. Simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results.

Energy resource system is a kind of complex nonlinear system. Over the last two decades, much attention has been paid to the chaos synchronization in this class system. Reference [

In many practical dynamic systems (including the energy resource demand-supply system), physical input saturation on hardware dictates that the magnitude of the control signal is always constrained. Saturation is a potential problem for actuators of control systems. It often severely limits system performance, giving rise to undesirable inaccuracy or leading instability [

Motivated by the above observations, an adaptive neural networks chaos synchronization method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assume that the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances. In the design, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on Lyapunov stability, an adaptive synchronization method is developed in order to make the states of two chaotic systems asymptotically synchronized. The new auxiliary design system is employed to address input constraints. Numerical simulations are provided to illustrate the effectiveness of the proposed approach.

Compared with the existing results, the main contributions of the proposed method are as follows: (i) the controlled response system of this paper contains unknown nonlinearities, and the proposed method can solve the unknown nonlinearity problem by neural networks, but the methods of [

The four-dimensional energy resource system can be expressed as follows (see [

When the system parameters are taken as the following values, this system exhibits chaotic behavior:

Three-dimensional view

Three-dimensional view

Three-dimensional view

In this section, a controller will be designed in order to make the response system track the drive system. The drive system with subscript 1 is written as

If no input saturation, uncertain nonlinearities, and unknown external stochastic disturbance (i.e.,

To design an adaptive controller, the following basic assumption is made for the system (

The disturbance covariance

To establish stochastic stability as a preliminary, consider a stochastic nonlinear system:

Consider (

In order to solve the unknown nonlinear

An RBFNN can approximate a continuous function

According to the literatures [

For different initial conditions of systems (

From (

In this section, we assume that all the parameters of the energy resource system are unknown. For convenience, similar to [

Similar to [

From (

In this section, external perturbations

The initial values are chosen as

The trajectory of

The trajectory of

The trajectory of

The trajectory of

The trajectories of

The trajectories of

The trajectories of

The trajectories of

It is worth pointing out that the method of [

This paper has solved the synchronization problems of a class of unknown parameters four-dimensional energy resource system. The main features of the proposed algorithm are that (i) the problems of the input constraint have been solved by employing a new auxiliary system; (ii) the unknown nonlinearities and stochastic disturbances that existed in the response system have been overcome by the neural networks and some special robust terms, respectively; (iii) the stability of the energy resource demand-supply system has been guaranteed based on stochastic Lyapunov theory.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 51308275) and the Foundation of Liaoning Educational Committee (no. L2012225).