This paper is concerned with the problem of the nonlinear dynamic surface control (DSC) of chaos based on the minimum weights of RBF neural network for the permanent magnet synchronous motor system (PMSM) wherein the unknown parameters, disturbances, and chaos are presented. RBF neural network is used to approximate the nonlinearities and an adaptive law is employed to estimate unknown parameters. Then, a simple and effective controller is designed by introducing dynamic surface control technique on the basis of firstorder filters. Asymptotically tracking stability in the sense of uniformly ultimate boundedness is achieved in a short time. Finally, the performance of the proposed controller is testified through simulation results.
The permanent magnet synchronous motor is widely used in the industrial applications [
For ameliorating the performance of the PMSM system, a large amount of literatures and control methods have been attempted to apply to the motor. For example, to improve the error convergence rate, the nonsingular fast terminal sliding mode control (SMC) [
Furthermore, the OGY method is a fundamental technology for controlling chaos [
Inspired by the work above, a new approach to design the nonlinear dynamic surface controller based on the minimum weights of RBF neural network is proposed for permanent magnet synchronous motor with the unknown parameters, disturbances, and chaos. During the controller design process, RBF neural network is employed to approximate unknown nonlinear functions. The main difficulty encountered in the controller design process is how to deal with the unknown control gain in the system. To overcome this difficulty, the adaptive method was also introduced to handle the problem. The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closedloop system. Furthermore, the suggested controller contains the minimum weights of RBF neural network. As a result, the computational burden of the scheme is greatly alleviated. This makes our design controller more suitable for practical applications.
It is well known that the PMSM is applied widely in the motor drives, servo systems, and household appliances owing to advantages, for instance, simple structure, high efficiency, high power density, and low manufacturing cost [
The denotations of the PMSM parameters are shown in Table
The denotation of the PMSM parameters.
Parameter  Denotation 


The directaxis currents (A) 

The velocity of the rotor (rad/s) 

The directaxis voltage (V) 

The load torque (Nm) 

The directaxis winding inductance (H) 

The permanent magnet flux (Wb) 

The polar moment of inertia (kgm^{2}) 

The quadratureaxis currents (A) 

The time (s) 

The quadratureaxis voltage (V) 

The stator winding resistance ( 

The quadratureaxis winding inductance (H) 

The viscous damping coefficient (N/rad/s) 

The number of pole pairs 
Then, the new normalized model for the PMSM is rewritten as
Figures
Threedimensional phase diagram with parameters
The chaotic time series of the PMSM.
The angular speed
The normalized
The normalized
It is obvious that the model of the PMSM has high nonlinearity because of the coupling between the speed and the currents. In addition, the indeterminate system parameters
For the sake of simplicity, the following symbols are introduced:
Then, the mathematic model of the PMSM can be represented as follows:
The unknown disturbance terms
The desired trajectory
The type of RBF neural network is considered as a twolayer network, which contains a hidden layer and an output layer. In this paper, the RBF neural network will be used to approximate the unknown continuous function
For given scalar
The approximation error
There exists a positive and bounded constant
In this section, the controller of dynamics surface control approach will be developed based on the minimum weights of RBF neural network. The design procedure consists of three steps. Then, the detail process will be given.
The first dynamic surface is defined as
The operating parameter
Substituting (
The virtual control and related adaptive laws can be designed as follows:
Introduce variables
Let
Define the filter error as
Then, the time derivative of
It is obtained that
Using Young’s inequality, one has
Substituting (
One has
Consider the Lyapunov function candidate as follows:
Then, the time derivative of
The second dynamic surface is given as
Then, differentiating
To facilitate engineering application, a minimumweightsbased RBF neural network will be employed to approximate the nonlinear function
Substituting (
Similarly, the relevant control law and adaptive law are provided in the following forms:
With (
One has
Choose the Lyapunov function candidate as follows:
Choose the last dynamic surface as
In the same way, there is a minimumweightsbased RBF neural network such that
Substituting (
At the present stage, the control input is designed as follows:
According to the mention above, the adaptive law is chosen as follows:
Similarly, (
There exists
Choose the Lyapunov function candidate as follows:
Up to now, the design procedure of proposed controller of the PMSM is completed. The proposed controller significantly reduces the computation complexity compared with traditional backstepping control and dynamics surface control. Based on previous procedure, the configuration of the proposed control system is depicted in Figure
Control schematic of the PMSM.
For any given
Suppose that the control laws in (
Define the Lyapunov function candidate as follows:
Consequently, one can obtain
Furthermore, (
The simulation is running to illustrate the effectiveness of the scheme presented in this paper under the assumption that the system parameters and nonlinear functions are uncertain. The initial conditions
To take into account the disturbance, the corresponding expressions are given as follows:
The simulation results are shown in Figures
The robustness analysis with 0.5sin(
The trajectory tracking
The rotor velocity tracking error
The periodic orbit
The
In this paper, an adaptive dynamic surface control method of chaos is applied to the permanent magnet synchronous motor based on the minimum weights of RBF neural network. The proposed controller guarantees the boundedness of all the signals in the closedloop system, while the tracking error eventually converges to a small neighborhood of the origin. Moreover, the suggested controller contains minimum weights of RBF neural network. This makes the design scheme easier to be implemented in practical applications. Simulation results are given to show the effectiveness and robustness of the proposed controller. Finally, some potential future research works are pointed out, such as the recursive filtering for timevarying nonlinear systems and sliding mode design for timeinvariant nonlinear systems.
The author declares that there is no conflict of interests regarding the publication of this paper.
This Project is supported by the National Natural Science Foundation of China (Grant no. 61203080) and Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ133202). The author also gratefully acknowledges the helpful comments and suggestions of the reviewers.