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Considering the system uncertainties, such as parameter changes, modeling error, and external uncertainties, a radial basis function neural network (RBFNN) controller using the direct inverse method with the satisfactory stability for improving universal function approximation ability, convergence, and disturbance attenuation capability is advanced in this paper. The weight adaptation rule of the RBFNN is obtained online by Lyapunov stability analysis method to guarantee the identification and tracking performances. The simulation example for the position tracking control of PMSM is studied to illustrate the effectiveness and the applicability of the proposed RBFNN-based direct inverse control method.

In engineering systems, various uncertainties exist including parameter uncertainties, unmodeled dynamics, and unknown external disturbances, which often bring adverse effects to stability and performance of the whole control systems. With growing interests of high-precision control systems, how to develop efficient control approaches to counteract the adverse effects, caused by various uncertainties, is an active topic in both the control theory and application [

On the basis of approachability, neural networks have been used to control unknown nonlinear dynamic systems, since it can be proved that a neural network can be trained to approximate any nonlinear function with the any given accuracy under certain condition [

However, effectively handling of the presence of disturbances is not well developed within the adaptive neural network control method. So reinforcing adaptive neural network controllers with disturbance attenuation capabilities still remains a challenging task in enormous practical applications. An initial approach is provided in [

Generally, weight matrix parameters of neural networks are adjusted with gradient method; however, there is currently no systematic way of ensuring when these methods will be successful. And analysis becomes very complicated when learning and control are attempted simultaneously, even the simplest control situation, such as a linear, time-invariant process, and a linear feedback control law, becomes a high-dimensional, coupled, nonlinear problem with the addition of online tuning of the neural network controller parameters [

The structure of this article is as follows. In Section

Let

A class of single-input single-output (SISO) nonlinear system with unknown disturbances can be described with the following Brunovsky form [

In the compact sets

The system state variable

The system external disturbance

The reference trajectory

The vectors

If the exact knowledge of the system dynamics and the external disturbances can be obtained precisely, that is, functions

Then, taking the derivative of

On the basis of the Lyapunov theorem, the stability of this closed-loop system can be ensured with control law (

The control law of MBIC in (

The MBIC method is unattractive to industry applications since the exact knowledge of system dynamics and lumped disturbances is hard to be obtained. But the MBIC method could not be neglected in designing the controller for uncertain nonlinear systems, as it is readily understood with good performance. An MBIC method with an appropriate compensative controller, proposed in [

The radial basis function neural network (RBFNN) with Gaussian activation functions is the most popular type of artificial neural network architectures. The RBFNN with desirable features of local adjustment of the weight and mathematical tractability has been successfully applied to various issues [

The structure of a radial basis function network in three-layers.

The RBFNN is a kind of feed-forward neural networks, which forms mappings from an input vector

It has been already proved that, under mild assumptions, the RBFNN can approximate any continuous function over a compact set to any degree of accuracy.

The output of the neural network

For an arbitrarily small positive number

Theoretical and numerical studies show that performance of RBFNN is highly dependent on the locations of centers. The structure of the RBFNN will be simpler with the less number of second layer. Unfortunately, satisfactory performance would not be easily obtained. On the contrary side, the identification precision is higher with the bigger number of hidden nodes, but the structure of RBFNN will be very complicated. So, we choose the nearest neighbor clustering algorithm to calculate the hidden nodes to fix the structure of the neural network, which is ignored here, similar to the authors’ the previous studies [

Following the above conclusions in Section

The structure of neural network-based direct inverse controller using RBF.

From (

The compact sets of

The parameters of weight coefficient matrix

The parameter

The estimation error

The output of control law (

Let

Although the imperfect neural network controller will generally lead to degradation of the tracking performance, the system can possess acceptable performance. If the neural network controller can approximate the ideal controller of the system, that is, the controller estimation

To synthesize an adaptive NBIC with convergence capability, guaranteed stability, and disturbance attenuation, it is first necessary to ensure the chosen architecture of the RBFNN is capable. It can be seen from the theoretical and numerical studies that the performance of RBFNN highly depends on the locations of centers

Along the trajectories of (

Noting that

There is

Substituting (

According to the characteristic of the

We can see that, in order to make

We can get

The closed-loop control system is supposed to be globally stable, since all the variables of the RBFNN are bounded. Better track performance can be received from (

The design steps about the proposed neural network-based direct inverse controller (NBIC) with disturbance rejection are summarized as follows:

Step 1. Specify the design parameters of the NBIC.

Specify

Step 2. Structure the proposed method.

Considering the characteristics of the PMSM, choose a small value as the central value of the hidden layer node

Step 3. Adapt the weight parameters.

Apply the control law (

In this section, a numerical simulation example will be performed to evaluate the effectiveness and applicability of the proposed RBFNN-based direct inverse control method. The dynamic equation of PMSM servo system can be described as

PMSM servo systems always confront load disturbances, friction force, and parameter uncertainties in some real industrial applications. The performance of the whole system will be degraded by these variations, disturbances, and uncertainties. Moreover, the control performance usually cannot be guaranteed with the fix control parameters. So, the adaptive position tracking control based on the proposed NBIC can be realized with as follows:

The PMSM position tracking control diagram based on the proposed NBIC.

Simulations on PMSM system have been performed to show the effectiveness of the proposed control method. For the purpose of comparison, two control methods, that is, model-based inverse controller (MBIC) and neural network-based adaptive direct inverse controller (NBIC), are applied for the position tracking control of PMSM servo system in the case of nominal model and the case under load disturbance and parameter variations, respectively.

The parameters of PMSM are given as rated torque

Nomenclature.

Symbol | Description (unit) | Symbol | Description (unit) |
---|---|---|---|

Trajectory of the matrix | |||

Real number set | Euclidean norm | ||

The minimum eigenvalue of matrix | Frobenius norm ( | ||

The maximum eigenvalue of matrix | Input | ||

State vector of ( |
Output | ||

Unknown continuous functions including internal uncertainties | Bounded reference trajectory | ||

External disturbances | Arbitrarily nonnegative constant | ||

Approximated by estimated function |
Approximated by estimated function | ||

Estimated weights | Input vector | ||

RBFNN controller output | Gaussian activation function of the hidden layer | ||

Weight | Central values of the | ||

Arbitrarily small positive number | Output of the neural network | ||

Number of the clusters | Estimation error | ||

Angular velocity | Torque constant (Nm/A) | ||

Viscous friction coefficient (Nms/rad) | Rotor position (rad) | ||

Current input | Inertia (kg·m^{2}) | ||

Reference position | Load torque (N·m) |

There are many literatures that show the superiority with respect to a standard approach, such as PID method and MPC. The RBFNN-based control methods with the nearest neighbor clustering algorithm, in the authors’ previous studies [

The position tracking performance is tested under no load disturbances and parameter variations in this part, that is, in case of a nominal model.

Figure

Position tracking and error performance in the nominal model case: (a) under the MBIC method and (b) under the NBIC method.

The output values of different control methods in the nominal model case: (a) the control output of MBIC and (b) the control output of NBIC.

The position tracking performance is tested under unknown load disturbances

Position tracking and error performance in case II: (a) the MBIC method and (b) the NBIC method.

Partial enlargement of position tracking under the NBIC: (a) enlargement at the beginning and (b) enlargement at the end.

The position tracking performance of PMSM using the MBIC is affected considerably by the load disturbances and parameter variations based on Figure

Considering the adaptive self-learning ability of neural networks, an adaptive neural network-based direct inverse controller (NBIC) for a nonlinear system with uncertain parameters and unknown external disturbances is presented to achieve satisfactory tracking performance in this paper. The proposed NBIC is realized by one RBFNN which has two outputs, one output of the RBFNN acts as the main controller to handle the parameter uncertainties, and the other output of RBFNN is used to handle external disturbances. The problems of the uncertainties and the ability of the self-adaptive control can all be handled in one single neural network framework, since the accuracy of the system identification and the ability of antidisturbance can satisfy the requirements of the system. Moreover, the stable online weight matrices adjustment mechanism of the RBFNN is determined by the Lyapunov theory to achieve the stability and guarantee attenuation of the disturbances. Simulation results of position tracking for the PMSM servo system illustrate that the proposed method NBIC has the robust and effective control performance with good disturbance rejection.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 61773335 and 51405428), Natural Science Foundation of Jiangsu Province (Grant no. BK20171289), Natural Science Fund for Excellent Young Scholars of Yangzhou City (Grant no. YZ2017099), Opening Foundation of Jiangsu Engineering Research Center on Meteorological Energy Using and Control (Grant no. KCMEIC03), Opening Foundation of Ministry Education Key Laboratory of MCCSE (Grant nos. MCCSE2015A01 and MCCSE2016A01), and a Project Funded by the High Talent Support Program of Yangzhou University.