Abstract

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.

1. Introduction

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 [1], for example, robust control [2], sliding model control [3, 4], adaptive control [5, 6], backstepping control [7, 8], and disturbance estimation-based compound control [8–12]. The effectiveness of these control methods has been proved by their applications in some industrial areas. Unfortunately, due to the need of exact knowledge of these control systems, the improvement of performance is limited. Moreover, some of these control methods lack online learning mechanism. The control performance cannot be guaranteed while the industrial systems are subjected to drastic internal and external disturbances. Considering the limitation of the kinds of disturbances estimate methods, some researches proposed many other disturbances estimation methods [13, 14]. In [13], an approximation-free funnel function is proposed to guarantee the transient and asymptotic behavior of the tracking performance. In order to approximate unknown nonlinearities and to dramatically diminish the computational costs, a novel high-order neural network with only a scalar weight is introduced in [14].

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 [15–19]. The use of neural network learning ability avoids complex mathematical analysis in solving the control problem of plant dynamics with high complexity and nonlinearity. It is commonly known that this ability of neural network is the obvious advantage compared with traditional control methods.

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 [20], since then, many works have been emerged in [21–27]. Recently, the output feedback control scheme combining a model-based controller with a neural network feed-forward compensator to model the unknown system dynamics is proposed in [23, 24]. For the purpose of enhancing the stability, an additional robust controller is needed to be introduced to solve the problems arising from approximation errors of the neural network. The RBFNN-based disturbance observer is proposed in [21, 22] to estimate the lumped disturbances, that is, external disturbance from uncertain external condition and internal disturbance caused by parameter variations or modeling errors. A method of indirect adaptive neural network control is presented in [23] to identify high-order nonlinear continuous plant. Moreover, control parameters will be updated with the identified model information to increase the control performance.

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 [28]. And for the purpose of stable and efficient online control, the sufficiently accurate identified information system by using the gradient method is a necessary prerequisite. That is to say, the off-line training which is time-consuming is needed to provide a good starting point for the online adaptive control. Application of gradient optimization methods contains instability mechanisms, since there exists parameter variations and internal and external disturbances. Some adaptive neural network control strategies employ the enhanced gradient algorithm or avoid the gradient method to obtain high performance, since the neural network trained by gradient algorithm may not exactly reconstruct a certain required nonlinear function. From the reliability point of view, adding more components to a system will involve a higher probability of malfunction. Motivated by the fact that the existence of a robust control Lyapunov function is a necessary and sufficient condition for robust stabilization via a suitable control law. In this paper, an adaptive neural network-based direct inverse controller (NBIC) with a RBFNN with guaranteed stability, convergence, and disturbance attenuation capabilities is investigated for the lumped disturbances, that is, the unknown nonlinear system with parameter variations, unmodeled dynamic response, and bounded external disturbances. The scheme does not need to design extra controllers but only using one RBFNN which acts not only on the feedback controller but also can compensate for the external disturbances. Thus, the structure of the control system is less complicated. The weights of RBFNN is tuned online based on the Lyapunov theory which will not only guarantee the given performance for this system but will also illuminate the relationships between performance and the parameters of the NBIC. Moreover, the control scheme not only guarantees the stabilities of the closed-loop system but also the tracking error will astringents to a small neighborhood of the origin. And the neural network controller can obtain the benefits of model-based control without a priori knowledge of system dynamics or without the computational burden of classical dynamic. Also, under the circumstance without disturbances, the proposed control strategy can ensure the uniform ultimate boundedness property if tracking error with respect to a compact set around the origin of any small area.

The structure of this article is as follows. In Section 2, some theoretical preliminaries are addressed, including mathematical notations, the description of the unknown nonlinear system under research, control objective, and the description of the RBFNN. In Section 3, the design procedure of adaptive neural network using a direct inverse controller is introduced and stability analysis is also given in this section. The application on the high-precision position tracking of PMSM servo system and the simulation comparisons and analysis with the model-based inverse controller are presented in Section 4. Finally, some conclusions are described in Section 5.

2. Preliminaries

Let and be the real number set and the -dimensional vector space, respectively. Define and let be the real matrix space. The parameters of and are the minimum and maximum eigenvalue of matrix , respectively. The symbol indicates the trajectory of the matrix , denotes the Euclidean norm, is the Frobenius norm ( norm). According to the particularity of norm, there are and

2.1. The Statement of the Problem

A class of single-input single-output (SISO) nonlinear system with unknown disturbances can be described with the following Brunovsky form [29]: where means the state vector of this system, and are the input and output, and are the unknown continuous functions including internal uncertainties, and denotes external disturbances. In practice, many systems such as chemical reactions, PMSMs, and robots are essentially nonlinear, whose input variables may enter in the systems nonlinearly as described by the above general form. In terms of nonlinear control literatures, these systems are feedback linearizable and have a relative degree equal to . The smooth function satisfies with , and it implies that the function is bounded away from zero with strictly positive or negative value. The control goal addressed here is to find a suitable control law , so that the system output can track a bounded reference trajectory with a satisfactory accuracy in the presence of internal disturbances caused by parameter uncertainties and external disturbances , while all involved variables, such as , , , and , should be bounded.

Assumption 1. In the compact sets , has inverse function and boundary, as , and is arbitrary nonnegative constant.

Assumption 2. The system state variable can be observable.

Assumption 3. The system external disturbance is defined by a known constant , that is .

Assumption 4. The reference trajectory is continuous and bounded known function of time with bounded known derivatives up to the nth order.

The vectors and the tracking error are defined as the following two equations.

If the exact knowledge of the system dynamics and the external disturbances can be obtained precisely, that is, functions , , and are known exactly, the following model-based inverse controller (MBIC) can be obtained as follows: where is the system coefficients related to transient performance of the closed-loop system. Therefore, the closed-loop system constituted by (1), (2), (3), and (4) can be redescribed as follows: where is the coefficient matrix. And this matrix should be reasonably chosen so that the roots of the Hurwitz polynomial are all in the open left-half plane. Furthermore, (5) can be overwritten as where is a stable matrix. By choosing Lyapunov, it represents as and assumes that the positive-definite matrices and satisfy the following:

Then, taking the derivative of along the trajectory of (6), the following can be obtained:

On the basis of the Lyapunov theorem, the stability of this closed-loop system can be ensured with control law (3). And the tracking error also can be astringented to zero. That is to say, the system state variable will asymptotically approximate the desired trajectory from any initial conditions, that is, .

The control law of MBIC in (3) depends highly on the exact knowledge of the nonlinear functions and and external disturbance of the nonlinear dynamics systems. So, precise parameters in the dynamic model in (1) have to be known. However, in many practical engineering projects, the perfect model of the system is difficult to obtain and external disturbances are impossible to ignore or impossible to measure directly or to obtain a precise mechanism model for them. In an effort to solve the problem of unknown nonlinearly parameterized and , adaptive control schemes employing function approximation techniques have been studied in [15, 18, 21, 27]. In these approaches, the nonlinear functions and , related to the system dynamics, are usually approximated by estimated function and with neural networks or fuzzy systems, respectively. The parameter denotes the estimated weights. Therefore, additional precautions are necessary to be made for avoiding possible singularities of the control action, that is, . In order to solve this problem, the initial values of the NN weights are chosen sufficiently close to the ideal values in [27]. Hence, offline training phases are needed before the controller is put into operation.

Remark 1. 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 [24], is valid for controlling uncertain nonlinear MIMO system with uncertainties and disturbances.

2.2. Description of RBFN

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 [17, 21, 22, 30–32]. And RBFNN has been proven that, for any given real continuous function, there exists an RBFNN that can uniformly approximate over a compact set with arbitrary accuracy. A schematic diagram of a simple type of RBFNN with three-layer is described in Figure 1.

Figure 1 

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 to an output vector . From Figure 1, the structure of the employed RBFNN, with inputs, two outputs, and hidden units, can be described by where is the input vector, represents Gaussian activation function of the hidden layer, and is the RBFNN controller output. The parameter is the weight, which connects the output unit with the radial basis function, is defined as the central values of the hidden layer node, and means the radius of Gaussian function, respectively. It can be seen that each hidden node in the RBFNN computes an output that depends on a radially symmetric function, and the sum of the hidden layer outputs export to three-layer with a linear weighted. It is common that stronger output can be obtained when the input is nearer at the centroid of the node.

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.

Assumption 5. The output of the neural network is continuous with respect to its arguments for all finite .

Assumption 6. For an arbitrarily small positive number , there is an optimal neural network output that makes .

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 [17, 21, 22]. The detailed process of this learning algorithm is also proposed in these researches.

3. Design of the RBFNN-Based Direct Inverse Control Scheme

Following the above conclusions in Section 2, the inverse controller based on (3) cannot be acted, since the parameter variation and external disturbances is hard to get directly. According to the superiorities of RBFNN, (3) can be realized online and adaptively by the RBFNN, which can be represented as (10). The structure of the novel neural network using direct inverse control system is shown in Figure 2.

Figure 2 

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

From (10) and Figure 2, we can see that the unknown nonlinear function and are parameterized by an RBFNN with output and , respectively. And the matrices of adjustable weights are described by the parameters and . By calling up the NN theory and Assumption 6, on a compact set , for every , there exists a Gaussian basis function and two weights vectors and such that

The compact sets of and are the known subsets of . Therefore, the optimal weights and , which can minimize the functions and , can be defined as

The parameters of weight coefficient matrix and are estimated by and . and represent a compact set of system states and a design parameter, respectively. The parameters and are constrained by and .

The parameter is described as the optimal estimation error of neural network, that is,

The estimation error is bounded by a finite constant , that is, . The error between (10) realize by the RBFNN and the ideal control law (3) can be described as where , , and . So, the designed control law of the proposed NBIC can be obtained as

The output of control law (15) is applied to (1), so the closed-loop system can be described as

Let . The closed-loop system (16) can be rewritten as follows combining with (4), (6), and (16):

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 and the parameter estimation error and , the tracking error can be achieved to zero, that is, as .

3.1. The Stability Analysis

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 . The radius determines the number of the clusters and affects the learning speed and accuracy. So, the choice of appropriate and is particularly significant. In order to make it easier to analyze, parameters and of RBFNN are learned by the nearest neighbor clustering algorithm and then kept fixed, since the weight matrix parameters and the structure of RBFNN can be adjusted by the nearest neighbor clustering algorithm, synchronously. The next work is to determine the adaptive weights of the NBIC according to Lyapunov stability analysis. And the Lyapunov function can be given as the follows: where is the learning rate of RBFNN. Assume is a stable matrix, so there are positive-definite matrices and satisfy the following

Along the trajectories of (17), the differential of Lyapunov function can be obtained as follows:

Noting that , , and , so the differential Lyapunov function can be re-described by

There is , if the weights adaptive update law of the RBFNN is chosen as

Substituting (22) into (21) yields

According to the characteristic of the norm in the preliminaries, (23) can be changed as follows:

We can see that, in order to make , the following inequality is presented or

We can get or

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 (27) when the characteristic value of is larger, and the smaller upper-bound of the estimation error of the RBFN, the eigenvalues of and the . That is to say, the convergence problem of the weights of neural network is solvable, and bounded neural network weights ensure bounded control input.

3.2. Design Procedure of Neural Network-Based Direct Inverse Controller with Disturbance Rejection

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 to get all roots in the left-half plane for the polynomial , solve the equation to obtain a symmetric matrix based on a positive-definite matrix , and choose the RBF neural network, specifying the inputs and the initial weights of RBFNN.

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 and an appropriate number of the cluster . Furthermore, calculate the hidden node parameters based on the nearest neighbor clustering algorithm and then obtain the structure of RBFNN. Calculate the hidden node output vector for input , based on (9); get the approximated control law from (10).

Step 3. Adapt the weight parameters.

Apply the control law (10) to the dynamic system expressed by (1). Use (22) to train the weight parameters and of RBFN, with the initial weight matrix parameters for subtle differences.

4. Position Tracking Control of PMSM Servo System with the Proposed NBIC Method

4.1. Control Design for PMSM Based on NBIC Method

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 where the coefficient means torque constant, represents the angular velocity, and refers to rotor position. The parameters and are the viscous friction coefficient and the moment of inertia which are always vary with the different working conditions, is the desired current input, and denotes load torque which can be seen as the external disturbances. The following state space can be obtained with , , , and :

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: where the parameter represents the reference position, and the position error between reference position and feedback position of the closed-loop system is . The details of the proposed method for PMSM position control can be seen from Figure 3.