The proposed PID AutoTuning controller using radial basis neural network based on relay feedback approach (ATNNRF) control structure is shown in Figure 1. At the first phase of control, the relay feedback control mode is activated. The ultimate frequency and ultimate gain are obtained by the peak detection and calculation of the times between zero crossings of the control input level.

At the second phase of control, a neural network with back propagation algorithm is used to adjust the PID gains continuously.

The ultimate gain and period of the system are determined by observing the ultimate frequency where the phase angle is -PI under a closed-loop relay feedback control.

Relay feedback control signal are shown in Figure 2.

The relay feedback control is carried out as following:

where is the control input of the system at current time; is the hysterical level at which the control input switch to the opposite position; is the position error of the control system at current time.

Applying relay feedback control (1) to the motion system, we obtain a periodic signal with amplitude and period. In order to identify the ultimate gain from the relay feedback control, the describing function analysis is used.

Because the amplitude and period of the oscillation are measured from the relay feedback control, the ultimate gain and ultimate frequency can be obtained as following:

Since neural networks can approximate any continuous function over a compact set with high accuracy and fast learning capability, they have been recognized as a powerful tool for the applications of control system. It is proven that the standard 2-layer neural network consists of two layers of weights, threshold, a hidden layer and an output layer has sufficient generality for closed-loop control purpose.

In particular, the radial basis function neural network (RBFNN) is a feed-forward 2-layer neural network and demonstrates good control performance in the presence of unmodeled dynamics. The RBFNN can be considered as a local approximation model of control system. Moreover, the capability of online learning and rapid convergence of RBFNN makes it feasible in the application of control system.

The general RBFNN shown in Figure 3 can be presented in matrix form:

where and are the weight vectors, is the input vector,

is the output vector, and denotes the activation function vector.

We know that the main disadvantage of the multi-layer neural network is highly nonlinear in parameter. Hence, we consider hereafter the neural network of fixed, which makes the neural network linearly parameterized. Defining, we have:

where is the generalized basis function vector. In this paper, we use the radial basis function with Gaussian form as following:

where is the node center of the i-th radial basis function, and is the width coefficient.

The proposed PID autotuning controller based on neural network and relay feedback (ATNNRF) is designed by the following procedures:

-Firstly, the relay feedback control is carried out to analyze the system dynamic and calculate the ultimate gain and ultimate frequency of the control system by measuring the output amplitude and period.

-Secondly, a RBFNN with gradient descent algorithm is designed to approximate the system dynamic. In this step, the Jacobian matrix, denotes the sensitivity of control system output to the control input is obtained.

-Lastly, the ultimate gain and ultimate frequency calculated in the first step are incorporated with the Jacobian matrix to adjust the coefficients related to the PID control parameters.

Consequently, the PID gains are tuned automatically by the self-learning capability of neural network. It is proved from the simulations results that the proposed controller has the strong robustness, high adaptibility and high performance compared with other current researches and commercial products.

The design of proposed ATNNRF is described as below.

The conventional PID controller can be represented with the discrete form as followings:

It is assumed that the closed-loop characteristics are invariant to the time constant.

Since the PID tuning parameters are uniform functions of ultimate gain and ultimate period as described in (2) and (3), they can be scaled for control system that has different time constant as followings:

where are the scale factors that define the tuning rule.

The output of PID controller in (7) can be rewritten as the following discrete form:

where is the error of closed-loop control system.

Applying the tuning rule from (8), (9), and (10), we obtain the update control output as follow:

Consider a RBFNN as in Figure 3. The input vector is presented as:

And the output of RBFNN is given by:

Define the performance cost function of RBFNN as:

where is denoted as the approximation error.

Apply the gradient descent method, we can obtain the updating parameters of weight vector, node centers and width coefficients of RBFNN as followings:

where is the learning rate of RBFNN, and is the momentum gene.

The Jacobian matrix for the model identification is obtained as follows:

Gradient descent is an iterative method that is given an initial point, and follows the negative of the gradient in order to move the point toward a desired local minimum.

In this paper, we assume that the approximation error in (21) converges to zero by gradient descent algorithm, then (27) can be rewrite as follows:

The PID control parameters can be obtained through the calculation of scale factor as described as followings:

Define the cost function of system as following:

According to the gradient descent method, the scale factors can be updated by the following rules:

From (17) and (27), we can obtain:

Consequently, the PID control parameters can be obtained and the system is automatically tuned.