DRNN Robust DTC for Induction Motor Drive Systems Using FSTPI

The direct torque control (DTC) technique is the most advanced strategy for three-phase induction motor (IM) drive systems in industrial applications. It was first proposed by Takahashi and Noguchi [1] and Depenbrock [2] in the mid of 1980s as a replacement of the popular field-oriented control technique. Today, DTC is one of the world's most advanced AC machine drive technologies. It is characterized by its simple and robust structure yet achieves a very fast dynamic torque response. This technique was successfully implemented to drive three-phase induction machine by performing a decoupled control of electromagnetic torque and stator flux of the motor. The induction machines are popular in industrial applications nowadays because they have a rigid structure which makes them reliable and cheap in terms of maintenance [3]. Ever since the DTC technique was introduced, many research studies to improve the conventional DTC strategy have been proposed. Most of the methods to improve the classical DTC that have been introduced are mainly focused on the variable switching frequency of the inverter and high torque ripples resulting from the electromagnetic torque and stator flux hysteresis comparators [4]. The space vector modulation method as proposed by Wang et al. [5-7] has been used to achieve a constant switching frequency and resulting to reduced electromagnetic torque ripples.

In most applications of the DTC technique, the voltage source inverter (VSI) driving the induction machine is a six-switch three-phase inverter (SSTPI) to achieve a high-performance drive system of the motor [3]. A DTC scheme where the induction machine is driven by a four-switch three-phase inverter (FSTPI) was proposed by Metwally [8]. The FSTPI fed drive system is aimed to develop power converters with reduced number of switches and losses which is economic for controlling the speed of a low power and low-cost induction machine drive system while employing the simple and robust DTC strategy [4]. In a FSTPI drive system, two legs of the IM are controlled by power switching devise and the third leg is connected in the middle point of the dc-link voltage. With only two phases of the IM being controlled in FSTPI drive system, the switching table, sector numbers of stator flux position and three-level electromagnetic torque hysteresis controller are reduced which in turn give a fast-processing time compared to a SSTPI fed drive system.

In DTC scheme with FSTPI fed drive system for IM, the optimal switching table is established with four space vectors of the FSTPI in accordance with four sectors to the stator flux position [8] which is less compared to the six space vectors and six sectors in SSTPI [9] fed drive system. The electromagnetic torque control is achieved by using a two-level hysteresis comparator. In most research proposal for both the SSTPI and FSTPI fed drive system for IM, the speed regulation is achieved by using a classical proportional integral (PI) controller to generate the torque command. However, a typical PI controller cannot easily achieve a rapid response and an accurate drive system over a wide range of speed, particularly when there are motor parameters uncertainties and unanticipated load changes [10].

Adaptive artificial intelligence technology to achieve high-performance control of the VSI-powered IM drive system has been developed to address the drawbacks associated with the PI regulator [11]. Artificial-intelligent techniques include, among others, neural network control, fuzzy logic control, and robust mode control. This paper presents a DTC strategy for the FSTPI-powered induction machine drive system, where the closed-loop speed regulator uses an adaptive diagonal recurrent neural network (DRNN) controller. The strategy implemented is based on the four space vector voltages of FSTPI, simply by deriving an effective voltage vector switching table controlled by two-level hysteresis controllers for stator flux and electromagnetic torque combined with stator flux location in the stationery reference frame on a four-level sector.

A detailed comparative analysis of the designed controller is evaluated, considering five different operating conditions of the IM, between the classical and neural network controllers. Fundamental principles of the outlined approaches are presented, and certain features of the designed algorithm's effectiveness are evaluated and demonstrated. The basic principles of the outlined approaches are presented and some features demonstrating the effectiveness of the designed algorithm are evaluated and verified.

3.1 Classical PI controller

The PI controller is used to generate the torque command for speed control. The design is based on Figure 7 and Eqns. (19)-(25) and the poles placement method is used to determine the controller gains.

7.png

Figure 7. Speed control block diagram

The dynamic equation is given as follows:

$\frac{d \omega_{m}}{d t}=-\frac{B}{J} \omega_{m}+\frac{T_{e}}{J}-\frac{1}{J} T_{L}$    (19)

and the transfer function using Laplace transform of the speed loop is given as follows:

$G_{p}(s)=G_{\omega_{m}}(s)=\frac{\omega_{m}(s)}{T_{e}(s)-T_{L}(s)}=\frac{1}{J s+B}$    (20)

The transfer function of the PI controller is defined as follows:

$G_{c}(s)=G_{P I}(s)=K_{p} s+\frac{K_{i}}{s}$    (21)

By considering the load torque $T_{L}$ as a disturbance. The overall transfer function of the speed control in open loop becomes:

$T_{o p}(s)=G_{c}(s) \times G_{\omega_{m}}(s)=\left(K_{p} s+\frac{K_{i}}{s}\right)\left(\frac{1}{J s+B}\right)$    (22)

The transfer function of the speed controller system in closed loop becomes:

$T_{c l}(s)=\frac{G_{c}(s) \times G_{p}(s)}{G_{c}(s) \times G_{p}(s)+1}=\frac{K_{p} s+K_{i}}{s^{2}+\frac{\left(K_{p}+B\right)}{J} s+\frac{K_{i}}{J}}$    (23)

Comparing the denominator of Eq. (23) with the denominator of canonical form of second order system given in Eq. (24):

$T(s)=\frac{1}{s^{2}+2 \xi \omega_{n} s+\omega_{n}^{2}}$    (24)

where, $\omega_{n}$ is the natural sampling frequency and ξ is the damping coefficient. The following can be concluded:

$\left.\begin{array}{c}K_{i}=J \omega_{n}^{2} \\ K_{p}=2 \xi \omega_{n}-B\end{array}\right\}$    (25)

The damping coefficient: $\xi=\frac{1}{\sqrt{2}}$.

3.2 DRNN-PI speed controller

The DTC structure speed controller is a non-linear system that has load inertia disturbances and variations of motor temperature parameters [14]. This makes it hard for the DTC control system to work adequately with the PI speed controller. The paper in hand proposes a diagonal recurrent neural network (DRNN) adaptive speed controller system, which is confirmed by simulation.

There are two parts to the NN-PI speed controller with DRNN algorithm One is the classic PI controller that controls the motor speed in the closed loop. The second part is DRNN that can adjust online the $K_{p}$ and $K_{i}$ parameters to ensure optimum performance. The NN-PI speed controller is schematically structured, as shown in Figure 8. This controller can react rapidly, adaptively and counter-interference [14].

8.png

Figure 8. Auto-tuning structure DRNN-PI speed controller

If r(k) is given speed reference $\omega_{m}^{*}$ and $y(k)$ is given actual output speed $\omega_{m}$, then the control error is:

$e(k)=r(k)-y(k)$    (26)

The two inputs of the PI controller are:

$x_{1}(k)=e(k)$    (27)

$x_{2}(k)=\sum_{i=1}^{k} e(k) \times T_{s}$    (28)

$T_{s}$ is sampling time. The control algorithm for tuning $K_{p}$ and $K_{i}$ online using DRNN is:

$u(k)=K_{p}(k) x_{1}(k)+K_{i}(k) x_{2}(k)$    (29)

The loss function using sum squared error can be defined as:

$E(k)=\frac{1}{2}(r(k)-y(k))^{2}=\frac{1}{2} e(k)^{2}$    (30)

The adjustment of $K_{p}$ and $K_{i}$ adopts the gradient descent method:

$D K_{p}=-\eta \frac{\partial E}{\partial K_{p}}=-\eta \frac{\partial E}{\partial y} \times \frac{\partial y}{\partial u} \times \frac{\partial u}{\partial K_{p}}=\eta e(k) \frac{\partial y}{\partial u} x_{1}(k)$    (31)

$D K_{i}=-\eta \frac{\partial E}{\partial K_{i}}=-\eta \frac{\partial E}{\partial y} \times \frac{\partial y}{\partial u} \times \frac{\partial u}{\partial K_{i}}=\eta e(k) \frac{\partial y}{\partial u} x_{2}(k)$    (32)

η is the learning rate, $\partial y / \partial u$ is the Jacobian information of controlled object, which can be obtained by using DRNN, which is trained by the dynamic backpropagation (DBP) algorithm [15].

The DRNN is made of three layers namely, input, hidden, and output layer and the network structure are shown in Figure 9.

9.png

Figure 9. Neural network structure

Let $s_{i}(k)(i=1,2, \ldots, h)$ where $h$ is the number of hidden neurons) represent the input sum of the recurrent neuron, $h_{i}(k)(i=1,2, \ldots, h)$ the output of hidden neuron, and $\hat{y}(k)$ the output of the network. Then the output of the ith hidden neuron can be written as:

$h_{i}(k)=f(x)=f\left(s_{i}(k)\right)$    (33)

$s_{i}(k)=w_{i}^{D} h_{i}(k-1)+\sum_{j=1}^{m+1} w_{i, j}^{I} I_{i}(k)$    (34)

where, $w_{i, j}^{I}$ is the weight connecting the $j$ th input to the ith hidden neuron, $w_{i}^{D}$ is the feedback weight of the ith hidden neuron, bias weight is zero, and f(x) is the sigmoid function. The network output can be written as:

$\hat{y}(k)=\sum_{j=1}^{h} w_{i}^{O} h_{i}(k)$    (35)

where, $w_{i}^{O}$ is the output layer weight connecting the jth hidden neuron and the output neuron.

The objective of the network training is to minimize the following error function:

$J(k)=\frac{1}{2}(d(k)-\hat{y}(k))^{2}=\frac{1}{2} e_{I}^{2}(k)$    (36)

d(k) is the desired network output.

Applying the DBP learning algorithm, the learning rules of the weights are obtained as follows:

(1) Output weights:

$w_{i}^{O}(k+1)=w_{i}^{O}(k)-\eta \frac{\partial J(k)}{w_{i}^{O}}=w_{i}^{O}(k)+\eta e_{I}(k) \frac{\partial \hat{y}(k)}{\partial w_{i}^{O}}$    (37)

$\frac{\partial \hat{y}(k)}{\partial w_{i}^{O}}=h_{i}(k-1), i=1,2, \ldots, h$    (38)

(2) Feedback weights:

$w_{i}^{D}(k+1)=w_{i}^{D}(k)-\eta \frac{\partial J(k)}{w_{i}^{D}}=w_{i}^{D}(k)+\eta e_{I}(k) \frac{\partial \hat{y}(k)}{\partial w_{i}^{D}}$    (39)

$\frac{\partial \hat{y}(k)}{\partial w_{i}^{D}}=w_{i}^{O} P_{i}(k-1), i=1,2, \ldots, h$    (40)

$P_{i}(k)=\frac{\partial h_{i}(k)}{\partial w_{i}^{D}} f^{\prime}\left(s_{i}(k)\right)\left(h_{i}(k-1)\right)+w_{i}^{D} P_{i}(k-1)$    (41)

(3) Input weights:

$w_{i, j}^{I}(k+1)=w_{i, j}^{I}(k)-\eta \frac{\partial J(k)}{w_{i, j}^{I}}=w_{i, j}^{I}(k)+\eta e_{I}(k) \frac{\partial \hat{y}(k)}{\partial w_{i, j}^{I}}$    (42)

$\frac{\partial \hat{y}(k)}{\partial w_{i, j}^{I}}=w_{i}^{O} Q_{i, j}(k-1) ; i=1,2, \ldots, h, \quad j=1,2, \ldots, m$    (43)

$Q_{i, j}(k)=\frac{\partial h_{i}(k)}{\partial w_{i, j}^{I}} f^{\prime}\left(s_{i}(k)\right)\left(I_{i}(k)\right)+w_{i}^{D} Q_{i, j}(k-1)$    (44)

The above training algorithm can be used both offline and online. An online weight update detects adjustments in the dynamic phase and therefore leads to adaptive regulation.

The Jacobian information of the controlled object value is:

$\frac{\partial y}{\partial u} \approx \frac{\partial \hat{y}}{\partial u}=\sum_{i} w_{i}^{O} f^{\prime}\left(s_{j}\right) w_{i, j}^{O}$    (45)

A development of a robust DTC controller for three-phase induction motor using a FSTPI with self-tuning PI controller based on DRNN for speed regulation has been presented. To minimize torque ripples compared with the classical DTC technique for FSTPI, the DTC look-up table implemented is based on the emulation of SSTPI operation by the FSTPI. The validity of the proposed motor drive control system is designed and simulated using Matlab/Simulink model. Comparative results between using classical PI and self-tuning PI speed regulator based on DRNN, shows that the robustness of the controller is enhanced with self-tuning PI controller. The DTC scheme with self-tuning PI speed controller based on DRNN provides a better response with reduced overshoot and faster settling time over using a classical PI controller.