A conventional function approximator should give a good prediction of output data, when a system is presented with a new input data. A conventional function approximator uses a mathematical model of the system as shown in Fig. 37.1A. Sometimes, it is not possible to have an accurate mathematical model of the system, in such a case; mathematical model-free approximators are required, as shown in Fig. 37.1B. A conventional function approximator can easily be replaced by a neural-network-based function approximator.

37.2.2 Neural Function Approximator

The function $y=f\left(x_1,x_2,\dots ,x_n\right)$ is the function to be approximated, and x₁,x₂,x₃,…x_n are the n variables (n inputs), and the approximator uses the sum of nonlinear functions g₁,g₂,g₃,g₄,…,g_i, where each of the g is nonlinear function of a single variable. Thus,

$\begin{array}{l}y=f\left(x_1,x_2,\dots ,x_n\right)=g_1+g_2+g_3+\cdots +g_{2n+1}\\ ={\displaystyle {\sum }_{i=1}^{2n+1}}{g}_i\end{array}$

where g_i is a real and continuous nonlinear function that depends only on a single-variable z_i. The output y can also be represented as

$y={\displaystyle \sum _{j=1}^n}{w}_{Mj}{x}_j=w_{M1}{x}_1+w_{M2}{x}_2+\cdots w_{Mn}{x}_n$

These equations can be represented by a network shown in Fig. 37.2. There are n input nodes in the input layer and M hidden nodes in the hidden layer.

Fig. 37.3 shows the schematic diagram of a neural approximation, and Fig. 37.4 shows the technique of its training. The error (e) between the desired nonlinear function (y) and the nonlinear function (ŷ) obtained by the neural estimator is the input to the learning algorithm for the artificial neural estimator. This type of learning is known as back propagation.

The output of a single neuron can be represented as

$a_i=f_i\left\{{\displaystyle \sum _{j=1}^n}{w}_{ij}{x}_j\left(t\right)+b_i\right\}$

where f_i is the activation function and b_i is the bias. Fig. 37.5 shows a number of possible activation functions in a neuron. The simplest of all is the linear activation function, where the output varies linearly with the input but saturates at $\pm 1$ as shown with a large magnitude of the input. The most commonly used activation functions are nonlinear, continuously varying types between two asymptotic values 0 and 1 or $-1$ and $+1$. These are, respectively, the sigmoidal function also called log-sigmoid and the hyperbolic tan function also called tan-sigmoid.

The learning process of an ANN is based on the training process. One of the most widely used training techniques is the error back-propagation technique, a scheme that is illustrated in Fig. 37.4. When this technique is employed, the ANN is provided with input and output training data, and the ANN configures its weights. The training process is then followed by supplying with the real input data, and the ANN then produces the required output data.

The total network error (sum of squared errors) can be expressed as

$E=\frac{1}{2}{\displaystyle \sum _{k=1}^P}{\displaystyle \sum _{j=1}^K}{\left(d_{kj}-o_{kj}\right)}^2$

where E is the total error, P is the number of patterns in the training data, k is the number of outputs in the network, d_kj is the target (desired) output for the pattern K, and $o_{kj}\left(=y_{kj}\right)$ is the jth output of the kth pattern. The minimization of the error can be arranged with different algorithms such as the gradient descent with momentum, Levenberg-Marquardt, reduced memory Levenberg-Marquardt, and Bayesian regularization.

37.3.1 Speed Estimation

In general, steady-state and transient analysis of induction motors is done using space-vector theory, with the mathematical model having the parameters of the motor. To estimate the various machine quantities such as stator- and rotor-flux linkages, rotor speed, and electromagnetic torque, the above mathematical model is normally used. However, these machine quantities could be estimated without the mathematical model by using an ANN. Here, no assumptions have to be made about any type of nonlinearity.

As an example, the rotor speed of an induction motor can be estimated from the direct and quadrature-axis stator voltages and currents in the stationary reference frame, as shown in Fig. 37.6. A three-layer feedforward neural network structure with $8\times 7\times 1$ (eight input, seven hidden, and one output) is used in this case. The input nodes were selected as equal to the number of input signals and the output nodes as equal to the number of output signals. The number of hidden layer neurons is generally taken as the mean of the input and output nodes. In this speed estimator ANN, seven hidden neurons were selected.

The results of an experiment conducted on a 1.1 kW, 415 V, 3 phase, 4 pole, 50 Hz induction motor is explained in this section. In order to investigate the case of speed estimation using ANN, a rotor-flux-oriented induction motor drive was set up in the laboratory, where the speed reference was changed in steps of 100 rpm and reversed every time the speed reached 1000 rpm. The load torque on the motor was kept constant at its full load rating. The stator voltages, stator currents, and rotor speed were measured for 5 s, and a data file was generated. The neural network was then trained using the trainlm algorithm with this data file. The training of the neural network converged after 48 epochs, and the error plot for this network training is shown in Fig. 37.7. The estimated speed was predicted with the trained neural network, and the result is shown in Fig. 37.8.

The noisy data in the plot are the estimated speed, and the continuous line is the speed measured with the encoder. The speed estimation was found to fail for speeds <100 rpm. If the trained neural network has to predict the speed under the complete range of operation of the drive, the data for training the neural network also have to be taken for the whole range. From this example investigated, it was found that the off-line training of the neural network could not produce satisfactory results, and it can be concluded that these off-line methods are not most suitable for these applications.

Artificial neural networks were also used for the estimation of the rotor speed of an induction motor together with the help of induction motor dynamic model. Though the technique gives a fairly good estimate of the speed, this technique lies more in the adaptive control area than in neural networks. The speed is not obtained at the output of a neural network; instead, the magnitude of one of the weights corresponds to the speed. The four quadrant operation of the drive was not possible for speeds <500 rpm. The motor was not able to follow the speed reference during the reversal for speeds <500 rpm. The drive worked satisfactorily for speeds above 500 rpm. Even though this method does not fall into a true neural network estimator, the results achieved with this type of implementation were very good except for lower speeds.

Alternately, the estimated speed can be made available at the output of a neural network as shown in Fig. 37.9. This speed estimator used a three-layer neural network with five input nodes, one hidden layer, and one output layer to give the estimated speed ${\hat{\omega }}_r\left(k\right)$ as shown in Fig. 37.9. The three inputs to the ANN are a reference model flux λ_r⁎, an adjustable model flux ${\hat{\mathrm{\lambda }}}_r$, and ${\hat{\omega }}_r\left(k-1\right)$, the time delayed estimated speed. The multilayer and recurrent structure of the network makes it robust to parameter variations and system noise. The main advantage of their ANN structure lies in the fact that they have used a recurrent structure that is robust to parameter variations and system noise. These authors were able to achieve a speed control error of 0.6% for a reference speed of 10 rpm. The speed control error dropped to 0.584% for a reference speed of 1000 rpm.

37.3.2 Flux and Torque Estimation

The same principle as described in Section 37.3.1 can also be extended for simultaneous estimation of more quantities such as torque and stator flux. When more quantities or variables have to be estimated, the complex ANN has to implement a complex nonlinear mapping.

The four feedback signals required for a direct field-oriented induction motor drive can be estimated using ANNs. A $4\times 20\times 4$ multilayer network has been used for the estimation of the rotor-flux magnitude, the electromagnetic torque, and the sine/cosine of the rotor-flux angle. It has been demonstrated both by modeling and experimental results that the above estimated quantities were almost equal to the same quantities computed by a DSP-based estimator. Both the estimated torque and rotor-flux signals using neural network were found to have higher ripple content compared with the DSP-based estimated quantities. It could be concluded that a properly trained ANN could totally eliminate the machine model equations as evident from the results reported.

In another application, an ANN with $5\times 8\times 8\times 2$ structure has been used to estimate the stator flux using the measured induction motor stator variable quantities. After successful training, the ANN was used in a direct field-oriented controlled drive. The rotor flux is computed from the stator flux estimate provided by the ANN and the stator current. This particular implementation also included an ANN-based decoupler which was used for the indirect field-oriented (IFO) drive. An ANN with a structure of $2\times 8\times 8\times 1$ was used for implementing the mapping between the flux and torque references and the stator current references. The estimated rotor flux using an ANN and a conventional FOC controller was shown to be equal. These authors have used experimental data for the ANN training, and thus, the effect of motor parameters was reduced. The authors were unable to use these estimated fluxes for controlling the induction motor in the experiment. The structure of the ANN used for this estimation is only that of a static ANN. It is preferable to have a dynamic neural network for this purpose.

37.3.4 Stator Resistance Estimation Using ANN

In this section, the capability of a neural network has been deployed to have online estimator for stator resistance in an RFOC induction motor drive. The stator resistance observer was realized with a recurrent neural network with feedback loops trained using the standard back-propagation learning algorithm. Such architecture with recurrent neural network is known to be a more desirable approach, and the implementation reported in this section confirms this.

The d-axis stator current in the stationary reference frame in the discrete form can be represented using the voltage and current model equations of the induction motor:

$\begin{array}{l}{i}_{ds}^{s^{⁎}}\left(k\right)=W_4{i}_{ds}^{s^{⁎}}\left(k-1\right)+W_5{\mathrm{\lambda }}_{dr}^{s^{im}}\left(k-1\right)+W_6{\omega }_r{\mathrm{\lambda }}_{qr}^{s^{im}}\left(k-1\right)\\ +W_7{v}_{ds}^s\left(k-1\right)\end{array}$

$\begin{array}{c}\mathrm{where},W_4=1-\frac{T_s}{\sigma L_s}\frac{L_m^2}{L_r{T}_r}-\frac{T_s}{\sigma L_s}{R}_s;W_5=\frac{T_s}{\sigma L_s}\frac{L_m}{L_r{T}_r}\\ W_6=\frac{T_s}{\sigma L_s}\frac{L_m}{L_r};W_7=\frac{T_s}{\sigma L_s}\end{array}$

The weights W₅, W₆, and W₇ are calculated using the induction motor parameters, the rotor speed ω_r, and the sampling interval used in the estimator T_s.

The relationship between stator current and stator resistance is nonlinear that could be easily mapped using a neural network.

When this Eq. (37.9) is represented graphically, it resembles a recurrent neural network as shown in Fig. 37.21. The standard back-propagation learning rule can then be employed for training this neural network. The weight W₄ is the result of training so as to minimize the cumulative error function E₂:

$E_2=\frac{1}{2}{{\overrightarrow{\varepsilon }}_2}^2\left(k\right)=\frac{1}{2}{\left\{i_{ds}^s\left(k\right)-i_{ds}^{s^{⁎}}\left(k\right)\right\}}^2$

To accelerate the convergence of the error back-propagation learning algorithm, the current weight adjustment is supplemented with a fraction of the most recent weight adjustment, as indicated in Eq. (37.11):

$W_4\left(k\right)=W_4\left(k-1\right)+{\eta }_2\mathrm{\Delta }{W}_4\left(k\right)+{\alpha }_2\mathrm{\Delta }{W}_4\left(k-1\right)$

where η₂ is the training coefficient and α₂ is a user-selected positive momentum constant.

Following a similar procedure, the q-axis stator current of the induction motor can be estimated using the discrete form as shown in Eq. (37.12):

$\begin{array}{l}{i}_{qs}^{s^{⁎}}\left(k\right)=W_4{i}_{qs}^{s^{⁎}}\left(k-1\right)+W_5{\mathrm{\lambda }}_{qr}^{s^{im}}\left(k-1\right)-{\omega }_r{W}_6{\mathrm{\lambda }}_{dr}^{s^{im}}\left(k-1\right)\\ +W_7{v}_{qs}^s\left(k-1\right)\end{array}$

Equation (37.12) can be represented by a neural network as shown in Fig. 37.22. The weight W₄ is updated with the training based on Eq. (37.12).

The stator resistance ${\hat{R}}_s$ of the induction motor can now be calculated using Eq. (37.13) as follows:

${\hat{R}}_s=\left\{1-W_4-\frac{T_s}{\sigma L_s}\frac{L_m^2{\hat{R}}_r}{L_r^2}\right\}\frac{\sigma L_s}{T_s}$

The stator resistance of an induction motor can be thus estimated from the stator current using the neural network system as indicated in Fig. 37.23.

37.3.4.1 Modeling Results of Stator Resistance Estimation Using ANN

The block diagram of a rotor-flux-oriented induction motor drive together with stator resistance identification is already shown in Fig. 37.14, where the stator resistance estimation is implemented by the stator resistance estimator (SRE) block.

The stator resistance estimation results are as shown in Fig. 37.24. It has three results (1) without both rotor and stator resistance estimators, (2) with only rotor resistance estimation, and (3) with both rotor and stator resistance estimations. As shown in the figure, both R_r and R_s were increased abruptly by 40% at 1.5 s. It can be seen that the estimated stator resistance ${\hat{R}}_s$ converges to ${\hat{R}}_s$ within 200 ms. It can be noted from this figure that the convergence of the stator resistance estimation is not affected by the convergence of the rotor resistance estimator.

37.3.4.2 Experimental Results of Stator Resistance Estimation Using ANN

The stator resistance estimation algorithm was tested together with the rotor-flux-oriented induction motor drive of Fig. 37.14 implemented in the laboratory. To test the stator resistance estimation, an additional 3.4 Ω per phase was added in series with the induction motor stator, with the motor running at 1000 rev/min and with a load torque of 7.4 Nm.

The estimated stator resistance together with the actual stator resistance is shown in Fig. 37.25. The estimated stator resistance converges to 9.4 Ω within <200 ms.

Fig. 37.26 shows both the measured d-axis stator current and the one estimated by the neural network model. The neural network model output $i_{ds}^{s^{⁎}}\left(k\right)$ follows the measured values i_ds^s(k), due to the online training of the neural network.

Another application of ANN is to identify and control the stator current of an induction motor. In one study, Burton et al. have used a current control strategy outlined by Wishart and Harley to train an ANN to control the induction motor stator currents. They have used a training algorithm named random weight change (RWC) that is reported to be slightly faster than back propagation. In RWC algorithm, the weights are perturbed by a fixed step size and a random sign. This is done for fixed number of trials, and after each trial, the error with the desired output is computed. Finally, the set of weight changes that result in the least error are chosen, and the whole process is repeated until convergence is reached. The modeling results reported in this scheme was excellent. Later, Burton et al. presented their practical implementation of this proposed stator current controller with a transputer controller card.

37.4.2 Induction Motor Control

Narendra and Parthasarathy proposed methods for identification and control of dynamic systems using ANNs. Wishart and Harley used the above basic principles to identify and control induction machines. A block diagram of the control scheme is shown in Fig. 37.27. For the induction motor, the nonlinear autoregressive moving average with exogenous input (NARMAX) model for the stationary frame stator current is derived and used for the identification of electromagnetic model. In its general form, the NARMAX model represents a system in terms of its delayed inputs and outputs. Random steps in the stator voltage are given for the purpose of identification. The neural network used is of the multilayer back-propagation type, and a quantity based on the rotor time constant is also computed as an extra weight. As opposed to the regular ANN architecture, this ANN has nonlinearity in the output layer, and the weighted sum of the inputs is used as the output. This gives an estimate of the rotor time constant and makes the system robust against variation of the parameters. Once, the identification is over, the ANN is used for current control. The stator currents predicted by the ANN are used to compute the input voltage for the induction motor, and the ANN output is made to track the reference currents by back propagating the error.

The rotor speed is also controlled in this system by identifying a NARMAX model for the speed increment rather than the absolute value of speed. To simplify the NARMAX model, the load torque is assumed to be a function of the motor speed, as is the case in a fan or pump type of load. For the current control case, the relationship between the control variable (voltage) and the controlled quantity (current) was linear. In the speed control case, this relationship is nonlinear, thus necessitating two ANNs, one for identification of speed and the other for control.

The identification ANN (N_i) predicts the value for the speed increment, which is compared with the actual speed increment, and the error (ɛ_i) is back propagated through the ANN. A PI controller is used for basic speed control, and the control ANN (N_c) produces the slip frequency, and the difference between the desired speed increment and actual speed increment (ɛ_c) is back propagated through the ANN. The induction motor drive therefore employs three ANNs as shown in Fig. 37.28.

37.4.3 Efficiency Optimization in Electric Drives

The efficiency improvement of induction motor drives via flux control can be classified into three groups: precomputed flux programs, real-time computation of losses, and online input-output efficiency optimization control. All of these methods target choosing a value of the motor excitation that optimizes the motor-converter losses. The main requirement of an input-output optimization is to achieve the flux optimization in a minimum number of search steps. A constant search step may take too long if the step is too small, or it may bypass the minimum power input if the step is too large. For each mechanical operating point of the motor, it is possible to find a combination of rotor-flux linkage and torque producing current i_q at which the dc-link power P_dc to the drive system is minimum.

One of the possible ANN efficiency optimizers reported is shown in Fig. 37.29. An ANN-based search algorithm is employed to operate as an efficiency optimizer. The inputs to the system are the dc power fed into the drive system at instant k and the change in the control variable at the previous instant $\mathrm{\Delta }c\left(k-1\right)$. The only output is the actual change in control variable Δc(k). Appropriate scaling from engineering units to the normalized interval [−1, 1] are implemented by the input and output interfaces, input scaling (IS), and output scaling (OS). The speed signal is used to generate the reference flux C(k)₀ corresponding to each speed. The mechanical steady state is detected by applying a moving average filtering to the speed variation Δω_m. The ANN efficiency optimizer is enabled only during the mechanical steady state.