Wavelet Neural Networks for Speed Control of BLDC Motor

3.1 Feedforward wavelet neural network (FFWNN)

The FFWNN has no feedback connection. That is, the output is calculated straightly from the input with feedforward connections [12]. There are two arrangements of feedforward wavelet networks:

3.1.1 Radial basis wavelet neural network (RBWNN)

Radial basis wavelet neural network (RBWNN) is the simplest form of the wavelet network [15, 16]. The arrangement of radial basis wavelet neural network (RBWNN) is shown in Figure 1 . This network approaches any required signal f(t) by simplifying a linear combination of a group of daughter wavelets ψa,b,where ψa,bare created by dilation a and translation b from mother wavelet ψ[11, 17].

ψa,b=ψx−baE1

The network output is specified as follows [10, 18]:

y=∑n=1NwNψaN,bNE2

where x is the input signal, N is the number of neuron in the hidden layer, and wNis the weights of the output. The network parameters wN,aN,andbNcan be training and optimizing by any optimization technique. In this chapter, the PSO algorithm is used to minimize the error according to the fitness function as will be demonstrated later.

3.1.2 Conventional wavelet neural network

The conventional WNN is a general form of radial basis wavelet neural network [19]. Figure 2 depicts the building of the conventional wavelet network, the number of hidden layers and neurons that are selected to create an appropriate WNN, and the parameters that are optimized by the PSO algorithm. The input layer can be represented by a vector x=x1x2…xM, the output layer represented by a vector y=y1y2…yK, and the activation function of hidden layer is the wavelet basis function. The output Yjcan be given as follows [11, 19]:

Yj=σuj=σ∑n=1Nwj,nψanbn∑m=1Mvn,mxm+gE3

where, j=1,2,3,…,K;Mis the number of inputs; K is the number of output layers; N is the number of hidden layers; g is the bias; and σuis the activation function of the output layer, the most common formula of activation function being sigmoid function which can be illustrated as follows [12]:

σu=11+e−uE4

3.2 Recurrent wavelet neural networks (RWNNs)

In recurrent WNNs, the output depends not only on the present inputs of the network but also on the prior outputs or conditions of the network [12, 15]. Recurrent networks have feedback and are also known as feedback networks. There are several types of recurrent networks that depend on the feedback connection [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

In the recurrent wavelet network structures, the wavelet network input involves delayed samples of the system output y(k). The number of inputs increases with the order of the system actuality demonstrated. Figure 3 depicts the structure of recurrent wavelet network. Hence, the output for each layer can be calculated as [20, 23, 24]:

where aNand bNare translation and dilation parameters of wavelets. The inputs of this layer for time n can be denoted as:

where ∅Nrepresents the weight of the self-feedback loop. The output of the network is given as follow:

where x is the input signal, N is the number of neuron in the hidden layer, wNis the output weight, Di,D0is the number of delay for the input and output network, and rnisthe weight of the output feedback loop.

4.1 Particle swarm optimization algorithm

The PSO algorithm is one of the evolutionary computation techniques to solve optimization troubles. In this algorithm, a swarm of individuals or entities called particles flies over the exploration space [29, 30]. Each particle acts as a probable solution to the optimization troubles. The position of a particle is influenced by the best position visited by itself, i.e., its own knowledge or experience and the position of the best particle in its knowledge of neighboring particles. When the neighborhood is the entire swarm, the best position in the neighborhood of the particle is denoted as the global best position and the resulting algorithm is referred to as the global best position PSO, where the finest prior position that gives the minimum fitness value of any particle is called local best position (lbest). The index of the best particle of all particles in the population is called global best position (gbest). The algorithm is generally referred to as the lbest PSO when smaller neighborhoods are used. For each particle, the performance is measured utilizing an objective function that differs depending on the optimization challenge. The basic PSO algorithm is given below according to the flow chart which is shown in Figure 5 [31, 32, 33, 34, 35].

Step 1. Generation of population particles

Create particles regularly distributed over x, then choose the number of particles, number of iterations, modification accelerating coefficients c1and c2, the inertia weight (w) and random numbers R1, and R2to start the optimum searching.

Step 2. The initialization for each particle

Initialize the present position xitand the velocitiyvitfor each particle. The particles are randomly produced among the minimum and maximum limits of parameter values. Each particle is treated as a point in a D-dimensional space. The ith particle is denoted as xi=xi1xi2…xiD. The velocity for the particle i is represented as vi=vi1vi2…viD, then the local best position (lbest) and the global best position (gbest) are initialization.

Step 3. Evaluation of fitness function

The overall performance (speed of convergence, efficiency, and optimization accuracy) of the PSO algorithm counts on the objective function that observes the optimization search. The objective function is chosen to minimize the reference constraints. The popular performance standards based on the error condition are integrated absolute error (IAE), integrated of time weight square error (ITSE), and integrated of square error (ISE) that can be estimated theoretically in the frequency domain [31, 32, 36]. In this chapter, multiobjective functions are utilized based on the integral of the squared error (ISE) criterion and overshoot (Mp)criterion as follow [37, 38]:

where

where y(i) is the system output and D(i) is the desired output, while n is the actual speed and nrefis the desired speed.

Step 4. Update the swarm

The updating of the velocity vitand the present position xitfor each particle in the swarm is done according to Eqs. (13) and (14). Then the main loop and the objective function are calculated for updating positions of particles. If the new value is improved than the previous lbest, the new value is fixed to lbest. Similarly, gbest value is also updated as the best lbest. Velocity of each agent can be modification by the following:

And, the present position can be modification by the following:

where xikis the present position of particle i at iteration k, vikis the velocity of particle i at iteration k, w is the inertia weight which can be represented in Eq. (15), c1, c2represent positive acceleration constants and R1, R2are random variables uniformly distributed in the range [0; 1].

where,wmin is the inital weight,wmax is the final weight, itermaxis the maximum iteration number.

Step 5. Stopping criteria

If the current iteration number reaches the predetermined maximum iteration number, then exit. Otherwise, execute another initialization for each particle and reiterate the process.

5.1 WNN-PID controller based on PSO

In this type of controller, the WNN is utilized with PID controller based on PSO algorithm. WNN-PID controller utilizes online learning by PSO algorithm, where the PSO learning algorithm is used to train the translation parameters akand bk, weights connection in the WNN, and the parameter (kp,ki,kd) of PID controller on-line with the model of BLDC motor to control the speed at the desired value. There are two major issues to implement any wavelet neural networks. First, the network architecture is used and second, the algorithm is used to learn the network by the PSO algorithm. Figure 6 depicts the block diagram of the BLDC motor with WNN-PID based on PSO algorithm. The structure and the design of the WNN-PID controller will be given in the next subsection.

5.2 Design of the structure of WNN-PID controller based on PSO training algorithm

1. 1. Design of PSO algorithm: the PSO algorithm is discussed in Section 5, where each particle parameters are initiated to make a population and then the algorithm is accomplished according to the flow chart given in Figure 6 , which includes training the parameters of this controller to guarantee the minimization of an objective function. The objective fitness is evaluated as follows:

1. where ISE is the integrated of square error and Mpis the maximum peak overshoot.

2. 2. Design of WNN-PID controller: to design the WNN-PID controller, the type of WNN must be selected as shown in Section 3 and also the number of layers and neurons and the wavelet function type must be selected [16]. In this chapter, the input layer has two inputs: the speed error and the change of this error. One hidden layer with four neurons is used. Three types of mother wavelet functions are used and they are: the Mexican hat function is [10, 11, 22, 37, 39]

1. The first partial derivative Mexican hat is

The Morlet's basic wavelet function is

where x is the desired signal and wis a variable value, which was adopted to satisfy the admissibility condition. w = 5 is chosen. The output layer contains one output which is the sum of PID controller and WNN outputs. The parameters values of the WNN-PID controller, such as the dilation factorsak′sand the translation factors bk′sof the mother wavelet function, the weights connection wk′sof the WNN, and PID parameters (kp, ki, kd), are optimized online in PSO algorithm. The results given in this chapter are for Mexican hat function only. The results for the rest functions are similar to that in Mexican hat and are not given.

6.1 Speed control based on feedforward WNN-PID controller

The feedforward WNN with PID controller (FWNN-PID) is utilized to control the speed of the BLDC motor as shown in the Simulink model in Figure 7 . The inputs of the WNN are the speed error and the change of this error, while the hidden layer has four neurons and one output in the output layer. The translation and dilation factors, weights connection for WNN, and PID parameters are learning on-line in PSO algorithm. The output of WNN is given by Eq. (2).

The PSO parameters are given in Table 1 . These parameters are chosen to get optimal parameters for the PID controller and the wavelet neural network; when it is tuned on-line in PSO algorithm and BLDC motor drive, the optimal values for the PID controller parameters and the WNN parameters (a's, b's, w's) are given in Tables 2 and 3 , respectively.

The BLDC motor drive is implemented in Simulink/Matlab program as shown in Figure 6 with the optimal values of PID controller parameters and the optimal values of WNN parameters. Figure 8 shows the speed response of the BLDC motor due to change in reference speed. The motor is started at a speed of 500 rpm and then is changed in step to a speed of 500 rpm for every 0.2 s. The actual speed of the motor is tracking the desired speed with a good response. The system starts at no load and suddenly a torque 2 N m (full load) is added at t = 0.4 s. Figure 9 shows the speed response of the BLDC motor at 2000 rpm during no load and load condition. The developed torque during no load and load condition is shown in Figure 10 . The position signal, the torque-speed characteristics, the phase current ia, Phase Back-emf eavoltage, and line voltage vabare given in Figures 11 – 15 , respectively.

6.2 Speed control based on proposed recurrent WNN-PID controller

The RWNN that is proposed here is similar to that of feedforward WNN with feedback connections. The RWNN consists of three layers, with two inputs in the input layer, the hidden layer has four neurons, with one output in the output layer and feedback connection for each layer. In this section, the feedback connection is called “Fully feedback.” Besides, the RWNN contains a number of delay samples in the input and output layers as shown in Figure 16 . The translation and dilation factors, weights and PID parameters are learning on-line to utilize PSO method in the same manner used in the previous subsection and the results are given in Tables 4 – 6 . The output of WNN is described by Eqs. (7) and (8).

The BLDC motor drive system with RWNN-PID controller is simulated in Matlab/Simulink program as shown in Figure 6 . The time period that is assumed in this model is 1 s. The WNN-PID controller can be utilized for speed control in a wide range between 0 and the rated value, with better performance and more flexibility in the controller. Figure 17 depicts the step change in speed of the BLDC. The motor is started at a speed of 500 rpm and then is changed in step to 500 rpm each 0.2 s. The actual speed of the motor is tracking the desired speed with a good response. The system starts at no load and suddenly a torque 2 N m (full load) is added at t = 0.4 s. Figure 18 shows the speed response of the BLDC motor at 2000 rpm during no load and load conditions. The developed torque during no load and load conditions is shown in Figure 19 . The position signal, the torque-speed characteristics, the phase current ia, Phase Back-emf eavoltage and line voltage vabare given in Figures 20 – 24 , respectively.