Optimization and comparative analysis of PID and FOPID controller for BLDC motor

Received Apr 22, 2019 Revised Jun 24, 2019 Accepted Jul 10, 2019 The FOPID and PID controller are designed to control the speed of the BLDC motor. The parameterskp,ki, kd, λ and μ of these controller are optimized based on genetic algorithm. The optimized coefficients keep in track with zero error signals. The output of the controller is given to the variable dc source which varies the input voltage to the three phase inverter depending on the input signal. The three phase inverter gives the voltage to the BLDC motor which enhances the stability of the system. The effectiveness of the controller is demonstrated by simulation.


INTRODUCTION
Brushless dc motor has windings in the stator and permanent magnet in the rotor. As it does not have brushes and commutator the efficiency of the motor increases with decrease in the ohmic loses. Brushless dc motor are available in single phase, two and three phase configuration. But three phase configuration are commonly used. Generally, Hall Effect sensors are used to sense the position of the rotor. But in this project the rotor position is determined from excitation current and voltage across the stator windings. The BLDC motor requires inverter as there is no brush and commutation arrangement. The inverter uses transistor for low drive application and Thyristor for high power drives. MOSFET/DIODE switches are used in the inverter. The commutation sequence to the MOSFET/DIODE depends on the rotor position. BLDC motor require less maintenance, generate less noise, have low inertia, have high efficiency, has long operating life.
The controller is used to improve the dynamic response of the system. The PID and FOPID controller parameters are tuned by genetic algorithm [1], more optimization techniques are used to tune the coefficients ofFOPID and PID controller [2]. In this paper the process to control is speed of the BLDC motor. The BLDC motor is modeled [3,4]. The FOPID controller transfer function is formed by using FOMCON toolbox [5,6]. The genetic algorithm optimization to tune PID and FOPID is done by suing optimization toolbox.

SPEED CONTROL OF BLDC MOTOR
The block diagram for speed control of BLDC motor is shown in Figure 1. The rotor position and speed of the BLDC motor is computed from the voltage and current supplied to the stator windings of BLDC motor. Each phase winding is excited in a sequence to run the BLDC motor. The phase winding of the BLDC motor is excited in a sequence by sequence commutation of the MOSFET/DIODE in the inverter [7,8]. That switching sequence is given to the inverter from position to gate signal block as shown in Figure 1. The speed of the BLDC motor is directly proportional to the voltage. Therefore, to control the speed of the BLDC motor, the input voltage must be controlled. The speed computed is compared with the reference speed by the comparator and the error signal is obtained. The error signal is given as input to the controller. The output of the controller is control signal. This control signal is given as input to the variable DC source. There by the input DC voltage is controlled based on the BLDC motor speed.

PID controller:
The PID controller consists of the proportional, integral and derivative controller. He proportional controller gives proportional response of the input error value. The integral controller is proportional to magnitude and the duration of the error. It improves the settling point and eliminates steady state error. The integral term gives response which also depends on the sum of previous error values. The sum of the previous error value is multiplied with the gain ki. The derivative control gives the slope of the error for specified duration and multiples it with the gain kd. The sum of this proportional, integral and derivative control gives the PID controller. The control signal of PID controller in time domain.
Taking Laplace transform of above equation, the transfer function of PID controller is given by, is an error signal. The block diagram of PID controller is shown in Figure 2. The parameter k p , k i ,k d have to be tuned by using genetic algorithm. The PID controller can implement in MATLAB/Simulink by PID block available in continuous time toolbox.

FOPID controller
Fractional order PID controller is commonly in the form PI λ D µ . In this the integrator and differentiator is in the order of λ and µ respectively [9]. The time domain representation of FOPID control signal is, Where, D −λ -Fractional order integrator, D µ -Fractional order differentiator. Taking Laplace transform for above equation, the transfer function of FOPID controller is given by, Where, λ -Order of integrator, µ -Order of differentiator. The block diagram of FOPID controller is shown in Figure 3.

DESIGN OF BLDC MOTOR
The rotor and shaft of the BLDC motor is assumed to be rigid [10]. The mode is assumed as having viscous friction model. Therefore the friction torque is proportional to the shaft angular velocity. Based on Newton second law and Kirchhoff's voltage law, three phase star connected BLDC motor can be described by following (1-4).
Where T e ,Ө m , b, J, T L ,V, R, L, I, e are electrical torque, mechanical rotational speed, viscous friction constant, the rotor inertia, mechanical load torque, phase to phase voltage, phase resistance, phase current, phase inductance and phase back emf respectively. The voltage and current equation is given by (5)(6), To simplify the modeling, only two voltage equation are need, they are (7-8).
The torque generated by the BLDC motor is given by (9), The trapezoidal back emf can be written as (10-12) : Where , Ө is the back emf constant and electrical angle respectively.

INVERTER TOPOLOGY
The inverter used here is a three phase inverter. It consist of three leg and six MOSFET/DIODE. It is implemented by using universal bridge available in toolbox. The gate signal in this universal bridge decides the MOSFET commutation. The inverter is shown in Figure 4 And for Circuit diagram of dc motor is shown in Figure 5.  Table 1.
The parameters are tuned for this system.

GENETIC ALGORITHM
Genetic algorithm is an optimization technique. This technique is used to tune the parameter of PID and FOPID controller [11,12]. The flowchart of the genetic algorithm is shown in Figure 6.

Figure 6. Flowchart of genetic algorithm
It is an iterative process with the population in each iteration called a generation. First the initial population is generated. The population may range from hundred to thousand. From the population the fitness solution to each individual is chosen by objective function. In the selection process the portion of the existing population is selected. In the crossover/mutation, a new generation is generated. Finally, it checks whether it meet the criteria. If it meets the criteria it gets terminate otherwise the iteration repeats.

Optimization of the PID and FOPID using genetic algorithm
The genetic algorithm can be implemented by using optimization toolbox. In FOPID tuning the parameter , , , λ, and µ should be tuned to the proposed system [13]. The fitness function should be called as @(x) fun name(x). The lower and upper limit of the cost function is shown in the Table 2. The parameter of genetic algorithm is chosen as in Table 3.  In PID tuning, , , are the cost function. The limit of PID controller is chosen as in Table 4. Similarly by choosing criteria in optimization toolbox the parameter are tunes as shown in Table 5. The graphical representation is given as shown in Figure 7.   The graph generated while tuning the parameter of FOPID controller is shown in Figure 8 and for PID controller is shown in Figure 9.

SIMULATION AND RESULT
The simulation model of the speed control of BLDC motor is shown in Figure 10. The PID controller is implemented by using PID block available in MATLAB. The transfer function of FOPID is generated by using FOMCON toolbox. The output speed response for PID controller is shown in Figure 10.
An external disturbance is applied to the BLDC motor at 0.5 second. The PID controller controls the input voltage and the BLDC motor reaches the steady state at 8.5 second. Thus the settling time of the BLDC motor using PID controller is 8 second. The output speed response of the system due to FOPID controller is shown in Figure 11.
An external disturbance is applied to the BLDC motor at 0.5 second. The FOPID controller controls the input voltage and the BLDC motor reaches the steady state at 1.3 second. Thus the settling time of the BLDC motor using FOPID controller is 0.8 second. Therefore the settling time of the speed due to the FOPID controller is 10.5 times faster than the PID controller. Comparison of PID and FOPID speed response is shown in Figure 12. In this section, it is explained the results of research and at the same time is given the comprehensive discussion. Results can be presented in Figures, graphs, Tables and others that make the reader understand easily [14,4]. The discussion can be made in several sub-chapters.

CONCLUSION
The FOPID controller enhances the dynamic response of the BLDC motor compared to the conventional PID controller. The settling time of the speed response is very fast when we use FOPID controller. So the performance of the FOPID based speed controller is better and efficient. Stability and robustness of the BLDC motor can be improved. So this proposed controller can be extended to other industrial control system.