performance comparison of ga-pid with fuzzy logic ...

PERFORMANCE COMPARISON OF GA-PID WITH FUZZY LOGIC CONTROLLER FOR POSITION TRACKING OF BRUSHLESS DC MOTOR LUKMAN A. YUSUF1, ABDURRASHID IBRAHIM SANKA2 NASIRU BELLO KADANDANI3 , BASHIR SADIQ USMAN4 1,2&3 4

Electrical Engineering Department, Bayero University Kano, Nigeria.

Electrical and Electronic Engineering Technology Department, Federal Polytechnic Damaturu, Yobe State, Nigeria

Abstract The brushless DC (BLDC) motor has been found to offer many advantages that includes simple to construct, high torque capability, small inertia, low noise etc. BLDC is an unstable and nonlinear system which implies that its internal parameter will change for different input conditions. Therefore, there is need for an excellent controller that will stabilize the position of BLDC motor, as most of conventional controller available lack accuracy, robustness and sharpness required. This paper considered two control schemes: Fuzzy logic controller (FLC) and PID optimized with Genetic Algorithm (GA-PID) controller, for position tracking of BLDC motor. The FLC scheme was designed with the joint angle error and its derivative as the input of the controller using Fuzzy logic toolbox in MATLAB Simulink environment. Meanwhile, a Matlab script for genetic algorithm was written with the aim of obtaining the optimum PID parameters that accurately track the position of BLDC by minimizing an objective function (Integral time average error ITAE). For the purpose of comparison between the two proposed schemes various position tracking reference are tested. The results show that GA-PID control schemes proved superiority when considering settling time and rise time under zero disturbance with values 0.08 seconds and 0.00 seconds respectively as compared with settling time of 0.42 seconds and rise time of 0.02 seconds for FLC. However, FLC proved better when a disturbance was introduced with settling time of 0.10 seconds and percentage overshoot of 0.00 as compared with settling time of 19.80 seconds and percentage overshoot of 48% for GA-PID. Therefore, both controllers can serve as valuable and effective controllers for the system. Keywords: Objective function, Fuzzy logic, brushless motor, Genetic Algorithm, Stability. 1.0 INTRODUCTION The industrial world of today is fast growing and so is the demand for precision and robustness. There are several applications today that demand high performance BLDC motor drives. Among the various motors, brushless dc motors are gaining widespread popularity in electric vehicles, aerospace, military equipments, hard disk drives,

HVAC industry, medical equipments etc., due to their well-known advantages like high efficiency, low maintenance and excellent speed-torque characteristics. The conventional controllers used in high performance drives are proportional integral (PI) or proportional integral derivative (PID) controllers. These are constant gain controllers and require accurate mathematical models or system response for

their design. The BLDC motor drive system is highly non-linear. It is often very difficult to obtain an accurate mathematical model for BLDC motor drive systems when the motor and load parameters are unknown and time-varying. The conventional controllers fail to give optimal performance during change in operating conditions like variations in parameters, saturation and noise propagations. This has resulted in an increased interest in intelligent and adaptive controllers. Fuzzy logic and GA-PID are some of the popular strategies to deal with uncertain control systems. These controllers can offer a number of attractive properties for industrial applications such as insensitivity to the parameter variations and external disturbances, in addition to quick in response and very high precision. There has been a significant and growing interest in the application of fuzzy logic and optimized PID (GA-PID) controllers to control complex, nonlinear systems. The design of fuzzy logic controller doesn’t require mathematical model of the system, but rules describing the behavior of the system have to be framed based on the knowledge of the system. It is possible to quickly develop and implement a fuzzy controller for nonlinear systems such as BLDC motor drives. GA is stochastic global search methods based on the mechanics of natural selection and natural genetic. They are iterative method widely used in optimization problems in general branches of science and technology. It was first proposed by Holland in 1976. GA offer some advantages over other search tools in the following ways [1]:  GAs search from a population of points not a single point  GAs use probabilistic transition rules not deterministic ones  GAs work on encoding parameters set rather than the parameter set itself (except where real-valued individuals are used)

 GAs do not require derivative information or other auxiliary knowledge; only the objective function and the corresponding fitness levels influence the directions of the research. To obtain a solution to a problem through genetic algorithms, the algorithm is started with a set of solutions (represented by chromosomes) termed as the population. This is Initialization. This is followed by selection, which means choosing random solutions of one population to forms a new population base on their evaluation on the objective function. This can be done either by Roulette wheel or Stochastic universal sampling. The former was used because it ensures that each parent chance of being selected is proportional to its fitness value but possibility also exists to choose the worst population member. The new population is formed assuming that the new one will be better than the old one. Parent solutions are selected from the population to form new solutions (offspring) based on their fitness measure through the application of genetic operators such as crossover (exchange of genes from parents), mutation (sudden change in genes, this should however be introduce on a minimum probability) etc. These processes are repeated over several iterations until a stopping criterion is reached [2]. Although both optimized PID (GA-PID) and FLC have been applied successfully in many applications, they also have some limitations such as non deterministic nature of GA and tax in tuning FL. This paper considered performance comparison of Fuzzy logic and GA-PID for position tracking of brushless dc motor. The information referred from various literatures for carrying out this work is as follows. The modeling of brushless dc motor, estimation of parameters and control schemes are discussed in [3,4,5]. The effect of change in

motor parameters, load disturbances on the performance of the brushless dc drive system is discussed in [6,7,8]. Several tuning methods for the PI and PID controllers are described in [9,10,11]. Design, implementation and performance analysis of sliding mode controllers for various applications such as AC motors, BLDC motor, etc., are presented in [12,13,14,15,16]. Design and implementation of fuzzy based controllers for improving the performance of dc motors and brushless dc drives under different operating conditions are discussed in [17,18,19,20,21]. Controllers design with artificial tunable gain is explained in [21,22,23]. The simulation software MATLAB/Simulink is explained in [24]. The rest of the paper is organized as follows. Section Two deals with modelling of BLDC motor drive, section three, dwells on the controllers design for the BLDC motor drives, Section four presents the comparison between the results obtained from simulation and the its discussion, and finally, Section five concludes the paper. 2.0 MODEL DESCRIPTION Consider the block diagram of a brushless dc motor in Figure 1. The simplified mathematical dynamic equation is obtained from the figure using Newton’s laws of motion as follows.

Figure 1 Block diagram of brushless dc motor The analysis of BLDC motor is based on the assumption that there is no loss in the torque for simplification and accuracy. The BLDC motor is type of unsaturated. To perform the simulation of the position control, an appropriate model needs to be established. Based on the equivalent circuit of BLDC motor shown in Figure 1, the dynamic equations of BLDC motor using the assumption can be derived as: The motor torque, Tm is related to the armature current I a by a constant factor of K Tm  KI a

(1)

The back emf, eb velocity, wm by;

eb  Kwm  K

is related to angular

 m t

(2)

From Figure 1, the following equations can be written based on Newton’s Second Law combined with Kirchhoff’s law; Newton’s Second Law,

Tm  J m

 2 m   Bm m 2 t  t

(3)

Substituting equation (1) into equation (3) to obtain;

Jm

 2 m   Bm m  KI a 2 t  t

(4)

Applying Khirchhoff’sLaw,

L

I a  I a R  Va  eb t

(5)

Substitute equation (2) into equation (5) to obtain;

L

I a   I a R  Va  K m t t

(6)

By taking Laplace Transform, equation (5) and (6) can be expressed in term of s as; J m s 2 m ( s )  Bm s m ( s )  KI a ( s )

(7)

LsI a ( s )  RI a ( s )  Va ( s )  Ks m ( s )

(8)

From equation (7), Ia(s) can be expressed as;

I a ( s) 

s m (s )[ J m s  Bm ] K

(9)

Substitute equation (9) into equation (8) to obtain;

Va ( s ) 

s m [ J m s  Bm ][ Ls  R]  K K

Thus, substituting these parameter values into equation (11), the transfer function of BLDC is as shown in equation (12)

 m (s ) 776.2076  2 Va s  6.4364 s

(12)

3.0 CONTROLLER DESIGN

2

(10)

Therefore, from the equation (10), he transfer function in term of position,  m as an output and the voltage, Va (s) as an input can be obtained;

m ( s) K  Va s[ J m s  Bm ][ Ls  R]  K 2

(11)

The constants value of voltage, Va, torque constant factor, K, rotor inertia, Jm, damping ratio, Bm, resistance R and inductance, L for BLDC motor must be known. The specifications of BLDC motor which will be used are described in the Table 1 [21]: Table 1 Parameters of BLDC motor

In this section, the two proposed controllers designs are carried out which are FLC and GA-PID based controllers. 3.1 FLC DESIGN Fuzzy logic design is presented as described in [25]. The two inputs of fuzzy logic are position angle error (e) and its derivative ( ̇ ) as shown in Figure 2. The control output signal is generated based on the magnitude of the input signals. A total of 25 possible control signals are sent to the system depending on the degree of variation of error angle and its derivative as shown in Table1. The membership function of the inputs are chosen to be five and that of output membership function was chosen to be seven for a better result, since accuracy depends greatly on the number of membership function, that is to have a tighter control [26,27]. The membership functions for each input are nb, ns, z, ps, pb, which represent Negative big, Negative small, Zero, Positive small, and Positive big respectively. The output membership functions are rb, rm, rs, lc, is, im, ib which correspond to Reduced big, Reduce

medium, Reduce small, leave constant, increase small, Increase medium, Increase big respectively. Note that the output from the fuzzy controller is the voltage as shown in Figure 2. Error Step

voltage In1

Out1

du/dt Derivative

Fuzzy Logic Controller

System Scope1

a non-fuzzy control that best represent the degree of certainty of an inferred fuzzy control action. They are several numbers of procedures of defuzzifying the rules aggregate for the Mamdani, methods such as Center of gravity, First of maxima, Middle of maxima, Center of sum etc. In this work Center of gravity was used because its considered as the most efficient in that it gives a defuzzification output which conveys the real meaning of the action that had to be taking at that instance [27].

Figure 2 Fuzzy Controller In order to achieve fuzzy control, the following steps are followed: Fuzzification Stage: In this stage, input values are mapped to domain of fuzzy variable (i.e the crisp inputs variable are assigned linguistic label). In this work, Symmetrical Triangular membership functions are used. Then the fuzzy rule base must be formed based on the expert knowledge of the system. For example, if angle error(e) is negative big(nb) and the derivative angle error( ̇ ) is negative big(nb) then reduce the voltage big (vrb). This is one of many possible fuzzy rules used in this work. Table 1 shows the rest of fuzzy base rules used. These fuzzy rules are then applied on fuzzy input variables to give fuzzy output variables. This process is called fuzzy inference stage. Fuzzy inference engine: They are two types of approach in designing of inference engine viz: Composition based inference and Individual rule-based inference. The former which make use MAX-MIN was used in this work as an inference engine to determine the degree of membership function of the output variables.

TABLE I Fuzzy Base rules e\ ̇ nb

ns

z

ps

pb

vrm vrs

vlc

nb

vrb

vrb

ns

vrb

vrm vrs

z

vrm vrs

ps

vrs

pb

vlc

vlc

vis

vlc

vis

vim

vlc

vis

vim vib

vis

vim vib

vib

3.2 GA-PID CONTROLLER DESIGN In this section the design of model base GAPID is presented. The GA-PID Controller is incorporated in the system as shown in Figure 4. The general transfer function of the controller is given as:   1 C  K p 1   sTd  (14)  sTi 

C  Kp 

Ki  Kd s s , , and

(15)

Where: are the controller gains. The objective function to be minimize is 

Defuzzification Stage: In this stage all the consequent were aggregated to obtained a crisp output. In fact, it is aimed at producing

ITAE  t e dt 0

(16)

two controller schemes in two decimal places. GA-PID and FLC responses of BLDC mtor 0.7

GA FLC

0.6

Figure 3 Block diagram of System with GAPID controller

A m p litu d e

0.5

0.4

0.3

0.2

Figure 3 shows the block diagram of GAPID. The error from the system is fed to GA for minimization. The flow chat for the GA process is shown in Figure 4.

0.1

0

0

1

2

3

4

5

6

7

8

9

10

Time(seconds)

Figure 5. Response of the controllers under step level of 0.5 From the Figure 5, it can be seen that the GA-PID settled much faster than the FLC as shown in Table II. Note that the following values are in two decimal places Table II Response under step level of 0.5 input

4.0 RESULTS AND DISCUSSION After the design and simulation of the controllers using Matlab, each of the controller were run under various input conditions and the results are plotted. The performance of each controller were evaluated using standard performance indices like (settling time, Overshoot, rise time and steady state error)

GA-PID

FLC

Settling Time (sec)

0.08

0.42

Overshoot

0.00

0.00

Rise time (sec)

0.00

0.02

Steady state error

0.00

0.00

GA-PID and FLC responses of BLDC motor 0.9 0.8 0.7

GA FLC

0.6

A m p litu d e

Figure 4. Flow Process in Application of GA

Performance index

0.5 0.4

The results from the two controller schemes are compared in this section. The responses of Fuzzy logic and GA-PID controller for position of brushless dc motor are shown in Figs (5, 6, 7 & 8) under various input steps value. TABLE (III, IV, V and VI) summarizes performance index used for the

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

Time(Seconds)

Figure 6. Response of the controllers under step level of 0.8

10

The response of the controller under step level of 0.8 is shown in Figure 6. The results shows that GA-PID still settled quickly. The summary of the result is shown in Table III.

Table IV Response under step input with disturbance Performance index

GA-PID

FLC

Table III Response under step level of 0.8 input

Settling Time (sec)

19.80

0.10

Overshoot

48%

0.00

Rise time (sec)

0.00

0.20

Steady state error

0.00

0.00

Performance index

GA-PID

FLC

Settling Time (sec)

0.05

0.64

Overshoot

0.00

0.00

Rise time (sec)

0.00

0.08

Steady state error

0.00

0.00

5.0 CONCLUSION

1.5

GA FLC

A m p litu d e

1

0.5

0

0

10

20

30

40

50

60

70

80

90

Time(seconds)

Figure 7. Response of the controllers under step level of unity with disturbance The response of the two controllers under step input with input disturbance is shown in Figure 7. FLC shows robustness as it rejects disturbance. This shows that FLC has proven to be better of the two controllers when a sinusoidal signal of 0.2 amplitude is used as the disturbance. The summary of the response is shown in Table IV.

100

It was observed that the two proposed control schemes performed well in the tracking of position of brushless motor, the GA-PID controller performs much better than the FLC controller when considering settling time and rise time with values 0.08 seconds and 0.00 seconds respectively as compare with settling time of 0.42 seconds and rise time of 0.02 seconds for FLC under various step input levels considered with no disturbance. However FLC proved better when a disturbance was introduced with settling time of 0.10 seconds and percentage overshoot of 0.00 as compared with settling time of 19.80 seconds and percentage overshoot of 48% for GA-PID. Therefore, GA-PID and FLC controllers can serve as valuable and effective controllers for the system.

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