PID PARAMETERS OPTIMIZATION USING ADAPTIVE PSO ALGORITHM FOR A DCSM POSITI.pdf

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PID PARAMETERS OPTIMIZATION USING ADAPTIVE PSO ALGORITHM FOR A DCSM POSITION CONTROL Ruba AL-MulaHumadi 1, Dr. NizarHadi Abbas2, WameedhHammadi3 1

(Mechanical Eng. Dept., College of Eng./ University of Baghdad, Iraq) 2 (Electrical Eng. Dept., College of Eng./ University of Baghdad, Iraq) 3 (University of Surrey, Guildford, UK)

ABSTRACT

This paper demonstrates a novel algorithm in particle swarm optimization, which is called Adaptive PSO (APSO), to optimize the gains of a proportional-integral-derivative (PID) controller concurrently to control the position of a DC servomotor (DCSM). The parameters of the PID controller (KP, Ki, and KD) are obtained, firstly, by using Ziegler-Nichols tuning method, secondly by using standard PSO algorithm, thirdly by using MPSO algorithm, and finally by using APSO algorithm. The transient response analysis has been done by comparing the performance of the four tuning methods. The results showed that the proposed algorithm gives better performance than the other optimization algorithms, and the DCSM reached very fast to the final position. The analysis and simulations have been made in MATLAB R2010a software environment. Keywords: DC Servo Motor, Position Control, PID Controller, Optimum PID Parameters, Adaptive PSO Algorithm. 1.

INTRODUCTION

Servomechanism means automatic control of a physical quantity (the angular displacement or the motor speed). It is used because of the human operator or worker suffers from many difficulties such as fatigue, slow reaction and limited power [1]. Servo motors work on close-loop mode (as shown in Fig. 1) unlike stepper motors work on open-loop mode [2]. The difference between the two modes is the feedback, which is existed in the close-loop, and it is not existed in the open-loop mode. In the closed-loop control, the information about the speed or the displacement is fed back in such a way that if the controlled quantity differs from the desired value. An error is observed and it will be transmitted to the control system, and the cycle is repeated [1]. 1

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME

Figure 1: Closed loop servo system [2]

DC servo motors can be used in the industrial areas that require different operating conditions. They are used in digitally controlled machines, weapon industry, full-automatic regulators to get fast and accurate start, industrial tools, and they are also used in controlling systems of computers. computers. All DCSMs have the same same structure with different parameters and different controlling methods. Many authors tried to improve the motor stability and performance in different ways. Some of them used a traditional P, PI and PID controllers because of their simple structure, and they are cheaper than any other types of controllers; others thought of another controlling method. During different operating conditions, speed and load changes causing negative and positive overshoots and oscillations. Therefore, in order to get rid of these problems M. Akar etal. [3], proposed a novel cascade fuzzy PI controller to dynamically control the speed and torque of DCSM. Their experimental results showed that the performance of the proposed controller was better than that of traditional PI controller at different operating speed and load conditions. M. Katal etal. [4], used genetic algorithms (GA) to optimize the gains of a PID controller for a Servo DC Motor. As comparing their results with Ziegler-Nichols, they found that GA gives less overshoot, less settling time and less rising time. Likewise, R. Bindu, and M. K. Namboothiripad[5], worked on GA-PID controller, and studied the position control of a DCSM. They found that their method was very flexible and fast for tuning the parameters of the PID controller. A. M. Yousef[6], worked experimentally and studied theoretically another approach in tuning the parameters of the PID controller for position control of DCSM, which was integral sliding mode (ISM). The goal of his study was to find the robustness and effectiveness of this method against the parameters' variations. In similar fashion, A.Kassem and A. M.Yousef [7], studied the position control of DC servo motor based on the the Sliding Mode (SM) approach and compared compared with integral Sliding Mode Controller (ISMC). The performances of the closed-loop system can be defined in terms of overshoot, rising time, settling time, and steady-state error. This paper deals with finding the optimal parameters of PID controller to control the position of DCSM based on the aforementioned transient specifications and by using four tuning methods namely:Ziegler-Nichols, SPSO, MPSO and APSO. This paper is structured as follows: Section 2, describes the mathematical model of DCSM. The concept of PID controller is presented in section 3. The four tuning methods of PID controller (ZN algorithm, SPSO algorithm, modified PSO algorithm, and Adaptive PSO algorithm) are described in section 4. Section 5, presents the results and discussion and finally, section 6, concludes the paper.

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME 2.

DCSM MATHEMATICAL MODEL

Servo motors can be either DC or AC operated unlike stepper motors are DC operated. There are three different types of servo motors [2]: • AC servo motor; based on induction motor designs. • DC servo motor (DCSM); based on DC motor designs; (which is considered in this paper). • DC or AC brushless servo motor. There are three different controlling schemes used to control DCSM: armature control, field control and permanent magnet [1]. In this paper, the armature control method is used; this means that the field is fixed while the voltage across the armature is controlled by a PID controller based on four different tuning methods; Ziegler-Nichols, SPSO, MPSO and APSO; all are presented in this research. The equivalent circuit of the armature controlled DCSM is shown in Fig.2.

+ Error signal

Constant field

r e l l o r t n o C

La Ra Ia

_

load

eb

θωbo

Jo

ea Figure 2: Motor equivalent circuit

The preliminary step in this research is modeling the armature controlled DC servo motor in order to obtain the open-loop transfer function using state-space representation method. Afterwards, the PID controller is including to change the system to a closed-loop system. Below are the steps of the motor modeling. The equations for armature controlled system The electrical equation is The mechanical equation is

௕ ൌ ଷ ሶ ௔ ൌ ଶ ௔

Where, Knowing that:

௔ ௔ ൅ ௔ ௗ௜ௗ௧ೌ ൅ ௕ ൌ ௔ ௢ ሶ ൅ ௢ ൌ ௔ ௔ ൌ ଓ௔ ሶ ,

Now let the states for the system be:

The input will be:

ൌ ሶ ൌ , ଶଶ ൌ ሷ ൌ ൌ ሶ ௔ ൌൌ ଵଶ ଓ௔ ሶሶ ൌൌ ଶଵሶሶ ௔ൌ 3

ሺ1ሻ ሺ2ሻ


In matrix form the sates equations will be:

൤ ଵଶሶሶ ൨ ൌ ൤െ ଶ௔⁄⁄௢௔ െെ ଷ௢⁄⁄ ௢௔൨ ቂ ଵଶቃ ൅ ቂ1⁄0 ௔ቃ ൌ ሾ0 1ሿ ቂ ଵଶቃ൅0 ൌ ൤െ ଶ௔⁄⁄௢௔ െെ ଷ௢⁄⁄ ௢௔൨, ൌቂ 1⁄0 ௔ቃ , ൌ ሾ0 1ሿ, ൌ 0 ሺ௔ሺ ሻሻ ൌ ൫ሺ ௔ ൅ ௔ ሻሺ ௢ଶ൅ ௢ ሻ ൅ ଶ ଷ൯

ሺ3ሻ ሺ4ሻ ሺ5ሻሺ5ሻ ሺ6ሻ

So the A, B, C, and D for the armature controlled DCSM will be:

After that the transfer function for position control DC servo motor c an be written as:

The physical parameters values for DCSM are illustrated in Table 1 [5].

Table 1: DCSM parameters Parameter

3.

Description

Value

Unit

Ra

Armature resistance

2.45

Ω

La

Armature inductance

0.035

H

bo

Viscous friction

Jo

Moment of inertia

0.022

Kg-m

k2

Torque Constant

1.2

N-m/A

k3

Back emf constant

1.2

Volt/rad/sec

0.5*10

-3

N-m/(rad/sec) 2

PID CONTROLLER

A controller is a unit designed to achieve some pre-specified performance requirements for a plant. The input to the controller is an error; e(t) based on the difference between the reference input; r(t) and the actual output; y(t). The controller then attempts to bring the actual output to track the reference. The mnemonic PID refers to be proportional, integral and derivative. So clearly that in this type of controller with three significant functions, each part can be selected to achieve distinct control action [8]. PID controller is vastly employed because of its simplicity in use, past record of success and wide availability. PID controller can be represented in different manners as depicted in Table 2.

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME Table 2: PID controllers’ representation Presentation type

Time domain

Laplace form (S domain) Transfer function MATLAB command

Equations

ሺ ሻ ൌ  ሺ ሻ൅ න௧ ሺ ሻ ௜ ൅ ௗ௜ ሺ ሻ ሺௗ ሻଶ ൌ൅  ሺ൅ሻ ൅௜ ൅ ௗ ሺ ሻ

notes

ሺሺ ሻሻ ൌൌ ሺ ሻ െ ሺ ሻ

output of PID

num=[ KdKP Ki] den=[1 0] g= tf(num,den)

4. TUNING METHODS OF PID CONTROLLER

The PID controller’s tuning methods refer to adapt its control parameters to attain the desired response of a DCSM position control. In this paper, four tuning methods are used; the description of each method is shown below: 4.1 ZIEGLER-NICHOLS TUNING METHOD Since it was discovered in 1942 [9], Ziegler-Nichols method was considered as the mother method used in selecting the parameters of the PID controller in order to meet given performance [10]. Recently, a simulation MATLAB program was made by S. Das etal. [11], the tuning method is made with the help of two functions Ziegler (key,vars) and the other function is WRITEPID( )[11].

The transfer function of DCSM position control is:

ሺ௔ሺ ሻሻ ൌ 0.00077 ଷ ൅0.1.5239 ଶ ൅1.440

ሺ7ሻ

Z-N results will be: Ku=84, Pu= 0.145 sec -1 PID parameters: KP=50, Ki=688.7sec ,Kd=0.59sec

4.2 STANDARD PARTICLE SWARM OPTIMIZATION TUNING METHOD Particle swarm optimization algorithm is a population based on computing technique. The idea behind this algorithm was the behavior of the swarm such as birds' flock or fish schools. It was introduced by J. Kennedy and R. Eberhartin 1995[12]. Many researchers [13, 14] described the standard PSO algorithm. In SPSO algorithm, birds' flock consists of a swarm of birds each bird has its' position and velocity; the aim of the flock is searching for food. Hence in SPSO algorithm; the swarm is the flock; each bird represents a particle in the swarm, and the aim of the f lock is the fitness function. At the beginning, the initial values for the position vector and velocity vector can be formulated as in Eqs.(8) &(9). 5


xx୧୧଴଴,,ଵଶ ൌൌ KK୫୧୫୧୮୧୫୧୬୬ ൅൅ ൫K൫K୫ୟ୶୫ୟ୶୮୧୫ୟ୶ െെ KK୫୧୫୧୮୧୫୧୬୬൯൯ rraanndd୧୧,,ଵଶൢ x୧଴,ଷ ൌ K୫୧ୢ଴ ୬ ൅ ൫K୫ୟ୶ୢ െ K୫୧ୢ ୬൯ rand୧,ଷ v୧଴,ଵ ൌ xx2୧୧଴,,ଵଶ v୧଴,ଶ ൌ x2୧଴,ଷ v୧଴,ଷ ൌ 2

ሺ8ሻ ሺ9ሻ

The initial personal best position vector will be considered P best and after that, the algorithm starts to search for the g best (gbest represents the most appropriate position vector that verifies the fitness function). Afterwards, velocity and position vectors can be updated according to the Eq. (10)&(11) then new P best and gbest will be obtained. The process will be repeated until reach the required number of iterations or a stopping criterion will be verified.

௜௧ାଵ,௠ ൌ ௜௧,௠ ൅ ଵ ଵ ൫ ௕௘௦௧ ௜,௠ െ ௜௧,௠ ൯ ൅ ଶ ଶ ൫ ௕௘௦௧ ௠ െ ௜௧,௠൯ ௜௧ାଵ,௠ ൌ ௜௧,௠ ൅ ∆ ௜௧ାଵ,௠

ൌ1, 2 , … …. , ൌ1, 2 , … …. ,

Where,

and

n: d : t :

;

Population size. Search space dimension. Current iteration. Current velocity of particle i at iteration t. Modified velocity of particle i. Inertia of the previous velocity. Acceleration constants. Uniformly generated random numbers in the range of [0,1]. Current position of particle i at iteration t. Modified position of particle i. Time step which is taken to be unity. The best previous position along the dimension of particle i in iteration t . The best previous position among all the particles along the dimensionin iteration t .

௜௜௧ାଵ௧,,௠௠ ଵ& ଵଵ& ௜௜௧ାଵ௧,,௠௠ ∆௕௘௦௧ ௜,௠ ௕௘௦௧ ௠ ௧௛ : :

:

:

: :

:

:

:

ሺ10ሻ ሺ1111ሻ

௧௛

ଵ& ଵ

The cognitive and social acceleration factors respectivelyare chosen to be 1.494 and the inertial weight to be 0.729 as recommended in Clerc’s PSO [15]. The fitness of each particle is measured based on Fitness and is representedbyEq.(12), representedbyEq.(12),

Fሺkሻൌ Jሺ1kሻ

And the objective function is given by Eq. (13)[16], 6

Fሺkሻ

ሺ12ሻ


J୫୧୫୧ሺ୬kሻൌሺ100Ess୫ୟ୶୫ୟ୶଴.ହ ൅ 5M୔ଶሻ ൅ ሺ10t10tୱ െ t୰ሻ KK୫୧୮୧ ୬ ൑൑ KK୮୧ ൑൑KK୫ୟ୶୧୮ K୫୧ୢ ୬ ൑ Kୢ ൑ K୫ୟ୶ୢ

Min Subject to:

ሺ13ሻ

Where k=[k pk ik d], E SS, MP, t S, and t r are the PID parameters, the steady state error, maximum overshoot, settling time and rising time respectively. They are the characteristics of the step response. In MATLAB they can be obtained from the command: step info(g),where g is the transfer function. 4.3

MODIFIED PARTICLE SWARM OPTIMIZATION TUNING METHOD In the latest researches, some modifications to the SPSO are introduced mainly to enhance the rate of convergence and reduced the consumed time to complete the global search, and this can be accomplished by altering the values of the PSO parameters, inertial weight ‘ w’, cognitive and social acceleration constants ‘ ’ and ‘ ’ respectively, therebygiving it the flexibility to optimize its particular performance. In this regards, a modified w has the same initial and final values as in the linear function, but is characterized by a sharp rate of decrease. A small value of w encourages local exploitation whereas a larger value of w supports global exploration, and its values can be computed as follows [17]:

ଵ

ଶ

௧ ൌ ሺ ௠௔௫ െ ௠௜௡ሻ

ቊെ൬ቊെ ൬ ௠௔௫

ଶ൰ ቋ ൅ ௠௜௡

ሺ14ሻ

The acceleration constants are adjusted to incorporate better compromise between the exploration and exploitation of each space in SPSO; time variant acceleration constants have been proposed in [18], and can be represented as follows:

ଵ௧ ൌ ൫ ଵ௙ െ ଵ௜൯ ௠௔௫ ൅ ଵ௜ ଶ௧ ൌ ൫ ଶ௙ െ ଶ௜൯ ௠௔௫ ൅ ଶ௜

Where,

Z: Constant equal to 2.2 tmax: Maximum number of iteration. t :Current iteration. wmin: Initial weights, is set to be 0.4. wmax:Final weights, is set to be 0.9. :Decreased from to :Decreased from to

ccଵ୲ଵଶ୲୲

ሺ15ሻ

ccଵ୧ଶ୧ൌ2.ൌ0.55 ccଵ୤ଵଶ୤୤ ൌ0.ൌ2.55

. .

ADAPTIVE PARTICLE SWARM OPTIMIZATION TUNING METHOD In this paper, an adaptive approach is proposed to adjust the particles' velocity and position to overcome the slow convergence problem that emerged in SPSO and MPSO algorithms. Thus, in the APSO, the particle position is updated such that the highly fitted particle moves slowly when compared to the lowly fitted particle. Therefore, in order to achieve the promising updating the following particles’ parameters should be adapted according to their objective function values[19, 20]:

4.4

7


1. The cognitive and social acceleration constants determine the step size of the particles' movements through the P best and gbest, respectively. In the SPSO, these step sizes are constant and for the whole population are the same and in order to achieve faster and more accurate movements, new step sizes can be introduced, which they should accelerate the convergence rate. In each generation, the value of the objective function is a gauge that presents the relative improvement of this movement with respect to the previous iteration movement. Thus, the difference between the values of the objective function in the different iterations can be selected as the accelerators. Therefore, the adaptive velocity and position formulas are clarified in Eqs. (16) and (17). 2. The adaptive inertia weight factor (AIWF) , is the improved version of Eq. E q. (14)to find out a compromised AIWF that satisfies both exploration (global search) and exploitation (local search). The AIWF is determined as in E q. (18). 3. The adaptive random numbers (ARNs) and , are proposed to increase the movement impact on the third term (swarm) and decrease the movement influence on the second term (individual) of Eq. (16). These ARNs are written as in Eq. (19). 4. The adaptive acceleration coefficients (AACs) and , they are used to award the efficient particle that has high fitness and punishes the not competent one. These AACs are formulated as in Eq. (20).

௜௧

Where,

௧ଵ,௜ ௧ଶ,௜ ௧ଵ,௜ ௧ଶ,௜ ௜௧ାଵ,௠ ൌ ௜௧ ௜௧,௠ ൅൅ ௧௧ଵ,ଶ,௜௜ ௧௧ଵ,ଶ,௜௜ ൫൫ ൫ሺ ௕௘௦௧௧௕௘௦௧௧ ௜௠,௠ሻ൯െെ ሺሺ ௜௧௜,௧௠,௠ሻሻ൯൯ ൫൫ ௕௘௦௧௧௕௘௦௧௧ ௠௜,௠െെ ௜௧,௠௜௧,௠൯ ൯ ሺ16ሻ ሺ ሻ 17 1 7 ௜௧ାଵ,௠ ൌ ௜௧,௠ ൅ ∆ ௜௧ାଵ,௠ ௧௜ ൌ ௠௜௡ ൅ ሺሺ ௕௘௦௧௧ିଵ௧ିଵ௜ ௜ሻሻ หห ሺሺ ௧ିଵ௜௧ିଵ௜ ሻሻെെ ሺሺ ௕௘௦௧௧ିଵ௕௘௦௧௧ିଵ ሻ௜ሻหห ሺ18ሻ ௧ଵ,௜ ൌ ൫ ଵ௙ െ ଵ௜൯ ௠௔௫ ൅ ඨ ሺሺ ௕௘௦௧௧ିଵ௧ିଵ௜ ௜ሻሻ ଵ௜ ሺ 1 9ሻ ௧ିଵ ௧ଶ,௜ ൌ ൫ ଶ௙ െ ଶ௜൯ ௠௔௫ ൅ ඨ ሺሺ ௕௘௦௧௧ିଵ௜ ሻሻ ଶ௜ ௧ଵ,௜ ൌ 1 െ ௕௘௦௧௧ିଵ௧ିଵ௜ ௜ ൅ ଵ ሺ20ሻ ௧ଶ,௜ ൌ 1 െ ௕௘௦௧௧ିଵ௧ିଵ௜ ൅ ଶ ௧௛ ൫ሺ ௜௕௘௦௧௧௧,௠ሻ௜,௠൯ ௧௛ ሺ ௕௘௦௧௧ ௠ሻ

: The best objective function value along al ong dimension of particle iin iteration t . : The objective function value of particle iin iteration t . : The global best objective function among all particles along the dimension in iteration t . The APSO algorithm as in the aforesaid tuning methods is used for determining the gains of PID controller for DCSM position control. Fig.3 shows the block diagram of the proposed SPSO/MPSO/APSO–PID controller and the pseudo code of APSO algorithm tuned PID controller is shown in Fig.4. 8

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME Transient Response Characteristics (t r, ts, MP, Ess) Objective Function

SPSO / MPSO / APSO Al orith orithm m

Kp

KI

KD Actual output

Reference input

+

-

PID Controller

Power System

Gc(s)

G(s) DC servo motor

Figure 3: Block diagram of proposed SPSO/MPSO/APSO-PID controller 01: Start 02: Define the limits of Kp, Ki, and Kd 03: Initialize particles swarm using Eqs. (8 & 9) 04: While (no. of iterations is not completed) 05: For i=1 to population size 06: For m=1 to space dimension 07: update c1 ,c2 ,w, r1 ,r2 values from Eqs.(18-20) 08: update the velocity (v) and the position (x) of particles fromEqs.(16 & 17). 09: Next m 10: Run the DCSM with PID controller for each set of parameters (i.e., Kp, Ki, and Kd) 11: Calculate the transient response specifications 12: Evaluate fitness for each particle in the swarm using Eqns. (12 & 13) 13: Obtain pbest 14: Next i 15: Obtain gbest 16: While (no. of iterations is not completed), repeat the cycle from step 05-15 17: end

Figure 4: Pseudo code of APSO algorithm tuned PID controller 5.

RESULTS AND DISCUSSIONS

The simulation results are performed using MATLAB R2010 a software environment run on Core(TM) i5, 2.5 GHz, and 4 GB RAM system. The number of particles is 25. The upper limits of the PID controller parameters (Kp, Ki and Kd) are [70, 750, 1], while the lower limits are [1, 50, 0.01]. The number of iterations is 10. The results were were rising time (t r), settling time (tS), maximum overshoot (M P), and steady-state error (E SS). The differences between PSO, MPSO and APSO are in the applied equations as they were presented in the previous sections. A series of simulations had been done to make some comparisons on the performance between the four tuning methods. Table 3, shows the results for the four presented tuning methods. It is clear from the table that the smallest settling time was with APSOPID tuning method, and the lowest overshoot was with MPSO-PID tuning method. 9

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME Table 3: Performance of the four tuning methods ZN-PID PSO-PID MPSO-PID APSO-PID KP KI KD tr

50 689 0.59 0.0341

24.9453 70 0.59 0.06

27.5007 60 0.7 0.0553

37.2333 112.8415 0.8925 0.0421

ts Mp Ess

0.672 72.8 0.0015

0.7204 10.2811 0.0061

0.7797 7.388 0.0019

0.5661 15.8455 0.0041

Fig 5, shows the step response of the DC servomotor with PID controller based on various tuning methods. The first upper shape shows the response using Ziegler-Nichols tuning method for the PID parameters. It can be seen that the response has high oscillations with high overshoot. The second upper shape shows the response using PSO tuning method; it can deduce that it is a better response as compared with the first one. A lowest overshoot is obtained by using the MPSO algorithm as shown in the first lower shape. The APSO algorithm should be employed in order to achieve smallest settling time; this case is demonstrated in the second lower shape. The four tuned methods are illustrated on the same sheet in Fig. 6.

Step Response

Step Response

2

1.5

e d u t i l p m A

PSO-PID

ZN-PID

1.5

1

e d u t i l p m A 0.5

1 0.5 0

0

0.5

0

1

0

Tim Time (s ec)

0.5 Time (sec)

Step Response

Step Response

1.5

1.5

MPSO-PID

APSO-PID

1

1


0

1


0

0.5

1

0

1.5

Tim Time (s ec)

0

0.5

1

Time (sec)

Figure 5: The response of the PID PI D controller driven DCSM using four tuning methods 10


Step Response 1.8 ZN-PID 1.6

PSO-PID MPSO-PID APSO-PID

1.4

1.2

1 e d u t i l p m 0.8 A 0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Figure 6: Compared response of the ZN, SPSO, MPSO, APSO-PID controller driven DCSM

6.

CONCLUSIONS

In this paper, a DCSM has been mathematically extracted and implemented using MATLAB software environment along with PID controller tuned by several algorithms. An Adaptive PSO algorithm was introduced to improve the performance of a position control DCSM with PID controller. The APSO tuning method was developed by incorporating four significant modifications into the standard PSO and modified PSO in order to create adaptive PSO algorithm that gives a superior response. The impact of the suggested optimization algorithm on the concerned system has been testified by computer simulations. Simulation results show that the proposed APSO-PID controller, if well-tuned, can provide prominent performance, with faster convergence speed, very good tracking, highly robustness, and minimum settling time than other characterized tuning methods.

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 4, July-August (2013), © IAEME REFERENCES

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PID PARAMETERS OPTIMIZATION USING ADAPTIVE PSO ALGORITHM FOR A DCSM POSITI.pdf

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