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Journal Title Proceedings Third International Conference on Emerging Trends in Engineering and Technology ICETET 2010 19-21 November 2010 Goa India I

Volume Issue MonthlYear nov 19-21 2010 Pages 422-425

Article Author

Article Title Mehra V Srivastava S Varshney P Fractional-Order PID Controller Design for Speed Control of DC Motor

ISSN OCLC 762162077

Lending String UMC WAU GATAZU

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NotesAlternate Delivery ILL allowed Print fax or secure electronic only

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This article was supplied by

Interlibrary Loan and Document Delivery Services University of Washington Libraries

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(206) 543-1878 or Toll Free (800) 324-5351 OCLC WAU - DOCLlNE WAUWAS

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This document is being supplied to you in accordance with United States copyright law (Title 17 US Code) It is intended only for your personal research or instructional use The document may not be copied or emailed to multiple sites Nor may it be posted to a mailing list or on the web without the express written consent of the copyright owner and payment of royalties Infringement of copyright law may subject the violator to civil fine andor criminal prosecution or both

Fractional-Order PID Controller Design for Speed Control of DC Motor

Abstractmdash This paper deals speed control of a DC motor using fractional-order control Fractional calculus provides novel and higher performance extensions for fractional order proportional integral and derivative (FOPID) controller In this paper the parameters of the FOPID controller are optimally learned by using Genetic Algorithm (GA) and the optimization performance target is chosen as the integral of the absolute error (IAE) Simulation results show that the FOPID controller performs better than the integer order PID controller

Keywords-Fractional calculus fractional-order controller genetic algorithm DC motor speed control

I INTRODUCTION DC motor is a power actuator which converts electrical

energy into rotational mechanical energy DC motors are widely used in industry and commercial application such as tape motor disk drive robotic manipulators and in numerous control applications Therefore its control is very important For the control of dc motor traditional controllers such as PI and PID controller have been used widely in literature [1] In this paper DC motor is controlled by a non-conventional control technique known as a fractional-order PID (FOPID) control This technique was developed during the last few decades and it has various practical applications viz flexible spacecraft attitude control Car suspension control temperature control motor control etc This idea of the fractional calculus application to control theory has been described in many other works ([2 4 11]) and its advantages are proved as well

This paper has following sections Sections 2 explains briefly about Fraction calculus and fractional-order controller Section 3 discusses the optimization technique using genetic algorithm Sections 4 and 5 show parameter optimization design process and simulation results Section 6 concludes the paper

II FRACTIONAL- ORDER CONTROLLER Fractional-order control systems are described by

fractional-order differential equations The FOPID controller is the expansion of the conventional PID controller based on fractional calculus FOPID has five parameters which are

responsible for adding more flexibility and robustness to the system

A Fractional calculus and Fractional-order controller Fractional calculus is a generalization of integration and

differentiation to non-integer (fractional) order fundamental

operator rD a t where a and t are the limits and (r ɛ R) is the

order of the operation The two definitions used for the

fractional differintegral rD a t are the Gruumlnwald-Letnikov

(GL) definition and the Riemann-Liouville (RL) definition [8 12] Further it has been mentioned in literature that for a wide class of functions these two definitions are equivalent [12] GL definition

t ah

jr r r ra t hh 0 h 0j 0

rD f(t) limh ( 1) f(t jh) limh f(t)

j

(1)

where [middot] means the integer part and rf(t)th is the

generalized finite difference of order r with step h RL definition

tnr

a t n r n 1a

1 d f( )D f(t) d(n a) dt (t )

(2)

for ((n-1)ltrltn) and Г() is the Eulerrsquos gamma function The Laplace transforms of the GL and RL fractional

derivativeintegral of signal f(t) at t=0 for order r is given by r r r

a tL D f(t) s L f(t) s F(s) (3) Using Laplace transformation

s F(s) I f(t) and ds F(s) f(t)dt

(4)

Then the fractional PID controller is written as P I DC(s) K K s K s 0 (5)

Selection of λ δ gives the classical controllers viz PD controller (λ =0) PI controller (δ =0) and PID controller (λ δ=1)

All the above-mentioned controllers are special case of the fractional PIλDδ controller Some advantages of PIλDδ controller include better control of dynamical systems (they are described by fractional-order mathematical models) and that they are less sensitive to parameter variations of a controlled plant

B Continuous-time Approximation of Fractional Differentiator Integrator Several approximation methods and techniques for

obtaining continuous-time and discrete-time fractional-order models in the form of IIR and FIR filters are developed in [9]

Vishal Mehra Smriti Srivastava and Pragya Varshney Department Of Instrumentation and Control Engg

Netaji Subhas Institute Of Technology New Delhi India 110078 vishalmehransitonlinein ssmritiyahoocom pragyavarediffmailcom

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

In this paper the Oustalouprsquos approximation [8] algorithm is used for simulation purposes This method is based on the approximation of a function

rH(s) s r R r 11

For the frequency range (ωl ωh) nk

k n k

sH(s) Ks

Using the following set of synthesis formulae the approximation for poles zeros and gain are obtained as follows k n 05 05r 2n 1

k l h lk n 05 05r 2n 1

k l h l

r n 2K k kh l k n

(6)

where ωh and ωl are the high and low transitional frequencies

This algorithm is implemented in MATLAB as a Function script lsquoora_focrsquo given in [7]

III GENETIC ALGORITHMS Genetic Algorithm (GA) is a stochastic global search

method that mimics the process of natural evolution The algorithm starts with no knowledge of the correct solution and depends entirely on responses from its environment and evolution operators to arrive at the best solution By starting at several independent points and searching in parallel the algorithm avoids local minima and converges to sub optimal solutions In this way GA has been shown to be capable of locating high performance areas in complex domains without experiencing the difficulties associated with high dimensionality

GA consists of three fundamental operators reproduction crossover and mutation These operators work with a number of artificial creatures called a generation By exchanging information from each individual in a population GA preserves a better individual and yields higher fitness generation such that the performance can be improved Given an optimization problem GA encodes the parameter designed into a finite bit string and then runs iteratively using the three operators in a random way but based on the fitness function evolution It performs the basic tasks of copying strings exchanging portions of string and changing some bits of strings Finally it finds and decodes the solution to the problem from the last pool of mature strings

IV DESIGN OF PID AND FOPID CONTROLLER USING GA FOR SPEED CONTROL OF DC MOTOR

According to control objective for a PID controller three parameters viz KP KI KD have to be approximated and for a FOPID controller five parameters KP KI KD λ and δ need to be approximated

The parameters that are used for the designing the controllers are listed in Table I

TABLE I PARAMETERS FOR DESIGN

1 Population size n 100

2 Crossover probability 065

3 Crossover function Arithmetic Crossover

4 Selection Function Stochastic uniform

5 Creation Function Uniform

6 Generation number 50

TABLE I

Fitness Function in this paper the fitness function minimized is the Integral of Absolute Error (IAE)

0

J e(t)dt (7)

Since it is not practical to integrate up to infinity we choose a value of T such that e (t) for t gt T is negligible

V SIMULATION AND EXPERIMENTAL RESULTS In this section a comparison of integer order PID and

FOPID is done using SIMULINK and by practical experimentation

A MS 15 DC motor module The LJ Technical Systems MS 15 DC motor module is

used for the experiment (Fig1) The angular velocity ω(t) of the motor is controlled by applied voltage Va A constant voltage applied to the DC motor produces a constant torque The motor runs at a constant speed This applied voltage is the plant input The motor speed is measured using a tachogenerator mounted on the same shaft as the motor A tachogenerator generates a voltage proportional to the motor speed The voltage from the tachogenerator is used as the plant output in this experiment The output voltage of tachogenerator is feedback to the plant The transfer function of the system has the following form

dcm0833G (s)

0267s 1 (8)

Figure 1 MS 15 DC motor module

B Real time control amp Hardware-in-loop experiment platform Schematic of the real-time hardware-in-loop

configuration is shown in Fig 2 A Data Acquisition and Control Board (DACB) is used to acquire the data from the plant PCI 1710 HG is used to provide feedback to the digital controller (computer) from the plant in the appropriate format

423423423

SIMULINK MATLAB Real-time windows target

ADVANTECH PCI 1710 HG

DAQ Card

DA Converter AD Converter

Plant HW loop

Figure 2 Schematic of the real-time hardware-in-loop experiment

platform

The Advantech PCI 1710 HG board has 8 analog inputs 2 analog outputs and 96 digital IO lines This is interfaced with computer and is realized in SIMULINK using the Real-time windows target tool of MATLAB The design consists of SIMULINK blocks Analog Input Analog Output and Digital InputOutput block Signals from the plant are given to the feedback using Analog input block and the control signal generated by the controller is inputted to the plant using Analog output block

Next the integer order PI PID and FOPID controllers are designed using GA for feedback control of MS 15 DC motor (i) For integer order PI with the following limits

P IK 010 K 030 By running GA for 50 generations the following results

were obtained P IK 00875 K 299820 Best value

IAE 1302 (Fig 3)

0 10 20 30 40 500

10

20

30

40

50

60

Generation

Fitn

ess

valu

e

Best 1302 Mean 13101

Best f itnessMean fitness

Figure 3 Fitness value Vs Generations for PI tuning

(ii) For integer order PID with the following limits P I DK 010 K 030 K 001

By running GA for 50 generations the following results were obtained

P I DK 79447 K 298703K 00498 Best value IAE 77082(Fig 4) (iii) For FO PID with the following limits

P I DK 010 K 030 K 001 01 and = 01 By running GA for 50 generations the following results

were obtained P I DK 57017 K 203722K 00487

09898 and =08324

And the best value of IAE is 10552 (Fig 5)

C Comparison in Simulation Fig1 shows the SIMULINK model for the feedback

control of DC motor where fractional-order controller is realized via lsquoNIPIDrsquo block proposed by D Valerio [9]

0 10 20 30 40 500

20

40

60

80

100

120

140

Generation

Fitn

ess

valu

e

Best 77082 Mean 78184

Best f itnessMean fitness

Figure 4 Fitness value Vs Generations for PID tuning

0 10 20 30 40 500

10

20

30

40

50

60

70

Generation

Fitn

ess

valu

e

Best 10552 Mean 12389

Best f itnessMean fitness

Figure 5 Fitness value Vs Generations for FOPID tuning

Figure 6 SIMULINK model feed back control of DC motor

Fig 6 shows the closed loop response of the DC motor for the square wave input with Integer order PI PID and FOPID controller The simulation results obtained from Fig 6 are tabulated in Table II

TABLE II SIMULATION RESULTS OF SIMULINK BASED SPEED CONTROL OF DC MOTOR

Parameters Integer order PI controlled process

Integer order PID controlled process

FOPID controlled process

Maximum overshot

50 9 4

Rise time 021s 0175s 02065s Settling time approx 15 s 053 s approx 03s Steady state error

0002 0003 0001

The closed loop response of the system for the different

types of control schemes is shown in Fig 7

D Experimental Verification in real time In any control system the final step in the design process

is the real time control experiment As can be observed from Figs 8-10 the performances of the controllers are in confirmation with the simulation results based on the identified model

424424424

0 2 4 6 8 10

0

05

1

15

2

25

3

35

TIME

OP

Refrence signalUncontrolled systemPI controllerPID controllerFOPID controller

Figure 7 Close loop response of system

0 5 10 150

05

1

15

2

25

3

35

4

TIME

Volta

ge

Real time closed loop response of PI controller

data1

data2

Refrence signal

OP signal

Figure 8 Real time closed loop response of PI controller

VI CONCLUSION In this paper a scheme for feedback speed control of DC

motor using a fractional-order PID controller using GA is presented The parameters of integer order PIPID and FOPID are optimally searched by genetic algorithm The results of both controllers are compared in simulation and verified in real time It is observed that the FOPID controller reduces the maximum over shoot settling time and steady state error as compared to the integer order PID controller and provides more robustness in parameter variation than conventional PID due to two extra degree of freedom

REFERENCES [1] I Griffin ldquoOn-line PID Controller Tuning using Genetic Algorithms

rdquo Masterrsquos Thesis DCU August 22 2003 [2] I Podlubny ldquoFractional-order systems and PIλDδ controllersrdquo IEEE

Trans On Automatic Control vol44 no 1 pp 208-213 1999 [3] I Petras L Dorcak and I Kostial ldquocontrol quality enhancement by

fractional order controllers rsquorsquoActa Montanistica Slovaca KosiceVol 3 No 2PP143-1481998

[4] I Petras ldquoThe fractional order controllers methods for their synthesis and application rsquorsquoJ Electrical Engineering Vol 50 No910 pp284-2881999

[5] Y Q Chen and K L Moore ldquoDiscretization Schemes for fractional-order differentiators and integratorsrdquo IEEE Trans On Circuits and Systems vol49 no 3 pp363-367 March 2002

[6] J Y Cao and BG Cao ldquoOptimization of fractional order controllers based on genetic algorithm rdquo Proceedings of the fourth International conference onmachine learning and cyber netics Guanghou18-21 august 2005

[7] Y Q Chen ldquoOustaloup - Recursive Approximation for Fractional Order Differentiators rdquoMath Works Inc August 2003

[8] httpwwwmathworkscommatlabcentralfileexchange3802 [9] Oustaloup A Levron F-Mathieu ldquoFrequency-Band Complex

Noninteger Differentiator Characterization and Synthesisrdquo IEEE Trans on Circuits and Systems I Fundamental Theory and Applications I 47 No 1(2000) 25ndash39

[10] B M Vinagre I Podlubny A Hernrsquoandez A Feliu ldquoSome Approximations of Fractional Order Operators used in Control Theory and Applicationsrdquo Fractional Calculus and Applied Analysis 3 No 3 (2000) 231ndash248

[11] DValerio Toolbox ninteger for Matlab v 23 (September2005) httpwebistutlptduartevalerionintegernintegerhtm [12] D Xue C Zhao Y Q ChenldquoFractional Order PID Control of a DC-

Motor with Elastic Shaft A Case Studyrdquo Procof the 2006 American Control Conference Minneapolis MinnesotaUSA June 14-16 2006 pp 3182ndash3187

[13] I Podlubny ldquoFractional Differential Equationsrdquo Academic Press San Diego 1999

[14] Matlab Genetic Algorithm Tool box

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real Time Closed loop response of PID controller

data1

data2OP signal

Refrence signal

Figure 9 Real time closed loop response of PID controller

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real time closed loop response of FOPID controller

data1

data2

Refrence signal

OP signal

Figure 10 Real time closed loop response of FOPID controller

425425425

Fractional-Order PID Controller Design for Speed Control of DC Motor

Abstractmdash This paper deals speed control of a DC motor using fractional-order control Fractional calculus provides novel and higher performance extensions for fractional order proportional integral and derivative (FOPID) controller In this paper the parameters of the FOPID controller are optimally learned by using Genetic Algorithm (GA) and the optimization performance target is chosen as the integral of the absolute error (IAE) Simulation results show that the FOPID controller performs better than the integer order PID controller

Keywords-Fractional calculus fractional-order controller genetic algorithm DC motor speed control

I INTRODUCTION DC motor is a power actuator which converts electrical

energy into rotational mechanical energy DC motors are widely used in industry and commercial application such as tape motor disk drive robotic manipulators and in numerous control applications Therefore its control is very important For the control of dc motor traditional controllers such as PI and PID controller have been used widely in literature [1] In this paper DC motor is controlled by a non-conventional control technique known as a fractional-order PID (FOPID) control This technique was developed during the last few decades and it has various practical applications viz flexible spacecraft attitude control Car suspension control temperature control motor control etc This idea of the fractional calculus application to control theory has been described in many other works ([2 4 11]) and its advantages are proved as well

This paper has following sections Sections 2 explains briefly about Fraction calculus and fractional-order controller Section 3 discusses the optimization technique using genetic algorithm Sections 4 and 5 show parameter optimization design process and simulation results Section 6 concludes the paper

II FRACTIONAL- ORDER CONTROLLER Fractional-order control systems are described by

fractional-order differential equations The FOPID controller is the expansion of the conventional PID controller based on fractional calculus FOPID has five parameters which are

responsible for adding more flexibility and robustness to the system

A Fractional calculus and Fractional-order controller Fractional calculus is a generalization of integration and

differentiation to non-integer (fractional) order fundamental

operator rD a t where a and t are the limits and (r ɛ R) is the

order of the operation The two definitions used for the

fractional differintegral rD a t are the Gruumlnwald-Letnikov

(GL) definition and the Riemann-Liouville (RL) definition [8 12] Further it has been mentioned in literature that for a wide class of functions these two definitions are equivalent [12] GL definition

t ah

jr r r ra t hh 0 h 0j 0

rD f(t) limh ( 1) f(t jh) limh f(t)

j

(1)

where [middot] means the integer part and rf(t)th is the

generalized finite difference of order r with step h RL definition

tnr

a t n r n 1a

1 d f( )D f(t) d(n a) dt (t )

(2)

for ((n-1)ltrltn) and Г() is the Eulerrsquos gamma function The Laplace transforms of the GL and RL fractional

derivativeintegral of signal f(t) at t=0 for order r is given by r r r

a tL D f(t) s L f(t) s F(s) (3) Using Laplace transformation

s F(s) I f(t) and ds F(s) f(t)dt

(4)

Then the fractional PID controller is written as P I DC(s) K K s K s 0 (5)

Selection of λ δ gives the classical controllers viz PD controller (λ =0) PI controller (δ =0) and PID controller (λ δ=1)

All the above-mentioned controllers are special case of the fractional PIλDδ controller Some advantages of PIλDδ controller include better control of dynamical systems (they are described by fractional-order mathematical models) and that they are less sensitive to parameter variations of a controlled plant

B Continuous-time Approximation of Fractional Differentiator Integrator Several approximation methods and techniques for

obtaining continuous-time and discrete-time fractional-order models in the form of IIR and FIR filters are developed in [9]

Vishal Mehra Smriti Srivastava and Pragya Varshney Department Of Instrumentation and Control Engg

Netaji Subhas Institute Of Technology New Delhi India 110078 vishalmehransitonlinein ssmritiyahoocom pragyavarediffmailcom

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

Third International Conference on Emerging Trends in Engineering and Technology

978-0-7695-4246-110 $2600 copy 2010 IEEE

DOI 101109ICETET2010123

422

In this paper the Oustalouprsquos approximation [8] algorithm is used for simulation purposes This method is based on the approximation of a function

rH(s) s r R r 11

For the frequency range (ωl ωh) nk

k n k

sH(s) Ks

Using the following set of synthesis formulae the approximation for poles zeros and gain are obtained as follows k n 05 05r 2n 1

k l h lk n 05 05r 2n 1

k l h l

r n 2K k kh l k n

(6)

where ωh and ωl are the high and low transitional frequencies

This algorithm is implemented in MATLAB as a Function script lsquoora_focrsquo given in [7]

III GENETIC ALGORITHMS Genetic Algorithm (GA) is a stochastic global search

method that mimics the process of natural evolution The algorithm starts with no knowledge of the correct solution and depends entirely on responses from its environment and evolution operators to arrive at the best solution By starting at several independent points and searching in parallel the algorithm avoids local minima and converges to sub optimal solutions In this way GA has been shown to be capable of locating high performance areas in complex domains without experiencing the difficulties associated with high dimensionality

GA consists of three fundamental operators reproduction crossover and mutation These operators work with a number of artificial creatures called a generation By exchanging information from each individual in a population GA preserves a better individual and yields higher fitness generation such that the performance can be improved Given an optimization problem GA encodes the parameter designed into a finite bit string and then runs iteratively using the three operators in a random way but based on the fitness function evolution It performs the basic tasks of copying strings exchanging portions of string and changing some bits of strings Finally it finds and decodes the solution to the problem from the last pool of mature strings

IV DESIGN OF PID AND FOPID CONTROLLER USING GA FOR SPEED CONTROL OF DC MOTOR

According to control objective for a PID controller three parameters viz KP KI KD have to be approximated and for a FOPID controller five parameters KP KI KD λ and δ need to be approximated

The parameters that are used for the designing the controllers are listed in Table I

TABLE I PARAMETERS FOR DESIGN

1 Population size n 100

2 Crossover probability 065

3 Crossover function Arithmetic Crossover

4 Selection Function Stochastic uniform

5 Creation Function Uniform

6 Generation number 50

TABLE I

Fitness Function in this paper the fitness function minimized is the Integral of Absolute Error (IAE)

0

J e(t)dt (7)

Since it is not practical to integrate up to infinity we choose a value of T such that e (t) for t gt T is negligible

V SIMULATION AND EXPERIMENTAL RESULTS In this section a comparison of integer order PID and

FOPID is done using SIMULINK and by practical experimentation

A MS 15 DC motor module The LJ Technical Systems MS 15 DC motor module is

used for the experiment (Fig1) The angular velocity ω(t) of the motor is controlled by applied voltage Va A constant voltage applied to the DC motor produces a constant torque The motor runs at a constant speed This applied voltage is the plant input The motor speed is measured using a tachogenerator mounted on the same shaft as the motor A tachogenerator generates a voltage proportional to the motor speed The voltage from the tachogenerator is used as the plant output in this experiment The output voltage of tachogenerator is feedback to the plant The transfer function of the system has the following form

dcm0833G (s)

0267s 1 (8)

Figure 1 MS 15 DC motor module

B Real time control amp Hardware-in-loop experiment platform Schematic of the real-time hardware-in-loop

configuration is shown in Fig 2 A Data Acquisition and Control Board (DACB) is used to acquire the data from the plant PCI 1710 HG is used to provide feedback to the digital controller (computer) from the plant in the appropriate format

423423423

SIMULINK MATLAB Real-time windows target

ADVANTECH PCI 1710 HG

DAQ Card

DA Converter AD Converter

Plant HW loop

Figure 2 Schematic of the real-time hardware-in-loop experiment

platform

The Advantech PCI 1710 HG board has 8 analog inputs 2 analog outputs and 96 digital IO lines This is interfaced with computer and is realized in SIMULINK using the Real-time windows target tool of MATLAB The design consists of SIMULINK blocks Analog Input Analog Output and Digital InputOutput block Signals from the plant are given to the feedback using Analog input block and the control signal generated by the controller is inputted to the plant using Analog output block

Next the integer order PI PID and FOPID controllers are designed using GA for feedback control of MS 15 DC motor (i) For integer order PI with the following limits

P IK 010 K 030 By running GA for 50 generations the following results

were obtained P IK 00875 K 299820 Best value

IAE 1302 (Fig 3)

0 10 20 30 40 500

10

20

30

40

50

60

Generation

Fitn

ess

valu

e

Best 1302 Mean 13101

Best f itnessMean fitness

Figure 3 Fitness value Vs Generations for PI tuning

(ii) For integer order PID with the following limits P I DK 010 K 030 K 001

By running GA for 50 generations the following results were obtained

P I DK 79447 K 298703K 00498 Best value IAE 77082(Fig 4) (iii) For FO PID with the following limits

P I DK 010 K 030 K 001 01 and = 01 By running GA for 50 generations the following results

were obtained P I DK 57017 K 203722K 00487

09898 and =08324

And the best value of IAE is 10552 (Fig 5)

C Comparison in Simulation Fig1 shows the SIMULINK model for the feedback

control of DC motor where fractional-order controller is realized via lsquoNIPIDrsquo block proposed by D Valerio [9]

0 10 20 30 40 500

20

40

60

80

100

120

140

Generation

Fitn

ess

valu

e

Best 77082 Mean 78184

Best f itnessMean fitness

Figure 4 Fitness value Vs Generations for PID tuning

0 10 20 30 40 500

10

20

30

40

50

60

70

Generation

Fitn

ess

valu

e

Best 10552 Mean 12389

Best f itnessMean fitness

Figure 5 Fitness value Vs Generations for FOPID tuning

Figure 6 SIMULINK model feed back control of DC motor

Fig 6 shows the closed loop response of the DC motor for the square wave input with Integer order PI PID and FOPID controller The simulation results obtained from Fig 6 are tabulated in Table II

TABLE II SIMULATION RESULTS OF SIMULINK BASED SPEED CONTROL OF DC MOTOR

Parameters Integer order PI controlled process

Integer order PID controlled process

FOPID controlled process

Maximum overshot

50 9 4

Rise time 021s 0175s 02065s Settling time approx 15 s 053 s approx 03s Steady state error

0002 0003 0001

The closed loop response of the system for the different

types of control schemes is shown in Fig 7

D Experimental Verification in real time In any control system the final step in the design process

is the real time control experiment As can be observed from Figs 8-10 the performances of the controllers are in confirmation with the simulation results based on the identified model

424424424

0 2 4 6 8 10

0

05

1

15

2

25

3

35

TIME

OP

Refrence signalUncontrolled systemPI controllerPID controllerFOPID controller

Figure 7 Close loop response of system

0 5 10 150

05

1

15

2

25

3

35

4

TIME

Volta

ge

Real time closed loop response of PI controller

data1

data2

Refrence signal

OP signal

Figure 8 Real time closed loop response of PI controller

VI CONCLUSION In this paper a scheme for feedback speed control of DC

motor using a fractional-order PID controller using GA is presented The parameters of integer order PIPID and FOPID are optimally searched by genetic algorithm The results of both controllers are compared in simulation and verified in real time It is observed that the FOPID controller reduces the maximum over shoot settling time and steady state error as compared to the integer order PID controller and provides more robustness in parameter variation than conventional PID due to two extra degree of freedom

REFERENCES [1] I Griffin ldquoOn-line PID Controller Tuning using Genetic Algorithms

rdquo Masterrsquos Thesis DCU August 22 2003 [2] I Podlubny ldquoFractional-order systems and PIλDδ controllersrdquo IEEE

Trans On Automatic Control vol44 no 1 pp 208-213 1999 [3] I Petras L Dorcak and I Kostial ldquocontrol quality enhancement by

fractional order controllers rsquorsquoActa Montanistica Slovaca KosiceVol 3 No 2PP143-1481998

[4] I Petras ldquoThe fractional order controllers methods for their synthesis and application rsquorsquoJ Electrical Engineering Vol 50 No910 pp284-2881999

[5] Y Q Chen and K L Moore ldquoDiscretization Schemes for fractional-order differentiators and integratorsrdquo IEEE Trans On Circuits and Systems vol49 no 3 pp363-367 March 2002

[6] J Y Cao and BG Cao ldquoOptimization of fractional order controllers based on genetic algorithm rdquo Proceedings of the fourth International conference onmachine learning and cyber netics Guanghou18-21 august 2005

[7] Y Q Chen ldquoOustaloup - Recursive Approximation for Fractional Order Differentiators rdquoMath Works Inc August 2003

[8] httpwwwmathworkscommatlabcentralfileexchange3802 [9] Oustaloup A Levron F-Mathieu ldquoFrequency-Band Complex

Noninteger Differentiator Characterization and Synthesisrdquo IEEE Trans on Circuits and Systems I Fundamental Theory and Applications I 47 No 1(2000) 25ndash39

[10] B M Vinagre I Podlubny A Hernrsquoandez A Feliu ldquoSome Approximations of Fractional Order Operators used in Control Theory and Applicationsrdquo Fractional Calculus and Applied Analysis 3 No 3 (2000) 231ndash248

[11] DValerio Toolbox ninteger for Matlab v 23 (September2005) httpwebistutlptduartevalerionintegernintegerhtm [12] D Xue C Zhao Y Q ChenldquoFractional Order PID Control of a DC-

Motor with Elastic Shaft A Case Studyrdquo Procof the 2006 American Control Conference Minneapolis MinnesotaUSA June 14-16 2006 pp 3182ndash3187

[13] I Podlubny ldquoFractional Differential Equationsrdquo Academic Press San Diego 1999

[14] Matlab Genetic Algorithm Tool box

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real Time Closed loop response of PID controller

data1

data2OP signal

Refrence signal

Figure 9 Real time closed loop response of PID controller

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real time closed loop response of FOPID controller

data1

data2

Refrence signal

OP signal

Figure 10 Real time closed loop response of FOPID controller

425425425

In this paper the Oustalouprsquos approximation [8] algorithm is used for simulation purposes This method is based on the approximation of a function

rH(s) s r R r 11

For the frequency range (ωl ωh) nk

k n k

sH(s) Ks

Using the following set of synthesis formulae the approximation for poles zeros and gain are obtained as follows k n 05 05r 2n 1

k l h lk n 05 05r 2n 1

k l h l

r n 2K k kh l k n

(6)

where ωh and ωl are the high and low transitional frequencies

This algorithm is implemented in MATLAB as a Function script lsquoora_focrsquo given in [7]

III GENETIC ALGORITHMS Genetic Algorithm (GA) is a stochastic global search

method that mimics the process of natural evolution The algorithm starts with no knowledge of the correct solution and depends entirely on responses from its environment and evolution operators to arrive at the best solution By starting at several independent points and searching in parallel the algorithm avoids local minima and converges to sub optimal solutions In this way GA has been shown to be capable of locating high performance areas in complex domains without experiencing the difficulties associated with high dimensionality

GA consists of three fundamental operators reproduction crossover and mutation These operators work with a number of artificial creatures called a generation By exchanging information from each individual in a population GA preserves a better individual and yields higher fitness generation such that the performance can be improved Given an optimization problem GA encodes the parameter designed into a finite bit string and then runs iteratively using the three operators in a random way but based on the fitness function evolution It performs the basic tasks of copying strings exchanging portions of string and changing some bits of strings Finally it finds and decodes the solution to the problem from the last pool of mature strings

IV DESIGN OF PID AND FOPID CONTROLLER USING GA FOR SPEED CONTROL OF DC MOTOR

According to control objective for a PID controller three parameters viz KP KI KD have to be approximated and for a FOPID controller five parameters KP KI KD λ and δ need to be approximated

The parameters that are used for the designing the controllers are listed in Table I

TABLE I PARAMETERS FOR DESIGN

1 Population size n 100

2 Crossover probability 065

3 Crossover function Arithmetic Crossover

4 Selection Function Stochastic uniform

5 Creation Function Uniform

6 Generation number 50

TABLE I

Fitness Function in this paper the fitness function minimized is the Integral of Absolute Error (IAE)

0

J e(t)dt (7)

Since it is not practical to integrate up to infinity we choose a value of T such that e (t) for t gt T is negligible

V SIMULATION AND EXPERIMENTAL RESULTS In this section a comparison of integer order PID and

FOPID is done using SIMULINK and by practical experimentation

A MS 15 DC motor module The LJ Technical Systems MS 15 DC motor module is

used for the experiment (Fig1) The angular velocity ω(t) of the motor is controlled by applied voltage Va A constant voltage applied to the DC motor produces a constant torque The motor runs at a constant speed This applied voltage is the plant input The motor speed is measured using a tachogenerator mounted on the same shaft as the motor A tachogenerator generates a voltage proportional to the motor speed The voltage from the tachogenerator is used as the plant output in this experiment The output voltage of tachogenerator is feedback to the plant The transfer function of the system has the following form

dcm0833G (s)

0267s 1 (8)

Figure 1 MS 15 DC motor module

B Real time control amp Hardware-in-loop experiment platform Schematic of the real-time hardware-in-loop

configuration is shown in Fig 2 A Data Acquisition and Control Board (DACB) is used to acquire the data from the plant PCI 1710 HG is used to provide feedback to the digital controller (computer) from the plant in the appropriate format

423423423

SIMULINK MATLAB Real-time windows target

ADVANTECH PCI 1710 HG

DAQ Card

DA Converter AD Converter

Plant HW loop

Figure 2 Schematic of the real-time hardware-in-loop experiment

platform

The Advantech PCI 1710 HG board has 8 analog inputs 2 analog outputs and 96 digital IO lines This is interfaced with computer and is realized in SIMULINK using the Real-time windows target tool of MATLAB The design consists of SIMULINK blocks Analog Input Analog Output and Digital InputOutput block Signals from the plant are given to the feedback using Analog input block and the control signal generated by the controller is inputted to the plant using Analog output block

Next the integer order PI PID and FOPID controllers are designed using GA for feedback control of MS 15 DC motor (i) For integer order PI with the following limits

P IK 010 K 030 By running GA for 50 generations the following results

were obtained P IK 00875 K 299820 Best value

IAE 1302 (Fig 3)

0 10 20 30 40 500

10

20

30

40

50

60

Generation

Fitn

ess

valu

e

Best 1302 Mean 13101

Best f itnessMean fitness

Figure 3 Fitness value Vs Generations for PI tuning

(ii) For integer order PID with the following limits P I DK 010 K 030 K 001

By running GA for 50 generations the following results were obtained

P I DK 79447 K 298703K 00498 Best value IAE 77082(Fig 4) (iii) For FO PID with the following limits

P I DK 010 K 030 K 001 01 and = 01 By running GA for 50 generations the following results

were obtained P I DK 57017 K 203722K 00487

09898 and =08324

And the best value of IAE is 10552 (Fig 5)

C Comparison in Simulation Fig1 shows the SIMULINK model for the feedback

control of DC motor where fractional-order controller is realized via lsquoNIPIDrsquo block proposed by D Valerio [9]

0 10 20 30 40 500

20

40

60

80

100

120

140

Generation

Fitn

ess

valu

e

Best 77082 Mean 78184

Best f itnessMean fitness

Figure 4 Fitness value Vs Generations for PID tuning

0 10 20 30 40 500

10

20

30

40

50

60

70

Generation

Fitn

ess

valu

e

Best 10552 Mean 12389

Best f itnessMean fitness

Figure 5 Fitness value Vs Generations for FOPID tuning

Figure 6 SIMULINK model feed back control of DC motor

Fig 6 shows the closed loop response of the DC motor for the square wave input with Integer order PI PID and FOPID controller The simulation results obtained from Fig 6 are tabulated in Table II

TABLE II SIMULATION RESULTS OF SIMULINK BASED SPEED CONTROL OF DC MOTOR

Parameters Integer order PI controlled process

Integer order PID controlled process

FOPID controlled process

Maximum overshot

50 9 4

Rise time 021s 0175s 02065s Settling time approx 15 s 053 s approx 03s Steady state error

0002 0003 0001

The closed loop response of the system for the different

types of control schemes is shown in Fig 7

D Experimental Verification in real time In any control system the final step in the design process

is the real time control experiment As can be observed from Figs 8-10 the performances of the controllers are in confirmation with the simulation results based on the identified model

424424424

0 2 4 6 8 10

0

05

1

15

2

25

3

35

TIME

OP

Refrence signalUncontrolled systemPI controllerPID controllerFOPID controller

Figure 7 Close loop response of system

0 5 10 150

05

1

15

2

25

3

35

4

TIME

Volta

ge

Real time closed loop response of PI controller

data1

data2

Refrence signal

OP signal

Figure 8 Real time closed loop response of PI controller

VI CONCLUSION In this paper a scheme for feedback speed control of DC

motor using a fractional-order PID controller using GA is presented The parameters of integer order PIPID and FOPID are optimally searched by genetic algorithm The results of both controllers are compared in simulation and verified in real time It is observed that the FOPID controller reduces the maximum over shoot settling time and steady state error as compared to the integer order PID controller and provides more robustness in parameter variation than conventional PID due to two extra degree of freedom

REFERENCES [1] I Griffin ldquoOn-line PID Controller Tuning using Genetic Algorithms

rdquo Masterrsquos Thesis DCU August 22 2003 [2] I Podlubny ldquoFractional-order systems and PIλDδ controllersrdquo IEEE

Trans On Automatic Control vol44 no 1 pp 208-213 1999 [3] I Petras L Dorcak and I Kostial ldquocontrol quality enhancement by

fractional order controllers rsquorsquoActa Montanistica Slovaca KosiceVol 3 No 2PP143-1481998

[4] I Petras ldquoThe fractional order controllers methods for their synthesis and application rsquorsquoJ Electrical Engineering Vol 50 No910 pp284-2881999

[5] Y Q Chen and K L Moore ldquoDiscretization Schemes for fractional-order differentiators and integratorsrdquo IEEE Trans On Circuits and Systems vol49 no 3 pp363-367 March 2002

[6] J Y Cao and BG Cao ldquoOptimization of fractional order controllers based on genetic algorithm rdquo Proceedings of the fourth International conference onmachine learning and cyber netics Guanghou18-21 august 2005

[7] Y Q Chen ldquoOustaloup - Recursive Approximation for Fractional Order Differentiators rdquoMath Works Inc August 2003

[8] httpwwwmathworkscommatlabcentralfileexchange3802 [9] Oustaloup A Levron F-Mathieu ldquoFrequency-Band Complex

Noninteger Differentiator Characterization and Synthesisrdquo IEEE Trans on Circuits and Systems I Fundamental Theory and Applications I 47 No 1(2000) 25ndash39

[10] B M Vinagre I Podlubny A Hernrsquoandez A Feliu ldquoSome Approximations of Fractional Order Operators used in Control Theory and Applicationsrdquo Fractional Calculus and Applied Analysis 3 No 3 (2000) 231ndash248

[11] DValerio Toolbox ninteger for Matlab v 23 (September2005) httpwebistutlptduartevalerionintegernintegerhtm [12] D Xue C Zhao Y Q ChenldquoFractional Order PID Control of a DC-

Motor with Elastic Shaft A Case Studyrdquo Procof the 2006 American Control Conference Minneapolis MinnesotaUSA June 14-16 2006 pp 3182ndash3187

[13] I Podlubny ldquoFractional Differential Equationsrdquo Academic Press San Diego 1999

[14] Matlab Genetic Algorithm Tool box

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real Time Closed loop response of PID controller

data1

data2OP signal

Refrence signal

Figure 9 Real time closed loop response of PID controller

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real time closed loop response of FOPID controller

data1

data2

Refrence signal

OP signal

Figure 10 Real time closed loop response of FOPID controller

425425425

SIMULINK MATLAB Real-time windows target

ADVANTECH PCI 1710 HG

DAQ Card

DA Converter AD Converter

Plant HW loop

Figure 2 Schematic of the real-time hardware-in-loop experiment

platform

The Advantech PCI 1710 HG board has 8 analog inputs 2 analog outputs and 96 digital IO lines This is interfaced with computer and is realized in SIMULINK using the Real-time windows target tool of MATLAB The design consists of SIMULINK blocks Analog Input Analog Output and Digital InputOutput block Signals from the plant are given to the feedback using Analog input block and the control signal generated by the controller is inputted to the plant using Analog output block

Next the integer order PI PID and FOPID controllers are designed using GA for feedback control of MS 15 DC motor (i) For integer order PI with the following limits

P IK 010 K 030 By running GA for 50 generations the following results

were obtained P IK 00875 K 299820 Best value

IAE 1302 (Fig 3)

0 10 20 30 40 500

10

20

30

40

50

60

Generation

Fitn

ess

valu

e

Best 1302 Mean 13101

Best f itnessMean fitness

Figure 3 Fitness value Vs Generations for PI tuning

(ii) For integer order PID with the following limits P I DK 010 K 030 K 001

By running GA for 50 generations the following results were obtained

P I DK 79447 K 298703K 00498 Best value IAE 77082(Fig 4) (iii) For FO PID with the following limits

P I DK 010 K 030 K 001 01 and = 01 By running GA for 50 generations the following results

were obtained P I DK 57017 K 203722K 00487

09898 and =08324

And the best value of IAE is 10552 (Fig 5)

C Comparison in Simulation Fig1 shows the SIMULINK model for the feedback

control of DC motor where fractional-order controller is realized via lsquoNIPIDrsquo block proposed by D Valerio [9]

0 10 20 30 40 500

20

40

60

80

100

120

140

Generation

Fitn

ess

valu

e

Best 77082 Mean 78184

Best f itnessMean fitness

Figure 4 Fitness value Vs Generations for PID tuning

0 10 20 30 40 500

10

20

30

40

50

60

70

Generation

Fitn

ess

valu

e

Best 10552 Mean 12389

Best f itnessMean fitness

Figure 5 Fitness value Vs Generations for FOPID tuning

Figure 6 SIMULINK model feed back control of DC motor

Fig 6 shows the closed loop response of the DC motor for the square wave input with Integer order PI PID and FOPID controller The simulation results obtained from Fig 6 are tabulated in Table II

TABLE II SIMULATION RESULTS OF SIMULINK BASED SPEED CONTROL OF DC MOTOR

Parameters Integer order PI controlled process

Integer order PID controlled process

FOPID controlled process

Maximum overshot

50 9 4

Rise time 021s 0175s 02065s Settling time approx 15 s 053 s approx 03s Steady state error

0002 0003 0001

The closed loop response of the system for the different

types of control schemes is shown in Fig 7

D Experimental Verification in real time In any control system the final step in the design process

is the real time control experiment As can be observed from Figs 8-10 the performances of the controllers are in confirmation with the simulation results based on the identified model

424424424

0 2 4 6 8 10

0

05

1

15

2

25

3

35

TIME

OP

Refrence signalUncontrolled systemPI controllerPID controllerFOPID controller

Figure 7 Close loop response of system

0 5 10 150

05

1

15

2

25

3

35

4

TIME

Volta

ge

Real time closed loop response of PI controller

data1

data2

Refrence signal

OP signal

Figure 8 Real time closed loop response of PI controller

VI CONCLUSION In this paper a scheme for feedback speed control of DC

motor using a fractional-order PID controller using GA is presented The parameters of integer order PIPID and FOPID are optimally searched by genetic algorithm The results of both controllers are compared in simulation and verified in real time It is observed that the FOPID controller reduces the maximum over shoot settling time and steady state error as compared to the integer order PID controller and provides more robustness in parameter variation than conventional PID due to two extra degree of freedom

REFERENCES [1] I Griffin ldquoOn-line PID Controller Tuning using Genetic Algorithms

rdquo Masterrsquos Thesis DCU August 22 2003 [2] I Podlubny ldquoFractional-order systems and PIλDδ controllersrdquo IEEE

Trans On Automatic Control vol44 no 1 pp 208-213 1999 [3] I Petras L Dorcak and I Kostial ldquocontrol quality enhancement by

fractional order controllers rsquorsquoActa Montanistica Slovaca KosiceVol 3 No 2PP143-1481998

[4] I Petras ldquoThe fractional order controllers methods for their synthesis and application rsquorsquoJ Electrical Engineering Vol 50 No910 pp284-2881999

[5] Y Q Chen and K L Moore ldquoDiscretization Schemes for fractional-order differentiators and integratorsrdquo IEEE Trans On Circuits and Systems vol49 no 3 pp363-367 March 2002

[6] J Y Cao and BG Cao ldquoOptimization of fractional order controllers based on genetic algorithm rdquo Proceedings of the fourth International conference onmachine learning and cyber netics Guanghou18-21 august 2005

[7] Y Q Chen ldquoOustaloup - Recursive Approximation for Fractional Order Differentiators rdquoMath Works Inc August 2003

[8] httpwwwmathworkscommatlabcentralfileexchange3802 [9] Oustaloup A Levron F-Mathieu ldquoFrequency-Band Complex

Noninteger Differentiator Characterization and Synthesisrdquo IEEE Trans on Circuits and Systems I Fundamental Theory and Applications I 47 No 1(2000) 25ndash39

[10] B M Vinagre I Podlubny A Hernrsquoandez A Feliu ldquoSome Approximations of Fractional Order Operators used in Control Theory and Applicationsrdquo Fractional Calculus and Applied Analysis 3 No 3 (2000) 231ndash248

[11] DValerio Toolbox ninteger for Matlab v 23 (September2005) httpwebistutlptduartevalerionintegernintegerhtm [12] D Xue C Zhao Y Q ChenldquoFractional Order PID Control of a DC-

Motor with Elastic Shaft A Case Studyrdquo Procof the 2006 American Control Conference Minneapolis MinnesotaUSA June 14-16 2006 pp 3182ndash3187

[13] I Podlubny ldquoFractional Differential Equationsrdquo Academic Press San Diego 1999

[14] Matlab Genetic Algorithm Tool box

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real Time Closed loop response of PID controller

data1

data2OP signal

Refrence signal

Figure 9 Real time closed loop response of PID controller

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real time closed loop response of FOPID controller

data1

data2

Refrence signal

OP signal

Figure 10 Real time closed loop response of FOPID controller

425425425

0 2 4 6 8 10

0

05

1

15

2

25

3

35

TIME

OP

Refrence signalUncontrolled systemPI controllerPID controllerFOPID controller

Figure 7 Close loop response of system

0 5 10 150

05

1

15

2

25

3

35

4

TIME

Volta

ge

Real time closed loop response of PI controller

data1

data2

Refrence signal

OP signal

Figure 8 Real time closed loop response of PI controller

VI CONCLUSION In this paper a scheme for feedback speed control of DC

motor using a fractional-order PID controller using GA is presented The parameters of integer order PIPID and FOPID are optimally searched by genetic algorithm The results of both controllers are compared in simulation and verified in real time It is observed that the FOPID controller reduces the maximum over shoot settling time and steady state error as compared to the integer order PID controller and provides more robustness in parameter variation than conventional PID due to two extra degree of freedom

REFERENCES [1] I Griffin ldquoOn-line PID Controller Tuning using Genetic Algorithms

rdquo Masterrsquos Thesis DCU August 22 2003 [2] I Podlubny ldquoFractional-order systems and PIλDδ controllersrdquo IEEE

Trans On Automatic Control vol44 no 1 pp 208-213 1999 [3] I Petras L Dorcak and I Kostial ldquocontrol quality enhancement by

fractional order controllers rsquorsquoActa Montanistica Slovaca KosiceVol 3 No 2PP143-1481998

[4] I Petras ldquoThe fractional order controllers methods for their synthesis and application rsquorsquoJ Electrical Engineering Vol 50 No910 pp284-2881999

[5] Y Q Chen and K L Moore ldquoDiscretization Schemes for fractional-order differentiators and integratorsrdquo IEEE Trans On Circuits and Systems vol49 no 3 pp363-367 March 2002

[6] J Y Cao and BG Cao ldquoOptimization of fractional order controllers based on genetic algorithm rdquo Proceedings of the fourth International conference onmachine learning and cyber netics Guanghou18-21 august 2005

[7] Y Q Chen ldquoOustaloup - Recursive Approximation for Fractional Order Differentiators rdquoMath Works Inc August 2003

[8] httpwwwmathworkscommatlabcentralfileexchange3802 [9] Oustaloup A Levron F-Mathieu ldquoFrequency-Band Complex

Noninteger Differentiator Characterization and Synthesisrdquo IEEE Trans on Circuits and Systems I Fundamental Theory and Applications I 47 No 1(2000) 25ndash39

[10] B M Vinagre I Podlubny A Hernrsquoandez A Feliu ldquoSome Approximations of Fractional Order Operators used in Control Theory and Applicationsrdquo Fractional Calculus and Applied Analysis 3 No 3 (2000) 231ndash248

[11] DValerio Toolbox ninteger for Matlab v 23 (September2005) httpwebistutlptduartevalerionintegernintegerhtm [12] D Xue C Zhao Y Q ChenldquoFractional Order PID Control of a DC-

Motor with Elastic Shaft A Case Studyrdquo Procof the 2006 American Control Conference Minneapolis MinnesotaUSA June 14-16 2006 pp 3182ndash3187

[13] I Podlubny ldquoFractional Differential Equationsrdquo Academic Press San Diego 1999

[14] Matlab Genetic Algorithm Tool box

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real Time Closed loop response of PID controller

data1

data2OP signal

Refrence signal

Figure 9 Real time closed loop response of PID controller

0 5 10 150

05

1

15

2

25

3

35

TIME

Volta

ge

Real time closed loop response of FOPID controller

data1

data2

Refrence signal

OP signal

Figure 10 Real time closed loop response of FOPID controller

425425425