application of combinatorial interaction design for dc servomotor

Research ArticleApplication of Combinatorial Interaction Design forDC Servomotor PID Controller Tuning

Mouayad A Sahib Bestoun S Ahmed and Moayad Y Potrus

Software Engineering Department Salahaddin University-Hawler (SUH) Kirkuk Road Erbil 44002 Kurdistan Iraq

Correspondence should be addressed to Bestoun S Ahmed bestoon82gmailcom

Received 22 January 2014 Revised 23 March 2014 Accepted 25 March 2014 Published 16 April 2014

Academic Editor Petko Petkov

Copyright copy 2014 Mouayad A Sahib et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Combinatorial optimization has been used in different research areas It has been employed successfully in software testing fieldsto construct minimum set of combinations (ie in terms of size) which in turn represents the minimum number of test cases Itwas also found to be a successful approach that can be applied to solve other similar problems in different fields of research In linewith this approach this paper presents a new application of the combinational optimization in the design of PID controller for DCservomotor The design of PID controller involves the determination of three parameters To find optimal initial PID parametersdifferent tuning methods have been proposed and designed in the literature The combinatorial design is concerned with thearrangement of finite set of elements into combinatorial set that satisfies some given constraints Consequently the proposedmethod takes the interaction of the input parameters as a constraint for constructing this combinatorial set The generated setsare then used in the proposed tuning method The method proved its effectiveness within a set of experiments in a simulatedenvironment

1 Introduction

Direct current (DC) motors have been used intensivelyin various industrial applications The significance of DCmotors is related to their efficient characteristics such as pre-ciseness fast adaptation smooth operation and high torquecapabilities From the control point of view DC motorsexhibit linear speed-torque characteristics therefore profi-cient control aspects can be achieved [1] During the recentdecades various controller structures have been proposedfor controlling the DC motor The most common controllerused is the proportional plus integral plus derivative (PID)due to its simple structure and robust performance [2] Thedesign of the PID controller involves the determination ofthree parameters which are the proportional integral andderivative gains Over the years various tuning methodshave been proposed to determine the PID gains The firstclassical tuning rule method was proposed by Ziegler andNichols [3] and also it is proposed by Cohen and Coon [4]The advantage of these classical tuning methods is that themodel of the system is not required to be known Howeverthe controller designed with these experimental methods can

have an acceptable but not optimum system responseThere-fore recently many artificial intelligence (AI) techniqueshave been employed to determine the optimal parametersand hence improve the controller performances Such AItechniques include differential evolution (DE) algorithm [5]multiobjective optimization [6] evolutionary algorithm [7]simulated annealing (SA) [8] artificial bee colony (ABC) [9]fuzzy systems [10] genetic algorithm (GA) [11] and manyoptimizing liaisons (MOL) [12]

In this paper combinatorial design is used as a newapproach to determine the optimal PID parameters In com-binatorial design the finite set of elements is arranged intopatterns (subsets words and arrays) according to specifiedrules [13] As a part of this theory the arrangement couldbe represented in the covering array (CA) form whichrepresents all the interaction combinations in one arrayThuscombination number can be reduced dramatically and at thesame time all of them are covered by the optimized set whichis represented in the CA

CAs have been used widely in different applications inthe literatureThey have been used mainly in software testingapplications (eg [14ndash16]) They have also been used in other

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2014 Article ID 576868 7 pageshttpdxdoiorg1011552014576868

2 Journal of Control Science and Engineering

applications such as hardware testing [17] gene expressionregulation [18] advance material testing [19] performanceevaluation of communication systems [20] and optimizationof dynamic voltage scaling (DVS) in high performanceprocessors [21] However CAs have not been used yet for PIDcontroller design thereby it is proposed here to determinethe optimal PID parameters

The rest of this paper is organized as follows Section 2gives an overview of the combinatorial design and its nota-tions Section 3 describes the PID controller design Section 4gives the details of the combinatorial set construction and theprocedure of its application on the system Section 5 presentsand discusses the evaluation results of the proposed approachfollowed by concluding remarks in Section 6

2 Combinatorial Optimization Background

Combinatorial design is used to effectively search for the bestsolution among a finite set of feasible solutions based on theinteractions It is essential in this method to cover all of thecombinations at least once CA has been introduced as amathematical object to represent all those combinations ACA120582(119873 119905 119896 V) represents an 119873 times 119896 array with V values suchthat every119873times 119905 subarray contains all ordered subsets from Vvalues of size 119905 at least 120582 times [22] where 119896 is the numberof components (parameters) When the target is optimalcombination-set it is essential that all 119905-combinations occurat least once In this case the value of 120582 = 1 and the notationbecomes CA (119873 119905 119896 V) [23] As the target is the optimal setthe size of119873 which is the size of the combination-set has tobe as minimum as possible119879-way testing strategies have been used widely to con-

struct CA and its variations such as mixed covering array(MCA) or variable strength covering array (VSCA) [24] forthe purpose of software interaction testingThere are differentmechanisms in the literature used with 119905-way strategies forgenerating the combinations These methods are lying underthe field of search based software engineering (SBSE) [25]

From the literature there are two methods for generatingthe combinations in general (1) algebraically which is basedon the generation of the combinations mathematically (2)computationally which is based on the generation of thecombinations using computational and iterative methods Inpractice the second method is mostly used because of theflexibility since the former method is more restrictive for thecombinations

There are different tools and strategies in the literaturefor generating the combinations of different inputs Eachtool uses a special algorithm for generation Among thoseimplemented algorithms the use of AI theories with therandom generation proves the generation of optimal combi-nation sizes This approach has been used for combinatorialoptimization in different researches (eg [14 26])

3 PID Controller Design

The objective of designing a controller for a DC servomotoris to make the system respond efficiently to the new desired

120579r(s) 120579a(s)

Combinatorialset constructor

PIDcontroller

DCservomotor+

+

minus

Figure 1 DC servomotor with PID controller tuned by combinato-rial set constructor

angular position This can be achieved by comparing theoutput angular position with the desired (reference) posi-tion to calculate an error (actuating) signal The controllermanipulates the error signal and generates an appropriateDC voltage that will cause the motor to respond to the newposition Proportional-integral-derivative (PID) controllerhas been widely used for such application because of itssimple structure and robust performance Figure 1 shows ablock diagram representation of a DC servomotor with PIDcontroller

The design of PID controller involves the computationof three parameters proportional (119870119901) integral (119870119894) andderivative (119870119889) constantsThe general transfer function of thePID controller is given by

119862 (119904) = 119870119901 +119870119894

119904+ 119904119870119889 (1)

The conventionalmethod of determining the PID parametersis a heuristic method called the Ziegler-Nichols method[27] This method is based on calculating the system criticalgain at which the output of the control loop oscillates alongwith the corresponding period of osculation The criticalgain and period of oscillation are used to set the P Iand D gains Later methods based on several optimizationtechniques have been introduced such as genetic algorithm[28] neural network [29] fuzzy based approach [30] andparticle swarm optimization techniques [31] Most systemsarising in practice have time varying parameters which willaffect the performance of the designed PID controller Insuch cases a supervisory system can be used such thatwhen the performance of the controller drops below a pre-scribed level of acceptable performance index the proposedoptimization procedure must be reengaged Therefore thetuning process has to be sufficiently satisfactory in terms ofconvergence speed and this can be achieved when reducingthe optimization searching space This work proposes a newmethod for determining the PID controller parameters usingcovering array arrangement based on combinatorial designapproach As shown in the block diagram of Figure 1 theldquocombinatorial set constructorrdquo feeds the PID controller withsets of parameters instead of the conventional techniquesThis will reduce the amount of the search space which inturn reduces the required time for the system to reach theoptimal value The following section illustrates how this

Journal of Control Science and Engineering 3

combinatorial set is generated and how it has been appliedto the model

4 Applying the Generated Combinatorial Set

Figure 3 summarizes the process of constructing the com-binatorial set and its application on the system For thepurpose of constructing the combinatorial set we have usedour previously developed strategy named PSTG [14 16]The strategy uses the particle swarm optimization (PSO) forconstructing and optimizing the combinatorial set

First of all the strategy takes the factors involved in PIDtuning process and then generates all the combinations of theinput parameters (ie exhaustive set) During this process anindividual random set is generated initially and involved intoan iterative path The combinations are optimized during theiterative path of selection using the PSO strategy Each row inthe random generated set is compared with the complete setof combinations iteratively to know how many combinationsit covers Then using the PSO mechanism for optimiza-tion the strategy updates the random set of combinationsaccording to an update rule The strategy continues to iterateuntil it finds one combination in the random set that couldcover more than one in the original set which optimizes theoriginal set to a smaller set of combinations The strategycollects all the optimized combinations in one set and it stopsuntil all the combinations are covered in the exhaustive setof combinationsThen it applies the generated set to the PIDcontroller (see Figure 1) Hence this set will be the source ofthe PID parameters instead of using other sources which arecoming by conventional techniques As a result the searchspacewill beminimized and the systemwill reach the optimalvalue in a satisfactory time

5 Simulation Results and Discussion

In this section the performance of the proposed PID tuningmethod is evaluated in MATLAB and applied to control aDC servomotor The DC motor system can be considered asa SISO system in which the input is the DC source voltage119864119886(119904) and the output is the shaft angular position 120579119886(119904)The dynamical behavior of the DC motor as a relationshipbetween the applied DC voltage source and the angularposition of the shaft can be described by the following transferfunction [32]

119866 (119904) =120579119886 (119904)

119864119886 (119904)

=119870119905

1198691198981198711198861199043 + (119877119886119869119898 + 119863119898119871119886) 119904

2 + (119863119898119877119886 + 119870119905119870119887) 119904

(2)

The parameters in (2) are defined in Table 1 [32 33]

Table 1 DC servomotor parameters

Parameter Value UnitTorque constant (119870119905) 001 NlowastmampEquivalent inertia (119869119898) 001 Kglowastm2

Armature inductance (119871119886) 05 HArmature resistance (119877119886) 1 OhmViscose damping (119863119898) 01 NlowastmlowastsecradBack emf constant (119870119887) 001 Voltlowastsecrad

With the parameters listed in Table 1 the DC servomotortransfer function will be

119866 (119904) =120579119886 (119904)

119864119886 (119904)

=001

119904 (00051199042 + 006 + 01001)

(3)

The unit step response of the DC motor without PIDcontroller is shown in Figure 3

It can be observed from Figure 3 that the DC motorsystem is overdampedwith a settling time equal to 3733 sec atwhich the response has settled to 98of the steady state valueIn the design of PID controller with combinatorial approach119870119901 values are selected in the range of 452 to 55 119870119894 in therange of 001 to 05 and 119870119889 in the range of 135 to 38 withstep sizes equal to 02 001 and 05 respectively Accordinglythe table of all combinations is produced to have a total of503= 125000 possible combinations It is observed that

within this range tuning is done effectively as can be shownlater Each PID parameter combination is evaluated using afitness function given by

119869 = 1205721

1003816100381610038161003816119905119903 minus 119905lowast

119903

1003816100381610038161003816

119905lowast119903

+ 1205722

1003816100381610038161003816119905119904 minus 119905lowast

119904

1003816100381610038161003816

119905lowast119904

+ 1205723

10038161003816100381610038161003816119905119901 minus 119905lowast

119901

10038161003816100381610038161003816

119905lowast119901

+ 1205724

10038161003816100381610038161003816119872119901 minus119872

lowast

119901

10038161003816100381610038161003816

119872lowast119901

+ 1205725119864119904119904

(4)

where 119905119903 119905119904 119905119901 and119872119901 are the rise time settling time peaktime and maximum overshoot variables which are used tospecify the systemrsquos response respectivelyThe same variablesassignedwith asterisk (lowast) are used in (4) to denote the desiredresponse specifications 119864119904119904 is the steady state error (1205721 minus 1205725)are weighting factors used to adjust the importance of theresponse specification variables and it is up to the designer tochoose the values of these weights In the simulation 119905lowast

119903 119905lowast119904

119905lowast

119901 and119872lowast

119901are selected to be 0475 0879 064 and 00455

respectively The weighting factors (1205721 minus 1205725) are all chosen tobe 1 and the unit step response is calculated between 0 and 8sec with a sampling time of 001 sec Thus (4) becomes

119869 =

1003816100381610038161003816119905119903 minus 04751003816100381610038161003816

0475+

1003816100381610038161003816119905119904 minus 08791003816100381610038161003816

0879

+

10038161003816100381610038161003816119905119901 minus 064

10038161003816100381610038161003816

064+

10038161003816100381610038161003816119872119901 minus 00455

10038161003816100381610038161003816

00455+ 119864119904119904

(5)


Start

Input the PIDfactors

Generate thecombinations

Allcombinations

Yes

Yes

No

No

Yes

No

Optimize thecombinations

using PSO

Allcombinations

covered

The optimizedset

Apply to thePID to find best

combination

Optimizedparameters

End

Figure 2 Process of combinatorial set construction

0 10 20 30 40 50 60 70 800

02

04

06

08

1Step response

Time (s)

Am

plitu

de

Figure 3 Step response of the DC motor without PID controller

For this objective function the value of 119869 has a uniqueminimum value equal to zero only when 119905119903 119905119904 119905119901 and119872119901 areequal to 0475 0879 064 and 00455 respectivelyThereforethe optimization problem is to design a PID controller whichhas optimum gains such that when this PID is used to controlthe DCmotor the overall system step response will exhibit 119905119903119905119904 119905119901 and119872119901 values equal to 0475 0879 064 and 00455respectively

Among all combinations 119870119901 = 50 119870119894 = 018 and 119870119889 =245 are found to be the global optimum PID parameterswhich can satisfy the desired response The step response oftheDCmotor controlled by the PID controller with optimumparameters is shown in Figure 4

From the unit step response shown in Figure 4 it canbe observed that 119905119903 = 04755 119905119904 = 08784 119905119901 = 06437and119872119901 = 45 The optimal PID parameters are optimized

0 10 20 30 40 50 60 70 800

02

04

06

08

1

12

Time (s)

Am

plitu

de

0 05 1 15 20

05

1

Without PID controllerWith optimum PID controller

Figure 4 Step response of the DC motor with optimum PIDcontroller parameters

to achieve minimum fitness value according to the desiredresponse specifications The minimum fitness value (119869min)achieved by the optimum PID parameters is 0009413

In conventional PID optimizationmethods such as ABCMOL or GA the optimization search is performed within anopen space of parameters set For each parameter a real rangeof values is defined such as the ranges selected in this sectionThe open space search delays the optimization process con-sumes its effort and may lead to local minimum problemsTherefore combinatorial interaction design reduces the spaceof search and thus achieves effective optimization process

The table of all PID parameters combinations is sup-plied to the process of combinatorial set constructionexplained in Figure 2 to produce 15 constructed sets of PID


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0009413

01

1

4653

Combinatorial sets

Fitn

ess f

unct

ion

(J)

Figure 5 Optimal fitness function values of each optimal PIDparameter combination corresponding to each constructed set

0 1 2 3 4 5 6 7 80

02

04

06

08

1

12

14

Time (s)

Am

plitu

de

J = 4653

J = 0009413

Worst combination (Kp = 55 Ki = 025 Kd = 135)

Global optimal combination (Kp = 50 Ki = 018 Kd = 245)

Response of system without PID controller

Figure 6 Step response of the optimal and worst combination PIDparameters

combinations Each constructed set consists of 2571 PIDparameters combinations which equals 2 of all PID param-eters combinations table size The parameters combinationsof each constructed set are tested in the control system usingthe objective function defined in (4)The optimal parameterscombination of each constructed set is the onewithminimumfitness value Therefore from the 15 constructed sets 15optimal combinations of PID parameters are determinedFigure 5 depicts the optimal fitness function values of eachoptimal PID parameter combination corresponding to eachconstructed set

The two indicated horizontal levels shown in Figure 5that is 0009413 and 4653 are the fitness function valuesof the global optimal combination parameters (119870119901 = 50119870119894 = 018 and 119870119889 = 245) and the worst combinationparameters (119870119901 = 55119870119894 = 025 and119870119889 = 135) respectivelyFigure 6 shows the step response of the optimal and worstcombination PID parameters as well as the step response ofthe DC motor without PID

It can be observed from Figure 6 that the fitness functionvalues of all the optimal PID parameters combinationscorresponding to each constructed set are close to the globaloptimal valueThis indicates that the combinatorial approachused to construct the subsets succeeded in reducing the

Table 2 Simulation results summary

119870119901 119870119894 119870119889 119905119903 119905119904 119905119901 119872119901 119869

Globaloptimal 50 018 245 04755 08784 06437 455 00094

Com

binatoria

lcon

structedsets

1 50 010 245 04756 08766 06438 453 001372 50 021 245 04755 08791 06437 456 001023 50 015 245 04756 08777 06438 455 001054 50 002 245 04758 08747 06438 452 001875 50 025 245 04754 08801 06437 456 001316 50 031 245 04753 08815 06437 458 001737 50 008 245 04757 08761 06438 453 001508 50 035 245 04753 08824 06436 458 002019 50 005 245 04757 08754 06438 453 0016810 50 002 245 04758 08747 06438 452 0018711 50 022 245 04755 08794 06437 456 0010912 50 018 245 04755 08784 06437 455 0009413 50 023 245 04754 08796 06437 456 0011614 50 030 245 04753 08812 06437 457 0016615 50 032 245 04753 08817 06436 458 00180

searching space by producing subset that assures the presenceof an optimal candidate to the optimization problem Table 2summarizes the simulation results for all the combinatorialconstructed sets

It can be observed from Table 2 that the timing param-eters observed from the step response of each optimal PIDparameter combination are almost equal to the correspond-ing desired values In addition the fitness values listed in therightmost column are close to the global optimal fitness value

6 Conclusion

In this paper the combinatorial optimization has beenemployed in the application of PID controller design forDC servomotor The objective of using the combinatorialdesign approach is to improve the PID tuning algorithm byreducing the search space of the optimization process Withdifferent constructed combinatorial sets simulation resultshave shown that in each constructed set there exists a com-bination of PID parameters with a fitness value close to theglobal optimal value In this way the proposed combinationoptimization method will ensure an optimal value findingby generating an effective reduced searching space Thusthe proposed optimization based on combinatorial approachcan be recommended to be adopted in conventional AIalgorithms such as ABC MOL and GA

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgment

This research is supported by the Information TechnologyUnit of the Engineering College Salahaddin University-Hawler (SUH)

References

[1] R Rahmani M S Mahmodian S Mekhilef and A A ShojaeildquoFuzzy logic controller optimized by particle swarm optimiza-tion for DC motor speed controlrdquo in Proceedings of the IEEEStudent Conference on Research and Development pp 109ndash1132012

[2] K OgataModern Control Engineering PrenticeHall NewYorkNY USA 2010

[3] J G Ziegler andN BNichols ldquoOptimum settings for automaticcontrollersrdquo Transactions of ASME vol 64 pp 759ndash768 1942

[4] G Cohen and G Coon ldquoTheoretical consideration of retardedcontrolrdquo Transactions of ASME vol 75 no 1 pp 827ndash834 1953

[5] S Panda ldquoDifferential evolution algorithm for SSSC-baseddamping controller design considering time delayrdquo Journal ofthe Franklin Institute vol 348 no 8 pp 1903ndash1926 2011

[6] S Panda ldquoMulti-objective PID controller tuning for a FACTS-based damping stabilizer using Non-dominated SortingGenetic Algorithm-IIrdquo International Journal of Electrical Powerand Energy Systems vol 33 no 7 pp 1296ndash1308 2011

[7] S Panda ldquoMulti-objective evolutionary algorithm for SSSC-based controller designrdquo Electric Power Systems Research vol79 no 6 pp 937ndash944 2009

[8] S J Ho L S Shu and S Y Ho ldquoOptimizing fuzzy neural net-works for tuning PID controllers using an orthogonal simulatedannealing algorithmOSArdquo IEEETransactions on Fuzzy Systemsvol 14 no 3 pp 421ndash434 2006

[9] HGozde andMC Taplamacioglu ldquoComparative performanceanalysis of artificial bee colony algorithm for automatic voltageregulator (AVR) systemrdquo Journal of the Franklin Institute vol348 no 8 pp 1927ndash1946 2011

[10] K A El-Metwally ldquoA fuzzy logic-based PID for power systemstabilizationrdquo Electric Power Components and Systems vol 29no 7 pp 659ndash669 2001

[11] S Panda and N P Padhy ldquoComparison of particle swarmoptimization and genetic algorithm for FACTS-based controllerdesignrdquo Applied Soft Computing Journal vol 8 no 4 pp 1418ndash1427 2008

[12] S Panda B K Sahu and P K Mohanty ldquoDesign and per-formance analysis of PID controller for an automatic voltageregulator system using simplified particle swarm optimizationrdquoJournal of the Franklin Institute vol 349 no 8 pp 2609ndash26252012

[13] D R Stinson Combinatorial Designs Constructions and Analy-sis Springer New York NY USA 2004

[14] B S Ahmed and K Z Zamli ldquoA variable strength interactiontest suites generation strategy using Particle Swarm Optimiza-tionrdquo Journal of Systems and Software vol 84 no 12 pp 2171ndash2185 2011

[15] B S Ahmed and K Z Zamli ldquoA review of covering arraysand their application to software testingrdquo Journal of ComputerScience vol 7 no 9 pp 1375ndash1385 2011

[16] B S Ahmed K Z Zamli and C P Lim ldquoApplication of ParticleSwarmOptimization to uniform and variable strength covering

array constructionrdquo Applied Soft Computing Journal vol 12 no4 pp 1330ndash1347 2012

[17] S Y Borodai and I S Grunskii ldquoRecursive generation of locallycomplete testsrdquo Cybernetics and Systems Analysis vol 28 no 4pp 504ndash508 1992

[18] D E Shasha A Y Kouranov L V Lejay M F Chou and GM Coruzzi ldquoUsing combinatorial design to study regulationby multiple input signals A tool for parsimony in the post-genomics erardquo Plant Physiology vol 127 no 4 pp 1590ndash15942001

[19] J N Cawse Experimental Design for Combinatorial and HighThroughput Materials Development Wiley-Interscience NewYork NY USA 2003

[20] D S Hoskins C J Colbourn andD CMontgomery ldquoSoftwareperformance testing usina covering arrays efficient screeningdesigns with categorical factorsrdquo in Proceedings of the 5thInternational Workshop on Software and Performance pp 131ndash136 ACM Palma Spain July 2005

[21] D R Sulaiman and B S Ahmed ldquoUsing the combinatorial opti-mization approach for DVS in high performance processorsrdquoin Proceedings of the International Conference on TechnologicalAdvances in Electrical Electronics and Computer Engineering(TAEECE rsquo13) pp 105ndash109 May 2013

[22] C Yilmaz M B Cohen and A Porter ldquoCovering arrays forefficient fault characterization in complex configuration spacesrdquoACM SIGSOFT Software Engineering Notes vol 29 no 4 pp45ndash54 2004

[23] A Hartman and L Raskin ldquoProblems and algorithms forcovering arraysrdquo Discrete Mathematics vol 284 no 1ndash3 pp149ndash156 2004

[24] M Grindal J Offutt and S F Andler ldquoCombination testingstrategies a surveyrdquo Software Testing Verification and Reliabilityvol 15 no 3 pp 167ndash199 2005

[25] P McMinn ldquoSearch-based software test data generation asurveyrdquo Software Testing Verification and Reliability vol 14 no2 pp 105ndash156 2004

[26] B S Ahmed and K Z Zamli ldquoComparison of metahuristic testgeneration strategies based on interaction elements coveragecriterionrdquo in Proceedings of the IEEE Symposium on IndustrialElectronics and Applications (ISIEA rsquo11) pp 550ndash554 IEEEComputer Society Langkawi Malaysia September 2011

[27] N B Nichols and J G Ziegler ldquoOptimum settings for automaticcontrollersrdquo Journal of Dynamic Systems Measurement andControl vol 115 no 2 pp 220ndash222 1993

[28] R A Krohling J P Rey R A Krohling and J P Rey ldquoDesignof optimal disturbance rejection PID controllers using geneticalgorithmsrdquo IEEE Transactions on Evolutionary Computationvol 5 no 1 pp 78ndash82 2001

[29] Q H Wu B W Hogg and G W Irwin ldquoA neural networkregulator for turbogeneratorsrdquo IEEE Transactions on NeuralNetworks vol 3 no 1 pp 95ndash100 1992

[30] A Visioli ldquoTuning of PID controllers with fuzzy logicrdquo IEEProceedings Control Theory and Applications vol 148 no 1 pp1ndash8 2001

[31] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009


[32] A Jalilvand A Kimiyaghalam A Ashouri and H KordldquoOptimal tuning of PID controller parameters on a DC motorbased on advanced particle swarm optimization algorithmrdquoInternational Journal on Technical and Physical Problems ofEngineering (IJTPE) vol 3 no 9 pp 10ndash17 2011

[33] M M R A Milani T Cavdar and V F Aghjehkand ldquoParticleswarm optimizationmdashbased determination of Ziegler-Nicholsparameters for PID controller of brushless DC motorsrdquo inProceedings of the International Symposium on Innovations inIntelligent Systems and Applications (INISTA rsquo12) 2012

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Navigation and Observation



DistributedSensor Networks



applications such as hardware testing [17] gene expressionregulation [18] advance material testing [19] performanceevaluation of communication systems [20] and optimizationof dynamic voltage scaling (DVS) in high performanceprocessors [21] However CAs have not been used yet for PIDcontroller design thereby it is proposed here to determinethe optimal PID parameters

The rest of this paper is organized as follows Section 2gives an overview of the combinatorial design and its nota-tions Section 3 describes the PID controller design Section 4gives the details of the combinatorial set construction and theprocedure of its application on the system Section 5 presentsand discusses the evaluation results of the proposed approachfollowed by concluding remarks in Section 6

2 Combinatorial Optimization Background

Combinatorial design is used to effectively search for the bestsolution among a finite set of feasible solutions based on theinteractions It is essential in this method to cover all of thecombinations at least once CA has been introduced as amathematical object to represent all those combinations ACA120582(119873 119905 119896 V) represents an 119873 times 119896 array with V values suchthat every119873times 119905 subarray contains all ordered subsets from Vvalues of size 119905 at least 120582 times [22] where 119896 is the numberof components (parameters) When the target is optimalcombination-set it is essential that all 119905-combinations occurat least once In this case the value of 120582 = 1 and the notationbecomes CA (119873 119905 119896 V) [23] As the target is the optimal setthe size of119873 which is the size of the combination-set has tobe as minimum as possible119879-way testing strategies have been used widely to con-

struct CA and its variations such as mixed covering array(MCA) or variable strength covering array (VSCA) [24] forthe purpose of software interaction testingThere are differentmechanisms in the literature used with 119905-way strategies forgenerating the combinations These methods are lying underthe field of search based software engineering (SBSE) [25]

From the literature there are two methods for generatingthe combinations in general (1) algebraically which is basedon the generation of the combinations mathematically (2)computationally which is based on the generation of thecombinations using computational and iterative methods Inpractice the second method is mostly used because of theflexibility since the former method is more restrictive for thecombinations

There are different tools and strategies in the literaturefor generating the combinations of different inputs Eachtool uses a special algorithm for generation Among thoseimplemented algorithms the use of AI theories with therandom generation proves the generation of optimal combi-nation sizes This approach has been used for combinatorialoptimization in different researches (eg [14 26])

3 PID Controller Design

The objective of designing a controller for a DC servomotoris to make the system respond efficiently to the new desired

120579r(s) 120579a(s)

Combinatorialset constructor

PIDcontroller

DCservomotor+

+

minus

Figure 1 DC servomotor with PID controller tuned by combinato-rial set constructor

angular position This can be achieved by comparing theoutput angular position with the desired (reference) posi-tion to calculate an error (actuating) signal The controllermanipulates the error signal and generates an appropriateDC voltage that will cause the motor to respond to the newposition Proportional-integral-derivative (PID) controllerhas been widely used for such application because of itssimple structure and robust performance Figure 1 shows ablock diagram representation of a DC servomotor with PIDcontroller

The design of PID controller involves the computationof three parameters proportional (119870119901) integral (119870119894) andderivative (119870119889) constantsThe general transfer function of thePID controller is given by

119862 (119904) = 119870119901 +119870119894

119904+ 119904119870119889 (1)

The conventionalmethod of determining the PID parametersis a heuristic method called the Ziegler-Nichols method[27] This method is based on calculating the system criticalgain at which the output of the control loop oscillates alongwith the corresponding period of osculation The criticalgain and period of oscillation are used to set the P Iand D gains Later methods based on several optimizationtechniques have been introduced such as genetic algorithm[28] neural network [29] fuzzy based approach [30] andparticle swarm optimization techniques [31] Most systemsarising in practice have time varying parameters which willaffect the performance of the designed PID controller Insuch cases a supervisory system can be used such thatwhen the performance of the controller drops below a pre-scribed level of acceptable performance index the proposedoptimization procedure must be reengaged Therefore thetuning process has to be sufficiently satisfactory in terms ofconvergence speed and this can be achieved when reducingthe optimization searching space This work proposes a newmethod for determining the PID controller parameters usingcovering array arrangement based on combinatorial designapproach As shown in the block diagram of Figure 1 theldquocombinatorial set constructorrdquo feeds the PID controller withsets of parameters instead of the conventional techniquesThis will reduce the amount of the search space which inturn reduces the required time for the system to reach theoptimal value The following section illustrates how this








119866 (119904) =120579119886 (119904)

119864119886 (119904)

=119870119905

1198691198981198711198861199043 + (119877119886119869119898 + 119863119898119871119886) 119904

2 + (119863119898119877119886 + 119870119905119870119887) 119904

(2)






119866 (119904) =120579119886 (119904)

119864119886 (119904)

=001

119904 (00051199042 + 006 + 01001)

(3)




119869 = 1205721

1003816100381610038161003816119905119903 minus 119905lowast

119903

1003816100381610038161003816

119905lowast119903

+ 1205722

1003816100381610038161003816119905119904 minus 119905lowast

119904

1003816100381610038161003816

119905lowast119904

+ 1205723

10038161003816100381610038161003816119905119901 minus 119905lowast

119901

10038161003816100381610038161003816

119905lowast119901

+ 1205724

10038161003816100381610038161003816119872119901 minus119872

lowast

119901

10038161003816100381610038161003816

119872lowast119901

+ 1205725119864119904119904

(4)


119903 119905lowast119904

119905lowast




119869 =

1003816100381610038161003816119905119903 minus 04751003816100381610038161003816

0475+

1003816100381610038161003816119905119904 minus 08791003816100381610038161003816

0879

+

10038161003816100381610038161003816119905119901 minus 064

10038161003816100381610038161003816

064+

10038161003816100381610038161003816119872119901 minus 00455

10038161003816100381610038161003816

00455+ 119864119904119904

(5)


Start



Allcombinations

Yes

Yes

No

No

Yes

No


using PSO

Allcombinations

covered

The optimizedset


combination

Optimizedparameters

End


0 10 20 30 40 50 60 70 800

02

04

06

08

1Step response

Time (s)

Am

plitu

de





0 10 20 30 40 50 60 70 800

02

04

06

08

1

12

Time (s)

Am

plitu

de

0 05 1 15 20

05

1







1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0009413

01

1

4653

Combinatorial sets

Fitn

ess f

unct

ion

(J)


0 1 2 3 4 5 6 7 80

02

04

06

08

1

12

14

Time (s)

Am

plitu

de

J = 4653

J = 0009413









119870119901 119870119894 119870119889 119905119903 119905119904 119905119901 119872119901 119869

Globaloptimal 50 018 245 04755 08784 06437 455 00094

Com

binatoria

lcon

structedsets

1 50 010 245 04756 08766 06438 453 001372 50 021 245 04755 08791 06437 456 001023 50 015 245 04756 08777 06438 455 001054 50 002 245 04758 08747 06438 452 001875 50 025 245 04754 08801 06437 456 001316 50 031 245 04753 08815 06437 458 001737 50 008 245 04757 08761 06438 453 001508 50 035 245 04753 08824 06436 458 002019 50 005 245 04757 08754 06438 453 0016810 50 002 245 04758 08747 06438 452 0018711 50 022 245 04755 08794 06437 456 0010912 50 018 245 04755 08784 06437 455 0009413 50 023 245 04754 08796 06437 456 0011614 50 030 245 04753 08812 06437 457 0016615 50 032 245 04753 08817 06436 458 00180



6 Conclusion





Acknowledgment


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119866 (119904) =120579119886 (119904)

119864119886 (119904)

=119870119905

1198691198981198711198861199043 + (119877119886119869119898 + 119863119898119871119886) 119904

2 + (119863119898119877119886 + 119870119905119870119887) 119904

(2)






119866 (119904) =120579119886 (119904)

119864119886 (119904)

=001

119904 (00051199042 + 006 + 01001)

(3)




119869 = 1205721

1003816100381610038161003816119905119903 minus 119905lowast

119903

1003816100381610038161003816

119905lowast119903

+ 1205722

1003816100381610038161003816119905119904 minus 119905lowast

119904

1003816100381610038161003816

119905lowast119904

+ 1205723

10038161003816100381610038161003816119905119901 minus 119905lowast

119901

10038161003816100381610038161003816

119905lowast119901

+ 1205724

10038161003816100381610038161003816119872119901 minus119872

lowast

119901

10038161003816100381610038161003816

119872lowast119901

+ 1205725119864119904119904

(4)


119903 119905lowast119904

119905lowast




119869 =

1003816100381610038161003816119905119903 minus 04751003816100381610038161003816

0475+

1003816100381610038161003816119905119904 minus 08791003816100381610038161003816

0879

+

10038161003816100381610038161003816119905119901 minus 064

10038161003816100381610038161003816

064+

10038161003816100381610038161003816119872119901 minus 00455

10038161003816100381610038161003816

00455+ 119864119904119904

(5)


Start



Allcombinations

Yes

Yes

No

No

Yes

No


using PSO

Allcombinations

covered

The optimizedset


combination

Optimizedparameters

End


0 10 20 30 40 50 60 70 800

02

04

06

08

1Step response

Time (s)

Am

plitu

de





0 10 20 30 40 50 60 70 800

02

04

06

08

1

12

Time (s)

Am

plitu

de

0 05 1 15 20

05

1







1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0009413

01

1

4653

Combinatorial sets

Fitn

ess f

unct

ion

(J)


0 1 2 3 4 5 6 7 80

02

04

06

08

1

12

14

Time (s)

Am

plitu

de

J = 4653

J = 0009413









119870119901 119870119894 119870119889 119905119903 119905119904 119905119901 119872119901 119869

Globaloptimal 50 018 245 04755 08784 06437 455 00094

Com

binatoria

lcon

structedsets

1 50 010 245 04756 08766 06438 453 001372 50 021 245 04755 08791 06437 456 001023 50 015 245 04756 08777 06438 455 001054 50 002 245 04758 08747 06438 452 001875 50 025 245 04754 08801 06437 456 001316 50 031 245 04753 08815 06437 458 001737 50 008 245 04757 08761 06438 453 001508 50 035 245 04753 08824 06436 458 002019 50 005 245 04757 08754 06438 453 0016810 50 002 245 04758 08747 06438 452 0018711 50 022 245 04755 08794 06437 456 0010912 50 018 245 04755 08784 06437 455 0009413 50 023 245 04754 08796 06437 456 0011614 50 030 245 04753 08812 06437 457 0016615 50 032 245 04753 08817 06436 458 00180



6 Conclusion





Acknowledgment


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start



Allcombinations

Yes

Yes

No

No

Yes

No


using PSO

Allcombinations

covered

The optimizedset


combination

Optimizedparameters

End


0 10 20 30 40 50 60 70 800

02

04

06

08

1Step response

Time (s)

Am

plitu

de





0 10 20 30 40 50 60 70 800

02

04

06

08

1

12

Time (s)

Am

plitu

de

0 05 1 15 20

05

1







1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0009413

01

1

4653

Combinatorial sets

Fitn

ess f

unct

ion

(J)


0 1 2 3 4 5 6 7 80

02

04

06

08

1

12

14

Time (s)

Am

plitu

de

J = 4653

J = 0009413









119870119901 119870119894 119870119889 119905119903 119905119904 119905119901 119872119901 119869

Globaloptimal 50 018 245 04755 08784 06437 455 00094

Com

binatoria

lcon

structedsets

1 50 010 245 04756 08766 06438 453 001372 50 021 245 04755 08791 06437 456 001023 50 015 245 04756 08777 06438 455 001054 50 002 245 04758 08747 06438 452 001875 50 025 245 04754 08801 06437 456 001316 50 031 245 04753 08815 06437 458 001737 50 008 245 04757 08761 06438 453 001508 50 035 245 04753 08824 06436 458 002019 50 005 245 04757 08754 06438 453 0016810 50 002 245 04758 08747 06438 452 0018711 50 022 245 04755 08794 06437 456 0010912 50 018 245 04755 08784 06437 455 0009413 50 023 245 04754 08796 06437 456 0011614 50 030 245 04753 08812 06437 457 0016615 50 032 245 04753 08817 06436 458 00180



6 Conclusion





Acknowledgment


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0009413

01

1

4653

Combinatorial sets

Fitn

ess f

unct

ion

(J)


0 1 2 3 4 5 6 7 80

02

04

06

08

1

12

14

Time (s)

Am

plitu

de

J = 4653

J = 0009413









119870119901 119870119894 119870119889 119905119903 119905119904 119905119901 119872119901 119869

Globaloptimal 50 018 245 04755 08784 06437 455 00094

Com

binatoria

lcon

structedsets

1 50 010 245 04756 08766 06438 453 001372 50 021 245 04755 08791 06437 456 001023 50 015 245 04756 08777 06438 455 001054 50 002 245 04758 08747 06438 452 001875 50 025 245 04754 08801 06437 456 001316 50 031 245 04753 08815 06437 458 001737 50 008 245 04757 08761 06438 453 001508 50 035 245 04753 08824 06436 458 002019 50 005 245 04757 08754 06438 453 0016810 50 002 245 04758 08747 06438 452 0018711 50 022 245 04755 08794 06437 456 0010912 50 018 245 04755 08784 06437 455 0009413 50 023 245 04754 08796 06437 456 0011614 50 030 245 04753 08812 06437 457 0016615 50 032 245 04753 08817 06436 458 00180



6 Conclusion





Acknowledgment


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgment


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









application of combinatorial interaction design for dc servomotor

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