Comparative Analysis of PSO Algorithms for PID Controller Tuning

CHINESE JOURNAL OF MECHANICAL ENGINEERING●928●Vol. 27, No. 5, 2014DOI: 10.3901/CIME.2014.0527.102, available online at www.springerlink .com; www.cjmenet.com; www.jmenet.com.cnComparative Analysis of PSO Algorithms for PID Controller TuningSTIMAC Goranka*, BRAUT Sanjin, and ZIGULIC RobertoFaculty of Engineering, University of Rijeka, Rijeka 51000, CroatiaReceived October 13, 2013; revised May 21, 2014; accepted May 27, 2014Abstract: The active magnetic bearing(AMB) suspends the rotating shaft and maintains it in levitated position by applying controlledelectromagnetic forces on the rotor in radial and axial directions. Although the development of various control methods is rapid, PIDcontrol strategy is still the most widely used control strategy in many applications, including AMBs. In order to tune PID controller, aparticle swarm optimization(PSO) method is applied. Therefore, a comparative analysis of particle swarm optimization(PSO) algorithmsis carried out, where two PSO algorithms, namely (1) PSO with linearly decreasing inertia weight(LDW-PSO), and (2) PSO algorithmwith constriction factor approach(CFA-PSO), are independently tested for different PID structures. The computer simulations are carriedout with the aim of minimizing the objective function defined as the integral of time multiplied by the absolute value of eror(ITAE). Inorder to validate the performance of the analyzed PSO algorithms, one-axis and two-axis radial rotor/active magnetic bearing systemsare examined. The results show that PSO algorithms are effective and easily implemented methods, providing stable convergence andgood computational efficiency of different PID structures for the rotor/AMB systems. Moreover, the PSO algorithms prove to be easilyused for controller tuning in case of both SISO and MIMO system, which consider the system delay and the interference among thehorizontal and vertical rotor axes.Keywords: PID gain tuning, rotor/AMB system, PID structures, particle swarm optimizationanalytical tuning rules, but which are known to give poor1 Introductionresults in many cases5especially when applied tosystems which involve higher order components,Active magnetic bearings(AMBs) present a magnetic nonlinearity and/or uncertainties.suspension technology which is used in a variety of rotatingHowever, modern tuning methods based on artificialmachines, such as turbo molecular pumps， flywheels, intelligence techniques, such as neural networks,machine tool spindles etc. Their unique advantages, such as fuzzy-logic'5 7] and neural-fuzzy, can also be applied. Incomplete elimination of oil lubrication systems, absence of addition, many methods based on evolutionary computationmechanical wear, low maintenance costs, programmable algorithm can be applied, such as genetic algorithml8,stiffness and damping, follow directly from their particle swarm optimization(PSO)9- 13] and ant colonycontactless suspension principle. However, these are optimizationl4.multivariable systems which require properly tunedPSO has gained wide recognition since its developmentfeedback loops in order to ensure stable rotor suspension.by KENNEDY and EBERHART in 1995, due to its abilityAlthough the development of processcontrol to provide solutions efficiently, requiring only minimalmethodologies is rapid, PID control still remains the most implementation efort!'s. Also, the ability of PSO to adaptwidely used feedback control strategy in industry and easily its components to a desired form, implied by theacademia in many applications, including AMBs'problem at hand, has placed PSO in a prominent positionOptimal control performances of PID controller can be among other itelligent optimization algorithmso. Due toachieved by identification of the set of the three adjustable the good properties of the PSO algorithm, it has evolvedgains, i.e. proportional gain, integral gain and derivative nowadays as a new optimization approach, which can begain. For that purpose various tuning methods have been applied in a variety of applications, such as in faultproposed, among which the most basic method is the identification, in estimation of the filter parameterslie, inZiegler-Nichols method, developed by Ziegler and Nichols control engineering for PID controller gain tuning9 -13] etcin 1942. It is an empiric method based on the well known Although the PSO approach has been widely investigatedin PID contro中国煤化工application to* Corresponding author. E-mail: gstimac@riteh.hrrotor/AMB syste[YHCNMHG found in theSupported by University of Rijeka, Croatia(Grant Nos. 13.09.1.2.11， published literature13.09.2.2.19)In this paper the performance analysis of the PSO◎Chinese Mechanical Engineering Society and SpringrVerag Berlin Heidelberg 2014CHINESE JIOURNAL OF MECHANICAL ENGINEERING●929●algorithms for PID controller tuning for a rotor/AMBEach particle tries to modify its current position andsystem is carried out. Two PSO algorithms, namely PSO velocity according to the distance between its currentwith linearly decreasing inertia weight(LDW-PSO) and position and its own best position, and the distance betweenPSO algorithm with constriction factor approach(CFA-PSO)，its current position and the global best position. If dare independently implemented and tested on three PID variables are optimized, particles move randomly over acontroller structures. In section 2 fiundamental theoretical d- dimensional search space in order to optimize anderivations and variants of the PSO algorithm are presented. objective function f(x), which is used as the criterion ofSection 3 briefly elaborates the one-axis and two-axis fitness of each particle. Therefore, for the search spaceRotor/AMB system models, in conjunction with the defined by j= 1,2,-,d, the velocities and positions of thefollowing three PID control structures: (1) conventional particles updated for the next iteration(k + 1) can be writtenparallel PID structure(PID-P), (2) PID controller structure in the following form:with set point on only I-controller(-PD), and (3) series PIDstructure(PID-S). Section 4 outlines their implementationand gain tuning using the PSO algorithms. Simulationresults for considered examples are elaborated in section 5,(+)x)=(2xy +(k+1)y,(1)which contains the transient performance response analysesand the convergence curves for the investigated controllerstructures. Finally, section 6 is the concluding section, in where w is the inertia weight, C1 and C2 are the weightingwhich the obtained results as well as the applied modeling factors known as the cognitive learning parameter and theand control procedures are summarized.social learning parameter, Ri and R2 are two uniformlydistributed random numbers from the interval [0, 1]. The2 PSO Algorithm and Its Variantsalgorithm defined in Eq. (1) is repeated iteratively until apredefined number of iterations are reached. Moreover, itPSO is a stochastic optimizaion algorithm based on was found out that larger inertia weights facilitate globalsocialsimulation models, whose development was based on exploration and prolong the convergence time and thatconcepts and rules of cllsionofree, synchronized moves smaller inertia weights facilitate local exploitation andthat govermn socially organized populations in nature, such ensure faster convergence, but possibly lead to local optima.as bird flocks, fish schools and animal herds. In rcent Therefore, some approaches were considered to improveyears, PSO has atracted a lot of atention because of its the performance of the presented PSO concept by variablenumerousadvantages, such as (i) the ease of inertia weight.implementation, (i) the algorithm does not use the gradientinformation of the objecive function, but its values, (i)it 2.1 Linearly deereasing inertia weight(LDW-PSO)can be applied for solving nonlinear, multiple optimum andThe concept of linearly decreasing inertia weight washigh dimensional problems, and (iv) its solution hardlyintroduced by EBERHART and SHI in 1998 resulting in andepends on initial states of the particles, which can be aimproved PSO variant1), using which the effect of velocitysignificant advantge in egneering design problems based fades linearly during the execuion of the algorithm. It ison optimization approaches.implemented as follows:PSO is a computational technique based on thmovement and intelligence of swarms. A“swarm”can bedefined as an apparently disorganized collection of moving()w=wm. - Wmax二Wmink,(2)kmaxparticles(population) that tend to cluster together, whileeach particle seems to be moving in a random direction.Consider a population containing N independent particles where Wmax and Wmim are the desirable maximum andthat move around in d-dimensional search space looking for minimum bounds of w and Kmax is the total number ofthe best solution. The initial positions and velocities of the iterations. At an early optimization stage a larger inertiaparticles are chosen randomly, usually in the interval [, 1]. weight factor is aplied to promote global exploration, afterwhich it is linearly decreased in order to facilitate localThe i-th particle at the k-th iteration has the positionexploitation. A very common choice is to set Wmax to a(x+,',xia)" and the velocity y, = (v>12,.,,nvalue of 1.2 and Wmin to a value of0.1.The best position of the particle achieved so far by itself isgiven as ()p,=(P+p,r,",Pua) and the global best 2.2 Constriction factor approach(CFA-PSO)position, i.e. the position with the lowest function valueBy a thorougl中国煤化宝CLERC andachieved so far by any particle of the entire swarm isKENNEDY(2002nt generalizedPSO models, deYHCNMHGrianthasbeen"p=(p中,唱，p唱)“, where the ltter “g”designatesintroduced!20].This variant introduces the constrictionthe global best.factor, which ensures better convergence. In this model, the●930●STIMAC Goranka, et al: Comparative Analysis of PSO Algorithms for PID Controller Tuningvelocity equation is calculated as follows:This implies that the rotor is reduced to a singleconcentrated mass suspended in the magnetic field, whichis an appropriate simplification for a preliminary study.where x is the constriction factor defined bymy+mx八 fm-rX=φ=c+cz>4. (4)2-φ-√φ2 - 40|RotorCommonly, both C1 and C2 are set to 2.05. Obviously, thisPSO model is algebraically equivalent with the inertia2weight model described in equation Eq. (2). However, ini年h:| Contoller Fliterature it is distinguished due to its theoretical propertiesthat point out the explicit selection of its parameters.3 Model of the Rotor/AMB SystemFig. 1. Cross-section of a typical radial AMBset of power amplifiers and a controller. They operate onTherefore, in this study two separate analyses are carriedthe principle of active magnetic suspension. Fig. 1ut. In the first, the rotor is modeled as a one degree ofillustrates a most common radial AMB structure with fourfreedom system, i.e. as a SISO system. In the second, thepole pairs, in which each AMB actuator consists of twointerference among the x- and y- axes is additionallypairs of electromagnets which operate in a differentialintroduced and the rotor is modeled as a two degree ofdriving mode. This means that one electromagnet in thefreedom system, i.e. as a 2x2 MIMO system.pair is driven with the sum of the bias current io and thecontrol curent(designated as ix in the x-direction and iy in 3.One-axis AMB suspension system modelthe y-direction) and the opposite one with their differencel .Fig. 2 shows the block diagram of a one-axis rotor/AMBAlthough rotations and transverse motions of the real system in a feedback loop, where Ym is the sensor output, urotor cannot be controlled by one pair of AMB electromagnets, is the control signal(voltage) applied to the plant, i is thethe basic properties of a magnetic bearing control loop can coil current and y is the actual rotor displacement. Eacheasily be investigated using only one degree of freedom component of the system is briefly elaborated in the(one-axis) or two degrees of freedom(two -axis) rotor models. following subsections.> Yrm VolageKampms'- ks1+2.ePID controlleramplifierID AMBDelaySensorFig. 2. Block diagram of a one-axis rotor/AMB system in a feedback loopG(<)=- 5(6)3.1.1 Magnetic actuator; power amplifierand sensor modelsFor magnetic actuators of the differential type， thewhere k; is the force-current coefficient, ks is the negativefollowing dynamic equation of motion applies: .force-displacement coefficient and m is the rotor mass.Power amplifiers and sensors are modeled as simple gainsmiy= fm,5)中国煤化工amics can beneglected.where fm =ksy+ki is the linearized magnetic force,MHCNMHGusing which Eq. (5) can be easily transformed into the 3.1.2 Delay modeltransfer function form:In actual magnetic suspension systems, the phase lagCHINESE JIOURNAL OF MECHANICAL ENGINEERING●931●delay is always present. It can be caused by many reasons, PSO algorithms.such as: iron losses in the actuator iron core, flux delaycaused by eddy currents, voltage saturation in the currentdriver, limited sensor and power amplifier frequencyresponse, the sampling period of the digital controller, etc.upID.PDIn order to simplify the analysis, but also to take intoaccount such influences, all possible phase lags will beincluded in the form of a single first order transfer function_YmGay(s)=7)Fig. 3. PID-P controller structure|1 -2nfosay ,r"I-PDKp Fwhere felay is the cut-off frequency measured in hertz(Hz).Kint FH出3.1.3 PID controller structures一KaThree PID controller structures will be examined. Thefirst is the parallel PID controller structure(PID-P controller)whose mathematical representation is given asFig. 4. I-PD controller structurede(t)up()= K.e()+ Kmfe()d1+KsdtKosmHOroswhere Kp is the proportional gain, Kint is the integral gainand Ks is the derivative gain. One disadvantage of this'mconfiguration is that a sudden change of the rotor positionFig. 5. PID-S contoller structure(and hence a large error e between the reference input r andthe measured position ym) will cause the derivative term to3.2 Two axis AMB suspension system modelbecome very large, causing large control signals as well.The generated radial magnetic forces are mostly alignedAccordingly, an alternative implementation isin two perpendicular axes which usually coincide with thegeometrical x and y axes. Under some circumstancesUL-p()==Kp2m()+ Kma J e()dt- KJdym(t)(9) (misalignment of the displacement sensor and theelectromagnet, flux due to eddy currents, the gyroscopiceffect, etc.) misalignment of the radial force can occur, i.e.where the proportional and derivative parts act on the the direction of the generated radial force can have anmeasured value and not on the error, i.e. only the angular error. As a consequence, force interference betweenI-controller acts on the set point, giving an I-PD controller.the two axes is induced. Fig. 6 shows two perpendicularThe third controller is obtained as a series connection ofthe axes x and y and two radial magnetic forces fm and JmyPD and PI cotolers, giving a series PID sruture(PID-S)，inclined by a small angle Q, wherefrom the total magneticforces can be calculated as follows:whose model can be presented as follows:fmr= fmcos0+ fmysinθ= far +fmr，upns()= K;e()+Ki, I e()dt,(10)fmy=-Jmsinθ+ fmycosθ= Jay + fm.e(t)= e(t)+KJDue to the small value of angle 0, cosθ can beapproximated by unity. The additional(interfering) forceswhere Kj， Iand Kj are the proportional, thecaused by the angular error are designated as fax and fay.integral and derivative controller gains of the seriesFinally， dynamic equations of motion of a rotorcontroller structure, respectively. The outlined controllersuspended by中国煤化工:plane arestructures are presented in Figs.3- 5.In order to tune the presented controllers(PID-P, I-PD,TYHCNMHGPID-S) and to achieve the desired responses, they will be(12)independently implemented and configured by using themjy= fmw.●932●STIMAC Goranka, et al: Comparative Analysis of PSO Algorithms for PID Controller Tuningsystem and p=2 for a two-axis system) and T is the time ofintegration. The required PID gains minimize the objectivefunction in the time domain, i.e. the performance indicators(overshoot, settling time, rise time and steady state error).05 Simulation Results and DiscussionTo identify the gains of the presented PID structures andto study the performances of the LDW-PSO and CFA-PSOalgorithms the simulation experiments were carried out on~Fmthe two AMB suspension systems. The input data for theseFig. 6. Angular error of the radial AMB forcesystems are presented in Table 1.Table 1. Characteristics of the rotor/AMB system4 Implementation of PID ControllerSystem propertyValueStructures Tuned by PSOMassm/kg12Force current cofficientk./(N.A")19Three PID controller structures tuned with LDW-PSOForce-displacement cofficient ks / (kN.m ')68and CFA-PSO algorithms were developed for a magneticPower amplifier gain Kmp /(V.A)suspension system. The algorithms are used to determineSensor gain Ksm/ (kV●m ")the three PID gains(Kp, Kimt, Ka), independently for each ofthe controller structures. Therefore, a three dimensionalThe system delay is modeled by the cut-off frequencysearch space is defined, in which each of the controllerfaelay=500Hz. In order to ensure convergence， thegains corresponds to one dimension, i.e. each particle in themaximum number of iterations kmax is set to 200 forsearch space represents a particular combination(Kp Kimt, LDW-PSO and to 250 for CFA-PSO. Unit step is applied asKa) for which a unique response can be obtained.a rotor position reference(r=1). The final time T is set toIn the framework of PSO, the quantities that need to be 0.1 s and each algorithm was repeated for 10 independentinitialized prior to the execution of the algorithm are the trials. In LDW-PSO, weighting factors c1 and Cr are definedinitial positions and velocities of each of the N particles inas C1=C2=1 and w is linearly decreasing from Wmax=1.2 tothe population. In this study the initial combination of gainsWmin=0.3. To study the performances of PID structures, the(Kp0, Kin0, Kao) is generated as a set of random valuesone-axis(example 1) and the two-axis AMB systemwithin the interval [0, 1]. These values were then scaled(example 2) are investigated.with the corresponding magnitudes obtained from theZiegler-Nichols tuning rules applied to one-axis AMB5.1 Example 1: One -axis rotor/AMB modelsystem. This procedure gives the following values:The one-axis rotor/AMB model is considered in thisexample. The best results, with respect to the transientKp0=3，Kimno=50, Kao = 0.01.(13) response performances, among the 10 independent trials areregistered andpresented as follows. The convergenceThe itial velocities are set to zero for all particles in all curves of the objective function are shown in Fig.7. Thethree dimensions. For thcaseof a two-axis magnetic gain convergence curves for each of the PID structures aresuspension model the aforementioned procedure is presented in Figs. 8 and 9. Finally, the performanceextended with the additional controller(two PID indicators in time domain are given in Table 2 and thecontrollers). It is assumed that the gains of both controllers corresponding unit step responses are ilustrated in Fig.10.are the same.Evaluation of a given set of contoller gains is achieved 5.2 Example 2: Two-axis rotor/AMB with interferenceby simulating a unit step response of the resulting closedAnalogous analyses are carried out for the two-axisloop system. In order to obtain a measure of the transientrotor/AMB model including three PID structures. Theresponse performance of the system, the integral of time angular error θ is defined as 50.multiplied by the absolute value of error(ITAE) is taken as .Simulation results were repeated for 10 independentthe objective function as follows:trials and LDW-PSO and CFA-PSO were employed. Theconvergence curves of the obective function are given inFig. 11. The PID中国煤化工r each controlJImxe =2fre(o)d,(14) structure are pr二corresponding unYHcNMHGtedinFigs.14.nd 15. By examining the obtained results some generalwhere p is the number of controllers(p= =1 for a one-axis conclusions can be drawn. The PSO algorithms can beCHINESE JOURNAL OF MECHANICAL ENGINEERING●933●successfully applied, regardless of the controller structure noticed that PID gains converge more steadily and in a lessand perform well in gain tuning of the presented systems. number of iterations in the case of the LDW-PSO approachThis means that various controller structures can be easily (Figs. 8, 12) then in the case of the CFA-PSO approachtuned using the presented PSO algorithms, which have (Figs. 9, 13). Moreover, PSO is identified as a robustshown ease of the tuning in case of both SISO and MIMO method in terms of its searching capabilities, computationalAMB systems. By analyzing the convergence curves it is efficiency and convergence properties.LDW-PSOCFA-PSO0~80-- PID-P. I-PD- I-PD60+, PID-SE60. PID-So50Iterations k15000Iterations A200 250(a) LDW-PSO approach(b)CFA-PSO approachFig. 7. Convergence curves of objective functions(example 1)6p0.05. PID-Pr1-PD之60这0.04--PID0.03品40.0.02i/2-司0.01-0.01-2050 100 150 200-0.025Iterations h(a) For Kp gain(b)For Kimt gain(c)For Ko gainFig. 8. Convergence curves of the PID control gains for the LDW-PSO algorithm(example 1)150-2.5p0.10--PID-P-1-PD>2.0一0.08PID-S高100夏1.5-国1.0-盒50F首0.5-0.02|50 100150200 250 .50 100 150 200 25050 100150200250(b) For Kint gain(c)For Ka gainFig. 9. Convergence curves of the PID control gains for the CFA-PSO algorithm(example 1)Table 2. Comparison of the obtained performance indicators in time domain(example 1: one-axis rotor/AMB model)LDW-_PSOParametre中国煤化工PID-PSetling time1s/s0.005 60.00690.006 80.MHCNMHG0.0043Rise timet/s0.000 50.00320.004 30.000 40.001 80.001 3Overshoot M, / %44.472.031.7540.772.720.45●934●STIMAC Goranka, et al: Comparative Analysis of PSO Algorithms for PID Controller TuningLDW-PSOCFA-PSO- PID-P. PID-P- I-PD一1-PIID-S- PID-S1.0H房0.50.5H0.01 0.020.030.04 0.05.0.010.020.03 0.040.o5Time 1/s(a) LDW-PSO approach(b)CFA-PSO approachFig. 10. Comparison of the step responses(example 1)-PID-P.I-PDo叶. PID-S一PID-S.6E 0.6。0.4g 0.4g 0.2510015205010150200 250Iterations k(a) LDW-PSO aprpachFig. 11. Convergence curves of objective functions(example 2)30-0.04「PID-PI-PD也0.03--PID-S其旨0.0210-自0.01oFN) 100 150 2000 100150200 -0.00 S0100 150 200Iterations 1(a) For Kp gain(b) For Kin gain(C) For Ks gainFig. 12. Convergence curves of the PID control gains for the LDW-PSO algorithm(example 2)0r2.50.04--PID-P .一1-PD? 30-足0.03-PID-S专1.5-20-0.02-g1.0占1s。10.58o.中国煤化工0 501001502002500 50100150200 25fYHCNMHG(b) For Kimt gain(C) For Kd gainFig. 13. Convergence curves of the PID control gains for the CFA-PSO algorithm(example 2)CHINESE JOURNAL OF MECHANICAL ENGINEERING●935●LDW-PSOCFA-PSO- PID-P-PID-P-I-PDM- PID-S-PID-S.s5昌0.5|0.010.02 0.03 0.04 0.050.02 一0.030.04 0.05Time t/sTime 1/s(a) LDW-PSO aproach(b) CFA-PSO approachFig. 14. Comparison of the step responses along the x axis(example 2)z 1.5- 1I-PD- I-PDw 0.50.01 0.02 0.03 0.04 0.05(a) LDW-PSO approachFig. 15. Comparison of the step responses along the y- axis(example 2)From Figs. 7 and 11, the same tendency of the objective(1) The PSO algorithms show ease of the controllerfunction can be observed in both examples. Although the tuning in case of both SISO and MIIMO rotor/AMBrise time and the settling time is smaller when using the systems, which consider also the system delay and theCFA-PSO approach, the LDW-PSO algorithm ensures interference among the horizontal and vertical rotor axes.faster convergence and almost aperiodic(i.e. non-oscillatory)(2) PSO is identified as a robust method both in terms ofresponse(Figs. 10, 14, 15).its searching capabilities and computational efficiency.Regarding to the controller structure, the PID-P structure(3) The CFA-PSO approach ensures faster rise time,provides the undesirably large overshoot values in each of while the LDW-PSO algorithm provides almost non-the considered cases, while the IPD and PID-S structures oscillatory system responses.perform almost equally well and ensure much better results(4) ITAE objective function is proved to be suitable forin comparison with the PID-P controller. As expected, this the optimal design of PID controllers.is a direct consequence of the inherent properties of the(5) Better performance of I-PD and PID-S controllersimplemented controller structures， which additionally over PID-P controller is observed, what is in directproves the efficiency of the presented PSO approaches.correlation with the nature of the implemented controllerstructure.6 ConclusionsIn future research, these investigations will be extendedto more complex rotor/AMB systems which involve rigidA comparative performance analysis of two PSO and, preferably, flexible rotors.algorithms for PID controller tuning of a rotor/AMBsystem is presented. The two PSO algorithms, theReferencesLDW-PSO and the CFA-PSO， are independentlyimplemented and tested for three PID structures(PID-P,[1] LEI s, PALAZZOLO A. Control of flexible rotor systems withactive magneti中国煤化工d Vbraion, 2008,I-PD and PID-S), for one-axis and two-axis rotor/AMB314: 19- -38.models. The PID controllers are designed considering2] STIMAC G I|YHcNMHGmspesssofminimization of the ITAE criterion. The main simulationflexible rotor using active magnetic bearings(AMB)[J].conclusions are outlined as follows:Transactions of Famena, 2011, 35:27- 38.●936●STIMAC Goranka, et al: Comparative Analysis of PSO Algorithms for PID Controller Tuning[3] ASTROM K J, HAGGLUND T. The future of PID control[]. [16] PARSOPOULOS K E， VRAHATIS M N. Particle swarmControl Engineering Practice, 2001, 9: 1163- 1175.optimization and intelligence: Advances and Applications[M]. New[4] ZIEGLER J B, NICHOLS N B. Optimum sttings for PIDYork: Information Science Reference, 2010.contolers[]. Transactions of ASME, 1942. 64: 759- -768.[17] XU H, CHEN G. An neleligent fault identifcation method of rlling[5] ZHAO z Y, TOMIZUKA M. Fuzzy gain scheduling of PIDbearings based on LSSVMoptimized by improved PSO[J].controllers[J]. IEEE Transactions on Systems， Man, andMechanical Systems and Signal Processing, 2012, 35: 167-175.Cybernetics, 1993, 23: 1392- 1398.[18] LIN Y L, CHANG W D, HSIEG J G A particle swarm optimization6] AL-ODIENAT A 1, AL-LAWAMA A A. The advantages of PIDapproach to nonlinear rational filter modeling[]. Exper Systemsfuzzy controllers over the conventional types[J]. American Journalwith Applications, 2008, 34: 1194 1199.of Applied Sciences, 2008, 5: 653- -658.[19] SHI Y, EBERHART R C. A modified particle swarm optimizer[(C]//[7] CHEN K Y, TUNG PC, TSAI M T, FAN Y H. A slftuning fuzryProceedings of the IEEE International Conference on EvolutionaryPID-type contoller design for unbalance compensation in an activemagnetic bearing[J]. Expert Systems with Applications, 2009, 36: [20] CLERC M, KENNEDY J. The particle swarm: explosion, stability8560- -8570.and convergence in a multi-dimensional complex space[J]. IEEE[8] CHEN H C, CHANG S H. Genetic algorithms based optimizationTransactions on Evolution Computation, 2004, 8: 204- -210.design of a PID controller for an active magnetic bearing[J]. [21] SCHWEITZER G, MASLEN Eric H. Magnetic bearings: theory,International Journal of Computer Science and Network Securiy,design and application 10 rotating machinery[M]. New York:2006, 6: 95- 99.Springer, 2009.[9] KHANDANI K, JALALI A A. PSO based optimal fractional PIDcontroller design for an active magnetic bearing suspension Bibliographical notessystem[//Proceedings of 18th Anmual Iternational Conference on STIMAC Goranka, bor in 1982 is currently a seniorMechanical Engineering, Iran, Teheran, May 11-13, 2010: 1-6.research/teaching assistant at Faculty of Engineering, University[10] KIM T H, MARUTA I, SUGIE T. Robust PID controller tuning of Rijeka, Croatia. She received her PhD degree from Faculty ofbased on the constrainedparticle swarm optimization[J]. Engineering, University of Rijeka, Croatia in 2012. Her researchAutomatica, 2008, 44: 1104- 11.1interests include rotordynamics, active magnetic bearings and[11] CHANG W D, SHIH s P. PID contollr design of nonlinear mechatronics.systems using an improved partile swarm optimization approach[J]. E-mail: gstimac@riteh.hrComunications in Nonlinear Science and Numerical Simulation,2010, 15: 3632 -3639.BRAUT Sanjin, born in 1973 is currently an associate professor at[12] FANG H, CHEN L, SHEN Z. Application of an improved PSO Faculty of Engineering, University of Rijeka, Croatia. He receivedalgorithm to optimal tuning of PID gains for water turbine his PhD degree from Faculty of Engineering, University of Rijeka,governor[J]. Energy Conversion and Management, 2011， 52: Croatia in 2006. His research and teaching interests include1763- 1770.dynamics，vibrations, dynamics of machines, rotordynamics,[13] MENHAS M I, WANG L, FEI M, et al. Comparative performance active magnetic bearings and mechatronics.analysis of various binary coded PSO algorithms in multivariable E-mail: sbraut@riteh.hrPID controller design[J]. Expert Systems with Applications, 2012,39: 4390- 4401.ZIGULIC Roberto, born in 1966 is currently a full professor at[14] CHIHA 1, LIOUANE N, BORNE P. Tuning PID controller using Faculty of Engineering, University of "Rijeka, Croatia. He receivedmultiobjective ant colony optimization[J]. Applied Computational his PhD degree from Faculty of Engineering, University of Rijeka,Intelligence and Soft Computing, 2012: 1-7.Croatia in 2001. His research and teaching interests include15] KENNEDY J, EBERHART R. Particle swarm optimization[C]// dynamics， vibrations, dynamics of machines, rotordynamics,Proceedings of IEEE International Conference on Neural Network, simulations of dynamical systems and robotics.Part IV, Perth, Australia, 1995: 1942-1948.E-mail: zigulic@riteh.hr中国煤化工MYHCNMH G