Open Access Research on PID Controller in Active Magnetic Levitation Based on Particle Swarm Optimization Algorithm

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1 Send Orders for Reprnts to 1870 The Open Automaton and Control Systems Journal, 2015, 7, Open Access Research on PID Controller n Actve Magnetc Levtaton Based on Partcle Swarm Optmzaton Algorthm Zhang Yanhong 1,2,*, Zheng Zhongqao 1,3, Zhang Jansheng 1 and Yn Le 1 1 School of Electrcal and Photoelectronc Engneerng, Changzhou Insttute of Technology, Changzhou, Jangsu, , Chna; 2 School of Electrcal and Informaton Engneerng, Jangsu Unversty, Zhenjang, Jangsu, , Chna; 3 School of Mechatronc Engneerng and Automaton, Shangha Unversty, BaoShan Dstrct, Shangha, , Chna Abstract: Actve magnetc levtaton control system s an open-loop and unstable control system. The parameters of PID controller are usually set accordng to the experence of the desgner. It s not only tme-consumng and tedous, but also can t be controlled accurately. So the partcle swarm optmzaton algorthm s used n ths paper, whch s used to optmze the parameters of PID controller by combnng wth PID controller. The proportonal coeffcent, ntegral coeffcent and dfferental coeffcent of PID controller are used as three partcles of the partcle swarm optmzaton algorthm. The accurate ntalzaton parameters are selected, and then the speed and poston nformaton s updated to realze the global optmzaton of partcle. Therefore, the three optmzed parameters of PID controller are obtaned, whch are used to smulate the actve magnetc levtaton control system. The smulaton result shows that the optmzed parameters can make the actve magnetc levtaton control system stable. Keywords: Actve magnetc levtaton, Global optmzaton, Partcle swarm optmzaton algorthm, PID controller, Stablty. 1. INTRODUCTION Actve magnetc levtaton control system has some advantages, such as no contract, no frcton, long lfe and controllable supportng force. So t has been wdely used n the ndustry. The core s controller of the actve magnetc levtaton control system. The structure of PID controller s smple, the desgn method s very mature and t has better control effect for the lnear control system. Therefore, PID controller s wdely used n ndustral control system. PID controller s also used n the actve magnetc levtaton control system, but the adjustment of the proporton coeffcent, ntegral coeffcent and dfferental coeffcent have always been a problem. If the parameter s not approprate, the actve magnetc levtaton control system wll be unstable. Therefore, how to select some optmzaton algorthms to make the three parameters of PID controller adjust onlne s a hot ssue n recent years. A knd of mproved PID controller n the actve magnetc levtaton has been studed n document [1]. The control performance of the system has been mproved. A knd of fuzzy and adaptve PID controller has been studed n document [2], and the parameters of PID controller are set by establshng fuzzy rules. A knd of PID controller based on the neural network has been studed n document [3], n whch the parameters of PID controller are set by neural network. Some better results have been ganed by dfferent methods n several documents. The partcle swarm optmzaton algorthm has been ntroduced and the three *Address correspondence to ths author at the School of Electrcal and Photoelectronc Engneerng, Changzhou Insttute of Technology, Changzhou, Jangsu, , Chna; Tel: ; Fax: ; E-mal: zhangyhczu@163.com parameters of PID controller are used as three partcles. The three optmzed parameters of PID controller are obtaned by updatng the poston and speed of the partcles. The smulaton model of the actve magnetc levtaton control system has been establshed by Smulnk. The three optmzed parameters of PID controller are transferred to the smulaton model, then the establshed system s smulated, whch realzes the stable control of the actve magnetc levtaton system. 2. THE COMPOSITION OF THE ACTIVE MAGNET- IC BEARING The structure of the actve magnetc bearng system s manly composed of the current dsplacement sensor, the rotor, the axal magnetc bearng, the radal magnetc bearng, the thrust plate, the drvng motor and the protecton bearng. The structure s shown n Fg. (1). The actve magnetc bearng system has fve degrees of freedom. They are respectvely two horzontal degrees of freedom, two vertcal degrees of freedom and one axal degree of freedom. The axal magnetc bearng s nstalled n the mddle of thrust plate, whch makes the system balance n the axal drecton. The two radal magnetc bearngs are respectvely nstalled at the left and the rght end of the rotor, and they make the rotor balance n the horzontal and vertcal drectons. The two protecton bearngs are located at both ends of the rotor. When the magnetc bearng system does not work, the rotatng shaft should be placed on the protectng bearng, whch reduces the ar gap between the protecton bearng and the rotor. When the actve magnetc levtaton bearng system s out of control, the protecton bearng supports the hgh-speed rotatng rotor and prevents / Bentham Open

2 Research on PID Controller n Actve Magnetc Levtaton The Open Automaton and Control Systems Journal, 2015, Volume Fg. (1). Structure of the actve levtaton bearng. Fg. (2). Structure of the actve magnetc levtaton system wth a degree of freedom. damagng the whole magnetc levtaton bearng system because of the collson of the rotor and the stator [4-6]. 3. MATHMATIC MODEL OF ACTIVE MAGNETIC LEVITATION WITH A DEGREE OF FREEDOM The structure of the actve magnetc bearng system wth a degree of freedom s shown as Fg. (2). The qualty of the rotor s m. The rotor locates between upper electromagnet and lower electromagnet, and only the bearng force n the vertcal drecton s consdered. When the rotor s n equlbrum, the followng assumpton s made. (1) The wndng leakage magnetc flux s gnored. (2) The reluctance of the ron core and the rotor are gnored, namely the magnetc potental s only n the ar gap. (3) The hysteress and the eddy current effect of the magnetc materal s gnored. Suppose that a dsturbance appears, whch makes the rotor devate the equlbrum poston x. In order to make the rotor return to ts equlbrum poston, a control current s produced by the electromagnet, whch reduces the electromagnetc force of the upper electromagnet and ncreases the electromagnetc force of the lower electromagnet, and the electromagnetc force s expressed as follows. F = F 1! F 2 = µ 0SN 2 4 [ ( 0! c! x! ( 0 + c + x ] (1) Where, µ 0 s the magnetc permeablty of vacuum, whch unt s H/m. S s the cross-sectonal area of the magnet, whch unt s mm 2. N s the number of turns of electromagnetc col, 0 s the bas current of the electromagnetc col, whch unt s A. x 0 s the ar gap between the rotor and the magnet when the rotor s n equlbrum, whch unt s mm. Accordng to Newton's second law, the force of the rotor s expressed as follows. F = ma = m d 2 x dt 2 (2) Where, F s the electromagnetc force, whch s the functon of the dsplacement and the current, m s the qualty of the rotor and x s the dsplacement. The formula (2) s substtuted nto the formula (1) and the followng formula can be obtaned.

3 1872 The Open Automaton and Control Systems Journal, 2015, Volume 7 Yanhong et al. µ 0 SN 2 4 [ ( 0! c! x! ( 0 + c + x ] = m d 2 x dt 2 (3) It can be seen from formula (3) that the mathematcal model of the magnetc bearng s a second order nonlnear dfferental equaton. It s dffcult to calculate the soluton of formula (3). In the practcal applcaton, the nonlnear control system s often solved by transformng nto the lnear control system. Therefore, the electromagnetc force s lnearzed by Taylor formula n the neghborhood of the equlbrum pont, the followng formula s obtaned. F( c, x)! "k # c + k x # x (4) k = µ 0SN 2 0,k 2 x = µ SN (5) x 0 x 0 Where, k s the current coeffcent, k x s the dsplacement coeffcent. It can be seen from formula (5), when the structural parameters of the system and the statc workng pont ( 0, x 0 ) are determned, where k and k x are both constants. It can be obtaned from formulas (2) and (4)!k " c + k x " x = m!!x (6) The Laplace transform s made n the formula (6), the transfer functon can be obtaned as follows. G(s) = X(s) I(s) =!k (7) ms 2! k x It can be seen from formula (7) that a characterstc root s located n the rght half plane, and the system s unstable, so the controller must be desgned. 4. PARTICLE SWARM OPTIMIZATION ALGO- RITHM Partcle swarm optmzaton algorthm (PSO) was proposed by Jm Kennedy and Eberhart n It s derved from the predatory behavor of brds. A group of brds search for food n the random, f there s only a pece of food, the most smple and effectve strategy of fndng the food s to search for the surroundng area of brds who are nearest to food. PSO algorthm s obtaned from ths model whch s used to solve the optmzaton problem [6-8]. In the partcle swarm algorthm, each ndvdual s called as one partcle, and each partcle represents a potental soluton. All partcles have a ftness value determned by the optmzaton functon, wth each partcle havng a speed, whch determnes the drecton and dstance of the partcle. Other partcles follow the optmal partcle to search for the extreme value n the soluton space. Accordng to the set ftness functon n advance, the current ftness value s calculated. The poston of the partcle s weghed by whether the ftness value s optmal. In each teraton, the partcles are updated by trackng two extreme values. One s the optmal soluton, whch s called as the ndvdual extreme value, namely pbest; the other s the best soluton of the whole partcle swarm currently, whch s called as the global extreme value, namely gbest. The ntalzaton of PSO s a group of random partcle, whch optmal soluton s found by teraton. The neghbors 2 of the part optmum partcles can be used and the extreme of all neghbors s a local extreme value. Because PSO algorthm s smple and easy to realze, t has been more and more appled n some felds, such as the functon optmzaton, the neural network tranng, the pattern classfcaton and the tradtonal optmzaton algorthm. The convergence speed of PSO algorthm s faster, but t s easy to fall nto the local optmum, whch s related to the selecton of the nerta factor!. Accordng to experence, the greater the nerta weght factor of the partcle s, the better the global optmum of the partcle s, snce the smaller! s better for the algorthm convergence. Therefore, at the begnnng of optmzaton, t s hoped that the larger nerta factor makes the scope of optmzaton wde. At the last stage of optmzaton, the smaller nerta factor s hoped to make the convergence better. The Flow of PSO algorthm s shown as Fg. (3). Fg. (3). Flow of PSO algorthm. Suppose that there s a populaton of m partcles, where the poston of partcle s p = (p 1, p 2,! p m ), the velocty s v = (v 1,v 2,!v m ), the updatng formula of poston and velocty s as follows. V k+1 =!V k + c 1 r 1 (P A " p k ) + c 2 r 2 (P B " p k ) (8) p k+1 = p k + v k Where, = 1,2,!,m, m s the scale of the partcle swarm. k k s the teratve number of algorthm, V represents the speed of partcle when the teraton number s k, p k represents the poston of partcle when the teraton number s k, r 1 and r 2 are the random numbers dstrbuted n the range of [0,1], c 1 and c 2 are the nonnegatve constants, whch are called as the acceleraton factor.! s the nerta factor. P A ndcates the optmal poston of a partcle n the partcle group, P B s the optmal poston of the partcle group. (9)

4 Research on PID Controller n Actve Magnetc Levtaton The Open Automaton and Control Systems Journal, 2015, Volume Fg. (4). Smulaton model of PID control system. 5. THE SIMULATION AND ANALYSIS In the actve magnetc levtaton system, the mass of the rotor s m=10 kg. The equlbrum poston of the rotor s x0 = m. The bas current of the electromagnetc col s I0 = 2.0A. The magnetc permeablty of the ar gap s µ0 = 4π 10-7 Vs/Am. The secton area of the magnetc pole s s0 = 340mm2. The number of turns of the electromagnetc col s N =190. These parameters are substtuted nto the equaton (7), so that the open-loop transfer functon of the actve magnetc levtaton control system can be obtaned as follows. G(s) =! 34 s 2! " 10 4 (10) The PID controller of ncomplete dfferental s adopted. The performance ndex s J = "! e(t) dt. The ntal parameters 0 of PSO are set. The sze of the partcle swarm s 100. The acceleraton constant s respectvely c1 = 1.5,c2 = 2. The range of the velocty s [-1,1]. The number of optmzed parameters s 3, namely three parameters of k p, k and kd. The range of the optmzed parameters s [0,100]. The nerta weght coeffcent s! = 0.8, and the maxmum of the teraton s 100. The PID controller based on PSO s realzed by MATLAB and the system model s establshed by Smulnk [9-11], whch s shown as Fg. (4). Fg. (6). The adaptve value of the best ndvdual. The values of three partcles are passed to k p, k and kd, the output value s observed from the osclloscope and the performance ndex s output from the output termnal. The change curves of k p, k and kd are shown as Fg. (5). The best ndvdual ftness value s shown n Fg. (6). When the parameters of PID controller are optmal, the output curve of the control system s shown as Fg. (7). Fg. (7). The output curve of the control system. Fg. (5). The optmzaton curve of PID parameters. It can be seen from (Fg. 5) that the optmal parameters of PID controller are kp=48.85, k= and kd= after 25 teratons. It s also seen from (Fg. 6) that the best ndvdual ftness value contnuously decreases at the begnnng, the value tends to be untl the number of the ter-

5 1874 The Open Automaton and Control Systems Journal, 2015, Volume 7 Yanhong et al. aton s 27. The parameters of PID controller are used n the smulaton model. It can been seen from Fg. (7) that the output of the control system tends to be stable, whch can be proved by that the PSO algorthm s effectve n the actve magnetc levtaton control system. CONCLUSION Because the actve magnetc levtaton system has some characterstcs, such as the uncertanty of parameters, nonlnearty etc. In order to solve the dffcult problem of the parameters tunng n the tradtonal PID controller, n the actve magnetc levtaton control system, the partcle swarm optmzaton algorthm s combned wth the conventonal PID controller. PID controller s desgned based on partcle swarm optmzaton algorthm. The proportonal coeffcent, ntegral coeffcent and dfferental coeffcent of PID controller are regarded as three partcles. At the same tme, the performance ndex of IAE s used to mnmze the poston error of the rotor. The parameters of PID controller are adjusted onlne. PID controller based on PSO can make the actve magnetc levtaton system more stable. It s found by the experment that the fxed nertal coeffcent of partcle swarm algorthm has a great nfluence on the optmal range of partcle. The larger the nerta coeffcent ω s, the better the global optmal of partcle s, and the smaller the nerta coeffcent ω wll be, whch lad the foundaton for subsequent research. CONFLICT OF INTEREST The authors confrm that ths artcle content has no conflct of nterest. ACKNOWLEDGEMENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna (Grant No and No ), the basc applcaton and research program of Changzhou (Grant No. CJ ). The Natural Scence Foundaton of Jangsu Provnce (Grant No. BK ). REFERENCES [1] X. Wang, and G. Zhang, Magnetc levtaton bearng control system smulaton based on varable unverse fuzzy PID algorthm, Instrument Technque and Sensor, vol. 12, pp , [2] J. Zhang, and L. Xu, Fuzzy adaptve PID control of maglev system, Journal of Computer Applcatons, vol. 29, no. 6, pp , [3] J. Zhang, X. Pe, and H. Xng, Improved RBF network control, Journal of Harbn Unversty of Scence and Technology, vol. 16, no. 1, pp , [4] K. Shen, and C. Sun, Research on an actve magnetc suspenson system wth Matlab smulaton, Industral Instrumentaton and Automaton Devces, vol. 4, pp. 7-11, [5] E. Shamel, M. B. Khamesee, and J. P. Issoon, Nonlnear controller desgn for a magnetc levtaton devce, Mcrosystem Technologes, vol. 13, no. 5, pp , [6] M. Zhang, Study on the applcaton of the mproved partcle swarm ntellgent algorthm and mult-objectve optmzaton, [unpublshed]. [7] Z. Lv, and Z. Hou, Partcle swarm optmzaton algorthm wth adaptve mutaton, Chnese Journal of Electroncs, vol. 32, no. 3, pp , [8] L. Zhang, H. Yu, and D. Chen, Analyss and mprovement of partcle swarm optmzaton algorthm, Informaton and Control, vol. 33, no. 5, pp , [9] F. Sh, H. Wang, and L. Guo, MATLAB Intellgent Algorthm, 1 st ed. Behang Unversty Press: Bejng, [10] J. Lu, Advanced PID Control and Matlab Smulaton, 1 st ed. Publshng House of Electroncs Industry: Bejng, [11] W. Luo, J. Han, and H. Lan, Advanced smulaton of fuzzy control Based on the Matlab (Smulnk) language, Computer Smulaton, vol. 18, no. 3, pp , Receved: May 26, 2015 Revsed: July 14, 2015 Accepted: August 10, 2015 Yanhong et al.; Lcensee Bentham Open. Ths s an open access artcle lcensed under the terms of the ( whch permts unrestrcted, noncommercal use, dstrbuton and reproducton n any medum, provded the work s properly cted.