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1 Available online at ScienceDirect rocedia Technology (16 ) th International onference Interdiscilinarity in Engineering, INTER-EN 15, 8-9 October 15, Tirg-Mres, Romania IM based ID ontrol of a Magnetic Levitation System Adrian-Vasile Da a, Mircea Dl a, Stelian-Emilian Oltean a, * a etr Maior University of Tg. Mre, No. 1 N.Iorga St., Tg.Mre, 5488, Romania Abstract Attraction tye magnetic levitation devices are nonlinear and nstable systems with fast dynamics. If a model of sch a system can be rodced, it cold be sed in the design rocess of a stabilizing controller. Internal Model ontrol (IM) rovides a strategy that exlicitly ses an existing model of the controlled rocess for develoing a sitable controller. In this aer, a linear model that reresents the nonlinear dynamics of the magnetic levitation system is first derived. Then, this model is sed in the design rocedre of an IM-based ID controller, which is sed for achieving stable levitation of a ferromagnetic object at redetermined distances with the hel of the magnetic field rodced by a coil. The reslts are shown by means of digital simlation, based on Simlin. 16 The Athors. blished by by Elsevier Ltd. Ltd. This is an oen access article nder the BY-N-ND license (htt://creativecommons.org/licenses/by-nc-nd/4./). eer-review nder resonsibility of the etr Maior University of Tirg-Mres, Faclty of Engineering. eer-review nder resonsibility of the etr Maior University of Tirg Mres, Faclty of Engineering Keywords: magnetic levitation; Internal Model ontrol; ID control. 1. Introdction Magnetic levitation is the rocess by which a ferromagnetic object is ssended in the air against gravity with the hel of a magnetic field generated by a coil. This rocess resents many ractical alications sch as: active magnetic bearings, vibration daming, ssension of wind tnnel models, transortation systems (e.g high seed assenger trains) etc. This aer investigates the develoment of a control system for a single degree of freedom magnetic levitation rocess who aims at obtaining stable levitation of a steel ball at redetermined distances, sing an IM-based ID controller. While the rincile of levitation is simle: by controlling the crrent throgh the coil an electromagnetic * orresonding athor. Tel.: address: adrian.da@ing.m.ro The Athors. blished by Elsevier Ltd. This is an oen access article nder the BY-N-ND license (htt://creativecommons.org/licenses/by-nc-nd/4./). eer-review nder resonsibility of the etr Maior University of Tirg Mres, Faclty of Engineering doi:1.116/j.rotcy

2 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) force is generated which is able to conteract the weight of a steel ball, the magnetic levitation systems (MLS) are nonlinear, nstable systems with fast dynamics. These roerties mae them good candidates as test beds for different control algorithms. Over the ast years, varios control strategies have been roosed for this tye of systems, sch as ID [,1,1], hase-lead comensation [16,19], feedbac linearization controllers [11,13,], sliding mode control [1,5], fzzy logic ID-tye controllers[9,1,1], neral networs [3] etc. In many cases, ID has roved itself to be an effective soltion in controlling these systems. It is easy to imlement, bt setting its arameters is somewhat difficlt, de to the natre of the MLS. The roblems reslt from the lant model ncertainties and the small oerating ranges of MLS. The IM control strategy is based on the fact that if a model of the controlled lant can be rodced, even if it is an aroximate one, it can be sed exlicitly in the design of the controller. For many lant models in indstry, classical ID controllers can be viewed as eqivalent arameterizations of IM controllers [14,18]. Using the IM secifications, the tning rocess of a ID controller is simlified and it is redced to changing jst one arameter, the closed loo time constant (the IM filter factor ), instead of three [4,14]. In [4,14,18] ID tye controllers are designed based on IM for different tyes of lants (first, second order, stable, nstable, dead time). For the MLS considered in this aer, first, a linear model is dedcted, which is written in the form of a second order transfer fnction with oles on both sides of the comlex lane. Then, as shown in section 3, this model is sed to design an IM-based ID controller for the lant. The effectiveness of the reslted controller is demonstrated by means of digital simlation sing Matlab/Simlin sing a nonlinear model of the lant, which was validated in [8].. lant model The mathematical model of the MLS, shown in eqation (1), is fond by alying Newton s law for the eqilibrim of forces. d x( m mg + f ( x, i, (1) e dt where: where f e (x,i, is the electromagnetic force that conteracts the weight of the ball, x( is the distance between the coil and the steel ball, i( is the crrent throgh the coil, m is the mass of the ball and g is the gravitational constant [1,19]. Fig. 1. The magnetic levitation system.

3 594 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) The electromagnetic force f e generated by crrent i( which flows throgh the coil is given by [7,19]: i( f e ( x, i, () x( where is a nonlinear arameter which is assmed to be constant for model simlification roses. In reality arameter deends on the levitation oint, as demonstrated in [8] and encaslates some of the system s nonlinearities which are very difficlt to model. For a given levitation distance (X ) this arameter was determined exerimentally. The linear model of the lant is determined sing system linearization abot an eqilibrim oint (I, X), where I is the crrent throgh the coil when the ball is at X, by exanding the Taylor series of () and reserving the first order terms. I I I f e ( x, i, ( ) + ( ) +... i I 3 x X (3) X X X In stationary regime, when levitation is achieved (i(i and x(x ) the electromagnetic force cancels gravity and the acceleration dx/dt, eqation (1) taes the following form [19]: I mg (4) X By combining (1),(3) and (4) we get: d xˆ I I m iˆ + xˆ (5) 3 dt X X where xˆ x X and iˆ i I. Eqation (5) reresents the linear eqation which describes the dynamic of the magnetic levitation system (lan. Based on this eqation, sing Lalace transform, we get the transfer fnction of the lant shown in eqation (6). H 1 ms Xˆ (6) Iˆ where 1 and are two constants deending on arameters, I and X. The dislacement of the steel ball arond the oerating oint X is determined sing a sensor system consisting of an infra-red LED and a hototransistor. The crrent throgh the coil is rodced with the hel of a crrent amlifier. Both the sensor and the crrent amlifier are linear elements which can be described by the roortional gain transfer fnctions K sens, K am. By considering these additional elements the following transfer fnction of the rocess is fond: ~ ' 1 1 K amk sens (7) ms ms

4 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) where 1 1 K am K sens The arameters in Table 1 characterize the rocess linearized abot the eqilibrim oint (X,I ). Table 1. MLS arameters. arameter Vale Observation m.11 g Mass of the levitated steel ball X.76 m Eqilibrim osition I.41 A Eqilibrim crrent Nm /A onstant, corresonding to the air (X, I ) (7.6mm, 41mA) N/A arameter of the transfer fnction in (6) N/m arameter of the transfer fnction in (6) K sens Arox V/m Sensor system transfer fnction (gain) K am.1 A/V rrent amlifier transfer fnction (gain) 3. IM-based ID control of the MLS 3.1. The IM-ID eqivalence Fig..a shows the standard IM strctre. By manilating the bloc diagram, an eqivalent feedbac control strctre can be obtained (Fig..b), where the feedbac controller is given by: ~ 1 where: form of a ID controller [18]. IM IM IM is the internal model controller and ~ is the internal model. In most cases has the (8) Fig.. (a) IM strctre; (b) Eqivalent feedbac strctre.

5 596 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) IM-based ID design for the MLS This section shows the design rocedre for a ID controller eqivalent to the IM for the MLS modeled in section. For this rose, the model of the lant of eqation (7) is rewritten in the following form: ~ ( τ s + 1)( τ s + 1) (9) where: ' / 1 is the rocess gain and τ τ m / are two ositive time constants. As seen in (9), the rocess has two oles, located on both sides of the comlex lane. s1/ reresents the nstable ole located in the right half art of the comlex lane, while s-1/ is the stable ole. The first ste in finding the ID controller is finding the internal model controller s transfer fnction IM (s), as shown in eqation (1). IM ~ 1 F() s As recommended in [4,15] this fnction incldes a low-ass filter: (1) F γs + 1 λ ( s + 1) n (11) where: n is the filter s order and is sally chosen to mae IM (s) roer or semiroer, is the filter s time constant and satisfies the filter reqirement of eqation (1). F s 1 1 τ (1) Becase we want to end with an ideal ID controller, we choose n, maing IM (s) imroer. According to [17,18] this action is allowed since IM (s) will have a zero excess of at most 1 which is needed for the ID. Solving eqation (1) for we find: λ γ λ + (13) τ In order to find the eqivalent feedbac controller, we se the transformation of eqation (8) as follows: ~ 1 IM IM 1 ( τ s + 1)( τ s + 1) γs + 1 ( λs + 1) ( τ s + 1)( τ s + 1) γs + 1 ( τ s + 1)( τ s + 1) ( λs + 1) (14) which leads to:

6 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) ( γ + τ ) γτ s + ( γ + τ ) τ s + 1 (15) λ ( γ + τ )s Recalling that the transfer fnction for a standard ID controller is: 1 TiTd s + Ti s Td s Ti s (16) Ti s we find the following relationshis for the ID arameters [4]: T T i d τ ( γ + τ ) λ γ + τ γτ γ + τ (17) As seen the ID arameters in eqation (17) deend on a single variable, the low-ass filter s time constant. 4. Simlation reslts In order to test the reslted controller, Matlab/Simlin simlations were erformed. These simlations were based on the following closed loo model of the magnetic levitation control system. Fig. 3. Schematic diagram illstrating the model sed for simlating the magnetic levitation control system sing the IM-based ID controller The arameters of Fig. 3 are: Xˆ - dislacement of the ball from the eqilibrim osition; V x sensor ott X ˆ ; V - voltage, roortional to Xˆ ; V s - setoint voltage needed to ee the ball at the desired osition sly voltage needed for I ; Vˆ - ott control voltage; I - electromagnet crrent. The ID controller reslted from the rocedre discssed in Section 3 is an ideal controller, which is imroer. The imlementation isses that it raises are addressed by considering a filter coefficient N1 for the derivative comonent, that sets the location of the ole in the derivative filter far away in left hand art of the comlex lane. Eqation (18) shows the transfer fnction of the considered ID controller. 1 T () d Ns s (18) Ti s s + N

7 598 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) Since is the only tning arameter of the controller (see eqation (17) and (13)), Fig. 4.a shows the closed loo ott resonses to a ste int for different vales of. All trajectories shown in the next figres are relative to the eqilibrim osition X. Fig. 4. (a) losed loo ott resonse to a ste int for different vales of ; (b) losed loo ott resonse to a ste int for different vales of, with setoint filter. Fig.4 shows that the closed loo resonse to ste int has qite significant overshoot. This is not accetable since the levitated object cold find itself otside the range of the osition sensor. A ste resonse withot overshoot can be obtained by sing a setoint filter of the following form [4]: F s 1 γ s + 1 (19) Fig. 4.b shows the closed loo ott resonses to a ste int for different vales of, considering the setoint filter of eqation (19). By choosing.8, almost erfect set-oint tracing caabilities (no overshoot, fast resonse time) are obtained, as shown in Fig. 5. Fig. 5. losed loo ott resonse for sqare trajectory tracing (.8), with setoint filter.

8 Adrian-Vasile Da et al. / rocedia Technology ( 16 ) onclsion This aer showed the design rocedre of an IM-based ID controller for a nonlinear nstable rocess. For the MLS sed in this aer, a second order linear model was fond that had one nstable ole. Using this model as internal model, by following the rocedre described in Section 3 and in [4], an ideal ID controller was determined for the rocess. The reslting controller had the advantage that its tning arameters ( c, T i, T d ) had an analytical reresentation which deended on model s arameters and on a single variable arameter (), which became the controller tning arameter. This simlified the tning rocess. The reslts in Section 4 show the closed loo system resonse to a ste change for varios vales of. By sing a standard choice as setoint filter the system s resonse is imroved and the initial overshoot is eliminated. hoosing a convenient vale for reslts in a damed resonse with short settling time, which is sitable for the limited oerating range of the osition sensor, bt also for assring erfect setoint tracing caabilities. References [1] Al-Mthair NF, Zribi M. Sliding Mode ontrol of a Magnetic Levitation System. Mathematical roblems in Engineering, no.,.93 17, 4 [] Ahmad I, Shahzad M, alensy. Otimal ID control of Magnetic Levitation System sing enetic Algorithm. Energy onference (ENERYON), 14 IEEE International, vol., no., , May 14 [3] Aliasghary M et al. Sliding Mode ontrol of Magnetic Levitation System Using Radial Basis Fnction Neral Networs. Robotics, Atomation and Mechatronics, 8 IEEE onference on, vol., no., , 1-4 Set. 8 [4] Beqette BW. rocess ontrol: Modeling, Design, and Simlation. rentice Hall rofessional, hater 9, , 3 [5] Bethox O, Floqet T, Barbot J. Advanced sliding mode stabilization of a Levitation system. Eroean ontrol onference (E), 3, vol., no.,.51-56, 1-4 Set. 3 [6] De-Sheng Li, Jie Li, Wen-Sen hang. Internal Model ontrol for Magnetic Ssension System. Machine Learning and ybernetics, 5. roceedings of 5 International onference on, vol.1, no., , 18-1 Ag. 5 [7] Dolga V, Dolga L. Modelling and Simlation of a Magnetic Levitation System, Annals of the Oradea University, Fascicle of Management and Technological Engineering, vol. VI (XVI), , 7 [8] Da AV. Modelling of an electromagnetic levitation system sing a neral networ, 1 IEEE International onference on Atomation, Qality and Testing Robotics (AQTR1), vol. I,.79-83, May 1 [9] olob M, Tovorni B. Modeling and control of the magnetic ssension system. ISA Transactions, vol. 4, isse 1,. 89 1, Jan. 3 [1] o QH, et al. Research on a Maglev Ball ontrol System Based on DS81, rogress In Electromagnetics Research Symosim (IERS) roceedings, , 4-8 Mar. 8 [11] Ishtiaq Ahmad, Mhammad Aram Javaid. Nonlinear model & controller design for magnetic levitation system. ISRA'1 roceedings of the 9th WSEAS international conference on Signal rocessing, robotics and atomation, , 1 [1 ]Jie Ma, Wenjn Fan, Fengha He. arameters self-adjsting fzzy ID control in magnetic levitation system. Systems and ontrol in Aerosace and Astronatics, 8. ISSAA 8. nd International Symosim on, vol., no.,.1-5, 1-1 Dec. 8 [13 ]Kmar T, et al. Modeling, simlation and control of single actator magnetic levitation system. Engineering and omtational Sciences (RAES), 14 Recent Advances in, vol., no.,.1-6, 6-8 Mar. 14 [14] Morari M, et al. Imlications of Internal Model ontrol for ID ontrollers American ontrol onference, vol., no., , 1984 [15] Morari M, Zafirio E. Robst rocess ontrol, rentice-hall, 1989 [16] Namovi MB, Veseli BR. Magnetic Levitation System in ontrol Engineering Edcation, FATA UNIVERSITATIS Series: Atomatic ontrol and Robotics, vol.7, no.1, , 8 [17] Rivera DE. Internal Model ontrol: A comrehensive view. Technical reort, Deartment of hemical, Bio and Materials Engineering, ollege of Engineering and Alied Sciences, Arizona State University, [18] Rivera DE, Morari M, Sogestad S. Internal Model ontrol. 4. ID controller design. Ind. Eng. hem. Des. Dev., 5:5-65, 1986 [19] Shiao YS. Design and Imlementation of a ontroller for a Magnetic Levitation System. roc. Natl. Sci. onc. vol. 11 no., , 1. [] Bhawna Tandon et al. Exlicit Feedbac Linearization of Magnetic Levitation System. World Academy of Science, Engineering and Technology, International Jornal of omter, ontrol, Qantm and Information Engineering, vol. 8, no. 1, , 14 [1] Yadav S, Tiwari J, Nagar SK. Digital ontrol of Magnetic Levitation System sing Fzzy Logic ontroller. International Jornal of omter Alications, vol.41, no.1,. -6, Mar. 1