A pressure output feedback control of turbo compressor surge with a thrust magnetic bearing actuator

Transcription

1 Journal of Mehanial Siene and Tehnology 23 (2009) 1406~1414 Journal of Mehanial Siene and Tehnology DOI /s y A ressure outut feedba ontrol of turbo omressor surge with a thrust magneti bearing atuator Hyeong-Joon Ahn 1,*, Min-Seo Par 2, Dorsa Sanadgol 3, In-Hwang Par 4, Dong-Chul Han 5 and Eri H. Maslen 6 1 Deartment of Mehanial Engineering, Soongsil University, 511 Sangdo-dong, Dongja-gu, Seoul, Korea, Constrution Equiment R&D Center, Doosan Infraore, Hwasu-Dong, Dong-Gu, Inheon, Korea Deartment of Physis and Engineering, Sweet Briar College, Sweet Briar, VA 24595, USA 4 BK21 Shool for Creative Engineering Design of Next Generation Mehanial and Aerosae Systems, Seoul National University 5 Shool of Mehanial and Aerosae Engineering, Seoul National University, Seoul, Korea 6 Deartment of Mehanial and Aerosae Engineering, University of Virginia, Charlottesville, Virginia 22904, USA (Manusrit Reeived Marh 6, 2008; Revised Otober 3, 2008; Aeted November 27, 2008) Abstrat This aer resents a ressure outut feedba ontrol of turbo omressor surge using ti learane atuation with a thrust magneti bearing atuator. First, stati and dynami omressor models were obtained for a ommerial turboharger, and the surge oint was found through loal stability analysis. Then, the effet of ti learane on the omressor ressure rise was derived, and Lyaunov analysis was used to establish a limit of stability with ti learane modulation. After that, a linear quadrati (LQ) state feedba ontrol was designed onsidering the limit established by the Lyaunov analysis. In addition, an extended Kalman filter (EKF) was designed to estimate the mass flow rate from the measured omressor ressure. Finally, the ressure outut feedba ontroller was built by ombining the LQ state feedba ontrol and EKF. Control simulation roved the effetiveness of the outut feedba ontroller. Keywords: Comressor surge; Ti learane modulation; Magneti bearings; Outut feedba ontroller Introdution The stable oeration of omressors is degraded by the onset of two flow instabilities: surge and rotating stall. These flow instabilities redue omressor effiieny and an ause severe damage to the omressor system. Hene, onsiderable researh has been direted toward ontrolling surge/rotating stall using different shemes to both revent damage and inrease the life san of the mahine [1]. Ative magneti bearings are widely used in turbomahinery aliations beause of their unique advan- This aer was reommended for ubliation in revised form by Assoiate Editor Dong Hwan Kim * Corresonding author. Tel.: , Fax.: address: ahj123@ssu.a.r KSME & Sringer 2009 tages: oil-free oeration, vibration ontrol and on-line monitoring. In addition, magneti bearings offer the otential for ative ontrol of the omressor flow instabilities. For turbo omressors suorted by magneti bearings, the blade ti learane an be adjusted by moving the shaft either radially or axially. Using radial magneti bearings as servo atuators, Saovsy et al. [2] obtained a 2.3% redution of the stalling mass flow. Sanadgol and Maslen [3-5] have rovided a theoretial basis to show that a magneti thrust bearing an be used as an atuator for ative surge ontrol of unshrouded entrifugal omressors. Through simulation, they have demonstrated that omressor surge an be stabilized desite extreme hanges in the throttle valve osition, whih results in an extended stable oerating region.

2 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~ Although a mass flow rate measurement is theoretially the best hoie for the ontrol of omressor instabilities, ressure measurement is used for the roortional feedba ontrol in most ases [1]. Comressor ressure is muh easier to measure than the mass flow rate, and ressure sensors usually have wider bandwidth than flow sensors. Therefore, better ontrol erformane an be ahieved if the omressor instability an be ontrolled by using dynami mass flow rates estimated from the measured ressures. Hene, ressure outut feedba ontrols have been reviously studied in [6, 7]. These studies, however, dealt with axial omressors and throttle valve ontrol without onsideration of roess and measurement noise in the design of the ontroller. This aer resents ressure outut feedba ontrol of omressor surge with a thrust magneti bearing atuator. First, stati and dynami omressor models were established for a ommerial turboharger and the surge oint was found through loal stability analysis. Then, a mathematial model of the ti learane effet on the omressor ressure rise was derived by extending the Greitzer omressor model [8], and the stabilizing feedba gain limit of the ti learane modulation was established by Lyaunov analysis. After that, a linear quadrati (LQ) state feedba ontrol was designed onsidering the stability limit. In addition, an extended Kalman filter (EKF) was designed to estimate mass flow rate from the measured omressor ressure. Pressure outut feedba ontrollers were onstruted by ombining the LQ state feedba ontrols with the EKF. A simulation was erformed using the omressor model of a ommerial turboharger, and the result roves the effetiveness of the designed ressure outut feedba ontroller for the omressor surge. 2. Comressor of a turboharger In this aer, an ative ontrol method of the omressor surge is investigated numerially by using the omressor model of a ommerial turboharger shown in Fig. 1(a). The turboharger onsists of a single-stage entrifugal omressor and a waste gate turbine. This omressor an be oerated u to 150,000 rm with floating ring journal bearings. For ative surge ontrol using axial ti learane atuation, the journal bearings of the turboharger are assumed to be relaed with magneti bearings, as shown in Fig. 1(b). (a) Turbo harger Fig. 1. A ommerial Turboharger. (b) AMB turboharger test rig Fig. 2. Aroximated omressor ma of the ommerial turboharger in Fig Stati omressor harateristi A omressor harateristi urve is aroximated ψ [9, 10]. by a ubi aroximation of ( ) φ 3 φ 2 F 1 φ 2 F 3 ψ ( φ ) = ψ ( 0) + H (1) Bau bearing Sensor where ψ (0) is the valley oint of the harateristi loated on the omressor ressure rise axis and φ = 2F orresonds to the dimensionless omressor mass flow rate at the to oint of the harateristi. The to oint ould be onsidered as the surge oint of the omressor. Then, the omressor harateristis at eah rotating seed ω an be formulated as a ubi olynomial of mass flow rate: (, ) ( ) ( ) ( ) ψ φ ω = ω + ω φ + ω φ (2) It is very diffiult to measure the omressor harateristi beyond the surge oint. The omressor harateristi at zero mass flow rate an be aroximated by using a theoretial alulation [8]. Therefore, the oeffiients 0 (ω), 1 (ω), and 2 (ω) of the entire omressor ma an be aroximated by data Radial bearings Thrust bearing Sensor Bau bearing

3 1408 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~1414 measured at the stable oerating oints and the alulated oint at zero mass flow rate, as shown in Fig Comressor dynamis Gravdahl derived a one-dimensional omressor system model desribed by Eqs. (3a)-(3) [11]. The first two equations are basially the same as Greitzer s model [12], and the last Eq. (3) balanes the torque on the rotating shaft, whih is ignored in this aer by assuming a onstant seed oeration. 2 a & = ( m& m& t) (3a) V && = A ( (, ) ) L ψ & ω (3b) m m 1 & ω = τ τ (3) I r ( ) d Fig. 3. Pole movements with equilibrium mass flow rate derease. Here, the throttle mass flow rate is given by: m& = t t 2.3 Loal stability analysis To find where the surge oints are loated, loal stability analysis is erformed. The loal stability near an equilibrium oint an be investigated by using a linearized lant of the nonlinear omressor harateristis. First, an equilibrium oint is determined by setting the derivatives of Eqs. (3) to zero. Then, the solution of the following equation (4) beomes the equilibrium oint. m& = ψ = 0 (4) te t e 0, The linearized lant in the equilibrium oint an be written as 2 2 a a t V & V (5) m&& ˆ mˆ A A ψ & L L m m= me ˆ 2 1 ˆ = The system is ontrollable and observable in the entire range. Aording to the hange of the equilibrium mass flow rate, the system ole movements are shown in Fig. 3. As exeted, the system beomes unstable as the mass flow rate dereases. However, e Fig. 4. Limit yle amlitudes of omressor surges. the omressor system has a stable limit yle in the hase ortrait even at the unstable region, whih is alled omressor surge. 2.4 Limit yle of the omressor surge The limit yles of omressor surge an be numerially determined. The system an be modeled with a differene equation and simulated by using the Euler forward method for its simliity. One erent roess noise of the omressor model is added during the numerial simulation. As mass flow rate is dereased, the limit yle amlitudes of omressor surges are alulated and shown in Fig. 4. The ea to ea amlitude of the limit yle inreases gradually as the mass flow dereases. The small amlitude limit yles right after the destabilization oint (between the two vertial marer lines) are alled a mild surge. However, fully develoed limit yles arise as the mass flow dereases further after the destabilization oint (left-most region in the figure), whih is alled a

4 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~ dee surge. 3. Control of omressor surge using ti learane atuation 3.1 Ti learane effet The ti learane between the blade tis and the adjaent shroud has a large effet on the omressor erformane. The reversal air flow from the ressure side to the sution side of the omressor blade is referred to as ti leaage. As the ti learane inreases, the indued leaage redues the energy transfer from the imeller to the fluid, whih onsequently rodues a substantial loss in effiieny and a ressure rise. If the ti learane at the best effiieny ondition is hanged, variations of the stati omressor harateristis are usually exressed as loss in effiieny. For the leaage loss in entrifugal omressors, a simle model was develoed by Senoo [13], shown in Eq. (6). The relative effiieny loss is nearly roortional to the ratio of the ti learane to the blade height at the imeller outlet rovided that this ratio is less than 0.1: Fig. 5. Ti learane effet on the ressure rise. η l = 0.25 (6) η b 2 The omressor stati isentroi effiieny an be alulated based on the erfet gas and the isentroi omression assumtions as: γ 1 γ T 1 η ( m& ) = (7) i T T ( ) Alying a quasi-steady aroximation for the effets of the ti learane based on Senoo s relation of Eq. (6) and the isentroi omression effiieny, Eq. (7), Sanadgol and Maslen [3] have derived that the omressor harateristi an be exressed as: γ γ 1 γ 1 γ 0 1 ( m, l ψ ψ & ) = l 1 + b 2 (8) The alulated ti learane effet on the ressure rise is shown in Fig. 5. The omressor ressure rise derease as ti learane inreases. Fig. 6. Error due to the ti learane aroximation. To establish a dynami model for the surge ontrol using ti learane atuation, the ti learane effet of Eq. (8) and Gravdahl s model of Eq. (3) are ombined and the resulting omressor dynami model is: γ γ 1 γ 1 A γ ψ 0 1 m&& = 1+ L 0.25 l 1+ b 2 (9) We assume that only the axial rotor motion modulates the instantaneous omression effiieny or the omressor ressure rise. Sine the ti learane modulation is usually small, the nonlinear dynami model of Eq. (9) an be aroximated by γ1 1 A A γ γ 0.25 m&& γ = ( ψ0) ψ 0 1ψ l (10) L L γ 1 b2

5 1410 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~1414 This model is a good aroximation of the original nonlinear system as shown in Fig. 6. The linearized equation (10) has very small relative error and an onfidently be used in a ontrol design. 3.2 Lyaunov analysis for stabilizing gain limit First, simle Lyaunov ontrol with mass flow rate feedba is alied to ontrol omressor surge by using ti learane atuation. Alying the following hange of oordinates to the system, the origin beomes the equilibrium under study (intersetion of the omressor and the throttle harateristis). m & ˆ = m & m & e, m& ˆ = m& m&, ˆ t t te = e, ψˆ = ψ ψ e and l ˆ = l l e (11) The model an be written in these new oordinates as: 2 a ˆ & = ( mˆ mˆt) V A m&& ˆ = mˆ l ˆ L ( ψ ˆ ˆ (, ) ) (12) Now, assume the use of mass flow rate feedba for ti learane atuation via the ontrol law: lˆ = mˆ (13) L A suffiient ondition for the equilibrium oint to be globally exonentially stable is that the gain L is hosen suh that: L ψˆ mˆ & (14) > ψˆ lˆ model: x & = Ax + Bu. T ( x Qx u Ru) T J = + dt (15) 0 The LQ ontrol an be designed by using the linearized system near an equilibrium oint. The LQ ontrol gain value is tuned onsidering the stabilizing gain limit of Eq. (14) established by Lyaunov analysis. Although a minimum energy ontrol that uses the least amount of ontrol effort is alied and Q in Eq. (15) beomes zero, the mass flow rate feedba gain of the LQ ontrol is still twie as high as the Lyaunov ontrol gain limit. Simulation results of both Lyaunov and LQ ontrols in a mild surge (equilibrium mass flow g/s) are shown in Fig. 7. Lyaunov ontrol gain is 0.68 m s/g, while the gain of the minimum energy LQ ontrol is m s/g (ressure feedba gain has a very small value of 2.02e-6 m/bar). The LQ ontrol shows faster stabilizing erformane due to the higher gain. It an be seen that the dynami harateristis of the losed-loo system are mainly determined by the mass flow rate feedba gain. 4. Outut feedba omntrol 4.1 Extended Kalman Filter (EKF) [14, 15] To build an outut feedba ontroller, a state estimator is required to estimate the mass flow rate from the ressure measurement. Although the Kalman filter has been most widely used to estimate states from noisy data in linear systems, the standard Kalman filter annot be alied to nonlinear systems suh as the omressor model in this aer. The EKF We will use the Lyaunov ontrol gain as a referene to tune the LQ ontrol gain. 3.3 LQ state feedba ontrol A state feedba ontroller is neessary in order to build an outut feedba ontroller. A simle LQ ontrol is used as a state feedba ontrol in this aer. For LQ ontrol, an otimal outut feedba gain is alulated to minimize the quadrati ost funtion, J in Eq. (15), of the ontinuous-time state-sae Fig. 7. LQ and Lyaunov ontrol.

6 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~ is the traditional form of nonlinear filtering and it essentially trunates the nonlinear dynamis at the first order to allow the use of linear Kalman filter equations. The EKF roedure is summarized in Eqs. (16-21). The EKF algorithm is done in two stes: redition and orretion. The redition ste involves rojeting or transforming the state in forward time-ste using the urrent nowledge of the model (Eq. (16)), as shown in Eq. (19). The seond ste is required to orret the redition by using the measurement at the new time ste (Eq. (17)), as shown in Eq. (20). The measurement is blended with the redition in an otimal manner in order to minimize the mean square error of the estimate. This blending fator is alled the Kalman gain in Eq. (21). (a) Phase ortraits Extended Kalman filter roedure State-udate equation: x 1 f ( x, u, v) Measurement equation: y g( x, n ) + = (16) = (17) T Initialize with xˆ0= E[ x0], P0 ( ˆ 0 0)( ˆ x = E x x x0x0) (18) For = {1, 2,, }, the time-udate equations of the extended Kalman filter are ( ) 1 T v T x 1 x 1 xˆ = f xˆ, u, v, P = A P A + B R B (19) and the measurement-udate equations are ( ) ( ) xˆ = xˆ + K y g xˆ, n P = I K C P x x ( ) 1 T T n T x x (20) K = P C C P C + D R D (21) where f ( x, u, v) f ( xˆ, u, v) A g x n C x x (, ) g( xˆ, n), B, x= xˆ v v= v and, D x= xˆ n n= n R v and R are the noise ovariane of v n and n, resetively. The noise means are denoted by n = E[] n and v = E v, and are usually assumed zero. [] A simulation in a mild surge (equilibrium mass flow g/s) was erformed to verify the estimation erformane of the designed EKF. Both 1% state and measurement noises (average ressure noise: 0.7 bar) were introdued during the simulation. (b) Time lot Fig. 8. Estimation of nonlinear omressor system using EKF. Estimation using full states (both ressure rise and mass flow rate) and estimation using outut (only ressure rise) signal are omared in Fig. 8. The KALMTOOL was used to imlement the extended Kalman filtering [15]. Noise ould be filtered signifiantly by the EKF. In addition, the EKF only with the ressure signal ould have omarable quality estimations of both ressure rise and mass flow rate, although a surge taes lae and the mass flow rate and ressure rise vibrate largely. 4.2 Simulations of outut feedba ontrols Control simulation of a mild surge (equilibrium mass flow is g/s) was erformed to test the ressure outut feedba ontrol using EKF. Both 1% roess and measurement noise (average ressure noise: 0.7 bar) were introdued. The LQ ontrol was used to stabilize the omressor surge. There is no

7 1412 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~1414 ontrol during the first 0.6 seond and after that, the ontrol turns on. The outut feedba ontrol using the EKF has omarable ontrol erformane to ontrol using mass flow rate signal, as shown in Fig. 9. The thrust magneti bearing has a limited learane of ±0.28 mm between the bearing and the housing. Thus, ti learane atuation is limited to a ±0.15 mm range. The ti learane variation during the ontrol is shown in Fig. 9(), whih illustrates that only a small variation of the ti learane is required in order to stabilize the surge. The simulation result shows that the designed outut feedba ontroller an stabilize omressor surge (a) Phase ortraits effetively. Although a surge henomenon begins, we an get stability of the omressor by using the outut feedba ontrol. Also, this result allows us to exet more effiieny when suorting the rotor of the turboharger by magneti bearings. 5. Conlusion We have disussed ressure outut feedba ontrols of omressor surge using ti learane atuation with a thrust magneti bearing atuator. First, we modeled the omressor harateristi of a ommerial turboharger and derived the ti learane effet on the omressor ressure rise by extending the Greitzer omressor model. Based on this model, Lyaunov analysis was erformed to establish the limit of the stable feedba gain. Then, an LQ state feedba ontroller was designed onsidering the stability limit of the feedba. In addition, an EKF was designed to estimate the mass flow rate by measuring the omressor ressure. Finally, ressure outut feedba ontrollers were designed by ombining the LQ feedba ontrol and the EKF. We verified with simulation that the outut feedba ontroller an stabilize the omressor surge effetively with axial magneti bearing. Anowledgement Authors wish to than KOSEF and KRF for their grant of the international ooerative researh rogram. Nomenlature (b) Time lot () Ti learane variations Fig. 9. The outut feedba ontrol simulations. a o1 : Seed of sound at the ambient ondition A and B : State equation for LQ ontrol A : Comressor dut area A, B, C and D : Linearized disrete state and measurement equation matries at th ste b 2 : Blade height at imeller outlet 0, 1 and 2 : Curve-fitted oeffiients of omressor harateristi olynomial : Seifi heat at onstant ressure F : Comressor harateristi semi width f : Nonlinear state equation for g : Nonlinear measurement equation H : Comressor harateristi semi height I r : Comressor rotor inertia J : Linear quadrati ost funtion

8 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~ K : Disrete Kalman gain at th ste t : Throttle arameter L : Lyaunov ontrol gain L : Comressor dut length l : Ti learane m& : Comressor mass flow rate m& : Throttle mass flow rate t n : Measurement noise at th ste P x0 : Covariane matrix of states o1 : Ambient ressure : Plenum ressure Q : State weighting matrix for LQ ontrol R : Control weighting matrix for LQ ontrol R v : Proess noise ovariane R n : Meausrement noise ovariane t : Time u :Control inut for LQ ontrol u : Control inut at th ste V : Plenum volume v : Proess noise x : State variable x : Disrete state variable at th ste y : Measurement signal at th ste φ : Comressor mass flow rate oeffiient Γ : Seifi heat ratio η : Effiieny of omressor η i : Isentroi effiieny of omressor τ : Comressor torque τ d : Drive torque ω : Rotating seed : Comressor stati ressure harateristi Ψ Referenes [1] F. Willems and B. De Jager, Modeling and ontrol of omressor flow instability, IEEE ontrol systems, (1999) [2] Z. S. Saovsy, J. D. Paduano, R. Larsonneur, A. Traxler and M. M. Bright, Ti Clearane Atuation with Magneti Bearing For High Seed Comressor Stall Control, ASME Turbo Exo (2000). [3] D. Sanadgol and E. H. Maslen, Sliding Mode Controller for Ative Control of Surge in Centrifugal Comressors with Magneti Thrust Bearing Atuation, Proeedings of the Ninth International Symosium on Magneti Bearings, Lexington, USA (2004). [4] D. Sanadgol and E. H. Maslen, Basteing for Ative Control of Surge in Unshrouded Centrifugal Comressors with Magneti Thrust Bearing Atuation, ASME TURBO EXPO (2005). [5] D. Sanadgol and E. H. Maslen, Effets of Atuator Dynamis in Ative Control of Surge with Magneti Thrust Bearing Atuation, IEEE/ASME International Conferene on Advaned Intelligent Mehatronis, Monterey, CA, USA, (2005). [6] J. R. Corrado, W. M. Haddad and A. Leonessa, Pressure feedba redued-order dynami omensation for axial flow omression systems, Proeedings of the 1999 IEEE International Conferene on Control Aliations, Kohala Coast, HI, USA, (1999) [7] M. Maggiore and K. Passino, Outut feedba ontrol of stabilizable and in omletely observable systems: Jet engine stall and surge ontrol, Proeedings of the 2000 Amerian Control Conferene, 5,, Chiago, IL, USA, (2000) [8] F. Willems, Modeling and Bounded Feedba Stabilization of Centrifugal Comressor Surge, PhD thesis, Det. of Mehanial Engineering, Eindhoven University of Tehnology, Netherlands, (2000). [9] F. Willems, B. De Jager, J. T. Gravdahl and O. Egeland, Modeling for surge ontrol of entrifugal omressors: omarison with exeriment, Pro. of the 39th IEEE onferene on Deision and Control (2000). [10] O. Egeland, J. T. Gravdahl and S. O. Vatland, Ative surge ontrol of entrifugal omressors using drive torque, Proeedings of the 40th IEEE Conferene on Deision and Control, (20). [11] F. Moore and E. M. Greitzer, A theory of oststall transients in axial omression systems: Part I- develoment of equations, ASME J. Engineering for Gas Turbines and Power, 108 (1) (1986) [12] D. A. Fin, N.A. Cumsty and E. M. Greitzer, Surge dynamis in a free-sool entrifugal omressor System, J. of Turbomahinery, 114 (1992) [13] Y. Senoo, Pressure losses and flow field distortion indued by ti learane of entrifugal and axial omressors, JSME Int. Journal, 30 (261) (1987) [14] M. E. Cambell and S. Brune, Nonlinear estimation of airraft models for on-line ontrol ustomization, Aerosae Conferene, 20, IEEE Proeedings, 2, Big Sy, MT, USA, (20) [15] M. Norgaard, KALMTOOL, State estimation for Nonlinear Systems, 2-edition, (20).

9 1414 H -J. Ahn et al. / Journal of Mehanial Siene and Tehnology 23 (2009) 1406~1414 Dr. Ahn earned Ph.D. from Seoul National University in 20. He was a researh assoiate of University of Virginia. He is urrently an assistant rofessor of deartment of mehanial engineering at Soongsil University and serving as an editor of international journal of rotating mahinery. His researh interests are rotordynamis, ontrol and mehatronis. Mr. Par is a junior researh engineer in Doosan infraore. He reeived his master from Seoul National University. His researh area is on dynamis and ontrol of rotating mahinery. Dr. Sanadgol is an assistant Professor of Physis and Engineering at Sweet Briar College. She earned her PhD in Mehanial and Aerosae Engineering with a fous in ontrols from the University of Virginia in Her researh interests are in ontrolling flow instabilities in omressors and aliation of nonlinear ontrol theories to mehatronis systems. Dr. Par reeived his PhD degree from the Seoul National University, Korea in He is urrently diretor of researh institute at KMB&SENSOR omany. His researh interests inlude the reision mahine design, rotor dynamis, and magneti atuators. Dr. Han reeived the Dil.-Ing. and Dr.-Ing. in mehanial engineering from University of Karsruhe, Germany in 1975 and 1979, resetively. In 1982, he joined the shool of mehanial and aerosae engineering, Seoul National University as an assistant rofessor. He is urrently an honorary rofessor of mehanial engineering. His researh interests are in mahine element design, magneti bearing, lubriation engineering and Bio-MEMS devies. Dr. Maslen is a Professor of Mehanial and Aerosae Engineering at the University of Virginia. He earned his Bahelor of Siene in 1980 from Cornell University and his dotorate from the University of Virginia in His researh fouses on aliation of automati ontrols to eletromehanial systems with a onentration in magneti bearings.