The Levitation Controller Design of an Electromagnetic Suspension Vehicle using Gain Scheduled Control

Transcription

1 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, 6 35 he Levitation Controller Design of an Electromagnetic Suspension Vehicle using Gain Scheule Control Min-Soo Kim, Yeun-Sub Byun, Young-Hoon Lee an Kwan-Sup Lee Maglev rain System Research eam Korea Railroa Research Institute 36-1 Woulam-Dong, Uiwang-City, Kyonggi-Do KOREA ms_kim@krri.re.kr ysbyun@krri.re.kr yhlee@krri.re.kr kslee@krri.re.kr Abstract: - Maglev vehicles constitute a new class of transport systems that has been constantly evelope an improve to become a new alternative of comfortable an secure transport. hese vehicles have suspension, propulsion an guiance systems base on magnetic forces. Electromagnetic suspension vehicle systems are highly nonlinear an essentially unstable systems. Levitation of an electromagnetically levitate vehicle prototype can be accomplishe with the action of attraction forces prouce by electromagnets. his paper escribes a robust controller esign metho of electromagnetic suspension systems using gain scheule control. For achieving the levitation control of the DC electromagnetic suspension system, we consiere gain scheuling metho. he propose gain scheuling is base on linear quaratic Gaussian(LQG) an continuous Kalman estimator for wiely varying nonlinear an parameter epenent ynamic system. Simulation results show that the propose gain scheuling metho base on LQG an continuous Kalman filter methoology robustly yiels uniform performance with aequate gap response over the mass variation range. Key-Wors:Electromagnetic Suspension System, Gain Scheuling, Linear Quaratic Gaussian 1 Introuction Maglev vehicles constitute a new class of transport systems that has been constantly evelope an improve to become a new alternative of comfortable an secure transport. hese vehicles have suspension, propulsion an guiance systems base on magnetic forces. A controlle DC electromagnetic suspension system is a highly nonlinear position regulator but an aequate insight into the esign requirements can be obtaine by consiering a linear moel. Different schemes of stabilization an control of single-egree of freeom suspension systems have been extensively stuie at recent years [1]~[9]. In this paper, an electromagnetically levitate vehicle prototype is consiere. Levitation can be accomplishe with the action of attraction forces prouce by electromagnets. In this type of suspension levitation may be attaine at any spee since the attraction force is inepenent of it. On the other han the levitation phenomenon is unstable an the electromagnetic force is nonlinear giving rise to ifficulties to get close-loop stability. Eight electromagnets are locate at the sie of the bogie to prouce the attraction forces an gap sensors are use to measure the gaps of the electromagnets. Recent efforts in the area of gain scheule control aim at eveloping esign techniques that aress the above gap between the existing theory an applications. he gain-scheuling controllers are generally confine to near equilibrium operation because they are esigne on the basis of the plant equilibrium linearization. A linear system is associate with every operating point of a nonlinear system. We suggest the following gain-scheuling esign proceure. 1. Selection of scheuling variables.. Linearization of the nonlinear plant moel base on the scheuling variables. 3. Determination of the control input. 4. Calculation of the control gain using Linear Quaratic Gaussian (LQG) base on the previous linear moel. 5. Design of the continuous Kalman filter to estimate state values.

2 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, he LQG regulator consists of an optimal state-feeback gain an a Kalman state estimator. he first esign step is to seek a state-feeback law that minimizes the cost function of regulation performance, which is measure by a quaratic performance criterion with user-specifie weighting matrices, an to efine the traeoff between regulation performance an control effort. he next esign step is to erive a state estimator using a Kalman filter because the optimal state feeback cannot be implemente without full state measurement. Since the Kalman filter is an optimal estimator when ealing with Gaussian white noise, it minimizes the asymptotic covariance of the estimation error[1][11]. his paper is organize as follows. Section iscusses briefly the single electromagnet levitation systems. Section 3 overviews the LQG controller esign. In section 4 contains the simulation results. he main conclusions are then summarize in section 5. Moeling of the Single Magnet Levitation Systems Each levitation electromagnet has an U shape as shown in Fig. 1 an its attraction force is given by µ N A i() t µ N Ai() t z() t vt () = Rit () + () z( t) t z ( t) t With ignoring the flux leakage an reluctance of electromagnet, we can get the nonlinear ynamic moel equation as equation (3) an (4). mz () t = F(, i z) + f () t + mg (3) ztit ()() zt () it () = + ( vt () Rit ()) (4) zt () µ N A where f () t is isturbance input, g is a gravity constant, A is cross-section area, an m is the mass of the suspene object. By choosing zt (), zt () an it () as the state variables, the nonlinear state-space representation of the above equation is where x () t = f[ x(), t u(), t f ()] t yt () = hxt [ ()] (5) x () t = x () t 1 1 µ N A x () t 1 x () t = + g+ f () t m x t m o 3 4 1( ) x () t x () t x ( t) x () t = + ( u() t Rx ()) t x1 () t µ on A yt () = x() t 1 (6) Fig. 1: Single Magnet Levitation System Fiz (, ) µ N Ai () t 4 z ( t) = (1) where i(t) is the current, z(t) is the air gap an N is number of turns. hus if R is the total resistance of the circuit then for an instantaneous voltage v(t) across the magnet wining, the excitation current is controlle by x () t = [ x (), t x (), t x ()] t = [ z(), t z (),()] t i t ut () = vt () Gain Scheule Control (7) 3.1 Linear Quaratic Gaussian Design Linear-quaratic-Gaussian (LQG) control is a moern state-space technique for esigning optimal ynamic regulators. It enables to trae off regulation performance an control effort, an to take into account process isturbances an measurement noise.

3 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, 6 35 Like pole placement, LQG esign requires a state-space moel of the plant. LQG esign aresses the regulation problem. he goal is to regulate the output y aroun zero. he plant is subject to isturbances an is riven by control inputs u. he regulator relies on the noisy measurements y = y+ n to generate these control inputs u. he plant state an measurement equations are of the form x = Ax+ Bu+ G (8) y = Cx+ Du+ H + n where an n are moele as white noise. he LQG regulator consists of an optimal state-feeback gain an a Kalman state estimator. hese two components are inepenently esigne output-feeback problem. his state estimate is generate by the Kalman filter x = Ax + Bu + L( y Cx Du) (1) t with control inputs u an measurements y. he noise covariance ata ( E ( ) = Q n, Enn ( ) = R n, E( n ) = N n ) etermines the Kalman gain L through an algebraic Riccati equation. he Kalman filter is an optimal estimator when ealing with Gaussian white noise. Specifically, it minimizes the asymptotic covariance lim E(( x x)( x x) ) of the estimation error t ( x x ). o form the LQG regulator, simply connect the Kalman filter an LQ-optimal gain K as shown below. 3. Optimal State-Feeback Gain In LQG control, the regulation performance is measure by a quaratic performance criterion of the form { } Ju ( ) = xqx+ xnu+ urut (9) he weighting matrices Q, N, an R are user specifie an efine the trae-off between regulation performance which is how fast x(t) goes to zero an control effort. he first esign step seeks a state-feeback law u=-kx that minimizes the cost function J(u). he minimizing gain matrix K is obtaine by solving an algebraic Riccati equation. his gain is calle the LQ-optimal gain. Fig. : LQG regulator his regulator has state-space equations x = [ A LC ( B LD ) K ] x + Ly t u = Kx (11) 3.3 Continuous Kalman Estimator he LQG solution is basically a state-feeback type of controller which it requires that all states be available for feeback. his is usually an unreasonable assumption an some form of state estimations necessary. As in the case of pole placement, the LQ-optimal state feeback u=-kx is not implementable without full state measurement. It is possible, however, to erive a state estimate x such that u = Kx remains optimal for the 4 Simulation 4.1 Controller Design Electromagnet use for this paper was previously use on a research vehicle. hey consist of 3 bogies which contain 4 electromagnets insulate copper wining woun on steel cores an esigne to lift a maximum loa of ten times of electromagnet mass at a nominal operating air gap of 1 [mm].

4 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, Disturbance f () t is selecte for scheuling variable which has limite bounary values an the isturbance estimator as equation (1) an (13), respectively. + F : = { f ( t) f ( t) [ f, f ], t } (1) µ N A it () f () t = m z() t + mg (13) 4 zt ( ) hen the linearization moe1 of the scheuling variable f () t is following equation: x () t = A( f ) x() t + B u() t + D f () t u( t) = K1 ( f) x( t) + K ( f) f( t) yt () = C xt () (14) z c δ = R K 13 ( ) From equation (16), we can calculate the controller gain: 11 = mg + f () t z 1 ( m g f t ) cm K () t c + () K 1 cm () t = (17) mg+ f () t c K13() t = R z Controller input u(t) calculate such as ut ( ) = K1 ( f( t)) xt ( ) x( f( t)) + u( f( t)) (18) where x() t = x() t x( f ) ut () = ut () u( f ) f () t = f () t f f A( f) = x( f), u( f), f x f B = x( f), u( f), f x f D = x( f), u( f), f x 4. Results he controller inputs v(t) is evaluate using the measure gaps z(t). We consier only single magnet levitation systems, but in practice the levitation of electromagnetic suspension vehicles has a large number of magnet an its actuators. he levitation control laws base on LQG an Kalman filter is use to improve system reliability. It is also expecte that the propose control scheme reuces the energy consumption require for levitation since power loss in each electromagnet is roughly proportional to its electromagnetic force. he characteristic equation of the close loop system of the linear moel ( 1 ) () s = et si A+ BK ( f ) = s + δ s + δ s+ δ 3 1 (15) where mg+ f 4 δ = K ( R K )( m g+ f ) cm cm δ = K mg+ f 1 1 cm (16) Fig. 4: Disturbance input he close-loop response for a reference gap of 1 [mm] from initial gap of 17 [mm] has been consiere.

5 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, A step isturbance (1% of the levitation force) between. [sec] an.5 [sec], an measurement noise (5% of the measurement values of gap sensor) have been applie to the plant by suenly laying a mass on the vehicle. Nominal air gap 1 [mm] an nominal mass 1,83 [kg] are use. he isturbance input is presente in Figure 4. Figure 5 shows that the transient gap response exhibits containing a isturbance an measurement noise. Fig. 7: Comparison of time responses over the mass variation range (nominal, 5%, an %) Fig. 5: Close-loop time response at the nominal point he isturbance has been completely rejecte in steay-state. Fig. 6 shows the control input as functions of time. Fig. 8: Comparison of control inputs over the mass variation range (nominal, 5%, an %) In these figures we can confirm that the time responses an control inputs are nearly invariant uner all mass variations. herefore we can conclue that the propose LGQ an Kalman filter base gain scheuling control system achieves the robust stability an performance uner mass variations. Fig. 6: Control input at the nominal point Figure 7 an figure 8 show the time response an the control inputs compare with the mass variation range which is nominal mass, 5% of mass, an % of mass, respectively. 5 Conclusion Gain scheuling is one of the most popular esign methos for a various control system. Control approach to the levitation of an electromagnetic suspension vehicle prototype base on gain scheuling coupling LQG an Kalman filter has been propose in this paper. he propose esign metho is very useful because it compensates the control gain while keeping the structure of the control system esigne by classical control metho. Simulation results have been shown that the propose gain scheuling metho base on LQG an continuous Kalman filter methoology robustly yiels

6 Proceeings of the 5th WSEAS Int. Conf. on CIRCUIS, SYSEMS, ELECRONICS, CONROL & SIGNAL PROCESSING, Dallas, USA, November 1-3, uniform performance with aequate gap response over the mass variation range. References: [1] Sinha, P. K., Electromagnetic suspension: ynamics an control, Peter Peregrinus Lt, Lonon, [] Sinha, P. K., an Pechev, A. N., Moel reference aaptive control of a maglev system with stable maximum escent criterion, Automatica35, 1999, pp [3] Jayawant, B. V., Electromagnetic suspension an levitation, Proceeings of the IEE, Vol.19, Pt.A, No.8, 199, pp [4] Huang, C. M., Yen, J. Y., an Chen, M. S., Aaptive nonlinear control of repulsive maglev suspension systems, Control Engineering Practice 8, No.1,, pp [5] Zheng, X. J., Wu, J. J., an Zhou, Y. H., Numerical Analyses on ynamic control of five-egree-of-freeom maglev vehicle moving on flexible guieways, Journal of soun an Vibration, Vol.35, No.1,, pp [6] Bohn, G. an Steinmetz, G., he electromagnetic levitation an guiance technology of the RANSRAPID test facility emslan, IEEE ransactions on Magnetics, Vol.MAG-, No.5, 1984, pp [7] Glatzel, K., Khurok G. an Rogg, D., he evelopment of the magnetically suspene transportation system in the feeral republic of germany, IEEE ransactions on Vehicular echnology, Vol.V-9, No.1, 198, pp [8] Meins, J., Miller, L. an Mayer, W. J., he high spee MAGLEV transportation system RANSRAPID, IEEE ransactions on Magnetics, Vol.MAG-4, No., 1988, pp [9] Moon, F. C., Superconucting Levitation: Applications to Bearings an Magnetic ransportation, John Wiley an Sons, [1] Athans, M., he role an use of the stochastic linear-quaratic-gaussian problem in control system esign, IEEE rans. on AC, Vol.AC16, No.6, [11] Athans, M., A tutorial on the LQG/LR Metho, Proc. ACC, 1986.