Nonlinear control of an active magnetic bearing with bias currents.pdf

3917 专用转塔车床回转盘部件设计

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Nonlinear control of an active magnetic bearing with bias currents:experimental studyThomas R. Grochmal and Alan F. LynchAbstractThis paper presents an experimental comparisonof position tracking controllers for a five degree-of-freedomactive magnetic bearing system. Two variations of a nonlineardesign are presented, each subject to a different actuatingconstraint: Constant Current Sum (CCS) uses bias currents andCurrent Almost Complementary (CAC) avoids bias currents.While both designs achieve accurate tracking for a non-rotatingshaft, a comparison of unbalance responses show that voltagesaturation can limit the dynamic response of the CAC-baseddesign. Performance of the nonlinear designs is also comparedto a decentralized PID controller.I. INTRODUCTIONActive Magnetic Bearings (AMBs) continue to receiveattention due to their unique characteristics. Contact freesuspension can lead to improved reliability and performanceover rolling-element bearings. AMBs provide variable stiff-ness and damping characteristics which allows for vibrationcompensation, force measurement, and precision motioncontrol 1.Linear feedback designs for AMBs have traditionallyutilized bias currents. These bias currents provide an op-erating point for the models linearization. Bias currents alsoprovide a higher dynamic force response 2. A number ofresearchers have considered zero- and low-bias controllers inan effort to minimize ohmic losses and reduce heating 3.Similar notions based on flux bias have been investigatedin 4 and 5. Time-varying, adaptive, and optimal bias ap-proaches have also been introduced to achieve a compromisebetween performance and low loss objectives 6, 7, 8.Nonlinear control techniques have been widely applied toAMBs and naturally lead to zero- and low-bias schemes9, 10, 11. In 10 L evine et al. design a nonlinearcontrol based on the flat property of the system and intro-duce the Current Almost Complementary (CAC) condition.This approach was experimentally validated for a machiningapplication in 12. Although the CAC condition can leadto low power losses, it can potentially require large inputvoltages 10. A variation of the flatness-based design usinga Constant Current Sum (CCS) bias condition is introducedin 13. This last work provides experimental validation ona five degree-of-freedom (DOF) system.This paper provides an experimental comparison betweenCAC and CCS-based nonlinear controllers. These designs areThis work was partially supported by the Natural Sciences and Engineer-ing Research Council of Canada (NSERC) under grant number 249681-02.Alan F. Lynch is with the Department of Electrical & ComputerEngineering, University of Alberta, Edmonton AB, T6G 2V4, CThomas R. Grochmal is a Ph.D. student under the supervision of A.F.Lynch.grochmalece.ualberta.cayzOFb,y,nFc,zFf,y,nFb,y,pFb,z,pFf,z,nFc,yFb,z,nxFf,z,pFf,y,pFx,pFx,nFig. 1.Shaft assembly, motor coupling, and radial bearing stators for a 5DOF AMB system.further compared to a decentralized PID controller. All threedesigns are implemented on a test stand manufactured bySKF Magnetic Bearings (Calgary, AB) that is interfaced tocontroller hardware developed at the University of Alberta.The contribution of this paper is to experimentally show thebenefits of a CCS-based nonlinear controller in providingaccurate trajectory tracking and robust stabilization underlow voltage limits.This paper is presented as follows. In Section II a dynamicmodel of the system is described. In Section III we identifythe parameters of the models magnetic force expressions.In Section IV the nonlinear feedback is formulated underthe CAC and CCS conditions. Finally, in Section V thetest bench is described, and in Section VI an experimentalcomparison of the controllers is provided.II. MODELINGThe five DOF AMB system consists of a horizontal shaftassembly coupled to a DC motor via a helical coupling.Figure 1 shows the shaft assembly and radial bearing stators.Figure 2 shows a cross section of the system. Assuming arigid shaft, the dynamic equations are 14m x = Fxm y = Fb,y+ Ff,y+ Fc,y+ mgym z = Fb,z+ Ff,z+ Fc,z+ mgzJz = (lb,a x)Fb,z+ (lf,a+ x)Ff,z Jx + lcFc,zJy = (lb,a x)Fb,y (lf,a+ x)Ff,y+ Jx lcFc,y(1)where x,y,z denote the coordinates of the center of masscmrelative to the origin O of the inertial frame. When theshaft is centered in all three bearings we have (x,y,z) = 0.The angles , denote the angular displacement of a bodyProceedings of the 2006 American Control ConferenceMinneapolis, Minnesota, USA, June 14-16, 2006FrA11.21-4244-0210-7/06/$20.00 2006 IEEE4558lb,alf,alb,slf,sW24b,yxyxaxial statorW13f,ycmOposition sensorposition sensorix,pix,nib,y,nif,y,nib,y,pif,y,nFig. 2.Cross section of Figure 1 in the x y plane. Motor coupling notshown.frame relative to the inertial frame. The angular velocityof the shaft about the x-axis is denoted by = and isassumed to be constant. The y- and z-axis components of thegravitational field are denoted by gyand gzrespectively. Theorientation of the radial bearings are such that gy= gz. Theshaft of mass m has principle moments of inertia denotedby Jx,Jy,Jz, and by the shafts symmetry Jy= Jz= J.The distances from the drive-end (subscript f) and non-drive-end (subscript b) stators to O are denoted lf,aand lb,arespectively. The motor coupling forces are denoted Fc,y/z1and are modeled as linear springsFc,y= K(y lc)Fc,z= K(z + lc)The distance from cmto the point at which Fc,y/zact is lc.The coupling spring constant is K. The axial bearing forceis denoted Fxand the drive-end and non-drive-end radialbearing forces are denoted Fb/f,y/z. Each of these forcesare the summation of positive (subscript p) and negative(subscript n) components generated by opposing coils. Theforce expressions areFx= Fx,p Fx,n=xi2x,p(x x)2xi2x,n(x+ x)2(2a)Ff/b,y/z= Ff/b,y/z,p Ff/b,y/z,n=f/b,y/zi2f/b,y/z,p( f/b,y/z)2f/b,y/zi2f/b,y/z,n( +f/b,y/z)2(2b)where (respectively x) is the nominal air gap betweenthe rotor and the radial (respectively axial) bearing stator.The bearing force constants are x,f/b,y/z, andf/b,y/zdenote the offset displacements of the shaft in the planesx = lb,aand x = lf,a(see Section III for further details onthis offset). The axial and radial bearing coils are driven bycurrents ix,p/n,if/b,y/z,p/nwhich we assume to be control1The shorthand expression Fc,y/zrefers to Fc,yand Fc,zinputs. In practice, an inner-loop current controller ensuresthe currents track their reference values sufficiently fast.III. FORCE PARAMETER IDENTIFICATIONIn this section we describe the identification of the radialbearing force parameters f/b,y/z. We also introduce mag-netic offsets f/b,y/zwhich represent the distances betweenthe center of the rotors and the center of the stators when(x,y,z,) = 0 15. When the position sensors arecalibrated the shaft is aligned to the center of the touchdownbearings and not the centers of the stators. If the touchdownbearings are not concentrically aligned to the stators, thenmagnetic offset results. See Figure 3. Mathematically, thisoffset enters asf/b,y/z= f/b,y/z+f/b,y/zwhere f/b,y/zare the displacements of the shaft in the planes x = lb,aandx = lf,a.Ob,yb,ystatorxytouchdown bearingbmagnetic centerFig. 3.Schematic of magnetic bearing system with magnetic offset.To obtain f/b,y/z,f/b,y/zwe consider the non-rotatingshaft decoupled from the motor and assume the system is inequilibrium with x, and equal to zero. Therefore model(1) becomes0 = Fx0 = Fb,y+ Ff,y+ mgy0 = Fb,z+ Ff,z+ mgy0 = lf,aFf,z lb,aFb,z0 = lb,aFb,y lf,aFf,ySolving for the equilibrium forces we obtainFb,y= Fb,z= mgy?lf,alb,a+ lf,a?= mbgy(3a)Ff,y= Ff,z= mgy?lb,alb,a+ lf,a?= mfgy(3b)where we have introduced effective masses mf/bsupportedby each radial bearing. For brevity, we present only the casefor the drive-end y-axis. From x = = = 0 we obtainf,y= y and substitute this into (2b). We impose the CCScondition where each coil is provided with a bias current anda differential current is added and subtracted to opposing4559coils. Therefore, the sum of currents from opposing coilsalways adds up to twice the bias current 1. We haveif,y,p= ib+ if,yif,y,n= ib if,ywhere ibis a constant bias and if,yis a differential current.Hence, we get the force expressionFf,y=f,y(ib+ if,y)2( f,y y)2f,y(ib if,y)2( + f,y+ y)2(4)and by combining (4) with (3b) we obtainmfgy=f,y(ib if,y)2( + f,y+ y)2f,y(ib+ if,y)2( f,y y)2Data is collected over a range of y and corresponding if,y.Defining the cost function?k(f,y,f,y) =f,y(ib if,y)2( + f,y+ y)2f,y(ib+ if,y)2( f,y y)2+ mfgywhere (ikf,y,yk),1 ? k ? N denotes the data set, we solvethe nonlinear least squares problemmin(f,y,f,y)UN?k=12kwhere U =?(x1,x2) R2: 0 x1, x2 0), both coils areenergized to prevent unbounded voltages at the zero forcepoint. For the x-axis the CAC condition isix,p=(x x)?FxxFx? (x x)?Fx+P2(Fx)x Fx 0Fx ix,n=0Fx? (x+x)xP(Fx) Fx (x+ x)?FxxFx The polynomial Pis chosen to ensure a smooth transitionat the switching instances Fx= 10.C. Constant Current Sum Condition (CCS)An alternate approach to inverting the force expressions(2a)-(2b) is to impose the CCS condition. For the axialbearing we express the force model asFx=x(ib+ ix)2(x x)2x(ib ix)2(x+ x)2(7)where ixis a differential current. Inversion of (7) gives thecontrol lawix=?x(x2+2x)ibx(x22x)Fxxx/x+i2b2xxxx ?= 0Fx2x/(4xib)x = 0(8)The expression for ixwhen x = 0 is determined by apply-ing lH opitals Rule. To ensure a non-negative discriminantin (8), it is sufficient to impose the limits4xi2b(x+ x)2 Fx4xi2b(x x)2This inequality is satisfied provided|ix| ibIn practice each coil is limited to a maximum current of Is.Setting ib= Is/2 provides the full range for ix,p/n.dSPACEmodularsystemCoil PWMamplifiers andsensor circuitryAMB test standSignalacquisitionFig. 4.Experimental setup.V. EXPERIMENTAL SETUPThe experimental setup is shown in Figure 4. The MBRo-tor Research Test Stand, available from SKF Magnetic Bear-ings, is used with the 305 mm long shaft configuration 15.The systems maximum rotational speed is 15000 rpm underno load. See Table III for bearing specifications.Specificationradial bearingaxial bearingstatic load cap.76 N205 Nsaturation current3.0 A2.8 Anominal gap525 m783 mstator ID35.1 mm38.6 mmstator OD82.8 mm71.4 mmstator length12.7 mm13.5 mmrotor OD34.3 mm66.0 mmTABLE IIIMAGNETIC BEARING SPECIFICATIONSBoth the position and current control loops are im-plemented using a modular dSPACE hardware system. AdSPACE DS1005 board performs real-time computations forcontrol at 10 kHz. Three DS2001 high-speed ADC boardssample ten coil currents and rotor displacement along fiveaxes. A DS3002 encoder board measures the rotationalvelocity of the shaft. A DS5101 board generates PWMwaveforms that drive the current control loops. The PWMswitching frequency is 10 kHz. A host PC provides a MAT-LAB/Simulink development platform and logs real-time datafrom the dSPACE system.VI. EXPERIMENTAL RESULTSA. Rotational StabilizationFigure 5 presents drive-end and non-drive-end orbitalplots of the shaft rotating at 14000 rpm. A comparison ismade between CCS-based nonlinear, CAC-based nonlinear,and PID control schemes. We remark that the unmodeledeffect of mass unbalance is significant because the shaftis manually assembled and no measures are taken to me-chanically balance it. For all controllers the feedback gainsare tuned to achieve a comparable performance over theoperating range of shaft speed. The controller gains are4561-50-40-30-20-1001020304050-50-40-30-20-1001020304050V13 mW13 mPIDCCS CAC -50-40-30-20-1001020304050-50-40-30-20-1001020304050V24 mW24 mPIDCCS CAC Fig. 5.Drive-end (left) and non-drive-end (right) orbital plot comparisonat 14000 rpm.00.0050.010.0150.00.811.21.4time scurrent ANonlinear CAC00.0050.010.0150.02-25-20-15-10-50510152025time svoltage VNonlinear CAC00.0050.010.0150.01.8time scurrent ANonlinear CCS00.0050.010.0150.02-4-20246810time svoltage VNonlinear CCS00.0050.010.0150.0200.511.522.5time scurrent APID00.0050.010.0150.02-20-15-10-50510152025time svoltage VPIDFig. 6.Non-drive-end z-axis currents ib,z,p(dashed line), ib,z,n(solidline) and voltages ub,z,p(dashed line), ub,z,n(solid line) for CAC (top),CCS (middle) and PID (bottom).given in Table IV along with the values of iband .From Figure 5 we observe the nonlinear tracking controllersprovide robust stabilization. They perform as well or betterthan PID which is often used for stabilization of high speedAMB applications 18. Figure 6 shows representative datak2s1k1s2k0s3 NCAC2501500001000000.1k2s1k1s2k0s3ibACCS3001500001000001.0kpA/mkiA s/mkdA/(m s)ibAPID800010000131.0TABLE IVFEEDBACK GAINS AND BIAS PARAMETERSfor the associated currents and voltages at 14000 rpm. Itis interesting to note that in the case of CAC, satisfactorystabilization performance may be achieved without biascurrents. This results in reduced power consumption, but theswitching action of opposingcoils requires high bandwidthinthe current controller output. Therefore, higher voltages arerequired by CAC in order to achieve the same stabilizationperformance as CCS. While the CCS achieves stability witha maximum voltage of approximately 10 V (not shown),we see that CAC coil voltages exceed 20 V. Therefore,lower voltage limits can limit the performance of zero-biasschemes. To further demonstrate the impact of low voltagelimits, Figure 7 shows the rotational performance of CACfor a voltage limit of 12 V. It is found that CAC is unableto stabilize the shaft for speeds greater than 5000 rpm. Thisresult cannot be improved by controller tuning. The tradeoffbetween biasing and maximum voltage requirements reflectsthe inductive nature of electromagnetic actuators which re-quire high voltages to achieve fast changes in current. Withvoltage limits zero-bias schemes are therefore generally lessrobust to disturbances than bias-based schemes due to currentslew rate limiting.9.81010.210.410.610.810050050100150y mtime s99.51010.500.511.522.5ib,y,p Atime s99.51010.500.511.522.5ib,y,n Atime s99.51010.515105051015ub,y,p Vtime s99.51010.515105051015ub,y,n Vtime sFig. 7.Destabilization at 5000 rpm using the CAC controller withsaturating voltage of 12 V. Top: shaft y coordinate. Middle: Non drive-end y-axis currents ib,y,p(left) and ib,y,n(right). Bottom: Correspondingvoltages ub,y,p(left) and ub,y,n(right).B. Tracking of a Non-rotating ShaftIn Figures 810 we demonstrate the tracking performanceof each controller to a time varying reference for the shaftscenter of mass. During these experiments the shaft speed is456200.00.25-100-50050100time sposition mZrefZYrefYX00.00.25-80-60-40-20020406080time sangle radFig. 8.Tracking performance of center of mass coordinates x,y,z (left)and , (right) for the CCS-based nonlinear controller.00.000.811.2time sif,y,p if,y,n A00.01.4time sif,y,p, if,y,n AFig. 9.Drive-end y-axis tracking currents for the CAC-based (left) andCCS-based (right) nonlinear controllers.zero, i.e., = 0. We use an elliptical trajectoryyr(t) = 50sin(20t) mzr(t) = 80cos(20t) mxr(t) = r(t) = r(t) = 0 mFigure 8 shows the performance of the CCS controller.Accurate tracking is observed: the tracking error of x,y,z isalways within 3 m and the angles , are stabilized towithin 45 rads. As expected, the tracking performance ofthe nonlinear controllers is unaffected by the choice of actu-ator condition, provided sufficient voltage head space exists.Figure 9 shows representative currents for each nonlineardesign. It is interesting to note that for CAC only the upperradial bearing coils need to be energized.Figure 10 shows the tracking performance for PID. De-centralized reference trajectories were generated using thetransformation (6). The performance is inadequate since theshaft overshoots its y and z coordinate trajectories and makescontact with the backup bearings. This contact accounts forthe distortion of the , signals. Varying the PID gains doesnot result in a noticeable improvement in performance.00.00.25-100-50050100time sposition mzrefzyrefyx00.00.25-80-60-40-20020406080time sangle radFig. 10.Tracking performance of center of mass coordinates x,y,z (left)and , (right) for the PID controller.VII. CONCLUSIONSThis paper presents an experimental study of nonlinearcontrol with and without bias currents. Under modest voltagelimits the nonlinear controller with bias demonstrates robuststabilization of a rotating shaft and tracks the position of anon-rotating shaft. By comparison, a zero-bias nonlinear de-sign requires larger voltages to achieve similar performance.Both nonlinear designs have a similar unbalance responseas a decentralized PID approach, but a PID controller can-not adequately track a time varying reference. Designs arevalidated on a commercially available five DOF test stand.REFERENCES1 G. Schweitzer, H. Bleuler, and A. Traxler, Active Magnetic Bearings.Switzerland: vdf Hochschulverlag AG an der ETH Zurich, 1994.2 E. Maslen, P. Hermann, M. Scott, and R. Humpris, “Practical limitsto the performance of magnetic bearings: Peak force, slew rate, anddisplacement sensitivity,” ASME J. Tribol., vol. 111, pp. 331336, Apr.1989.3 T. Nakamura, M. Hirata, and K. Nonami, “Zero bias Hcontrol ofactive magnetic bearings for energy storage flywheel systems,” in Proc.9thISMB, Lexington, KY, Aug. 2004, paper No. 134.4 L. Li, “Linearizing magnetic bearing actuators by constant currentsum, constant voltage sum, and constant flux sum,” IEEE Trans.Magn., vol. 35, no. 1, pp. 528535, Jan. 1999.5 P. Tsiotras and B. Wilson, “Zero and low-bias control designs foractive magnetic bearings,” IEEE Trans. Contr. Syst. Technol., vol. 11,no. 6, pp. 889904, Nov. 2003.6 M. B. J. Ghosh, D. Mukherjee and B. Paden, “Nonlinear control of abenchmark beam balance experiment using variable hyperbolic bias,”in Proc. 2000 ACC, Chicago, IL, Jun. 2000, pp. 21492153.7 D. Johnson and G. Brown, “Adaptive variable bias magnetic bearingcontrol,” in Proc. 1998 ACC, Philadelphia, PA, Jun. 1998, pp. 22172223.
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