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

Nonlinear control of an active magnetic bearing with bias currents: experimental study Thomas R. Grochmal and Alan F. Lynch AbstractThis paper presents an experimental comparison of position tracking controllers for a fi ve degree-of-freedom active magnetic bearing system. Two variations of a nonlinear design are presented, each subject to a different actuating constraint: Constant Current Sum (CCS) uses bias currents and Current Almost Complementary (CAC) avoids bias currents. While both designs achieve accurate tracking for a non-rotating shaft, a comparison of unbalance responses show that voltage saturation can limit the dynamic response of the CAC-based design. Performance of the nonlinear designs is also compared to a decentralized PID controller. I. INTRODUCTION Active Magnetic Bearings (AMBs) continue to receive attention due to their unique characteristics. Contact free suspension can lead to improved reliability and performance over rolling-element bearings. AMBs provide variable stiff- ness and damping characteristics which allows for vibration compensation, force measurement, and precision motion control 1. Linear feedback designs for AMBs have traditionally utilized bias currents. These bias currents provide an op- erating point for the models linearization. Bias currents also provide a higher dynamic force response 2. A number of researchers have considered zero- and low-bias controllers in an effort to minimize ohmic losses and reduce heating 3. Similar notions based on fl ux bias have been investigated in 4 and 5. Time-varying, adaptive, and optimal bias ap- proaches have also been introduced to achieve a compromise between performance and low loss objectives 6, 7, 8. Nonlinear control techniques have been widely applied to AMBs and naturally lead to zero- and low-bias schemes 9, 10, 11. In 10 L evine et al. design a nonlinear control based on the fl at property of the system and intro- duce the Current Almost Complementary (CAC) condition. This approach was experimentally validated for a machining application in 12. Although the CAC condition can lead to low power losses, it can potentially require large input voltages 10. A variation of the fl atness-based design using a Constant Current Sum (CCS) bias condition is introduced in 13. This last work provides experimental validation on a fi ve degree-of-freedom (DOF) system. This paper provides an experimental comparison between CAC and CCS-based nonlinear controllers. These designs are This 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 & Computer Engineering, University of Alberta, Edmonton AB, T6G 2V4, Canada Thomas R. Grochmal is a Ph.D. student under the supervision of A.F. Lynch.grochmalece.ualberta.ca y z O Fb,y,n Fc,z Ff,y,n Fb,y,p Fb,z,p Ff,z,n Fc,y Fb,z,n x Ff,z,p Ff,y,p Fx,p Fx,n Fig. 1.Shaft assembly, motor coupling, and radial bearing stators for a 5 DOF AMB system. further compared to a decentralized PID controller. All three designs are implemented on a test stand manufactured by SKF Magnetic Bearings (Calgary, AB) that is interfaced to controller hardware developed at the University of Alberta. The contribution of this paper is to experimentally show the benefi ts of a CCS-based nonlinear controller in providing accurate trajectory tracking and robust stabilization under low voltage limits. This paper is presented as follows. In Section II a dynamic model of the system is described. In Section III we identify the parameters of the models magnetic force expressions. In Section IV the nonlinear feedback is formulated under the CAC and CCS conditions. Finally, in Section V the test bench is described, and in Section VI an experimental comparison of the controllers is provided. II. MODELING The fi ve DOF AMB system consists of a horizontal shaft assembly 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 a rigid shaft, the dynamic equations are 14 m x = Fx m y = Fb,y+ Ff,y+ Fc,y+ mgy m z = Fb,z+ Ff,z+ Fc,z+ mgz Jz = (lb,a x)Fb,z+ (lf,a+ x)Ff,z Jx + lcFc,z Jy = (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 mass cmrelative to the origin O of the inertial frame. When the shaft is centered in all three bearings we have (x,y,z) = 0. The angles , denote the angular displacement of a body Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006 FrA11.2 1-4244-0210-7/06/$20.00 2006 IEEE4558 lb,a lf,a lb,slf,s W24 b,y x y x axial stator W13 f,y cm O position sensorposition sensor ix,pix,n ib,y,nif,y,n ib,y,p if,y,n Fig. 2.Cross section of Figure 1 in the x y plane. Motor coupling not shown. frame relative to the inertial frame. The angular velocity of the shaft about the x-axis is denoted by = and is assumed to be constant. The y- and z-axis components of the gravitational fi eld are denoted by gyand gzrespectively. The orientation of the radial bearings are such that gy= gz. The shaft of mass m has principle moments of inertia denoted by 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,a respectively. The motor coupling forces are denoted Fc,y/z1 and are modeled as linear springs Fc,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 force is denoted Fxand the drive-end and non-drive-end radial bearing forces are denoted Fb/f,y/z. Each of these forces are the summation of positive (subscript p) and negative (subscript n) components generated by opposing coils. The force expressions are Fx= Fx,p Fx,n= xi2 x,p (x x)2 xi2 x,n (x+ x)2 (2a) Ff/b,y/z= Ff/b,y/z,p Ff/b,y/z,n = f/b,y/zi2 f/b,y/z,p ( f/b,y/z)2 f/b,y/zi2 f/b,y/z,n ( +f/b,y/z)2 (2b) where (respectively x) is the nominal air gap between the rotor and the radial (respectively axial) bearing stator. The bearing force constants are x,f/b,y/z, and f/b,y/z denote the offset displacements of the shaft in the planes x = lb,aand x = lf,a(see Section III for further details on this offset). The axial and radial bearing coils are driven by currents ix,p/n,if/b,y/z,p/nwhich we assume to be control 1The shorthand expression Fc,y/z refers to Fc,yand Fc,z inputs. In practice, an inner-loop current controller ensures the currents track their reference values suffi ciently fast. III. FORCE PARAMETER IDENTIFICATION In this section we describe the identifi cation of the radial bearing force parameters f/b,y/z. We also introduce mag- netic offsets f/b,y/zwhich represent the distances between the center of the rotors and the center of the stators when (x,y,z,) = 0 15. When the position sensors are calibrated the shaft is aligned to the center of the touchdown bearings and not the centers of the stators. If the touchdown bearings are not concentrically aligned to the stators, then magnetic offset results. See Figure 3. Mathematically, this offset enters as f/b,y/z= f/b,y/z+f/b,y/zwhere f/b,y/z are the displacements of the shaft in the planes x = lb,aand x = lf,a. O b,y b,y stator x y touchdown bearing b magnetic center Fig. 3.Schematic of magnetic bearing system with magnetic offset. To obtain f/b,y/z,f/b,y/zwe consider the non-rotating shaft decoupled from the motor and assume the system is in equilibrium with x, and equal to zero. Therefore model (1) becomes 0 = Fx 0 = Fb,y+ Ff,y+ mgy 0 = Fb,z+ Ff,z+ mgy 0 = lf,aFf,z lb,aFb,z 0 = lb,aFb,y lf,aFf,y Solving for the equilibrium forces we obtain Fb,y= Fb,z= mgy ? lf,a lb,a+ lf,a ? = mbgy(3a) Ff,y= Ff,z= mgy ? lb,a lb,a+ lf,a ? = mfgy(3b) where we have introduced effective masses mf/bsupported by each radial bearing. For brevity, we present only the case for the drive-end y-axis. From x = = = 0 we obtain f,y= y and substitute this into (2b). We impose the CCS condition where each coil is provided with a bias current and a differential current is added and subtracted to opposing 4559 coils. Therefore, the sum of currents from opposing coils always adds up to twice the bias current 1. We have if,y,p= ib+ if,yif,y,n= ib if,y where ibis a constant bias and if,yis a differential current. Hence, we get the force expression Ff,y= f,y(ib+ if,y)2 ( f,y y)2 f,y(ib if,y)2 ( + f,y+ y)2 (4) and by combining (4) with (3b) we obtain mfgy= f,y(ib if,y)2 ( + f,y+ y)2 f,y(ib+ if,y)2 ( f,y y)2 Data is collected over a range of y and corresponding if,y. Defi ning the cost function ?k(f,y,f,y) = f,y(ib if,y)2 ( + f,y+ y)2 f,y(ib+ if,y)2 ( f,y y)2 + mfgy where (ik f,y,y k),1 ? k ? N denotes the data set, we solve the nonlinear least squares problem min (f,y,f,y)U N ? k=1 2 k where U = ?(x 1,x2) R2: 0 0), both coils are energized to prevent unbounded voltages at the zero force point. For the x-axis the CAC condition is ix,p= (x x) ? Fx x Fx? (x x) ? Fx+P2 (Fx) x Fx 0Fx ix,n= 0Fx? (x+x) x P(Fx) Fx (x+ x) ? Fx x Fx The polynomial Pis chosen to ensure a smooth transition at 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 axial bearing we express the force model as Fx= x(ib+ ix)2 (x x)2 x(ib ix)2 (x+ x)2 (7) where ixis a differential current. Inversion of (7) gives the control law ix= ? x(x2+2 x)ibx(x 22 x)Fxxx/x+i2b 2xxx x ?= 0 Fx2 x/(4xib) x = 0 (8) The expression for ixwhen x = 0 is determined by apply- ing lH opitals Rule. To ensure a non-negative discriminant in (8), it is suffi cient to impose the limits 4xi2 b (x+ x)2 Fx 4xi2 b (x x)2 This inequality is satisfi ed provided |ix| ib In practice each coil is limited to a maximum current of Is. Setting ib= Is/2 provides the full range for ix,p/n. dSPACE modular system Coil PWM amplifiers and sensor circuitry AMB test stand Signal acquisition Fig. 4.Experimental setup. V. EXPERIMENTAL SETUP The 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 confi guration 15. The systems maximum rotational speed is 15000 rpm under no load. See Table III for bearing specifi cations. Specifi cationradial bearingaxial bearing static load cap.76 N205 N saturation current3.0 A2.8 A nominal gap525 m783 m stator ID35.1 mm38.6 mm stator OD82.8 mm71.4 mm stator length12.7 mm13.5 mm rotor OD34.3 mm66.0 mm TABLE III MAGNETIC BEARING SPECIFICATIONS Both the position and current control loops are im- plemented using a modular dSPACE hardware system. A dSPACE DS1005 board performs real-time computations for control at 10 kHz. Three DS2001 high-speed ADC boards sample ten coil currents and rotor displacement along fi ve axes. A DS3002 encoder board measures the rotational velocity of the shaft. A DS5101 board generates PWM waveforms that drive the current control loops. The PWM switching frequency is 10 kHz. A host PC provides a MAT- LAB/Simulink development platform and logs real-time data from the dSPACE system. VI. EXPERIMENTAL RESULTS A. Rotational Stabilization Figure 5 presents drive-end and non-drive-end orbital plots of the shaft rotating at 14000 rpm. A comparison is made between CCS-based nonlinear, CAC-based nonlinear, and PID control schemes. We remark that the unmodeled effect of mass unbalance is signifi cant because the shaft is manually assembled and no measures are taken to me- chanically balance it. For all controllers the feedback gains are tuned to achieve a comparable performance over the operating range of shaft speed. The controller gains are 4561 -50-40-30-20-1001020304050 -50 -40 -30 -20 -10 0 10 20 30 40 50 V13 m W13 m PID CCS CAC -50-40-30-20-1001020304050 -50 -40 -30 -20 -10 0 10 20 30 40 50 V24 m W24 m PID CCS CAC Fig. 5.Drive-end (left) and non-drive-end (right) orbital plot comparison at 14000 rpm. 00.0050.010.0150.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time s current A Nonlinear CAC 00.0050.010.0150.02 -25 -20 -15 -10 -5 0 5 10 15 20 25 time s voltage V Nonlinear CAC 00.0050.010.0150.02 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time s current A Nonlinear CCS 00.0050.010.0150.02 -4 -2 0 2 4 6 8 10 time s voltage V Nonlinear CCS 00.0050.010.0150.02 0 0.5 1 1.5 2 2.5 time s current A PID 00.0050.010.0150.02 -20 -15 -10 -5 0 5 10 15 20 25 time s voltage V PID Fig. 6.Non-drive-end z-axis currents ib,z,p(dashed line), ib,z,n(solid line) 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 controllers provide robust stabilization. They perform as well or better than PID which is often used for stabilization of high speed AMB applications 18. Figure 6 shows representative data k2s1k1s2k0s3 N CAC2501500001000000.1 k2s1k1s2k0s3ibA CCS3001500001000001.0 kpA/mkiA s/mkdA/(m s)ibA PID800010000131.0 TABLE IV FEEDBACK GAINS AND BIAS PARAMETERS for the associated currents and voltages at 14000 rpm. It is interesting to note that in the case of CAC, satisfactory stabilization performance may be achieved without bias currents. This results in reduced power consumption, but the switching action of opposingcoils requires high bandwidthin the current controller output. Therefore, higher voltages are required by CAC in order to achieve the same stabilization performance as CCS. While the CCS achieves stability with a 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-bias schemes. To further demonstrate the impact of low voltage limits, Figure 7 shows the rotational performance of CAC for a voltage limit of 12 V. It is found that CAC is unable to stabilize the shaft for speeds greater than 5000 rpm. This result cannot be improved by controller tuning. The tradeoff between biasing and maximum voltage requirements refl ects the inductive nature of electromagnetic actuators which re- quire high voltages to achieve fast changes in current. With voltage limits zero-bias schemes are therefore generally less robust to disturbances than bias-based schemes due to current slew rate limiting. 9.81010.210.410.610.8 100 50 0 50 100 150 y m time s 99.51010.5 0 0.5 1 1.5 2 2.5 ib,y,p A time s 99.51010.5 0 0.5 1 1.5 2 2.5 ib,y,n A time s 99.51010.5 15 10 5 0 5 10 15 ub,y,p V time s 99.51010.5 15 10 5 0 5 10 15 ub,y,n V time s Fig. 7.Destabilization at 5000 rpm using the CAC controller with saturating 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: Corresponding voltages ub,y,p(left) and ub,y,n(right). B. Tracking of a Non-rotating Shaft In Figures 810 we demonstrate the tracking performance of each controller to a time varying reference for the shafts center of mass. During these experiments the shaft speed is 4562 00.00.25 -100 -50 0 50 100 time s position m Zref Z Yref Y X 00.00.25 -80 -60 -40 -20 0 20 40 60 80 time s angle rad Fig. 8.Tracking performance of center of mass coordinates x,y,z (left) and , (right) for the CCS-based nonlinear controller. 00.00.250.3 0.2 0 0.2 0.4 0.6 0.8 1 1.2 time s if,y,p if,y,n A 00.00.250.3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 time s if,y,p, if,y,n A Fig. 9.Drive-end y-axis tracking currents for the CAC-based (left) and CCS-based (right) nonlinear controllers. zero, i.e., = 0. We use an elliptical trajectory yr(t) = 50sin(20t) m zr(t) = 80cos(20t) m xr(t) = r(t) = r(t) = 0 m Figure 8 shows the performance of the CCS controller. Accurate tracking is observed: the tracking error of x,y,z is always within 3 m and the angles , are stabilized to within 45 rads. As expected, the tracking performance of the nonlinear controllers is unaffected by the choice of actu- ator condition, provided suffi cient voltage head space exists. Figure 9 shows representative currents for each nonlinear design. It is interesting to note that for CAC only the upper radial bearing coils need to be energized. Figure 10 shows the tracking performance for PID. De- centralized reference trajectories were generated using the transformation (6). The performance is inadequate since the shaft overshoots its y and z coordinate trajectories and makes contact with the backup bearings. This contact accounts for the distortion of the , signals. Varying the PID gains does not result in a noticeable improvement in performance. 00.00.25 -100 -50 0 50 100 time s position m zref z yref y x 00.00.25 -80 -60 -40 -20 0 20 40 60 80 time s angle rad Fig. 10.Tracking performance of center of mass coordinates x,y,z (left) and , (right) for the PID controller. VII. CONCLUSIONS This paper presents an experimental study of nonlinear control with and without bias currents. Under modest voltage limits the nonlinear controller with bias demonstrates robust stabilization of a rotating shaft and tracks the position of a non-rotating shaft. By comparison, a zero-bias nonlinear de- sign requires larger voltages to achieve similar performance. Both nonlinear designs have a similar unbalance response as a decentralized PID approach, but a PID controller can- not adequately track a time varying reference. Designs are validated