Synchronous Sensor Runout and Unbalance Compensation in Active Magnetic Bearings Using Bias Current Excitation

Transcription

1 Joga D. Setiawan Graduate Student Ranjan Mukherjee Associate Professor Michigan State University, East Lansing, MI Eric H. Maslen Associate Professor, Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA Synchronous Sensor Runout and Unbalance Compensation in Active Magnetic Bearings Using Bias Current Excitation Sensor runout and unbalance are the dominant sources of periodic disturbances in magnetic bearings. Many algorithms have been investigated for individual runout and unbalance compensation but the problem of simultaneous compensation of the two disturbances remains essentially unsolved. The problem stems fundamentally from a lack of observability of disturbances with the same frequency content and is critical for applications where the rotor needs to be stabilized about its geometric center. A credible way to distinguish between the synchronous disturbances is to vary rotor speed but speed variation is not acceptable for many applications. We present in this paper a new method for simultaneous identification and compensation of synchronous runout and unbalance at constant rotor speed. Based on traditional adaptive control design, our method guarantees geometric center stabilization of the rotor through persistency of excitation generated by bearing stiffness variation. The variation in magnetic stiffness is achieved through perturbation of the bias currents in opposing electromagnetic coils in a manner that does not alter the equilibrium condition of the rotor. Our theoretical results are first validated through numerical simulations and then experiments on a laboratory test-rig. The experimental results adequately demonstrate efficacy of our approach and provide clues for future research directions. DOI: / I Introduction The dominant sources of periodic disturbances in magnetic bearings are sensor runout and mass unbalance. Unbalance, which results from a lack of alignment between the principal axis of inertia and the geometric axis of the rotor, generates a force disturbance synchronous with rotor angular speed. Runout originates from nonuniform electrical and magnetic properties around the sensing surface and lack of concentricity of the sensing surface. It generates a disturbance in rotor position at multiple harmonics of the frequency of rotation. Both runout and unbalance, which are unavoidable since they result from manufacturing imperfections, cause rotor vibration, degrade performance, and can lead to instability if they are not adequately compensated. Although the problem of simultaneous compensation of unbalance and sensor runout has appeared in the literature only recently 1 4, a large volume of research exists on compensation of the individual disturbances. Some of the early work on unbalance compensation is based on insertion of a notch filter in the control loop 5. The drawback of this approach stems from negative phase of the notch transfer function which can reduce the stability margin of the closed-loop system and lead to instability 6,7. Another popular approach is adaptive feedforward control 8,9, where Fourier coefficients of the disturbance are estimated and cancelled online. Operationally, these controllers resemble notch filters 7 and can result in instability if designed without considering the underlying structure of the closed-loop system. To preserve stability, Herzog et al. 10 developed the generalized notch filter and Na and Park 7 proposed variation of the least mean square algorithm. Other approaches that compensate for unbalance while ensuring stability include adaptive auto-centering 11 Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division September 18, Associate Editor: E. Misawa. and output regulation with internal stability 12. Both of these approaches stabilize the rotor about the mass center. Though unbalance compensation has been widely studied with the objective of stabilization about the mass center, most commercial applications require geometric centering to avoid seal wear. The problem of geometric center stabilization has been addressed by a few researchers 13,14, but more general results 15,16 establish that mass or geometric center stabilization can be achieved through cancellation of the disturbance in current or displacement signal, respectively. In a general approach for disturbance attenuation, Knospe et al. 17,18 claimed that any form of vibration, which can be measured, can be attenuated using a pseudo-inverse of the pre-computed influence coefficient matrix. The performance of the algorithm amidst uncertainties was investigated and experiments used to demonstrate effectiveness. The method decouples the problem into two independent tasks, and while it has been demonstrated to work successfully, there is no theoretical basis for stability of the two interacting processes. Other approaches employed for disturbance compensation include robust control designs 19 21, Q-parameterization control 22, and off-line adaptation 23. Among them, the work by Kim and Lee 23 and Setiawan et al. 21 address the problem of sensor runout compensation. Unfortunately, none of the above approaches lend themselves to unbalance compensation in the presence of significant sensor runout. This problem, widely acknowledged in the literature but essentially unsolved, stems from lack of observability of disturbances with the same frequency content. A credible way to distinguish between these disturbances is to perturb the operating conditions of the plant or its parameters, but recent research 1,2,4 that proposes rotor speed variation is unacceptable for many applications. In this paper we propose a method for individual identification of synchronous runout and unbalance at constant rotor speed through persistency of excitation generated by bearing stiffness variation. Using bias current excitation for varying bearing 14 Õ Vol. 124, MARCH 2002 Copyright 2002 by ASME Transactions of the ASME

2 stiffness and an adaptive control framework, we mathematically establish and experimentally demonstrate geometric center stabilization. Our solution to the problem is based on the important assumption that geometric and magnetic centers of the rotor coincide. In the absence of this assumption, geometric center stabilization is still an open problem. Our paper is organized as follows: We present background material in Section II and formulate the problem in Section III. The variable stiffness approach to synchronous disturbance compensation and rotor stabilization about the geometric center is presented in Section IV. Section V contains procedures for implementation and experimental results. Future research directions and concluding remarks are provided in Section VI. II Background II-A Mathematical Modeling: Sensor Runout and Unbalance. Consider the magnetically levitated rigid rotor in Fig. 1, which has two degrees of freedom along the x and y axes. The dynamics of the rotor along these axes are similar, but decoupled. Along the x axis, one may write mẍ F mḡ f u, ḡ g/& (1) where m is the rotor mass, x is the position of the rotor geometric center, F is the magnetic force, g is the acceleration due to gravity, and f u is the unbalance force. Since the magnetic center is assumed to coincide with the geometric center, the magnetic force F can be expressed as 25 F k i 10 I l x 2 i 2 20 I l x (2) where k is the magnetic force constant, l is the nominal air gap, i 10, i 20 are the bias currents in the upper and lower electromagnets, and I is the control current. By linearizing Eq. 2 about x 0, I 0, and through proper choice of bias currents, Eq. 1 can be written as mẍ K s x f c f u, f c K c I, K s 2k i 2 10 i 2 20 /l 3, K c 2k i 10 i 20 /l 2 (3) where f c is the control force, and K s and K c are the magnetic stiffness and actuator gain of the magnetic bearing, respectively. The unbalance force due to mass eccentricity can be modeled as f u m 2 cos t u m 2 p sin t q cos t (4) where p sin u, q cos u, u is the phase of unbalance, is the eccentricity of the rotor, and is the rotor angular speed. The true location of the rotor geometric center is not available for a magnetic bearing with sensor runout. Instead, the gap sensors provide the signal x s x s x d (5) Fig. 1 A magnetic bearing system with unbalance and sensor runout where, d, the sensor runout disturbance, can be expressed by the Fourier series n d a 0 i 1 a i sin i t b i cos i t (6) In the above expression, n is the number of harmonics, a 0 is the DC component, and a i, b i, i 1,2,,n, are the harmonic Fourier coefficients. II-B Manual Sensor Runout Identification. In this section we present a method for manual identification of sensor runout. This method, which will be used to verify the performance of our adaptive algorithm, is in line with our assumption that magnetic and geometric centers of the rotor are coincident. Manual runout identification which has to be performed separately for each axis of the bearing, will require the rotor to be spun at low speed to avoid the effects of unbalance. We will first levitate the rotor using a proportional plus derivative controller, as shown in Fig. 2. Using a digital signal processor to generate the function E 0 A 0, we will close the feedback loop using the signal (x s E 0 ). We will then adjust the magnitude of A 0 such that (x s E 0 ) has zero mean. Once this is accomplished, we will have identified the DC component of runout. Next, we will generate the signal E 1 A 1 sin( 1 ), where t is the rotor angular position obtained from the shaft encoder. We will select 1 such that E 1 is in phase with the first harmonic of (x s E 0 ). The feedback signal will be subsequently changed to (x s E 0 E 1 ) and A 1 selected by trial and error to eliminate the first harmonic of (x s E 0 E 1 ). Having identified the first harmonic, we will proceed to identify the second harmonic E 2 A 2 sin(2 2 ), and higher harmonics, if necessary. The complete runout signal E E 0 E 1 E 2 A 0 A 1 sin 1 A 2 sin 2 2 (7) will be subtracted from the sensor signal to recover the position of the geometric center, E s x. II-C Unbalance Verification Using a Trial Mass. In this section we outline the procedure for verification of rotor unbalance estimated by our adaptive algorithm. Let U, U be the estimated magnitude and phase of the initial unbalance. We will add a trial mass m T to the rotor and re-estimate the unbalance using our algorithm. If T, T are the magnitude and phase of the trial mass, we can verify the efficacy of our algorithm if the new unbalance vector R, R is a vector sum of the initial unbalance and the unbalance due to the trial mass. This is explained with the help of Fig. 3. III Fig. 2 Manual sensor runout identification Problem Formulation III-A Framework for Adaptive Compensation. We adopt an adaptive control framework for rotor stabilization through compensation of both unbalance and runout. The adaptive controller, shown in Fig. 4, is comprised of a feedback law for geometric center stabilization and adaptation laws for individual cancellation of the synchronous periodic disturbances. For the purpose of feedback, the geometric center is estimated as x x s d (8) Journal of Dynamic Systems, Measurement, and Control MARCH 2002, Vol. 124 Õ 15

3 Fig. 3 Unbalance verification using trial mass Fig. 4 Framework for adaptive compensation where d is the estimated runout, given by the relation n d ā 0 i 1 ā i sin i t b i cos i t (9) In the above expression, ā 0 is the estimated value of a 0, and ā i, b i, are estimated values of a i,b i, respectively, for i 1,2,,n. Using Eqs. 5, 6, 8, and 9, the estimated geometric center can be expressed as x x d, d d d Y T (10) where d is the error in the estimated value of runout, and Y and are the regressor vector and the vector of Fourier coefficient estimation errors, defined as Y 1 sin t cos t sin 2 t cos 2 t sin n t cos n t T ã 0 T T T, ã 1 b 1 T, ã 2 b 2 ã n b n T (11) The estimation errors in the above expression are given by the relations ã 0 (a 0 ā 0 ), ã i (a i ā i ), b i (b i b i), i 1,2,,n. If these errors are converged to zero, the estimated geometric center, x, will converge to the true geometric center, x. For compensation, the unbalance force in Eq. 4 is estimated as f u Y u T u, Y u m 2 sin t cos t T, u p q T (12) where p,q, are estimates of the Fourier coefficients p, q, respectively. If we define errors in these estimates as p (p p ), q (q q ), the net unbalance force acting on the rotor will be f u f u Y u T u, u p q T (13) If the errors in the estimates of the unbalance Fourier coefficients p,q are converged to zero, the unbalance force will be cancelled through feedforward compensation, in accordance with Fig. 4. III-B Multiple Angular Speed Approach. In this section we review the multiple angular speed approach for synchronous disturbance identification, proposed by Setiawan et al. 2. This discussion provides background and motivation for bias current excitation as the means for synchronous disturbance compensation, which will be presented in the next section. With the objective of stabilizing the rotor geometric center through individual identification and cancellation of sensor runout and unbalance, we propose the control action f c K s x m x cē m 2 p sin t q cos t along with the adaptation laws (14) 8 Y m ē, 8 u u Y u ē (15) In Eqs. 14 and 15, ē is the weighted sum of the estimated position and velocity of the rotor geometric center, defined as ē x x (16) where is a positive constant, R (2n 1) (2n 1) and u R 2 2 are positive definite constant adaptation gain matrices, given by the relations diag 0, 1, 1, 2, 2,, n, n, u diag p, q c is a positive constant, and Y m is the derived regressor vector Y m K s Y mÿ (17) 16 Õ Vol. 124, MARCH 2002 Transactions of the ASME

4 Furthermore, in Eq. 15, the sensor runout adaptation gains, i, i 0,1,2,,n, are chosen such that the time-invariant constant, defined below, satisfies 0 1. n Y T Y m i 0 i K s mi 2 2 (18) The dynamic behavior of the closed-loop system is obtained by substituting Eq. 14 into Eq. 3, and simplifying using Eqs. 10 and 13 Y K s m 2 sin t cos t T Y E K s Y T T E ã 0 T T Ks m 2 2 sin 2 t K s m 2 2 cos 2 t Y ] K s m n 2 sin n t K s m n cos n t T 2 (25) mẍ K s Y T m x cē Y u T u Using the relations x ẍ d from Eq. 10 and e x x from Eq. 16, this can be rewritten as me md Ks Y T cē Y u T u (19) To simplify further, we examine the expression for d from Eq. 10 and substitute Eq. 15 therein d Ÿ T 2Ẏ T 8 Y T Ÿ T 2Ẏ T Y m ē Y T Ẏ m ē Y T Y m e (20) Using Eq. 18, and the identities Ẏ T Y m Y T Ẏ m 0 which can be established from Eqs. 11 and 17, Eq. 20 can be expressed as d Ÿ T e. Substitution of this relation in Eq. 19 and restatement of the adaptation laws in Eq. 15, results in the closedloop system dynamics where, have been defined earlier in Eq. 11. It can be shown that there are some positive constants 1, 2 and T 0, such that 2 I tt T 0YE Y E T d 1 I where I is the identity matrix of dimension (2n 1). Therefore Y E is a persistently exciting signal 24. This implies from Eq. 24b that E 0 ast. From the definition of and E in Eqs. 11 and 25 we conclude that the estimated values of all harmonics of sensor runout, except the first, converge to their true values, i.e., ã 0 0 and ã i, b i 0 for i 2,3,,n. The estimated values of harmonic unbalance and the first harmonics of runout, however, do not converge to their true values. This is evident from the following equations m 1 e Y m T cē Y u T u, Y m ē (21a) (21b) ã 1 p 0 b 1 q 0 a 1 ā 1 p p 0 b 1 b 1 q q 0 m 2 K s m 2 (26) 8 u u Y u ē (21c) with (ē,, u) (0,0,0) as an equilibrium. Indeed, it can be verified from Eq. 21 that (ē,, u) (0,0,0) implies (e, 8, 8 u ) (0,0,0). To study the stability of the equilibrium, we define the Lyapunov function candidate V ē,, u mē2 T 1 ut u 1 u (22) which is positive definite since and u are positive definite matrices and satisfies 0 1. Furthermore, V is continuously differentiable and its derivative along the system trajectories in Eq. 21 is given by the relation V m 1 ēe T 1 8 ut u 1 8 u cē 2 0 Since V is negative semi-definite, (ē,, u) (0,0,0) is a stable equilibrium. In addition, since V is uniformly continuous, we deduce from Barbalat s lemma 24 that V 0 ast. This implies ē 0 as t. By differentiating Eq. 21a, we can show e e (ē,, u,t) is bounded. This implies that e is uniformly continuous. Since ē 0 ast, we use Barbalat s lemma 24 to deduce e 0 ast. From Eq. 21a we now claim Y m T Y u T u 0 (23) Using the orthogonality property of the harmonic components, we separate Y m into Y R 2 and Y E R (2n 1), and rewrite Eq. 23 as Y T Y u T u 0 Y E E 0 (24a) (24b) which can be obtained from Eq. 24a using Eqs. 11, 12, and 25. To converge both and u in Eq. 24a to zero, we need to distinguish between the harmonics of unbalance and the first harmonics of runout. To this end, Setiawan et al. 2 proposed that the rotor speed be switched between two different values, which effectively generates two sets of Eq. 26 corresponding to two distinct values of. These four equations can be used to individually identify the four unknowns a 1, b 1, p, and q. The estimates of harmonic unbalance, ā 1, b 1, and first harmonics of runout, p, q, can be subsequently converged to their true values using innocuous modification of the adaptation laws in Eq. 15. Subsequent analysis of the closed-loop system can establish asymptotic stability of the rotor geometric center about the origin. Similar analysis will be presented in Section IV-A and is therefore skipped here. In implementation, the multiple angular speed approach suffers from two main drawbacks. First, numerical computation of a 1, b 1, p, q, using Eq. 26, is very sensitive to the value of. Since uncertainty in the values of m, K s, and result in large errors in calculating, it is difficult to identify the correct values of a 1, b 1, p, and q. Second, the approach requires rotor operation at two sufficiently well separated operating speeds in order that the four algebraic equations are well-conditioned. In many applications, speed variation may not be desirable, or even permitted. In such cases the above approach cannot be used. IV Adaptive Compensation Using Bias Current Excitation IV-A Persistent Excitation Through Variation of Magnetic Stiffness. From our discussion in the last section it is clear that an attempt to identify both unbalance and runout at constant rotor speed results in Eq. 23, which can be written as Journal of Dynamic Systems, Measurement, and Control MARCH 2002, Vol. 124 Õ 17

5 Y T m Y T T u u Y mu u 0 K s K s m 2 sin t K s m 2 cos t ] Y mu K s m n 2 sin n t K s m n 2 cos n t m 2 sin t m 2 cos t R 2n 3 (27) In the above equation, Y mu is composed of n frequency components and a DC term, and as such fails to satisfy the persistent excitation condition 24. To overcome this problem, we propose variation of magnetic stiffness of the bearing, K s, through sinusoidal excitation of bias currents. In the Appendix we will show that bias current excitation at an appropriate frequency can indeed guarantee persistency of excitation. In this section we discuss the procedure for bias current variation in the opposite coils. The bias currents in opposite coils are nominally chosen to provide the force that cancels the weight of the rotor when the rotor is geometrically centered. Therefore, the bias currents nominally satisfy k i 2 10 i 2 20 mḡl 2 (28) We propose small excitations 1, 2, in the bias currents of the top and bottom coils according to the relations i 10 i 10 * 1, i 20 i 20 * 2 (29) where i 10 * and i 20 * are constants, and 2 1, Substitution of Eq. 29 into Eq. 28 yields k i 10 * 2 i 20 * 2 2k i 10 * 1 i 20 * 2 mḡl 2 (30) In order to prevent rotor oscillation due to bias current variation, we choose 1 and 2 according to the relation 2 i 10 * /i 20 * 1, 1 A sin e t, e (31) where A, e, are the amplitude and frequency of bias current excitation. Indeed, substitution of the above equation into Eq. 30 indicates that rotor equilibrium is maintained for i 10 i 10 * and i 20 i 20 *. When 1 are 2 satisfy Eq. 31, the magnetic stiffness and actuator gain of the bearing are given by the expressions K s K s * 8k l 3 i 10 * 1, K s * 2k l 3 i 10 * 2 i 20 * 2 K c K c * 2k l 2 1 i 10 * /i 20 * 1, K c * 2k l 2 i 10 * i 20 * (32) where K s * and K c * are constants. In the sequel, K s and K c will be treated as variables. IV-B The Adaptive Algorithm. In order to individually identify and compensate the synchronous disturbances due to runout and unbalance using bias current excitation, and stabilize the rotor geometric center to the origin, we propose the control action f c K s x m x c 1 2 m ē m 2 p sin t q cos t (33) The above control action is very similar to the one proposed in Eq. 14, except for the additional term involving. Although was defined as a constant in Eq. 18, it varies when bias currents are excited. The expression for can be obtained using Eqs. 18, 31, and 32 as follows n K s i 1 i 8 n ek l 3 A cos e t i 10 * i 1 i Along with the control law in Eq. 33, we propose the adaptation laws 8 Y m ē, 8 u u Y u ē (34) which are exactly the same as in Eq. 15. The dynamics of the rotor is obtained by substituting Eq. 33 into Eq. 3, and simplifying using Eqs. 10 and 13 mẍ K s Y T m x cē 1/2 m ē Y u T u (35) Using the relations x ẍ d from Eq. 10 and e x x from Eq. 16, Eq. 35 can be rewritten as me md Ks Y T cē 1/2 m ē Y u T u (36) To simplify, we examine the expression for d from Eq. 20. Using Eq. 18, and the identity Ẏ T Y m 0 which can be established from Eqs. 11 and 17, we can rewrite Eq. 20 as d Ÿ T ē e (37) Substitution of Eq. 37 in Eq. 36, and restatement of Eqs. 16 and 34 provides the following closed-loop system dynamics x x ē (38a) m 1 e c 1/2 m ē Y m T Y u T u 8 Y m ē (38b) (38c) 8 u u Y u ē (38d) The following observations can now be made with respect to the closed-loop system. Theorem 1: Consider the subsystem described by Eqs. 38b, 38c, and 38d. For this subsystem, the equilibrium (ē,, u) (0,0,0) is asymptotically stable. Proof: From Eqs. 38b, 38c and 38d first notice that (ē,, u) (0,0,0) implies (e, 8, 8 u ) (0,0,0). Therefore, (ē,, u) (0,0,0) is an equilibrium point. To show that this equilibrium is asymptotically stable, we consider the Lyapunov function candidate in Eq. 22 V ē,, u mē2 T 1 ut u 1 u The derivative of V along the system trajectories in Eq. 38 is given by the relation V m 1 ēe 1 2 m ē 2 T 1 8 ut u 1 8 u cē 2 0 Since V is negative semi-definite, (ē,, u) (0,0,0) is stable. In addition, since V is uniformly continuous, we can use Barbalat s lemma 24 to claim V 0 as t. This implies ē 0 as t. By differentiating Eq. 38b, we can show that e e (t,ē,, u) is bounded. This implies that e is uniformly continuous. Since ē 0 ast, we again use Barbalat s lemma 24 to deduce e 0 ast. Knowing ē, e 0 ast, we conclude from Eq. 38b Y T m Y T T u u Y mu u 0 (39) 18 Õ Vol. 124, MARCH 2002 Transactions of the ASME

6 In the Appendix we have shown that Y mu is persistently exciting 24 under sinusoidal bias current excitation, for a range of excitation frequency. Therefore, for an appropriate choice of excitation frequency, we can claim, u 0 ast. This concludes our proof. Lemma 1: The origin of the closed-loop system in Eq. 38, (x,ē,, u) (0,0,0,0), is an asymptotically stable equilibrium point. Proof: The closed-loop system in Eq. 38 is an interconnected system of the form ż 1 f 1 t,z 1,z 2 (40a) ż 2 f 2 t,z 2 (40b) where z 1 x, and z 2 (ē T ut ) T are the state variables of the two sub-systems. From Theorem 1 we know that z 2 0 is an asymptotically stable equilibrium of the subsystem in Eq. 40b. Also, ż 1 f 1 (t,z 1,0) has an asymptotically stable equilibrium point at z 1 0. This can be established from Eqs. 38a and 40a. Using the asymptotic stability theorem for cascaded systems 24, we conclude (x,ē,, u) (0,0,0,0), is an asymptotically stable equilibrium. Theorem 2: The coordinate (x,ẋ,, u) (0,0,0,0) is an asymptotically stable equilibrium point for the closed loop system defined by Eqs. 3, 4, 33, and 34. Proof: Using Eqs. 10, 18, 38a, and 38c, we find that at (x,ẋ,, u) (0,0,0,0), we have d Y T 0, x x d 0, d8 Ẏ T Y T 8 Y T Y m ē ē x x x Also, at (x,ẋ,, u) (0,0,0,0), d8 (x ẋ) x. Comparing with the expression for d8 above, we have d8 ē x 0, since 1. From Eqs. 34 and 35, it follows that (ẋ,ẍ, 8, 8 u ) (0,0,0,0). Therefore, (x,ẋ,, u) (0,0,0,0) is an equilibrium point. The fact that (x,ẋ,, u) (0,0,0,0) is asymptotically stable can now be deduced from a (x,ē,, u) (0,0,0,0) is an asymptotically stable equilibrium follows from Lemma 1, b the transformation matrix P that maps (x,ē,, u) to (x,ẋ,, u) 1 0 P YT 0 1 Ẏ T E 2n E where E (2n 1), E 2 are identity matrices of dimension (2n 1) and 2, respectively, is well defined and upper bounded, and c the inverse transformation P 1 exists, and P 1 is also upper bounded. Theorem 2 establishes that our adaptive controller, in conjunction with sinusoidal excitation of the bias currents, guarantees Table 1 Magnetic bearing test rig parameters geometric center stabilization of the rotor through identification and cancellation of synchronous runout and unbalance. IV-C Simulation Results. We present simulation results to demonstrate the role of bias current excitation in adaptive compensation of synchronous unbalance and runout. The simulation was performed using the nonlinear model of the plant, with parameters chosen to match ones in our experimental hardware, provided in Table 1. The rotor speed was assumed to be 1500 rpm 25 Hz and the frequency and amplitude of excitation were chosen to be 10 Hz and 0.2 Amps, respectively. The initial conditions of the rotor were assumed to be x(0) m and ẋ(0) 0.0 m/s. The Fourier coefficients of runout and unbalance were chosen as a a cos b sin a cos b sin p cos (41a) q sin (41b) where the units are in micrometers. The simulation results for error gains 400 s 1 and c 1200 kg/s, runout adaptation gains diag(1.4,3,3,3,3) 10 7 m/n, and unbalance adaptation gains u diag(3,3) 10 5 m/n are shown in Figs. 5, 6, and 7. In all the figures, dotted lines are used to denote the time interval over which bias currents are excited, namely, 5 t 35 s. Under adaptation without excitation, ā 0, ā 2, b 2 converge to their true values, but ā 1, b 1, p, q do not. This is evident from the steady state behavior of the estimated coefficients in Fig. 6, over the time interval 0 t 5 s. As a result, rotor geometric centering is not achieved. With 0.2 Amp approx. 10% of nominal bias currents excitation amplitude, synchronous unbalance and runout identification requires approximately 30 s. The excitation is terminated at t 35 after geometric centering is achieved through feedforward cancellation of both unbalance and runout. This reduces the cyclic stress of the power amplifiers which now produce only the harmonic component required to cancel the unbalance force. The harmonic component of the control current is visible in Fig. 7, for t 35 s. Fig. 5 Plots of geometric center and sensor signal Journal of Dynamic Systems, Measurement, and Control MARCH 2002, Vol. 124 Õ 19

7 Fig. 6 Estimated Fourier coefficients of runout and unbalance, true values are shown in dashed lines Fig. 7 Currents in top and bottom coils V Experimental Verification V-A Apparatus. We performed experiments to validate the efficacy of our algorithm on synchronous runout and unbalance compensation. The schematic of our test rig is shown in Fig. 8. We used a steel rotor, 43.2 cm in length and 2.5 cm in diameter, Fig. 8 Schematic of experimental test-rig with a balance disk for adding trial masses for unbalance. The rotor was quite rigid with the first flexible mode frequency at approximately 450 Hz, which was six times higher than the bandwidth of the closed-loop system. At one end, the rotor was connected to an absolute encoder using a bellows-type torsionally rigid coupling. Without introducing significant radial forces on the rotor, the coupling accommodates lateral misalignments. The encoder output was used in generating the feedforward terms in our adaptive algorithm. At the other end, our rotor was connected to a motor via a flexible rubber coupling. An optical speed sensor was used to provide feedback to an analog controller unit to maintain the speed of the rotor at a constant desired value. The rotor was levitated using two bearings, A and B. Among them, both axes of bearing B were controlled using analog PD controllers. Although both axes of bearing A were computer-controlled, unbalance and runout was compensated in one of the axes. A PD controller was used to control the rotor along the other axis. The currents in the electromagnets of both bearings were driven by switching power amplifiers, operating with a bandwidth of 1.6 KHz. The physical parameters and operating conditions of bearing A are provided in Table 1. The rotor mass enumerated in this table pertains to that of the whole rotor. We programmed our adaptive algorithm for synchronous runout and unbalance compensation in Matlab/ 20 Õ Vol. 124, MARCH 2002 Transactions of the ASME

8 Table 2 Experimental results with the balance disk located at rotor midspan Simulink environment and downloaded it to a Digital Signal Processor DSP board, manufactured by dspace. The sampling rate of the board was set at 13 KHz for on-line identification and control. A separate DSP board sampling at 5 KHz, along with suitable analog circuits, was used for manual identification of runout. The manually identified runout was used to determine the position of the rotor geometric center from the sensor signal. V-B Implementation Procedure. The discussion in this section and the next pertains to the single axis of bearing A in which synchronous unbalance and runout were compensated using our adaptive algorithm. Before implementation of our algorithm, we levitated the rotor using a PD controller and manually identified runout following the procedures outlined in section II-B. Although the first harmonic of runout was significant, higher harmonics of runout were negligible. On the basis of these results, we set n 1 in our algorithm for estimation of runout. We performed experiments with the balance disk at two different locations, shown in Fig. 8. For each location, we implemented out algorithm three times. In the first experiment, Expt. 1, we did not introduce any unbalance but estimated the initial unbalance of the rotor. In line with our discussion in section II-C, we added a trial mass in the second experiment, Expt. 2, and re-estimated unbalance. The third experiment, Expt. 3, was performed by introducing the trial mass at a different phases angle. Although a trial mass was added to the balance disk, unbalance was compensated only in bearing A. Since bearing B did not have unbalance compensation, we conducted two sets of experiments with the balance disk at two different locations to gain a high level of confidence in out results. We performed our experiments at constant rotor speed of 1500 rpm 25 Hz. We used the control law in Eq. 33 and the adaptation laws in Eq. 34 with the following choice of gains 400 s 1, c 1200 kg/s, diag 1.4,3,3,3, m/n, u diag 3, m/n The derivative term x in the control law was numerically computed using the transfer function 2500 s/(s 2500). This eliminates potential problems arising from infiltration of wideband noise into the sensor signal. During adaptation, the top bias current was excited using sin(20 t) Amperes. The excitation frequency was therefore less than half of the rotor frequency. After estimated parameters reached steady state, adaptation, and bias current excitation were both discontinued. In our algorithm, bias currents are excited concurrently with estimation. This eliminates drift in the estimated Fourier coefficients of unbalance and the first harmonics of runout in the absence of persistent excitation. Compared to standard implementation, our algorithm requires an extra D/A channel for every axis of implementation since both coils of each axis are excited independently. V-C Experimental Results. We first performed experiments with the balance disk located at rotor midspan. The results are provided in Table 2. The first column of data in Table 2 Expt.0 pertains to the manually identified values of sensor runout. This data includes the DC component and the first harmonics of runout only since second and higher harmonics were found to be negligible. The phase of the first harmonic was set to zero through encoder calibration. The second column of data Expt.1 corresponds to our experiment performed without a trial mass. This data includes the DC component and first harmonics of runout, and the harmonics of initial unbalance of the rotor. The next two columns of data pertain to our experiment with the addition of a trial mass of eccentricity T 91.1 m 1 and phase T 56. Of these two columns, the left column Expt.2 provides experimentally obtained values of runout and unbalance. The right column provides computed values of unbalance solely due to the trial mass. The computed values were obtained in line with our discussion in Section II-C, as follows T T R R U U (42) The last two columns of data in Table 2 pertain to experimental results obtained with the same trial mass, located at the same radial distance, but at the new phase angle T 146. Among these two columns, the left column provides Fourier coefficients of runout and unbalance obtained through experiments Expt.3. 1 The trial mass had a mass of 10 gms and was placed at a radial distance of cms. Since the mass of the rotor was 4.87 kgs, T 0.01/( ) m. Fig. 9 Time history of rotor geometric center position, x, and position sensor signal, x s Journal of Dynamic Systems, Measurement, and Control MARCH 2002, Vol. 124 Õ 21

9 Table 3 Experimental results with the balance disk located closer to Bearing B The right column provides computed values of unbalance solely due to the trial mass. This data was obtained as follows T T R R U U (43) The time history of the rotor geometric center position, x, and sensor signal, x s, are provided in Fig. 9 for one of the experiments, Expt.2. The geometric center position was evaluated from the sensor signal through cancellation of manually identified runout. The time scale in Fig. 9 is divided into three distinct regions: a t 0, where runout and unbalance were not compensated, b 0 t 300, where runout and unbalance were adaptively estimated and compensated, and c t 300, where runout and unbalance were completely compensated and bias current excitation terminated. Due to the relatively long duration of the experiment, we acquired data over the subintervals 0.2 t 0.3, t 120.5, and t The time trajectories of the estimated Fourier coefficients of runout and unbalance are shown in Fig. 10, with final values of the coefficients shown with dashed lines. The sensor runout coefficients show larger fluctuations than those of unbalance. This can be primarily attributed to the difference in scale of the plots. V-D Interpretation of Results. First consider the Fourier coefficients of sensor runout presented in Table 2. These values, identified by our algorithm, are very similar for Expts.1, 2, and 3, performed with varying degrees of unbalance. Furthermore, the identified coefficients closely match the manually identified values of runout, Expt.0. We can therefore claim that sensor runout has been correctly identified. Next, investigate the estimated magnitude and phase of unbalance due to the trial mass alone, for the two cases in Table 2. The estimated magnitudes, 75.9 m and 74.7 m, are similar and therefore consistent, and their respective phases, 56.4 and 150.6, compare very well with the true values, 56 and 146, respectively. The average value of the estimated magnitudes of unbalance, 75.3 m, is approximately 82 percent of the trial mass eccentricity of 91.1 m, added at rotor midspan. Other than this percentage factor, which will be discussed later, the above data indicates that our algorithm determines the phase of unbalance accurately and provides consistent estimates for eccentricity over repeated trials. Now consider the plot of the rotor geometric center position, x, in Fig. 9. Although this plot specifically pertains to Expt.2, it is representative of the general behavior of the rotor geometric center with our algorithm. It can be seen from Fig. 9 that the geometric center initially fluctuates about a nonzero mean value but this fluctuation is virtually eliminated with our algorithm. The stabilization of the rotor geometric center to the origin convinces us that both mass unbalance and sensor runout have been correctly estimated and compensated. A second set of experiments were performed with the balance disk closer to Bearing B, as shown in Fig. 8. The results of these experiments, provided in Table 3, are very consistent with the results in Table 2. Specifically, the estimated values of sensor runout are very similar to the values in Table 2, and closely match the manually identified values. The magnitude of estimated unbalance is consistent over repeated trials and the phase of unbalance closely match the phase of the trial mass for both experiments. Fig. 10 Time trace of estimated Fourier coefficients of runout and unbalance for Expt.2 in Table 2 22 Õ Vol. 124, MARCH 2002 Transactions of the ASME

10 The plots of the rotor geometric center, not shown here due to their similarity with the plot in Fig. 9, also indicate geometric center stabilization. The ratio between the average magnitude of estimated unbalance and trial mass eccentricity is 0.51 for the experimental results in Table 3. Although this value is less than the 0.82 ratio obtained with the balance disk at rotor midspan, as one would expect, both values are higher than expected. An explanation of the higher values would require further analysis that takes into consideration: a characteristics of the support provided by Bearing B under PD control, in the absence of unbalance and runout compensation; b performance of our adaptive algorithm, formally developed for a single degree-of-freedom rotor with collocated sensor and actuator, in our experimental test-rig; and c additional stiffness and unbalance introduced by the couplers at the two ends of the rotor. It will, however, not be worthwhile pursuing such analyses since our adaptive algorithm will have to be extended to a complete rotor model before it can be implemented in any industrial rig. Our experimental results amply demonstrate the basic feasibility of our algorithm but significant work remains to be done before it can be adopted by commercial vendors. We conclude this section with our comments on the time taken for synchronous disturbance compensation. It can be seen from Figs. 5 and 9 that compensation in simulation requires a shorter time than compensation in experiments. This can be attributed to the fact that only one of the bearings in our experimental setup was compensating the disturbances. The other bearing, in the absence of unbalance and runout compensation, acted as a source of additional periodic disturbances. We expect the time to reduce significantly when both bearings compensate for disturbances, and amplitude and frequency of bias current excitation are chosen optimally. The time taken for compensation in our experiments, nevertheless, should not be construed as significant. This time will be required during rotor spinup only. During steady-state operation, adaptation and bias current excitation will be implemented for a few seconds periodically to account for possible drift in runout and unbalance. Depending upon the type of application, periodic implementation may occur few times every hour to once every few hours. VI Concluding Remarks In most commercial applications for magnetic bearings, the control objective is to stabilize the rotor about its geometric center. This objective requires simultaneous compensation of sensor runout and mass unbalance, which are periodic disturbances with similar frequency content. In the absence of on-line compensation algorithms, commercial vendors use off-line techniques or a combination of off-line and on-line techniques requiring significant tuning during rotor spinup. A few researchers have proposed online estimation algorithms based on variation of rotor speed but applications where speed variation is not permissible preclude their use. In this paper we present a novel approach for synchronous runout and unbalance identification and compensation at constant rotor speed. Our approach is based on the assumption of coincidence of magnetic and geometric centers of the rotor and methodical excitation of bias currents in opposing electromagnetic coils that enhances observability of the system without altering the equilibrium condition. Our approach is mathematically developed for a single degree-of-freedom rotor model and validated through numerical simulations and laboratory experiments. The experimental results adequately demonstrate the capability of our algorithm to correctly identify runout and unbalance and stabilize the rotor to its geometric center. Future research problems include extension of our algorithm to a four degree-of-freedom rigid rotor model and optimal selection of amplitude and frequency of bias current excitation for faster estimation and convergence. The robustness of our algorithm to variation and uncertainty of system parameters also needs to be investigated although some degree of robustness is already established through successful experimental implementation. Importantly, the problem of geometric center stabilization needs to be revisited to address the case where magnetic and geometric centers of the rotor do not coincide. Appendix Persistent Excitation. Under bias current excitation, the stiffness of the magnetic bearing varies according to the relation K s K s * sin e t, 8ki 10 * A/l 3 (A1) This follows from Eqs. 31 and 32. Substituting the above expression in Eq. 27, we compute matrix Q, defined as follows Q 1 0Ymu tt T Y T T mu d 0 By choosing T 0 as the least common multiple of the time periods of rotation and excitation, and e in the range 0 e, with e /2, we find that Q R (2n 3) (2n 3) has the structure C C Q 1 ] ] ] ] ] ] ] ] ] ] ] n n n (A2) n C m C m 2 4 where Journal of Dynamic Systems, Measurement, and Control MARCH 2002, Vol. 124 Õ 23

11 C 1 m 2 K s * m 2 i K s * m i , i 0,1,2,,n Let Q i R i i, i 1,2,,2n 3, denote the (2n 3) upper left square submatrices of 2Q. Since Q 2n 1 is a diagonal matrix with strictly positive entries, we claim Det Q i 0, i 1,2,,(2n 1). Furthermore, using the relations DetM DetA Det D CA 1 B M 1 A 1 A 1 B D CA 1 B 1 CA 1 A 1 B D CA 1 B M 1 A B D CA 1 B 1 CA 1 D CA 1 B 1 C D we can show DetQ 2n 2 DetQ 2n 1 m C Q 2n 1 0 C T 4 K DetQ 2n 1 m 2 s * m K s * m DetQ 2n 3 DetQ 2n 2 m C Q 2n C T 4 K DetQ 2n 2 m 2 s * m K s * m Since Det Q i 0, i 1,2,..., (2n 3), we claim that Q is positive definite. This, along with the fact that all entries of Q are bounded, enables us to verify that Q satisfies 2 I Q 1 I, for some positive constants 1, 2, when I is the identity matrix of dimension of (2n 3). This proves that Y mu is persistently exciting 24. References 1 Kanemitsu, Y., Kijimoto, S., Matsuda, K., and Jin, P. T., 1999, Identification and Control of Unbalance and Sensor Runout on Rigid Rotor by Active Magnetic Bearing Systems, 5th Int. Symp. on Magnetic Suspension Tech., Santa Barbara, CA. 2 Setiawan, J. D., Mukherjee, R., Maslen, E. H., and Song, G., 1999 Adaptive Compensation of Sensor Runout and Mass Unbalance in Magnetic Bearing Systems, IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Atlanta, GA. 3 Setiawan, J. D., Mukherjee, R., and Maslen, E. H., 2000, Variable Magnetic Stiffness Approach for Simultaneous Sensor Runout and Mass Unbalance Compensation in Active Magnetic Bearings, 7th Int. Symp. on Magnetic Bearings, ETH Zurich, Switzerland. 4 Sortore, C. K., 1999, Observer Based Critical Response in Rotating Machinery, Ph.D. dissertation, University of Virginia, Charlottesville, VA. 5 Beatty, R., 1988, Notch Filter Control of Magnetic Bearings, MS thesis, Massachussetts Institute of Technology, Cambridge, MA. 6 Bleuler, H., Gahler, C., Herzog, R., Larsonneur, R., Mizuno, T., Siegwart, R., and Woo, S.-J., 1994, Application of Digital Signal Processors for Industrial Magnetic Bearings, IEEE Trans. Control Syst. Technol., 2, No. 4, pp Na, H.-S., and Park, Y., 1997, An Adaptive Feedforward Controller for Rejection of Periodic Disturbances, J. Sound Vib., 201, No. 4, pp Hu, J., and Tomizuka, M., 1993, A New Plug-In Adaptive Controller for Rejection of Periodic Disturbances, ASME J. Dyn. Syst., Meas., Control, 115, pp Shafai, B., Beale, S., Larocca, P., and Cusson, E., 1994, Magnetic Bearing Control Systems and Adaptive Forced Balancing, IEEE Control Systems, 14, pp Herzog, R., Buhler, P., Gahler, C., and Larsonneur, R., 1996, Unbalance Compensation using Generalized Notch Filters in Multivariable Feedback of Magnetic Bearings, IEEE Trans. Control Syst. Technol., 4, No. 5, pp Lum, K.-Y., Coppola, V., and Bernstein, D., 1996, Adaptive Autocentering Control for An Active Magnetic Bearing Supporting a Rotor With Unknown Mass Imbalance, IEEE Trans. Control Syst. Technol., 4, No. 5, pp Matsumura, F., Fujita, M., and Okawa, K., 1990, Modeling and Control of Magnetic Bearing Systems Achieving a Rotation Around the Axis of Inertia, 2nd Int. Symp. on Magnetic Bearings, Tokyo, Japan, pp Hisatani, M., and Koizumi, T., 1994, Adaptive Filtering for Unbalance Vibration Suppression, 4th Int. Symp. on Magnetic Bearings, ETH Zurich, Switzerland. 14 Song, G., and Mukherjee, R., 1996, Integrated Adaptive Robust Control of Active Magnetic Bearings, IEEE Int. Conf. on System, Man, and Cybernetics, Beijing, China. 15 Reining, K. D., and Desrochers, A. A., 1986, Disturbance Accomodating Controllers for Rotating Mechanical Systems, ASME J. Dyn. Syst., Meas., Control, 108, pp Mizuno, T., 1998, An Unified Approach to Controls for Unbalance Compensation in Active Magnetic Bearings, IEEE Int. Conf. on Control Applications, Italy. 17 Knospe, C. R., Hope, R. W., Tamer, S. M., and Fedigan, S. J., 1996, Robustness of Adaptive Unbalance Control of Rotors with Magnetic Bearings, J. Vib. Control, 2, pp Knospe, C. R., Tamer, S. M., and Fedigan, S. J., 1997, Robustness of Adaptive Rotor Vibration Control to Structured Uncertainty, ASME J. Dyn. Syst., Meas., Control, 119, pp Fujita, M., Hatake, K., Matsumura, F., and Uchida, K., 1993, Experiment on the Loop Shaping Based H-Infinity Control of Magnetic Bearing, Proc. American Control Conference. 20 Rutland, N. K., Keogh, P. S., and Burrows, C. R., 1994, Comparison of Controller Designs for Attenuation of Vibration in a Rotor Bearing System under Synchronous and Transient Conditions, 4th Int. Symp. on Magnetic Bearings, ETH Zurich, Switzerland, pp Setiawan, J. D., Mukherjee, R., and Maslen, E. H., 2001, Adaptive Compensation of Sensor Runout for Magnetic Bearings with Uncertain Parameters: Theory and Experiments, ASME J. Dyn. Syst., Meas., Control, 123, No. 2, pp Mohamed, A. M., Matsumura, M., Namerikawa, T., and Lee, J., 1997, Qparameterization Control of Vibrations in a Variable Speed Magnetic Bearing, IEEE Int. Conf. on Control Applications, Hartford, CT. 23 Kim, C.-S., and Lee, C.-W., 1997, In Situ Runout Identification in Active Magnetic Bearing System by Extended Influence Coefficient Method, IEEE/ ASME Trans. Mechatron., 2, No. 1, pp Khalil, H., 1996, Nonlinear Systems, 2nd Ed., Prentice Hall, Upper Saddle River, NJ. 25 Siegwart, R., 1992, Design and Applications of Active Magnetic Bearings AMB for Vibration Control, Von Karman Institute Fluid Dynamics Lecture Series , Vibration and Rotor Dynamics. 24 Õ Vol. 124, MARCH 2002 Transactions of the ASME