1995_Experiments in the control of unbalance response using magnetic bearings
Pergamon 09574158 (95) 000151
Mechatronics Vol. 5, No. 4, pp. 385400, 1995 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved. 09574158/95 $9.50+0.00
EXPERIMENTS IN THE CONTROL OF UNBALANCE RESPONSE USING MAGNETIC BEARINGS
CARL R. KNOSPE, R. WINSTON HOPE, STEPHEN J. FEDIGAN and
R O N A L D D. W I L L I A M S Center for Magnetic Bearings, Department of Mechanical, Aerospace, and Nuclear Engineering, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 25 August 1994; accepted 13 January 1995)
AbstractUnbalance response is a common vibration problem associated with rotating machinery. For several years, researchers have demonstrated that this vibration could be greatly alleviated for machines using active magnetic bearings through active control. Many of the control strategies employed fall into a class which the authors have termed adaptive open loop control. In this paper, three algorithms in this class are presented and their performances are examined experimentally. These algorithms are (1) a nonrecursive control law with simultaneous estimation, (2) a recursive control law with simultaneous estimation, and (3) a recursive control law with gain scheduling according to operating speed. Each algorithm was coded in C and executed on a highspeed, multitasking digital controller. The advantages and disadvantages of each algorithm are illustrated by examining experimental results from a laboratory magnetic bearing rotor rig. These results deafly demonstrate the high degree of synchronous vibration attenuation (over 30 dB) which can be achieved with adaptive open loop methods. The response of these algorithms to a sudden change in "simulated imbalance" is used to evaluate their relative transient performances. These results indicate the benefits of recursive control laws in adapting the synchronous open loop control currents to cancel the vibration. The ability of each of the algorithms to adapt the open loop control during changes in rotor speed is also examined. On this test, the recursive gain scheduled algorithm shows superior performance: rotor midspan vibration is almost completely eliminated over the operating speed range. However, surprisingly, the nonrecursive control law shows better performance than the recursive law with simultaneous estimation. This result is explained in terms of the stability of the adaptation process. 1. INTRODUCTION Active m a g n e t i c bearings p r o v i d e a n u m b e r of advantages over conventional bearings for a variety of practical industrial applications. These include elimination of the lubrication system, friction free o p e r a t i o n , decreased p o w e r c o n s u m p t i o n , o p e r a t i o n at t e m p e r a t u r e extremes, and vibration control. R e c e n t l y there has b e e n a great deal of interest in digital control of magnetic bearing systems. Digital control offers several m a j o r benefits over analog control systems for magnetic bearing s u p p o r t e d rotors: quick tuning of a magnetic bearing system during installation i m p l e m e n t a t i o n of some simple but powerful control strategies, scheduling application o f fault tolerant controller architectures builtin m o n i t o r i n g and diagnostic capabilities. such as gain
A s the results p r e s e n t e d here indicate, digital control provides capabilities for 385
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adaptive control which can be used to greatly alleviate the unbalance vibration of rotating machinery. This is often the worst vibration problem encountered during operation. The source of this vibration is the discrepancy between the geometric axis of the rotor and its inertial axis. When the rotor is spinning, this imbalance results in a centrifugal force which causes synchronous vibration throughout the machine. This problem is managed on conventional machinery through mechanical balancingthe addition or removal of a small amount of mass from the shaft to reduce the residual imbalance. Rotor balancing in the field, unfortunately, is usually time consuming and costly. The downtime incurred can also be very expensive in terms of lost production. Also, for some machines where the imbalance changes often during operation, such as centrifuges, mechanical balancing will have a limited benefit. Magnetic bearings, being active devices, offer the capability to establish new and beneficial relationships between rotor and casing vibration and applied bearing force. This capability has been employed by a number of researchers [110] investigating the control of unbalance response. One method to achieve unbalance response attenuation is through design of the feedback compensation. This has been achieved via the addition of filters to stabilizing controllers [1, 8] or through the addition of pseudostates in observerbased controllers [2, 3]. Other researchers [410] have employed methods which the authors refer to as adaptive open loop control. These methods, as pointed out by Larsonneur [7, 8] and Shafai et al. [9], have the advantage that they may be added to feedback controllers that have been designed for optimum transient response without altering system stability or performance. These methods were first employed on a magnetic bearing supported rotor by Burrows and Sahinkaya [4] who solved a leastsquaresbalancing problem for the proper forces to apply using an offline theoretical model. They later extended this work to obtain an estimate of an influence coefficient matrix through trial forces and the use of a recursive control law [5]. Higuchi et al. [6] applied an adaptive open loop method (periodic learning control) that employed an estimate of the inverse transfer function in a recursive update equation. This method as presented can only be employed on systems with square influence coefficient matrices (number of actuators equals number of vibration sensors). Both refs [5] and [6] are very similar to the convergent control algorithm presented by Knospe et al. [10] which uses a lookup table of influence coefficients obtained through offline testing. Shafai et al. [9] employ a very different method of adapting the open loop forces to cancel a synchronous signal. Here, only one Fourier coefficient of the open loop signal is changed per adaptation cycle in such a fashion as to decrease the residual error. This method, originally developed for SISO systems, was extended to square MIMO systems. Stability and performance robustness of this method (convergence to optimal open loop control) is guaranteed. This is in contrast to most of the modelbased methods [47] where stability and performance robustness has generally not been examined. Also, the transient performance of the proposed adaptive open loop algorithms [49] to changes in imbalance or rotor speed has not been considered. In this paper, three adaptive vibration control algorithms are evaluated experimentally. These algorithms were coded in C and executed on a digital controller designed and built at the University of Virginia [11]. Feedback and adaptive unbalance control algorithms are coded and executed as separate tasks under a multitasking, realtime operating system written at the University [12].
Control of unbalance response using magnetic bearings
2. ADAPTIVE OPEN LOOP CONTROL
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The paradigm used for design of the control algorithms presented is that of adaptive open loop control. Ideally, an open loop controller preschedules the bearing forces to be applied to minimize the vibration. An advantage of this method is that if the scheduled forces are correct, nearly perfect performance can be realized since stability considerations impose no constraint upon achievable performance for an open loop controller. However, this control strategy relies on the accuracy of the precomputed schedule; any error in calculating the scheduled forces cannot be corrected. Also, to calculate the proper schedule of forces, future disturbances to the rotor sytem must be known in advance. This imposes serious limitations on the type of rotor vibration problems which can be tackled with open loop control. The first problem of open loop control methods, the inability to fix an incorrect schedule, can be overcome by recomputing the schedule periodically. If the length of the schedule is much longer than the largest time constant of the system, this adaptation of the open loop control does not appreciably affect the rotor's transient response and the system's dynamics may be considered to be unaltered. The adaptive open loop controller may also be viewed as a very slow, nonlinear feedback controller. However, this paradigm is not very useful in designing such controllers or in analyzing their behavior. The second problem of open loop control methods, the requirement of knowing future disturbances so as to compute a schedule, restricts the successful application of adaptive open loop control to only a few disturbance sources, such as rotor imbalance, where future disturbances are known to be synchronous sinusoids.
3. ADAPTIVE OPEN LOOP CONTROL ALGORITHMS The forced response of a rotor supported in magnetic bearings at any particular operating speed co can be described by the following equation: X = TU + w2Q, (1)
where X is a 2n x 1 real vector of the synchronous Fourier coefficients of vibration (or signals to be attenuated) at n points of interest (perhaps, along the rotor and in the housing), U is a 2m x 1 real vector of the synchronous Fourier coefficients of the open loop control currents applied to the m magnetic actuators, T is a 2n x 2m real matrix of influence coefficients relating the open loop control forces to the synchronous vibrations, and Q is a 2n x 1 real vector of the synchronous Fourier coefficients of uncontrolled vibration at the n points of interest normalized by the square of the operating speed w. Here, each point of interest consists of only one direction of vibration (e.g. vertical). Also, two magnetic actuators (e.g. horizontal and vertical) constitute one magnetic bearing. Note that the vector of synchronous Fourier coefficients U determines the amplitude and phase for each of the synchronus open loop control currents and thus determines a schedule of applied open loop control forces. The vector X is determined during operation through a discrete time convolution of the vibration signals with 1 x sine and cosine waves. The influence coefficient matrix T is a function of the operating speed and is determined by the
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dynamics of the rotor, bearings, and housing. All the algorithms explored in this paper are designed to minimize a weighted quadratic performance function: J = X1WX, (2)
where W is a diagonal weighting matrix. This form of performance function was chosen since it has an analytic solution for the optimal control and generally yields good performance. The optimal control vector that minimizes this performance index is U*
=
~o2(TTWT) '1 TTWQ.
(3)
The model given in Eqn (1) is rewritten here with subscripts to reflect that a sequence of open loop control schedules is applied to the rotor: X, : TU, + ~o~Q. (4)
Here the subscript i has been added to indicate that the values of the quantities are those during the ith open loop schedule applied. This quasisteady model of the machine's synchronous response assumes that the rotor speed is changing slowly enough that (1) the unbalance response can be considered to have reached its steady state value and (2) the Fourier coefficients of the vibration can be properly calculated. This model also assumes that any transients from the last update of the control vector (from schedule i  1 to schedule i) have decayed before the Fourier coefficients of the next (schedule i) vibration vector are calculated. This imposes a constraint on the update rate of the control force schedule. The three adaptive open loop control algorithms examined in this paper fall into two categories based on whether continuous online estimation is used: Simultaneous Estimation and Control Algorithms (SEC) (1) a nonrecursive control law that employs online estimation (NRSEC) (2) a recursive control law that employs online estimation (RSEC) Gain Scheduled Algorithm (GS or Convergent Control) (3) a recursive control law that employs a gain schedule (RGS). The SEC algorithms repeatedly estimate the model's parameters (typically, T and Q) while applying control based on these estimates. In contrast, the RGS algorithm does not update its estimate of the model's parameters but instead relies on the recursion of the control law to adapt the open loop control schedule. These three algorithms are now examined in detail. 3.1. Nonrecursive simultaneous estimation and control with leastsquares estimation The nonrecursive optimal control law is derived by minimizing the 2norm of the vibration vector and substituting the estimates (7"i and Qi) into the control law for actual parameters (T and Q). It should be noted that this substitution, known as certaintyequivalence, yields a different result than if the uncertainty about the parameter estimates was used in deriving the control law, Using the model given by Eqn (4), the optimal control law that minimizes the
Control of unbalance response using magnetic bearings weighted quadratic norm of the vibration vector is
^ U/+I =  o i +2t ( T ^ T W T i ) i ^ ^ 1TTWQ i
389
(5)
(see Appendix). Note that the next control schedule applied is only dependent on the previous schedule in an indirect manner (through the estimation process). Therefore, this control law is referred to as nonrecursive. The estimates Ti and Qi are computed using a moving batch leastsquares estimator: [0, T,] = [X]i[U]r~([U]i[u]T) ~, (6)
where [U]i and [X]i are matrices composed of the p most recent augmented controls and responses
[Uli
(~0i
O~i 1
. .
...
O.)ip+ 1
[Xl

[xi
X _l
...
(7)
This identification algorithm may have numerical problems when the rotor is operating at a constant speed. In this case, the control vector is nearly constant and the columns of [U]i will be nearly identical. Therefore, this matrix will be rank deficient and the inverse in Eqn (6) will be poorly conditioned. To remedy this, the identification algorithm has been slightly modified. If the difference between the most recently applied control vector Ui and the most recent vector in the moving batch Uk as measured by
is greater than a predetermined constant ~ then the oldest vector in the batch is removed and the vector U/is added to the batch. If e < ~, then the most recent vector in the batch Uk is removed and the new vector U~ is added to the batch. The same operations are also performed on the corresponding response vectors in the batch [X]i. This modification prevents the control batch [U]i from having rank less than its column dimension 2m + 1.
3.2. Recursive simultaneous estimation and control with leastsquares estimation
To minimize the performance index Xi+lrWXi+l, the recursive algorithm employed calculates the new control vector using U/+l = U/  (TTW ~ i )  I ~ T w x i (8)
(see Appendix). Note that since Xi is measured, only T needs to be estimated for this control law. The leastsquares estimate of T is given by Ti = [AX]i[A U] f(tA U],[A U] r)l, where the batches [A U ] i and [AX]i are defined by
[AU]i ~ [AU/ AU/_ 1 ... AU/_p+I]
(9)
(lO)
[aXl,IaX,
ax,_
...
Ax,_
+d
and
AXi= Xi  Xia
A G ~ U/  U/1.
(11)
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Note that substitution of Eqn (4) into Eqn (8) yields ~+~ = [I  ( T / W T , ) 1TirWT~] ~  c ~ ( T / W T , )  ' T~WQ. (12)
This clearly shows the direct recursive relationship between one control schedule and the next. The stability of this algorithm, while difficult to analyze because of the updating of the estimates, is obviously related to the eigenvalues of the matrix

For both the SEC controllers, a batch of p previously applied control vectors and responses must be available to begin estimation. For the estimation of the NRSEC algorithm, the batch size p must be greater than or equal to 2m + 1. For the estimation of the RSEC algorithm, the batch size must be at least 2m. Therefore, in both these cases, the adaptive open loop control algorithm must be started with a series of at least 2m + 1 test schedules. 3.3. Recursive gain scheduled algorithm (convergent control) The recursive gain scheduled control algorithm is very similar to the RSEC controller, Eqn (8), but with the RGS control algorithm T is not estimated during adaptation of the control vector. The RGS algorithm consists of iteratively applying the control law U,+l = U,  A,,Xi with
~T A,, = (T,,,T,,,) 1 T,,,. = ~l~
(13)
Here, T,~ is the estimate of T for the rotor operating speed ~o. The A,,, matrix is determined prior to the iterative application of Eqn (13). If the influence coefficient matrix T is square (i.e. the number of actuators is equal to the number of vibration sensors), the control vector will converge to U* if T~o is sufficiently close to T. 3.4. Performance robustness for square T If the error in the estimate of a square T matrix is expressed as an additive error matrix E~ T,,, = T + E, (14)
then the control law of Eqn (13) will converge to U* and X will converge to zero provided that O(Ea) < ~(T,,) (15) (a sufficient condition) where O(e) and g ( e ) indicate the maximum and minimum singular values respectively (see Appendix). With an error in the estimate of a nonsquare T matrix, it is easy to demonstrate that the control vector generally will not converge to the optimal vector U*. A bound on the degradation in the performance due to a certain size error O(Ea) is difficult to obtain (a bound will be presented by the authors in a future paper). However, it is easy to determine the stability robustness properties of the adaptation.
Control of unbalance response using magnetic bearings
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3.5. Stability robustness of adaptation for nonsquare T
If the error in the estimate of a nonsquare T matrix is expressed by an additive error matrix as in Eqn (14), the RGS adaptation algorithm will be stable (i.e. U will converge to some finite U~s)if
O(ea) <  
1
O(A)
(16)
(see Appendix). Our experience indicates that the RGS algorithm has good performance robustness for nonsquare T matrices as well as the indicated stability robustness. For the results presented in this paper, the matrix To, is estimated using the least squares estimator of Eqns (9)(11) for a set of speeds f~  {to1, to2. . . . . toq} covering the operating speed range. Gain matrices Ak are then computed for each speed tok in the set f~ and these are stored in a lookup table for use during operation. The matrix Ao to be used at any particular speed to during operation is determined by elementwise linear interpolation between the matrices for the two nearest operating speeds in the table, tok and tok+l
A'°= ( tok+lto tok/)Ak+l+ __ _tok_
\tok+l_:k)Ak" (tok+l
(17)
4. E X P E R I M E N T A L RESULTS
A laboratory test rig with two radial magnetic bearings, shown in Fig. 1, was used to examine the efficacy of the adaptive balancing algorithms using a multitasking digital controller. The rotor of this rig has a 12.7 mm (0.5 in.) diameter and a 508 mm (20in.) bearing span. Eddy current position sensors are located vertically and horizontally 32 mm (1.25 in.) outboard from the center of each bearing and 51 mm (2 in.) inboard from the center of the midspan disk. Bearing housing asymmetry resulted in unequal horizontal and vertical support stiffness. The rotor is supported using decentralized proportionalderivative control. The first critical speed of this rotor is at approximately 2700 rpm. In the experiments, both bearings (m = 4) were used to reduce the vibration at the inboard, outboard, and midspan locations (n  6). The control algorithms described in the previous section were implemented on a digital controller designed and built at the University of Virginia's Center for Magnetic Bearings. The digital controller is a 32 bit floating point machine using a Texas Instruments TMS320C30 digital signal processor [11]. Both the feedback control and the adaptive open loop control algorithms were coded in C and executed under a multitasking realtime operating system written at the University of Virginia [12]. The feedback algorithm executes as the highest priority task. The adaptive open loop algorithm executes during spare time between feedback updates. For all results presented herein, unless otherwise noted the following test conditions were used: the midspan vibration was weighted a factor of 6.3 greater than the inboard and outboard vibration
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Fig. I. Magnetic bearing test rig.
the batch size used in least square estimation, p, was 14 the Fourier coefficients of the vibration were computed over 15 revolutions.
4.1. Constant speed vibration reduction2700 rpm
The rootmeansquare synchronous midspan, inboard, and outboard vibration amplitudes without adaptive open loop control and with the NRSEC, RSEC, and RGS algorithms are shown in Fig. 2. Note that all three algorithms achieve excellent attenuation of the midspan vibration, over 36 dB. The inboard vibration has increased slightly with the use of the adaptive controllers while the outboard vibration has decreased. The authors do not view the slight differences between the steady state performance of the three algorithms as significant. These differences fall within the variation that is normally seen on experiments with this test rig.
4.2. Sudden change in imbalance
A sudden change in the imbalance condition of the rotor was experimentally simulated by the addition of synchronous perturbation currents to the bearings. These currents act to produce an unknown force which rotates with the shaft much like a physical imbalance. For these tests, the rotor was operated at 2200 rpm. Figure 3 shows the midspan synchronous response of the rotor when adaptive open loop control is not used. For the first 10 updates, the rotor vibrates due to its physical imbalance. Then, the "balance condition" of the rotor changes suddenly due to the
Control of unbalance response using magnetic bearings
35O 300 I '~ 250

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= 200
e~
o
150
o
.~ 100 rm 50 0 No Bal Global Local CC Fig. 2. RMS synchronous vibration amplitudes without and with the NRSEC, RSEC, and RGS adaptive open loop control algorithms, 2700 rpm.
1000 900 = 800 o .,~ 700 " ~" 600
t
0 5


I
I I
I
_
....
i
n
.~
=o g ,o u =
500
400 300 200 100 0 10 15 20 25 30 35 Time (seconds)
Fig. 3. Synchronous midspan vibration amplitude without adaptive open loop control during the simulated change in balance, 2200 rpm.
applied synchronous perturbation currents. Figures 4, 5, and 6 show the transient response of the NRSEC, RSEC, and RGS algorithms, respectively, during this test. The NRSEC responds rather poorly to the sudden change in imbalance. It updates its values of T and Q from the input and output data that has been collected. However, much of this information describes the system as it existed before the change. The estimator finds the parameters with the "best fit" to the available data, but performs poorly until all the old information (predating the change) has moved out of the batch. Then, excellent vibration attenuation is once again recovered. The RSEC algorithm recovers much more quickly after the sudden change in imbalance than the NRSEC algorithm. This quick recovery is due to the following. (1) The data collected before the sudden change is still useful in determining the T
394
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C . R . KNOSPE et al.
co
>~
800 700 'i 600 ! 500 i
l i
"'
gE
I,. "  e., e/3
400 , 300
200 100
0
,, I1!
i
I i i I
I'1, / 'i I
Im'
mmmaalnemnill libel mnl~uilillll
0
5
10
15
20
25
30
35
Time (seconds)
Fig. 4. Synchronous midspan vibration amplitude with the NRSEC algorithm during the simulated change in balance, 2200 rpm.
1000
c' .o ~
> ~
900 800 700 600
I
i
7,,
O ~.__. soo g E 400
:~ r/3
300 200 100
O
4 !
ilnlllDililll+
iilililNliilnNnl!iNlnlUgllinl
10 15 20 25 30 35 Time (seconds)
0
5
Fig. 5. Synchronous midspan vibration amplitude with the RSEC algorithm during the simulated change in balance, 2200 rpm.
matrix since this has not changed. Only a single set of measurements ( A X and A U) in the batch straddles the sudden change and is therefore misleading in estimation. (2) The convergent nature of the recursive algorithm used. Since the RSEC is essentially a RGS algorithm with the T matrix updated continuously, it has essentially the convergence property indicated in Eqns (15) and (16). Therefore, the algorithm can produce good performance even when the estimate of T is in error due to the misleading m e a s u r e m e n t set. As shown in Fig. 6, the RGS algorithm has a very quick recovery to the sudden change in imbalance. This high performance is due to the following. (1) The algorithm does not estimate the T matrix. Since A , , is calculated from a
Control of unbalance response using magnetic bearings 1000 900 800
700
395
e.
o
,.3 O
~'= 600
5oo
g
= (.)
era3
400 300 200 100 0 0 5 10 15 20 25 30 35 Time (seconds)
Fig. 6. Synchronousmidspan vibration amplitude with the RGS algorithm during the simulated change in balance, 2200rpm. lookup table, the algorithm assumes it changes only as a function of operating speed. A misleading data set (AX and A U) does not corrupt the estimate of T and hence the gain matrix Ao,. (2) The convergent nature of the recursive algorithm. Note that the convergence property of this algorithm permits good vibration attenuation even if T changes, although the RGS algorithm does not track these changes through online estimation as do the SEC algorithms. In summary, the recursive algorithms respond best to a change in the vector Q. The performance of the three algorithms during a change in the T matrix is examined next.
4.3. Changing rotor speed
As the rotor speed changes, the influence coefficient matrix T changes. Here, the performance of the three algorithms is examined during a 30 second runup from 1200 to 3000 rpm. Figure 7 shows the amplitude of the synchronous midspan vibration without adaptive open loop control. The first critical speed is clearly visible at 2700 rpm. The performance of the NRSEC, RSEC, and RGS algorithms during the runup is shown in Figs 8, 9, and 10, respectively. The NRSEC is effective in reducing the midspan vibration from 1200 to 2500 rpm. Over most of this speed range, the influence coefficient matrix is changing slowly and the batch leastsquares estimator can track these changes. Near the critical speed, the T matrix changes quickly and the estimator has difficulty in following this variation. Note that for the estimation performed at 2700 rpm in this test, the oldest data in the batch is from 2050 rpm. The RSEC algorithm also performs well from 1200 to 2500 rpm. However, as with the NRSEC, the estimates of the system parameters are increasingly in error near the critical speed. For the RSEC algorithm, however, this results in adaptation instability as the poor estimate of T results in an eigenvalue of the matrix in Eqn (12)
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300 p 250
.m
C.R. KNOSPE et al.
o
E
0
Midspan
200 150
>
100 ,
G} ep
50
I
k
o
1200
P
t
1400
1600
1800
2000
2200
2400
2600
2800
3000
Rotor Speed (rpm)
Fig. 7. Synchronous vibration amplitudes without adaptive open loop control during a 30 second runup
from 1200 to 3000 rpm.
300
i
250 +
/
= 200
i
= ~ ~*~
Inboard Outboard Midspan
0
15()
e
o
l ()0 i
5(l
1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2l)00 2 2 0 0 24(}0 26(}0 2 8 0 0 3 0 0 0
Rotor Spccd ( r p m )
Fig. 8. Synchronous vibration amplitudes with the NRSEC algorithm during a 30 second runup from 1200 to 3000 rpm.
being outside the unit circle. This is the cause of the loss of rotor support just below
2600 rpm.
The RGS algorithm performs very well during changes in rotor speed as Fig. 10 illustrates. This algorithm quickly adapts to the changes in rotor speed using the lookup table. Since the RGS algorithm is computationally much simpler than either SEC algorithm, it executes faster and more often. Thus, the RGS algorithm updates its control vector 50 times in 30 seconds while the NRSEC updates only 36 times. Considering that a significant proportion of the time between updates is spent
Control of unbalance response using magnetic bearings .... 300 ~250
397
.=~
>
O
200 150 100
Inboard
   0   Outboard *Midspan
0
~"~~'~"
~
~~I'~"~*"4~'~*'~"~'~"
;
~
1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Rotor Speed (rpm) Fig. 9. Synchronous vibration amplitudes with the RSEC algorithm during a 30 second runup from 1200 to 3000 rpm.
300
~250
O .~.
= 200
Inboard Outboard
>. 150 100
50
°
Midspan
3"DOOCe~I~243G~
o 12oo 1400 1600 1800 2000 2200 2400 2600 2800 3000
Rotor Speed (rpm)
Fig. 10. Synchronous vibration amplitudes with the RGS algorithm during a 30 second runup from 1200 to 3000 rpm.
performing a discrete time convolution to obtain the Fourier coefficients (which is the same for all three algorithms), this is a significant increase in update rate. An increased update rate yields better tracking and vibration rejection.
5. DISCUSSION The best algorithm for an industrial application m a y be a hybrid of the R  G S and SEC algorithms. This hybrid might use p e r f o r m a n c e measures to determine when the estimates need to be updated. U n d e r most conditions, it would function as recursive gain scheduled algorithm and would have a quick response during changes in unbalance response or rotor operating speed. H o w e v e r , when its performance
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differed significantly from that expected (e.g. divergence of control action), the controller would begin to update its estimates. When the performance returned to being acceptable, the estimation procedure would be halted. It would be important for this hybrid algorithm to distinguish between sudden changes in the imbalance condition resulting in momentary poor performance and controller divergence due to error in the estimate of T. In the first case, the proper action is to continue to employ the RGS algorithm. In the second case, a new estimate of T needs to be obtained from recent online measurements. Thus, the criterion used to determine which mode of operation to adopt cannot be based solely on vibration level. A measure of the accuracy of the T matrix employed should also be used. This measure could be a norm of a propagated covariance matrix as used in discretetime Kalman filters [13].
6. CONCLUSIONS
All three adaptive open loop control algorithms presented in this paper have been shown to provide excellent synchronous vibration attenuation in steady state operation. The performance of the three algorithms, however, differed greatly during the transient tests. The nonrecursive SEC algorithm had sluggish vibration rejection after the sudden change in imbalance. G o o d performance was not recovered until all the measurements previous to the change had m o v e d out of the batch. The recursive SEC algorithm performed much better at this since the corruption of its estimates was not severe enough to cause adaptation instability. The recursive gain scheduled algorithm had the best performance during this test. The midspan vibration was canceled after one adaptation. During the rotor runup test, the N R  S E C algorithm had poor performance near the critical speed since the estimator could not track the changes in the influence coefficient matrix. This performance was better, however, than that of the RSEC algorithm. Poor tracking with this algorithm resulted in the adaptation process instability. The performance of the RGS algorithm during this test was excellent; the midspan vibration was canceled over the entire operating speed range.
REFERENCES
1. Habermann H. and Brunet M., The active magnetic bearing enables optimum damping of flexible rotors. ASME Paper 84GT117 (1984). 2. Higuchi T., Mizuno T. and Tsukamoto M., Digital control system for magnetic bearings with automatic balancing. Proceedings of the 2nd International Symposium on Magnetic Bearings, Tokyo, 1214 July (1990). 3. Matsumura F., Fujita M. and Okawa K., Modeling and control of magnetic bearing systems achieving a rotation around the axis of inertia. Proceedings o f the 2rid International Symposium on Magnetic Bearings, Tokyo, 1214 July (1990), 4. Burrows C. R. and Sahinkaya M. N., Vibration control of multimode rotorbearing systems. Proc. R. Soc. Lond. 386, 7794 (1983). 5. Burrows C. R., Sahinkaya M. N. and Clements S., Active vibration control of flexible rotors: an experimental and theoretical study. Proc. R. Soc. Lond. 422, 123146 (1989). 6. Higuchi T., Otsuka M., Mizuno T. and Ide T., Application of periodic learning control with inverse transfer function compensation in totally active magnetic bearings. Proceedings o f the 2nd International Symposium on Magnetic Bearings, Tokyo, 1214 July (1990).
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7. Larsonneur R., Design and control of active magnetic bearing systems for high speed rotation. Ph.D. thesis, Swiss Federal Institute of Technology, ETH Zurich, Switzerland (1988). 8. Larsonneur R. and Herzog R., Feedforward compensation of unbalance: new results. IUTAM Symposium, Bath, U.K., September (1994). 9. Shafai B., Beale S., LaRocca P. and Cusson E., Magnetic bearing control systems and adaptive forced balancing. IEEE Contr. Syst. 14(2), 413 (1994). 10. Knospe C. R., Hope R. W., Fedigan S, J. and Williams R. D., Adaptive online rotor balancing using digital control. In Proceedings of MAG '93 Magnetic Bearings, Magnetic Drives, and Dry Gas Seals Conference (Edited by Allaire P. E.), pp. 153164. Technomic, Lancaster, PA (1993). 11. Daddio J. K., An integrated magnetic bearing controller. M.S. thesis, University of Virginia, Charlottesville, VA (1992). 12. Fedigan S. J., A realtime operating system for a magnetic bearing digital controller. M.S. thesis, University of Virginia, Charlottesville, VA (1993). 13. Mendel J., Lessons in Digital Estimation Theory. PrenticeHall, Englewood Cliffs, NJ (1987).
APPENDIX
A.1. Control laws A.I.1. Nonrecursive SEC. The NRSEC control law minimizes the performance index J = E{X~IWXiH}, (A1) where E{e} is the expected value operator. Substitution of Eqn (4) into this and changing the index to i + 1 yields J = E((T~+I + O)2i+lQi+l)r(TUi+l + o)2+lQi+l)}. Assuming T is known (certainty equivalence assumption), this can be simplified to 2 T ^ J = UT+lf'TWTUi+I + ogi+lUi+lT r WE(Qi+I} + {terms not including Url}. (A3) Since there is no a priori information available about Qi+l, the expected value of this vector is assumed to be equal to that experienced in cycle i (a posteriori information), that is E{Qi+I} = E{Qi} = O.i. (A4) Here, it is assumed that the estimate produced by the leastsquares estimator is unbiased. Substituting Eqn (A4) into Eqn (A3) and setting the first variation of J with respect to UfH equal to zero yields 0 = f"TW~'iUiH + co2i+lTTWO.i. Now, solving for U/H yields the optimal control law UiH  w~+l[7"~W:Fi]lTr, WOi. A.1.2. Recursive SEC. Solving Eqn (4) for Qi yields (A5) (A6) (A2)
Qi = ~(x;
Substituting (A7) into (A4) yields Ef Xior
 TU;).
(A7)
tTJ
TUi)
:
O,
(A8)
(once again, using a certainty equivalence assumption). Substituting this expression into Eqn (A6) yields Ui+l = (wi+~l]2(Ui [ T f W f i l H f f W X i } . '~ ~Oi l This control law was simplified for implementation by assuming that tOi+l = wi, yielding Eqn (8). (A10)

(Xi  TiUi) = Q.i
i
(A9)
400
C.R.
K N O S P E et al.
A.2. Stability and performance robusmess o f R  G S algorithm A.2.1. Stability and performance robustness for square T. Assuming the rotor is at constant speed o) and that the uncontrolled vibration is constant, the vibration during schedule i + 1 can be related to that during schedule i via X,+j = X, + /(U,I  U/). Substituting Eqns (13) and (14) into this expression yields X,+, = {1 [7",r,,WT,,,I'I'J,W(7",,, E , ) I X , . For square T. this simplifies to X,.l [Ti,,lf:~jXi. If Cr('l'L~E,,)1 (AI4) then the 2norm of the vibration vector will decrease with each adaptation, and therefore converges to zero indicating both stability and optimal performance. Condition (A14) will be satisfied if O(7"S,,~)O(E.) Using the identity o(M I ) _ a ( M ) l Eqn (AI5) can be written as O(E,) < o(~/',,,). A.2.2. Stabili O' robustness jbr nonsquare T. Substituting Eqns (4) and (14) into Eqn (13) yields U,.i = &)  A,,,{(f',,, E,)U, + toPOi}. (AI8) (A17) (A16)
<
(All) (A12)
(AI3)
I.
(A15)
The equilibrium solution Ueq to this difference equation must satisfy U,,q = U,,q  A,,,{(T,., Subtracting Eqn (A19) from (A18) yields Oh',_, = {I  A,,,~r., + A,,,E.IbU,, (A20) E,)U,,q + ~o~Qi}. (A19)
where 6U, is the difference between Ui and the equilibrium solution. Therefore, this difference becomes smaller with each iteration if 0(I Since A,,,T,o = l, this equation simplifies to O ( A , , , E , , ) . 1. Condition (A22) will be satisfied if O(A,,,IO(E.) < 1. Therefore, a sufficient condition for stability of the adaptation process is O(E.) ~ . . . . o( A ,,,) " Under this condition the control vector will converge to equilibrium solution of Eqn (AI9) lYl,q  ,t~[I  A,,,Ea] 1A,,,Qi and the adaptation process will be stable.
1
A,,,T., + A,~E,,) < 1.
(A21)
(A22)
(A23)
(A24)
(A25)
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