Stabilization of Centrifuges with Instabilities Due to Fluid-Structure Interactions : Various Control Approaches

Partially filled centrifuges exhibit unstable behavior over particular rotational speed 
ranges. To remove this unstable behavior, three different control approaches are 
presented which are experimentally implemented using a magnetic bearing. First, by 
means of a PD-controller the system stiffness is changed leading to a shift of the instability 
region. Second, the instability region is removed by a cross coupled feed back 
of the rotor vibration velocities. This cross coupling directly counteracts the fluid excitation 
mechanism. Third, a disturbance observer is presented which allows to estimate 
and compensate the not exactly known fluid force.


INTRODUCTION
In industry partially filled centrifuges are used in different fields, e.g., as sugar-or dirty-water- centrifuges or separators for the separation of fluids with different densities.In certain rotational speed ranges couplings between the motion of the rotor and the motion of the rotating fluid lead to an unstable behavior of the centrifuge.Since passive measures do not offer any or only un- satisfying solutions and do not allow to change the system parameters in operation, the objective of the present research is to actively remove the instability using a magnetic bearing.For the design of suitable approaches for stabilizing the fluid-rotor-system a model was presented in (Ulbrich et al., 1990; Ulbrich et al.,  1998 and Ahaus, 1999).In (Ulbrich et al., 1996)   the fluid viscosity was neglected leading to an un- stable behavior in the simulation over the whole operating range when external damping was con- sidered.This discrepancy to the experimental results is eliminated when the fluid viscosity is taken into account (Ulbrich et al., 1998; Ahaus,   Corresponding author.Tel.: 49-201-183-2905, Fax: 49-201-183-2871, e-mail: ulbrich@uni-essen.de1999).Besides these research works, the viscous fluid model was investigated by (Hachmann, 1989;  Riedel, 1992) and others.Although the non- viscous and viscous models yield similar results, the viscous fluid model renders a more exact stability statement.However, the disadvantage of the viscous fluid model is that the equation of motion can not be described in the time domain (Ahaus, 1999).Based on these models the three control approaches described in this paper are designed and experimentally verified on a test rig.Control approaches for the stabilization of a partially filled centrifuge have been presented by (Takagi et al., 1993 and Matsushita et al., 1988).
Takagi used a PD-controller which, allows to shift the instability region to another speed range.He determined two different pairs of PD-values for two non-overlapping instability regions of a centrifuge modeled as a rigid body and supported by two active magnetic bearings.By switching between these two different PD-values the system remains stable throughout the considered speed range.Matsushita removed the instability by damping a specific forward natural motion that he identified as the unstable natural motion of the system.He achieved this by a cross coupled feedback of the rotor deflection.Because he need- ed only the components of the deflection signal related to the unstable natural motion, these com- ponents have been determined by a tuning filter that was manually adjusted to the respective rota- tional speed.
The first control concept presented in this paper, the PD-controller, is based upon the approach presented by Takagi.The PD-controller is applied to the present test rig and its effect is illustrated with respect to design parameters.
The second control approach, the velocity cross coupling, is derived from investigating the excita- tion mechanism of the instability.As the induced fluid force is proportional to the rotor vibration velocity and acts perpendicular to it, the system can be stabilized by a cross coupled velocity feedback.
In the third concept the fluid force is regarded as a disturbance force which is estimated by a disturbance observer and compensated by the active magnetic bearing.

MODEL DESCRIPTION
In the model of the investigated centrifuge the fluid container is mounted on top of the vertical shaft.
The shaft is supported by a fixed bearing at the bottom and by an elastically mounted bearing in the middle of the shaft (see Fig. 1).Because of the low stiffness of the upper bearing (compared with the rotor stiffness), at first the rotor was considered as a rigid body.
During the verification process it became evident that the magnetic bearing excites not only the rigid body eigenmode, but also the first two bending eigenmodes of the shaft.Therefore, the rotor model presented in (Ulbrich et al., 1996 and  Ulbrich et al., 1998) is extended using the FE- program MADYN.Introducing the control vec- tor u [Ux, uy] r and the vector of fluid effects fF [FFx, FFy, MFx, MFy] T the equations of motion are given by II + (1; + 0) + t MU + FfF (1)   where is the vector of the position coordinates.
Using the modal matrix , where and considering only the first three eigenmodes both for the xz-and for the yz-plane, Eq. ( 1) is reduced to ij + (I-IG + D)b + Kq JMU + JFfF. (3) The vector q contains the six modal coordinates (three for each plane), the vectors u and fF account for the control forces and fluid effects.The three eigenmodes included in the reduced model are depicted in Figure 1.
For the magnetic force in x-direction, fMx, the following linear relationship is assumed: fMx ksx / kii.The same applies to the y-direction.In Eq. (3), the negative stiffness ks and the control current are already accounted in the stiffness-matrix K and in the control vector u, respectively.The vector of the fluid effects is described depending on the modal position vector fF FJFrq, (5) where F represents the viscous fluid model and shall not be described in more detail here because of its complex structure.A detailed derivation is given in (Ahaus, 1999).It should be remarked that this matrix shows the following dependence on the eigenvalues of Eq. ( 3): (6) Additionally, F depends on other parameters like the rotational speed D,, the viscosity u of the fluid and the filling ratio f, which is the quotient of the radius b of the free liquid surface and the inner radius a of the casing (f= b/a).As the fluid matrix F contains square roots of the eigenvalues, Eq. (3)   can not be evaluated in the time domain, but only in the frequency domain.With the formulation q l e'xt, (7) substituting Eqs. ( 5) and (7) into Eq.( 3) yields 8)   In this case, Eq. ( 8) describes the homogeneous system.This equation can be extended by the control vector u as shown in Eq. ( 12).The eigen- values of Eqs. ( 8) or ( 12) are numerically deter- mined using standard optimization methods (e.g., the Levenbergor Gauss-Newton-method).
As already mentioned, the influence of the con- trol vector u on the modal position vector q is determined by the input matrix J M. On the other hand, the output matrix ']sen transforms the modal coordinates into physical coordinates: y Y--Jsenq. (9) The matrices J M and ']sen are calibrated such that the voltages in y are proportional to the rotor deflections and the voltages in u are proportional to the control currents.The position of the control and measurement plane is represented in Figure 1.
The control and measurement unit can be de- scribed as an ideal proportional feed back element in the considered frequency range.

TEST RIG
The test rig was already presented in (Ulbrich et al.,  1996 and Ulbrich et al., 1998).Here, only the es- sential features are listed: The upper roller bearing's stiffness is adjustable.
The rotational speed range extends from 0 to 50 Hz.
The rotor deflections in x-and y-direction are measured contact-free with two eddy current sensors.
The control forces affect the centrifuge contact- free and are induced by an active magnetic bearing (max.force 350N, nominal air gap 0.8 mm).
The uncontrolled (passive) test rig has an un- stable region for rotor frequencies from 7 to 13 Hz (Ulbrich et al., 1998).
4. CONTROL APPROACHES 4.1.PD-controller The PD-controller is implemented analogously to Takagi's concept.Like in Takagi's case, the PD-controller controls the magnetic bearing (compensation of the negative stiffness ks) and re- locates the instability region.The resulting control Using Eqs. ( 10), ( 3) and ( 7), Eq. ( 8) can be extended to (IA + (fG + D)Ai + K JFFJF r -+-JMRpJsen -+-JMRDJsen/i)ti O. (12)   According to Eq. ( 12), p could be determined such that the instability region lies above the one of the passive system.The quasi-stationary startup behavior of both systems is represented in Figure 2. Two non-overlapping instability regions exist whereby the upper instability region's boundary of the PD-controlled system lies outside of the realizable speed range of the test rig.The ampli- tudes of this upper boundary of the PD-controlled system illustrated in Figure 2 are estimated.When the PD-controller is turned on during the startup just before the instability region of the passive system is reached, the instability region is shifted and the startup is readily carried out.If the passive system's unstable region is passed the PD-con- troller is switched off in order to stay out of the instability region of the PD-controlled system.
Analogously to the behavior of the PD-controlled system, the position of the instability region of the passive system depends on the upper roller bearing's stiffness and thus on the system stiffness.When increasing the system stiffness, the instabil- ity region is shifted to higher rotational speeds and extends over a greater speed range (see Fig. 3 the boundary lines are not closed for _f2= 0 and 1-f2= because the model does not apply to an empty or totally filled container).As the system stiffness can be influenced by the PD-controller, too, one can move the instability region in an equivalent way either by changing the stiffness of the upper bearing or by changing the P-gain.If the design parameters of a centrifuge are known, the instability region could be positioned by a suitable choice of the system stiffness such that it lied above the operating speed range, thus providing stability in the entire speed range.To that end, however, the plant might have to be de- signed very stiff.In this case the operating speed would be below the instability region, but the plant would have to be operated subcritically, which led to higher vibration amplitudes and thus to a higher load in the plant.Hence, the advantage of the active magnetic bearing is that the system stiffness can be varied as desired (restricted by the saturation of the control force).Thus, the instability region can be passed and the plant can be operated supercritically.

Velocity Cross Coupling
The drawback of the PD-controller is that the instability region might not be shifted high enough for certain system configurations so that the instability region of the uncontrolled and controlled system overlap.In this case, other control approaches are needed to stabilize the centrifuge.
An alternative control concept is derived from the characteristics of those fluid force components that lead to an excitation of the rotor vibrations.
For that purpose, a partially liquid-filled container rotating with 1 =const.around its axis is con- sidered which is excited harmonically along its x-axis and y-axis, respectively (no DOFs).Consid-  ering the phase of the vibration-exciting compo- nents of the transfer function matrix fv/ft that account for the vibration excitation, the charac- teristics shown in Figure 4 are obtained.The phases jump to 90 and + 90 , respectively, in a wide range of the excitation frequency; the corre- sponding fluid force components are shown in where FFx, FFy are the fluid forces and JCF, F are the velocities of the center of mass of the fluid container in x-respectively in y-direction.
In order to counteract the vibration-exciting fluid force components, a velocity cross coupling control law is derived: The coupling factor g depends on D, and f and is determined as to provide stability for the parti- cular operating point.To that end, the coupling factor g is increased until the real part of the un- stable eigenvalue becomes negative and a suffi- cient stability reserve is obtained.Hence, only one coupling factor is sufficient for larger ranges of the rotational speed and the filling ratio.
In the continuous case, the cross-coupling factor can be increased without loosing stability until saturation of the control unit is reached.As the control is implemented on a digital controller board, the system might turn unstable for high coupling factors due to e.g., truncation effects or a time delay in the system.This effect holds for any filling ratio (for an empty container, too: comp.Ahrens, 1996) and was also verified on the test rig.By varying the feed back factor d the transition to the unstable behavior can be influenced.For the rotor without fluid, Figure 6 shows the boundary of the instability region in the discrete case as a function of the damping parameter d and the coupling factor g for a rotational speed /2r 12.8 Hz.
Hence, for the choice of the coupling factor, both of the destabilizing effects (fluid excitation and truncation effects) have to be taken into consideration.
The effectiveness of the velocity cross coupling is represented in Figure 7 by a time plot of the x-deflection for a representative operating point (g 0.0025 s, 1]/2-12.8Hz and f= 0.894, corre- sponding to a filling of the casing of 20%).Until 3 s the rotor deflection increases exponentially.
The pulsing of the amplitude (beat) is caused by the superposition of two oscillations: The rotor motion corresponding to the rotational speed and the destabilizing natural motion of the fluid whose frequency is slightly lower than the rotor speed frequency (comp.Ulbrich et al., 1998 and Fig.  8).
Once the controller is switched on, the system is stabilized.The deflection is reduced and the pulsing of the amplitude decreases, i.e., the fluid excitation is compensated.
The velocity cross coupling is a robust control approach, which directly counteracts the fluid forces.Similar to the PD-controller, only one coupling factor suffices for stabilizing large ranges of rotational speeds and filling ratios.

Disturbance Observer
The disturbance observer provides another possi- bility for the stabilization of the centrifuge.The basic idea consists in regarding the influence of the fluid as a disturbance which is not exactly known, but which can be observed based upon a simple model and then be compensated.
This system can be described using state space representation of Eq. ( 3):  For observing the disturbance vector z, a disturbance model is developed which describes the significant features of the disturbance.One characteristic feature is the "fluid frequency" cor- responding to the unstable natural motion of the fluid (see Fig. 8).
The fluid frequency can be determined from Eq. ( 8) and depends on the filling ratio f and the rotor speed 13,,.For a certain operating point, the following simplified fluid model is introduced where CO F is the characteristic fluid frequency and 5 F is suitably chosen in order to obtain a robust system behavior of the disturbance observer when implementing it on the test rig.
Transforming Eq. ( 16) into state space repre- sentation Jdis AdisXdis, --KF --DF : z-[I 0 ]Xdi ff Z--CdisXdi (17) The observer is designed using the differential equation (F611inger, 1992) Xe (Ae LeCe):e q-Beu q-Ley, ( where :e is the estimated state vector and Le is the observer matrix which is determined by pole placement.The pole radii of the observer poles are chosen slightly greater than the pole radii of the plant.However, the real parts of the observer poles are smaller than those one of the plant poles.Thus, the observer is more damped than the plant.To compensate the observed disturbance dis, an additional control voltage Ildi has to be deter- mined and applied to the plant such that according to Eq. ( 15) the following condition has to be BUdis + BFCdisdis e O. (20) Multiplying the left hand side with the pseudo inverse B + leads to ildi -B+BFCdisdis, (21) which requires the control matrix B to have full rank.
Using Eq. (2!), the disturbance compensation Ildi contains not only frequency components of the fluid frequency, but also components different from the fluid frequency.Therefore, Ildi as cal- culated in Eq. ( 21) is filtered with a band-pass filter (2nd order Butterworth).
The effect of the disturbance compensation can be seen from Figure 9 which shows the x-deflection for the same representative point of the instability region as in Section 4.2.The increasing deflection is stopped by switching on the disturbance ob- server in a similar way as by the use of the velo- city cross coupling.The deflection is reduced and the fluid excitation removed.
With respect to the other two approaches, the disadvantage of this controller is that for every operating point (D_,,f) another observer model is 0.8 time in FIGURE 9 Measurement of the x-deflection: Effect of the disturbance compensation (/2-= 12.8 Hz, _f2= 20%).needed and that not only the deflection, but also the filling ratio f and the rotor speed 1 have to be measured in order to determine the current operating point.

CONCLUSIONS
For removing the unstable behavior of a partially filled centrifuge, three control approaches are pre- sented in this paper.
First, by means of the PD-controller, the instability region is shifted such that by appro- priate switching between different PD-gains, the unstable region can be passed through during the startup.The advantage of this approach consists in obtaining a stable behavior of the centrifuge, regardless of the filling ratio and the rotational speed.The disadvantage is that the instability re- gion might possibly not be shifted to sufficiently high rotor speeds due to the saturation of the control force.
Second, a velocity cross coupled feed back is proposed which stabilizes the centrifuge by directly counteracting the fluid force.For awide range of the instability region only one coupling factor g is required so that the controller depends on the filling ratio and the rotational speed only to a small degree.Due to truncation effects, the cou- pling factor can not be chosen arbitrarily high.Therefore, the design has to be carried out with respect to both, the cross coupling g and the velo- city feed back factor d, which suppresses the destabilization effects caused by truncation.
As for the last control approach, a disturbance observer is presented.The fluid force which does not have to be known exactly is represented by a simplified model.After designing a suitable ob- server, the fluid force can be estimated and com- pensated.The disadvantage of this approach is that for every operating point a new disturbance model is required and that not only the deflection, but also the filling ratio and the rotational speed have to be measured in order to determine the current operating point.inner radius of the casing in m inner radius of the free liquid surface in m velocity feed back factor in s filling ratio magnetic bearing's force in N velocity coupling factor in s force-current-factor in N/A negative bearing stiffness in N/m deflection feed back factor displacements of the fluid container in x-respectively in y-direction velocities of the fluid container in x-re- spectively in y-direction eigenvalue rotational speed in 1/s vector of the fluid effects vector of position coordinates reduced modal vector of position coordi- FIGURESketch of the centrifuge and its relevant eigenmodes. 325

FIGURE 2 FIGURE 3
FIGURE 2 Startup behavior of the passive and the PD-controlled system (p 1.4, d= 0.0015 s).

FIGURE 5
FIGURE 4 Phase frequency characteristics for a lateral harmonic excitation of the casing.

FIGURE 6
FIGURE 6 Instability region by use of digital velocity cross-coupling, depending on g and d (l/2r 12.8 Hz).
matrix of the fluid effects input matrix of the magnetic bearing's force reduced modal input matrix of the mag- netic bearing's force reduced modal output matrix stiffness, matrix reduced modal stiffness matrix observer matrix of the extended model mass matrix velocity feed back matrix feed back matrix of the coupling factor deflection feed back matrix modal matrix