Linearization of Force Characteristics of ActiveMagnetic Bearings for the FPGA-based LQ-controller

Rafal Jastrzebski, Riku Pollanen, Olli PyrhonenDepartment of Electrical Engineering

Lappeenranta University of Technology, Lappeenranta, FinlandRafal.Jastrzebskiglut.fi, Riku.Pollanenglut.fi, Olli.Pyrhonenglut.fi

Abstract- This paper discusses a linear-quadratic (LQ) control Most components of the studied AMB-rotor system can beand nonlinearities compensation of a rotor suspended by Active effectively linearized, therefore they comprise a plant suitableMagnetic Bearings (AMBs). The control structure is based on a for linear control methods. However, the essential nonlinearitystate-space controller with additional integrative feedback,current estimator and disturbance estimator. An effective ofthe AMBs, i.e. the force-current-displacement characteristiccompensation of performance variations in dynamic force of the electromagnets, should be compensated, if the highestcharacteristics of AMBs is performed using two nonlinear performance is required. The nonlinear performanceprinciples. Firstly, the usage of the inverse nonlinearity, a well- characteristics of the radial AMB can be sufficientlyknown method in control engineering is studied. Secondly, the determined by numerical computation methods [4]. Especiallynovel model reference based method is introduced and validated.It is shown that the proposed compensation provides certain accurate solution of the magnetic field equations is provided byadvantages over conventional linearized model based controllers. the 3-D Finite Element Method (FEM) analysis, which takesThe overall control is tested with the flexible rotor model and the into account the end-field effects.nonlinear AMBs. Finally, we present an integration of the The increased performance demands for the AMBproposed compensation into the digital controller. Accuratemodeling of the AMB nonlinearities in a Field Programmable applicatios caues mp inin the impementedGate Array (FPGA) based controller is achieved by using controllers. However, in the real-time embedded controlmultivariable interpolation and look-up table. systems the minimization of an input-to-output controller delay

is of major importance. Therefore, functional and softwareintegration of more complex control methods, which comprise

I. INTRODUCTION fast dynamics and nonlinearities, into the real-time embeddedThe invention of the wheel is one of the major and widely controllers requires fast signal processing and flexible

known breakthroughs in the history of engineering. The hardware platforms.invention of the bearing, which is used to reduce a friction in The main objective of the paper is to propose an effectivethe wheel, is however mostly underestimated. The most compensation of performance variations in the forceadvanced concept among all known bearing types is an Active characteristics of the AMBs. Firstly, the design of theMagnetic Bearing (AMB). The AMBs as a technical solution centralized linear-quadratic (LQ) controller, state observer andare based on computer science, control, electrical, electronic disturbance observer are described. Secondly, the proposedand mechanical engineering. Many advantages over the state controller and estimator are combined with two nonlinearconventional bearings such as absence of friction and control concepts: a traditionally applied inverse nonlinearitylubrication, precise position control and vibration isolation control method ([5], [6], [7]) and a novel model referencemake the AMBs particularly appropriate for high performance based approach. Furthermore, the proposed nonlinearitymachines. In the AMB-rotor system considered in this work compensation methods and the overall linear control are testedtwo radial bearings and one axial bearing control five degrees by simulations and compared to their linearized equivalents. Inof freedom of a rotor of the high performance electrical the simulations a more accurate plant model [8] includingmachine. The rotational speed is controlled by an independent nonlinearities, high frequency dynamics, couple unbalancedriving motor. As the magnetic actuators two eight-pole radial forces and noise is used. Finally, the integration of thebearings with a differential driving mode [1] are employed. proposed nonlinearities compensation, into the digitalThe axial suspension is considered separately. For this controller, is studied. It is shown that accurate modeling of theparticular system, our choice for the starting point of a position system nonlinearities can be achieved by using a multivariablecontrol design is an optimal state-space controller with interpolation and a look-up table. The algorithms based onadditional integrative feedback [2]. The position control serves three different interpolation principles are tested and comparedas an external controller to the internal current control loops, in terms of performance, computational complexity andwhich are designed to cover the bandwidth of the centralized resources usage ofthe digital implementation.position control as in [2] and [3].

1-4244-0726-5/06/$20.OO '2006 IEEE 2420

11. DYNAMICS OF THE ROTOR-MAGNETIC BEARING SYSTEM component, e.g. in the x-axis of the first radial bearing, can belinearized in the vicinity of the operating point as

A. Rotor DynamicsThe magnetically suspended rotor considered in this work is Fix (x, 'Cx) = + kx (3)

supported radially by two bearings (Fig. 1). The axial where k. and kx are the current stiffness and position stiffness,suspension can be treated separately from the radial one. The respectively. The linear model is a good approximation of therotor parameters are calculated from the real rotor geometry. force-field close to the linearization point. However, theThe rotor model [8] used in this work is a coupled system of deviations from the linearized solution appear for high valuesfour displacement variables, 4-6 additional modal coordinates of current and displacement (saturation regions). Theseand their derivatives. Rotational symmetry of the shaft and differences increase also when a reduced premagnetizationconstant rotational speed Q are assumed. The equation of current is applied to decrease power losses. The accurate force-motion of a linear non-externally excited rotating system is field characteristics in the whole operating range can be

Mz + (G + D)z + Kz = 0 (1) obtained with FEMs or Reluctance Network Methods (RNMs).The force-position-current characteristics of the studied radial

where M, G, D and K are the mass, gyroscopic, damping, and bearing were obtained with the use of a RNM presented in [4].stiffness matrices, respectively. These parameters are obtained Its solution region, depicted in Fig. 3, is two-dimensional andusing the finite element modeling, modal reduction technique takes into account the magnetic saturation, cross coupling andand experimental analysis. Based on the equation of motion, a leakage flux over the stator slots. The currents in the actuatorscontinuous-time state-space model of form are controlled by the internal, in regard to the external position

x=Ax+Bu, y=Cx+Du (2) control, current control loops. The proportional andfeedforward controllers control the coil currents. The fast

is obtained, as in [9], [10]. The parameters x, u, y are vectors current feedback compensates the inductive voltage drop andof state variables, input variables and outputs, respectively. variations in the coil inductance. The feedforward gainB. Actuator Dynamics compensates the effect of a resistive voltage drop.

Before synthesizing the position controller, the dynamics of III. LQ-CONTROL DESIGNthe electromechanical actuator have to be added to the From the control point of view the AMB-rotor system ismechanical model in (2). The actuator features an eight-pole unstable, nonlinear, multivariable plant with changingradial bearing with paired coils. The four electromagnets are 'controlled in a differential driving mode (Fig. 2). The control pajraters And unmodeledsyamics.eIp i yearste

majority of AMB systems were stabilized by decentralizedof the attractive force is performed with the control currents PID-blx, PID-based controllers. This tendency continues in nowadaysicy. A premagnetization current io (bias current) is applied to all AMB applications [11]. Recently, however, due to increasedcoils. The current and position dependent magnetic force capacity of digital signal processing devices, more complex

bearing I x& a bearing 2 model-based control solutions are selected. The physical,:)3_ z limitations of the bearing forces, currents and voltages point at

- - - ---- the formulation of the control strategy as an LQ optimizationa b problem. The LQ optimal control, as in [1], [12] seems to be a

Fig. 1. Geometryof studied rotor particularly suitable linear control method for the AMBsystem. The presented solution differs from the above ones inthe usage of a full current state observer, disturbance observerand utilization of Bryson's rules [5]. The schematic of theoverall close-loop system is depicted in Fig. 4. The continuous-time model is considered for the sake of the clear presentation.

iio+i.A. LQ-Regulator Design

ff X 11Yt *-Xt0<'\ Disturbances, external reference commands and variations in, , r

the plant parameters cause steady-state errors in plantr ov h_ _0,*\V/zUresponses. The use of integral control eliminates these errors.

To calculate the feedback gains for the original state vector andadditional integral feedback the model of the plant isaugmented with the integral states, as in [5]. In the augmentedmodel, the state-feedback law minimizes the quadratic integral

~~~~~~~performance index

Fig. 2. Schematic diagram of eight pole radial active magneticbearing, its J = F[xTQtx+uTQ2uldt(4)coils and currents

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position measurements, the controller performance can beincreased by assembling the rigid body states from the positionmeasurements and estimated rotor states (the block marked Xemin Fig. 4).

z 1.5 .B. LQ-EtimatorDesign05.~ ......; In reality, all system states, particularly velocities, are not

05 <; 1: :, : . available as direct measurements. It is, however, possible to......................... ~obtain the estimated states by using the dynamic model of the

15 plant. Additionally, the full-order observer acts as a low-pass25 ~~~~~~~~~~~~~filterto the measurement noise. In the preparation of the LQ-

0.3 estimator we have to define the stochastic properties of the0.1 _r4 process and measurement noise

40.2 2 Rv = eye(4,4)y2, Rw = eye(4,4)S2 (9)Displacement [mEr] -0.3 6 C4 tl st [A]

3 ~~~~~~~~~~~~~~whereRw, Rv are the covariance matrices of unbiased processFig. 3. Force versus current and position characteristics obtained by RNM noise wan aremennovie vatheYs statesfor ersnoise w and measurement noise v. The ym,s states for the rmswhere the weighting matrices Q and*Q2 can be based on the accuracy of the position measurements and scaling factor S-,real physical constraints. Using a guideline, sometimes called equals to the rms accuracy of the control input.Bryson's rules [5], one can assure that the states enter the cost However, when this kind of observer is employed and thevia the important outputs. The weighting matrices are non-stochastic disturbances are acting in the process inputs the

estimated states may become biased due to disturbances. The= CTQ1Cs, 2diag[1 Ui1jax .. (5) cancellation of the predictable part of the disturbance forces

and unbiasing of the state estimates can be arranged by addingQ diag LX.1,2 1 ,2 0,0,0,0,...] (6) a disturbance observer. The disturbance is described by an

assumed dynamic model and augmented in the state observer.where Si is the scaling factor for the integral states, and Yma,

sumdyn icoelndug ntdnthsaeobrv.where Smax isethescalingmfactormforthevtiontg sututes, andmaxn The negated disturbance estimate is then used as a feedforwardUmax, 'max are the maximum deviations of outputs, inputs and control to cancel the effect of the real disturbances. For thecontrol currents, respectively. The Csi matrix selects which plant modelstates have to be kept under close regulation. We have to addadditional weights for the integral states in Csi. Its diagonal i=Ax+Bu+Bww+Bdwd (7)elements, which correspond to the currents, positions andposition integrals, are equal to ones. Furthermore, the flexible whreW is the non-stochasticdturbain vector, thestates that are actively controlled are also scaled in (6) and comPutaselected in Cs. in the same way as the positions, i.e., the disturbance estimator gain matrix Ld is based on the systemeigenvectors in the flexible rotor model are scaled so that the equations with the augmented disturbance modelmaximum rotor deformation of each mode equals 1 m. Finally, Fx ABAdCd x B u+Bthe optimal gains KR and K1 for the control plant are solved L 0edL° Ad ILWed 1+0 w (8)with the built-in procedures of the Matlab Control SystemToolbox. The augmented system is uncontrollable, but the principle is to

In the case of fast and stable (not delayed and not erroneous) use the estimated disturbance vector Wed in a feedforwardx manner and not to control it. Assuming that the disturbances

Fd+w enter the plant identically to the process input, we get BdCd=B,x~~~~~~~~~~~~~~~~~~~X.o-~Assemble and thus the unitary disturbance feedback gain (Kd=I) cancels,em 0-I ic Integrated r F Rotor y the load disturbances. Moreover, one can scale the process1F-R[ (<F +AMB +

I|Actuator Dynamics related covariance matrix based on the control input. When-K -.computing the disturbance estimator gain matrix Ld, Su is

W'd equated to the expected maximal disturbance value.Distubanc Obs Connecting the disturbance estimate as a control input, onestae Observer Antaliasing adds additional integral dynamics to the close-loop system.

Filter Instead, we prefer to use Kd=O and the additional integrativex_ T feedback.

Fig. 4. Block diagram of overall close-loop control system, where Xr, Xm, Xe Similarly to the LQ-regulator, the LQ-observer can beare reference position, measured position and estimated rotor's state vectors, designed using Matlab's built-in functions, which return therespectively; Fd, F, w, v, ic and ie are disturbance force, total force, process otmlosre anmtiedh rdco tt nnoise, measurement noise, reference current and estimated current vectors, pmaobevrgm atcsL,L.Thpedtrsaeanrespectively disturbance observers are described by

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x Ax+B(wed+U)+LpF, (10)

Wed=AdWed d (11

where x is the state estimate vector provided by the predictor f4observer. In the predictor estimator, the current value of the O 2control depends on the delayed value of the observation. In --general, a better regulation of outputs is obtained by using a 2cuffrent estimator that provides the state estimates based on the0-4current measurements. The equation for the current estimatorrelated to the predicted estimate (if the A matrix is notsingular) is 6

O = 2=x + Lc [y - Cxl Lc =A'Lp (12) -0.2 -4 -2

Displacement [mm] -0.3 -6 Control curent [A]where x is the state estimate provided by the current observer. Fig. 6. Compensation surface based on the inverse nonlinearity

IV. ACTUATOR LINEARIZATION nonlinear actuator is directly incorporated into the controller asthe nonlinearity inverse approximation, the control input u

nonlineaofithesidfmsp ular thods for comeinges becomes force instead of current, and the position stiffness kxnonlinearities in dIfferent actuators iS that of the inverse is canceled out from the close-loop system dynamics

beunonlnearities. ITh asvrsumes nonlinearity istdiertlAns can The compensation of the actuator is also possible withoutbe undone. The inverse nonlinearity method for AMBS was any changes in the upper control. Firstly, the force reference isstudied in [6],where the analytical m odsuwere employed, calculated from (3). Secondly, it is combined with the invertedand in [7] and [13], where a polynomial formulation was used. nolnert int th copnainsrac,wihcnbThe reported deviation in approximated force with second

odepolynomials in realized as the single look-up table, shown in Fig. 6.Order polynomals in [7] and fourth-fifth order The inverse nonlinearity compensation principle is presented[13] were less than 100% and 500O of the maximum force value,iFg.7respectively.A. Inverse Nonlinearity B. Model Reference BasedApproach

We propose an alternative solution for the actuatorIn order to compensate actuator nonlinearities we introduce learon, an utiedelythe for chaactrist

the force nonlinear relation into the AMB controller as a look- .anitis apied parallewith theexisti contrer.stheup table and multivariable interpolation. First the inverted cand it is applbed parallel with the existng controller. The

relation of the aforementioned force-current-displacement c t

characte c hs to be dreference approximations is an intrinsic part of the presentedcharacteristics has to be determined and stored in theoti solution (Fig. 8). The control current ic and rotor displacementappropriate table. The force nonlinearity is inverted to obtain . .

=

--1/(x,F. the graph of which is presentedinFig. 5. Theare utilized in computation of the electromagnetic forces. The

c ( r t forces are calculated using both the linearized andvalue of modeled function can be obtained from the table ofstored values. The closest entry or other various types of iterpolation-based nonlinear bearing models. The differenceinterpolations can be used. When the compensation of the between these two forces is employed in a feedforward control

Inverse -Control Plant

= 2 2 ..... .. _!7 .......................Fig. 7. Compensation of nonlinear actuator using approximate inverse| . . . _ ~~~~~~~~~~~~~~~nonlinearity, where P, Xr, Fr are position control, position reference and force0 q. li 0 ~~~~~~~~~~~~~~reference,respectively

Nonlinearity No Nonlin arity x

xF-4 1 ~ t t>,,fxi) Dynamics

0 S _ .... X . iV; i~- _f:0f0:- 4i

6 ) L

Displacement [nun] 02 -0.3 -2Lo~e[*N] Fig. 8. Model reference based compensaton of electromagnetc forces,Fig. 5. Inverted relation of force versus current and position characteristics where ic, it rare control and compensation currents, respectively

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manner as a compensation current icp1 The first choice for the TABLE Ifeedforward control gain is the inverted nominal current AccuRAcY AND COMPUTATIONAL COMPLEXITY OF SELECTEDstiffness of the bearing. The possible variations of this INTERPOLATION METHODScompensation include the use of the measured or control Method rms error [%] ROM size Arithmeticcurrents. In addition, the negated compensation currents can be (a) (b) [elements] Operationssubtracted from the observer inputs instead of adding them to 1. Sum of areas 0.34 0.68 30x50 15the control inputs. However, the solution presented in Fig. 8provides the fastest compensation and best performance. The 2. First order 0.36 0.68 30x30x2 8practical realization can be reduced to a single look-up table as polynomialswith the inverse nonlinearity method. 3. Second order 0.18 0.68 30x3Ox3 16

polynomials

V. INTEGRATION OF THE NONLINEARITY COMPENSATION INTO 4. Second order 0.40 0.25 30x30x4 8THE FPGA-BASED CONTROLLER surfaces

The signal processing of the compensation method was respectively 50 and 30 elements, whereas in (lb) the vectorintegrated into the controller of the actuators. Our first step to lengths are reversed. For the methods (2a) and (3a), which arethe integration was the selection of the suitable interpolation based on the single variable polynomials, the better results givealgorithm. polynomial approximations of the force values. In the method

A. Interpolation Methodsfor the Hardware Implementation (4b) smaller rms error is achieved with the parabolic force and

Basic examples of fast methods for evaluating elementary linear position, than when the arrangement is reversed. The

single variable dependent functions in hardware can be found approximation with second order surfaces (method 4) deviatesin* Al The . based from the non-changed force-field (fF(xji) in Fig. 3) less than.loku tal an poyoma approximation 0.240o of the maximum force value. The deviation increases toalgorithms were applied in [15] for emulating the AMB-rotorsystem in FPGA. The same principles can be used for 0.36% after the HDL implementation with fixed-point numberimplementing the compensation function. format is carried out. From the implementation point of view,The most straightforward interpolation method is based on the best choices are: method 4 (the smallest number of the

the original but sparser matrix, which contains the general arithmetic operations) and method 1 (small memory* ^ * r 1 ~~~~~~~requirements data points used directly - no approximationcompensation surface. The method is referred as a sum of quy

areas and is based on the sum of the weighted four closest required).entry values. C The Integrated Current Controller, Modulator Logic andThe second tested method is the piecewise interpolation with Compensation

low order single variable polynomials. The interpolated The embedded controllers of the actuators and accuratevalue is obtained from the evaluated for the first input two modulation schemes are more often implemented in FPGAs orclose entry polynomials, which are scaled proportional to the ASICs, in order to integrate these tasks into the hardware layer.distances between the second input and the corresponding The presented implementation is the extension of the solutionlook-up table breakpoints. described in [16], which uses a carrier based PWM scheme.The third and the most memory consuming method is the The major advantages over the previous design include the

piecewise interpolation with low order surfaces. The second variable carrier frequency and the integration of theorder surface was chosen for comparison with other methods. nonlinearity compensation. The schematic model of the AMB

B. Comparison ofAlgorithms current controllers, modulator logic and the look-up tablebased compensator is depicted in Fig. 9. The design is coded inThe accuracy of particular methods iS strongly dependent on V.LItuiie. ieie aa lwadotmzdfxd

data and arrangement of the input arguments, e.g. direction ofpolynomial fitting. The actual choice of algorithm and point arithmetic. The operational principle is as follows.

Firstly, the control currents are compensated according to theupolyntmiableqamutationisalcradeomplbexitye o

i oftarox ion compensation surface in the interpolation block. Secondly,upalg The totaleor o mplemente algorimaist each compensated currlent is changed into two positivealgorithm. Te tl e. or of te ireference currents, such as ir,(2n-1)=iO+ic,(n), ir,(2n)=iO-ic,(n), forsum of the interpolation error and the digitalization error.

introuced bthe fixtedpolation daformandthedigitalizat.oneffo n=1... 5. Thirdly, based on the reference and measured currentsTheroducompartheison -pofacr reatedatothemaximal cont ten voltage references are computed and passed to thec Th valueandcomparisonofaccutacyrelatiol c exit of seetedl modulator logic. Finally, the voltages in the modulator buffer

cuff ent value an opttare updated twice at the selected carrier frequency fc. Theinterpolation methods is listed in Table I. The results modulator logic comprises of two carrier counters and twentycorrespond to the interpolation of the inverted force field. The . .listed~~~~~ ~nubeofaiheirprtosi qa otesmo comparators, which utilize the common carrier signal. The

,.. , ~~~~master counter controls the sign ofthe fractional increment Culttotal required multiplications and additions per control currentanthse-stloiofhelvecuerTef anbchannel. In the case (1la) the force and position vectors have

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measurements are given in Fig. 11 and Fig. 12. In theseoo-up Buffers 10- ,f 20 simulations the plant features the flexible rotor model and

table-basedx spatial P t _: Sw actuators from [8]. Additionally, the mass unbalance is

interpolation Proportional Carrier (S) included. The anti-aliasing filter from Fig. 4 eliminates theand Counteri,,, . feedforward (int/frac) Comparators observation spillover from the residual modes and prevents

controllerInverted nonlinearity Nonlinearity rejection

200 .....200 .....

N ~~~~~~~Carrier (M) piuso set(Counter us EE

Reset 100 ... 1 0 ..... .Iset Logic 15 ....5056Fig. 9. Integrated current controller, PWM modulator logic and compensation I

adjusted on-line (between 2.5kHz and 80kHz) by two input 0 ol 62 0 61 62 6.parameters time [s] time [s]

linearized control linearized controlN =fl(2fc), Cint (Ac 2)/N, (13) 200 ....-....20...

whereAc stands for the constant amplitude of the carrier signal E .......... 150 ......-and f I OOMHz is the design clock frequency. The time Eresolution of the modulation is IOns, but a relative resolution _ |1-6ofthe comparators varies with Ci,,t. The implementation details S1501.... 0 |............. 50-..............are collected in Table II. 0 ________The overall input-output delay is minimized if the inputs 0 __ ________

update rate is synchronized with the reference voltages update tm1 020] time [rate in the modulator. Additionally, the variable modulationdelay should be accounted in the position control loop. maximal control effort maximal control effort

VI. SIMULATION RESULTS -35 .. ......

The comparative simulation results of two types ofnonlinearity compensations are presented in Fig. 10. Just to test = 4the compensations the equivalent tuning of the controllers, the nonlinear nonlinearsame values of the step reference position and step disturbance -5 5 linearized -5.5 linearizedforce of the studied AMB system and the measured states are 0.26 0.28 03 026 028 0E3assumed. The proposed nonlinear controllers prove successful time [sl time [s]operation of the control concepts and their advantage over the Fig. 10. The comparison of the nonlinear control concepts is depicted. In the

first column the response of the control system based on the inverseconventional linearized-model-based controllers. However, nonlinearity is compared to its linearized equivalent controller. In the secondnoise limits the achievable bandwidth of a close-loop system column the performance ofthe model reference based approach is shown.and may decrease the effectiveness of the nonlinearitycompensative control [5]. Also in the non-collocated systemthe effectiveness of the compensation is decreased. The 250simulation results of the close-loop control system with a band-limited white noise introduced to the actuators and to the 200

IMPLEMENTATION DETAILS OF ACTUATOR CONTROLLER 150

Implementation target Virtex2P(xc2vp3O-6) o100Sum of areas!Interpolation method / Breakpoints x Coefficients (30x50)x I 50

Maximum frequency 118.2MHz0 A

Maximum data rate per clock cycle ic /2clk, ir /lclk O I 0 0 0

Fig. 11. The simulation of the closed-loop control system with the couple

14 /9 unbalance (150 gmm, the rotor mass equals 52.8 kg) and a band-limitedTotal latency / Interpolation latency [clock cycle] white noise, introduced to the position measurements. The linear control_____________________________________ ______________ - (dashed line) is compared to the nonlinear one (solid line).

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controllers for the switched power amplifiers of the AMB-rotorsystem were presented.

0S- The overall closed-loop control system with the integratedcompensators proved successful operation of the control

-0.3 concepts. The presented compensation methods were sufficientfor the studied application. Their importance would increase

-0.6 k ll | for even more reduced premagnetization current. The stableoperation range of the nonlinear controllers increased when

-0.9 compared to their linearized equivalents. The effectiveness ofthe compensation was particularly visible during transient

-1.2 l Icp nIN conditions. In the steady state conditions, the fast constant| I|P in--MR disturbance estimator compensated the force differences. The

-1.5 ______________________________________ ,effect of the compensation and performance of the LQ-control,0 0.1 0.2 0.3 0.4 0.5 in the close-loop system, were tested in the presence of the

tune [S] residual modes, disturbance forces, unbalance and noise.Fig. 12. Comparison between compensation current icp, which results frominverted nonlinearity (IN) method and model reference (MR) method during REFERENCESboth tests

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