2-DOF Controller Design for Precise Positioning a Spindle Levitated with Active Magnetic Bearings

European Journal of Control (2012)2:194–206© 2012 EUCADOI:10.3166/EJC.18.194–206

2-DOF Controller Design for Precise Positioning a Spindle Levitatedwith Active Magnetic Bearingsg

Iñigo Arredondo∗, Josu Jugo∗∗

D. Electricity and Electronics, F. Science and Technology, Basque Country University (UPV/EHU), Barrio Sarriena s/n, 48940,Leioa (Bizkaia) Spain

This article describes the design of a high performance2-DOF controller for an electromechanical system con-sisting of a rotary flexible spindle hovered by ActiveMagnetic Bearings (AMB). The purpose of such a con-troller is to achieve the better positioning possible of thespindle by means of a good stabilization mechanism andby counteracting the vibrations generated naturally by therotation. The implementation of this controller is based onthree main blocks, a stabilization block, a vibration mini-mization block and, if needed, a dynamics decoupler blockbetween the previous ones. The stabilization controller isdesigned utilizing a MIMO decoupling technique. Withthe aim of proving its effectiveness, two different stabi-lization controllers (PID and H∞) are presented. On theother hand, an adaptive feedforward algorithm which isin charge of the vibration minimization is implemented.In order to not feedback the vibration and, therefore, todecouple the controller’s dynamics, the implementationof a variable Notch filter is discussed. It is demonstratedthat this optional block can enhance or impoverish thebehavior of the system depending on several factors suchas resonances or rotational speed measurement.

Experimental results and simulations are provided toshow the effectiveness of the complete design process.

Keywords: 2-DOF control, adaptive vibration controller,MIMO decoupling, active magnetic bearings, machining

∗Correspondence to: I. Arredondo, E-mail: [email protected]∗∗E-mail: [email protected]

1. Introduction

In the machining field the increasing development of theelectromechanical devices allows more precision in theproducts obtained. However, the mechanical accuracy canbe limited by the vibrations and an active control can beconsidered to achieve better results.Particularly, several applications depend considerably

on the positioning of the end of the spindle, but its dynam-ics is perturbed by the forces generated inherently dueto the rotation and the inevitable unbalance of the rotor.Although there are mass balancing techniques [21, 18],better results can be obtained combining them with anactive control.The implementation of the active control into the sys-

tems can be performed in several ways, such as theinclusion of dampers in a ball bearings based machine [5].Nowadays, the use of Active Magnetic Bearings (AMBs)may be a good choice, knowing their main advantages (nolubrication, no cooling, longer life,...) [20]. This technol-ogy is based on the control of magnetic forces to maintainthe rotor in levitation. Therefore, it is possible to placeaccurately the spindle and to counter undesirable effectsby choosing properly the bearings’ generated forces. Toaccomplish this objective several control techniques canbe applied such as robust (QFT , LQR, H∞) [12, 19] oradaptive ones [17, 22].The positioning challenge in these type of machines

can be separated into two problems, leading to a two

Received 15 October 2009; Accepted 9 December 2010Recommended by S.M. Savaresi, E.F. Camacho

2-DOF Position Controller for an AMB System 195

degrees of freedom (2-DOF) problem (for further informa-tion about 2-DOF control, [7, 16, 23]). On the one hand,the robust stabilization which guarantees the levitationand, on the other hand, the suppression of the above men-tioned vibrations generated by the rotation and the spindleunbalance.In this article a precise positioning of a rotor hovered

by two AMBs is performed using a controller which com-bines three specific purpose blocks, following the 2-DOFsstrategy. Firstly, a dynamics decoupling can be consid-ered to split the vibration from the hovering and, then,the remaining two blocks are in charge, one for the sta-bilization of the spindle and the other for the vibrationrejection.A typical dynamics decoupler is a spindle’s speed fre-

quency variable Notch filter. As it will be discussedthroughout the article, the inclusion of the dynamicsdecoupler is not essential if some conditions are satisfied.However, its inclusion changes the dynamics of the closedloop of the system and, depending on the device, it canenhance or impoverish its behavior.The robust stabilization block is designed using a

MIMO decoupling technique based on the symmetry ofthe system. This MIMO to SISO conversion techniqueallows to utilize several control methods easily. In this par-ticular case, PID and H∞ based controllers are designed.The PID controller has been obtained by root locus tech-nique plus an experimental gain tuning, following thestandard ISO 14839-3 [9]. This tuned PID controller hasbeen implemented in the laboratory set-up to demonstratethe methodology effectiveness.The vibration minimization block has been designed by

mean of an adaptive feedforward algorithm. Its objectiveis to drive the bearings to generate forces in opposi-tion with the vibration, reducing it. The experimentallyimplemented algorithm includes a Frequency LockedLoop (FLL, [2]) based synchronization technique foravoiding measurement errors from the rotational speedsensor.Experimental results performed with a laboratory set-

up cored with the MBC500 Rotor Dynamics corroboratethe effectiveness of the proposed 2-DOF controller. A ∼80% of vibration reduction is achieved when the spindleis rotating at 2100rpm.Summarizing, the main contributions of this article

are:

1) The design and experimental implementation of a 2-DOFcontrollerwith vibrationminimization in anAMBbasedmachine for precise positioning, which has a richand complex flexible dynamics.

2) The utilization of the symmetry (typical in these typeof systems) to facilitate the design of the stabilizationcontroller. The design of two stabilizing controllers

(PID+tuning and H∞) is performed in order to showthe effectiveness of the method.

3) The study of dynamics decoupling in AMB baseddevices with and without a decoupling block. Theeffect of the rotational speed and the effect of noiseare analysed.

4) The solution proposed includes features which allowsa more economical AMB based system, facilitating itsimplementation in industrial applications.

This article is organized as follows. Firstly, a systemdescription and problem statement are presented. Here,the characteristicswhich constrain the control are detailed.In section 3, the 2-DOF scheme and the methodologyused to design the controllers, based on the system sym-metry, are explained. Then, the design of the differentelements of the controller is performed, describing thedecoupling, stabilizing andvibrationminimizationblocks.Finally, the experimental results and the conclusions arepresented.

2. System Description and ProblemStatement

The AMB system under study is based on the MBC500Rotor Dynamics of LaunchPoint technologies [14], whichis a laboratory device specially designed for research pur-poses [13]. It is composed of two AMBs and a rotorwhich includes an air turbine drive, allowing speeds upto 22000rpm.As is schematically shown in Fig. [1], the shaft position

is measured by Hall effect sensors and the currents, whichcause the forces in the bearings to maintain the hoveringstate, are driven by voltage amplifiers. Thus, the systeminputs are the voltages given to the amplifiers and the out-puts are the voltages provided by the position sensors.Finally, the scheme is completed by closing the loop witha stabilizing controller, thus appropriately feedingback thesystem response. The implementation of the real-time dis-crete controller requires a suitable hardware. In this case,the Digital Signal Processor (DSP) DS1003 of dSPACE isused.The modeling of the device is performed combining

a Finite Element Method (FEM) with sensors and actu-ators behavior and the dynamics provoked by the massunbalance (explained later in this section). Then, reductionmodel techniques are used, leading to a linear 4x4 MIMOcomplex system (9th order), due to its unavoidable flex-ible modes. It is characterized by the strong mechanicalresonances corresponding to the two first flexible modesof the spindle.The system has a very nonlinear nature, due to many

factors, but the magnetic force (2.1) can be considered the

196 I. Arredondo and J. Jugo

X

Rotor Sensors

Air turbine

BearingsAmplifiers

Vi

Controller

Rotation EffectsOutputs

Inputs

r

Ω

d(t)

F

Fig. 1. MBC 500 rotor dynamics scheme.

most relevant source of this characteristic [11]. Concretely,this force can be modeled in this way,

Fi = K

(i0 + ici

)2(xi − xg

)2 − K

(i0 − ici

)2(xi + xg

)2 , i = 1,2 (2.1)

where K = 2.8 × 10−7N · m2/A2 is a geometric con-stant depending on the bearing, i0 is a bias current andxg = 0.0004m is the effective distance between the bear-ing and the rotor. This expression is reached because,in this case, the AMBs are disposed in a differentialconfiguration [20].On the other hand, the AMB based systems are charac-

terized by their inherent instability. It makes compulsorythe implementation of a controller and conditions the studyof the device’s dynamics. The controller design is facili-tated through a linearizedversionof themodel. In addition,the largest linear range of operation is achieved when thespindle is centered. For this reason, the linearization isperformed around the origin of coordinates.The resulting linearized model is characterized by the

symmetry in the matrix function which represents it. Infact, the system, once simplified and linearized for a con-stant rotational speed, can be written as follows for one ofits axis, using the external representation.

[Vsensor1Vsensor2

]=

[As Bs

Bs As

][Vcontrol1Vcontrol2

](2.2)

with As and Bs transfer functions with order 9.From the equation (2.2) it could be stated that the system

may be modeled with a symmetric structure. This featurepermits to decompose the original 4x4MIMO system into4 independent SISO problems (Appendix) [10]1 for whichto simplify the design of the controllers. Concretely, the

1 In [10], another advantages of the symmetry are stated, such as a controlindependent closed-loop identification method.

order of the systems is eighteen for the MIMO problemand five for each SISO one. Moreover, since the dynamicsof the x and y axis are very similar, the problem can bereduced to design a controller for only 2 independent SISOsystems.When a symmetric controller is chosen, the character-

istic equation can be split into these two,

(1+ (Cs + Ds)(As + Bs)) = 0 (2.3)

(1+ (Cs − Ds)(As − Bs)) = 0

where As and Bs are the transfer functions of the system(2.2) and Cs and Ds the transfer functions of a symmetric

controller[

Cs DsDs Cs

].

The stabilizing block of the controller, either the PID orthe H∞ (section 3.2), is designed taking advantage of thisproperty.On the other hand, the rotation of the spindle has a

very important influence on the system’s dynamics sinceit causes the forward and backward whirls and, indeed,makes possible the creation of the combination reso-nances [15], which are all related with the aforementionedmechanical resonance modes of the system. These combi-nation resonances are provoked by the nonlinear behaviorand take part in the stability range limitation. Hence, agood centering of the rotor can be beneficial, because thenonlinearity becomes more noticeable and the combina-tion resonances are more relevant, if the rotor is displacedfrom its equilibrium point.The rotor dynamics is excited by one of the most impor-

tant characteristics of the systems with rotary elements, asthe AMBs based ones: the mass unbalance in the spindle.Its influence could be more or less noticeable dependingon several factors. On one hand, the deviation magnitudebetween the mass center and the principal axis of iner-tia (ei with i = x,y depending on the axis) and on theother hand, the cross elements of the inertia matrix (Iklwith k, l = x,y,z depending on the term). The main conse-quence of this unbalance is the appearance of a centrifugal


force associated to the rotational speed (�),

⎡⎢⎢⎣

Fctfxxi

τctf θxi

Fctfyyi

τctf θyi

⎤⎥⎥⎦ = �2

⎡⎢⎢⎣

−meyi mexi

Iyzi Ixzi

mexi meyi

−Ixzi −Iyzi

⎤⎥⎥⎦

[sin�tcos�t

],

(2.4)

where Fctf represents a centrifugal force and τctf a torque.

2.1. Problem Statement

As can be appreciable in the expression (2.4), the cen-trifugal force is larger when the rotational speed increasesand then, if a high rotational frequency performance isdesired, the vibration could be large. Therefore, if a pre-cise positioning is required, countering this vibration isdesirable or, inmany applications, necessary. For instance,the unbalance provoked vibration needs particular atten-tion in machining, since vibrations are undesired for goodmaterial finishing and, in addition, feeds the activation ofchatter.One of the most common techniques to achieve this

purpose is to balance the spindle adding masses preciselyi.e. mechanically balancing the spindle better for differentrotating speed ranges. This task is very time consumingbut normally it is applied as a necessary first improvement.Another technique usually applied is to use an appro-

priate control in systems in which the active control ispossible. Different control solutions are possible basedon 1-DOF strategies, for instance using robust schemes.Another very extended possibility is the utilisation ofa 2-DOF strategy, dedicating an adaptive feedforwardblock of the controller to reduce vibrations in the spindle(section 3.3).The last approach is applied in this article, but focusing

the attention in the interaction between the two controlschemes implemented and, then, evaluating the actualdegree of freedom between both control loops. Betterunderstanding of the decoupling will allow the obtainingof a enhanced 2-DOF strategy. Here, the stability prob-lem and the vibrations reduction problem are effectivelydisconnected, resulting in an overall more simple design.In conclusion, better control solutions and improvedbehavior are expected.

3. 2-DOF Controller

The main purpose of the controller design (see Fig. [1])is to improve the positioning of the spindle, rotating atthe largest speed possible. Due to the nature of the AMBbased machines, two challenges have to be overcome.

Firstly stabilize the naturally unstable system with themaximum rigidity and, secondly, counter the centrifugalforces generated by the rotation.Two blocks are in charge of these tasks, the stabilization

block and the vibration minimization block. In general,attending to the time constants of each block, it is possibleto decouple their dynamics, i.e. making the stabilizationresponse much more faster than the vibration minimiza-tion. In this manner, when the vibration minimizationtakes part in the controlling action, the system can be con-sidered in the stationary regime and, then, its behavior ismaintained bounded. Since one of the objectives of thisarticle is also to study the dynamics coupling, a generalaccepted solution to lessen the coupling between the twoblocks, the dynamics decoupler block, is discussed andadded to the scheme as is shown in Fig. [2]. In this figure,the dynamics decoupler has been drawn with a dotted linebecause it can be removed from the scheme.The stability block is designed (section 3.2) utilizing

the aforementionedMIMOsplitting techniquewhich takesadvantage of the symmetry of the system, while to accom-plish the vibration minimization, an adaptive feedforwardcontrol is tried (section 3.3). Finally, to implement thedynamics decoupler, a rotational speed dependent Notchfilter is studied. Note that, if the proposed blocks are uti-lized, the inputs of the minimization and the stabilizationblocks are the position of the spindle (measured by the halleffect sensors) filtered by the decoupler.One of the most important features when the design of

a device controller is carried out, is to make the systemimpervious to perturbations, thus augmenting its robust-ness. In this manner, an specific tuning method of thecontroller is convenient to analyze the performance of theAMBbasedmachine. In this article, an experimental anal-ysis following the ISO 14839-3 [9] and some theoreticalconsiderations are presented.

r(k)

v(k)

Controller

(k)

PID H

MBC500 RotorDynamics

DynamicsDecoupler

Vibration

Ω

Stabilization

Minimization

8

Fig. 2. Control block scheme.


3.1. Decoupling Block

The main function of this block is to become the stabil-ity block independent to the vibration and, therefore, thefeedforward adaptive block is only utilized to counteractthe disturbance provoked by the unbalance.Hence, the purpose of the block is to maintain the vibra-

tion in open loop (OL), so a filtering of the frequencyband associated with the rotational speed is required. Withthis aim, and taking into account that this frequency bandis narrow and known, a Notch filter design is proposed.There are other options which are valid too, as the use ofa Kalman filter, but from the point of view of the imple-mentation and computing cost, the first option seems to bethe best choice in this particular case.The frequency filtered by the Notch filter has to be the

same as the rotational speed (�). In this manner, its trans-fer functionwill be dependent on this parameter as follows:

N(�) = s2 + 2ζ1�s +�2

s2 + 2ζ2�s +�2. (3.1)

In the equation (3.1), the factors ζ1 and ζ2 are related tothe bandwidth and the sharpness of the filter magnitude.These parameters define the quality factor of the filter.One of the benefits of this decoupling is that the con-

troller effort is reduced because it is not controlling thevibration [3]. However, depending on the frequency range,this can lead to an increase of the vibration magnitude. Infact, when the rotational speed coincides with some closedloop resonance, the Notch filter can enhance the behaviornotoriously, since the resonance changes the place whenopening the loop [8].Taking this into account, the effect of the disturbance

can be more noticeable when the decoupler is included inthe loop, in a part of the operation speed range.Simulations of the MBC500 Rotor Dynamics system

comparing the behavior of the machine with and withoutthe decoupler are presented in Fig. [3]. The results showthat the closed loop (CL) resonances, giving the vibrationin OL, occur at a lower frequency than without the notchfilter. Hence, the CL resonance at∼ 500Hz (in simulation)changes its frequency to ∼ 415Hz and then, it does not fitwith the rotor’s mechanical resonance.Including the Notch filter, the behavior at low frequen-

cies (until ∼ 24000rpm) is better since the amplitude ofthe vibration is lower than without it. Not including theNotch filter, this vibration increases its magnitude but thespeed which can be achieved is greater, leading to a largerstability range. Thus, depending on the application, theimplementation of the decoupling block could be a goodchoice.On the other hand, the actual rotational frequency is

compulsory to achieve a good behavior of the systemwhen

0 6000 12000 18000 24000 30000 36000

0

1

2

3

4

5x 10

5

Ω (rpm)

posi

tion

(m)

Position vs frequency comparison with and without notch decoupling

normalnotch

Fig. 3. Comparison of the simulations of the system’s rotational behaviorfor the full speed frequency range of the system with and without thedecoupling block.

0 3000 6000 9000 12000 15000 18000 21000 24000 27000 30000

0

1

2

3

4

5x 10

5

Ω (rpm)

posi

tion

(m)

Position vs frequency comparison with and without notch decoupling including noise in frequency measurement

normalnotch

Fig. 4. Comparison of the simulations of the speed frequency rotationbehavior of the system with and without the decoupling block when thefrequency sensor is affected by random and mean noise.

the Notch filter is included. In fact, the measurement errorof this signal can decrease the stability range of the deviceas is shown in Fig. [4]. In this graph, a simulation of themachine includingwhite noise andmean error in the speedmeasurement is performed. A reduction of a 25% of themaximum stable rotational frequency is observed.Summing up, Figs. [3, 4] show how the dynamics

decoupler improves the behaviour of the system understudy, unless it is affected by sufficient noise.In the theoretical and experimental results obtained for

this article the dynamics decoupler makes the plant unsta-ble in a range of frequencies of operation, mostly becauseof themean error and noise present in the actual speed sen-sor of the laboratory device. For this reason, it has not been


implemented experimentally in the final controller. How-ever, this solution is included into the discussion, becauseit achieves good results in other cases [3], [8].

3.2. Stabilizing Block

As is stated in section 2, the system dynamics can bedecoupled due to the symmetry and, hence, the stabiliza-tion controller can be considered as a 2-DOF controller,turning the whole controller into a 3-DOF one.On the other hand, performing this MIMO to SISO

decoupling, the design of the stabilizing block is facili-tated. Following this designing way, several SISO controltechniques may be applied providing great flexibility tothe designer. In order to show this feature, two designstrategies are followed in this article. Firstly, a root locusdesigned PID, tuned experimentally by applying the ISO14839-3 (specific for AMB systems) and, secondly, a con-troller designed by means of H∞ technique. The aim ofthis article is not to provide an exhaustive developmentof the techniques and the features achieved with eachone. Therefore, they are briefly explained in the followingsections.

Experimentally tuned PID option: Applying the rootlocus technique to the expression (2.3), but considering thesymmetric controllerwithDs = 0, the PID controller (3.2),which is able to stabilize the two characteristic equationsin (2.3), is designed.

CPID = 6.5(z − 0.999)(z − 0.9)

(z − 1)(z − 0.45)(3.2)

The chosen poles correspond to the requirement of aderivative action to achieve the stability (compulsory inAMB based devices) and an integral action to maintainthe spindle well positioned. As it has been stated in thesection 2, this centering leads to a more linear behaviorand to a larger stability speed range.The stability margin, which is a measure of the per-

mitted deviation from some stable nominal state, is,obviously, strongly related with the robustness of the sys-tem. In the case of AMB systems, the classic stabilitymargin criterion, such as the gain margin and the phasemargin, are not sufficient condition to guarantee a goodoperation. This is the reasonwhy the output sensitive func-tion is utilized in the specific standards [9] for AMB baseddevices.One of the advantages of this methodology is that

it can be applied experimentally in real time with theappropriate hardware. This allows to tune manually a pre-viously designed controller minimizing the maximum ofthe MIMO sensitivity function.In particular, the ISO 14839-3 establishes a classifica-

tion in which each peak sensitivity level has associated aperformance zone as it is presented in Table 1.

TheA andB zones are considered good enough to permita safety unrestricted long-term operation while C and Dare inappropriate for this action. Actually, the machinecan operate in the C zone for a limited period. However,the machine could be severely damaged operating in theD zone, since collisions with the retainer bearings can beproduced due to large vibrations or instability.The sensitivity function depicted in Fig. [5] is obtained

following the procedure of the standard at several rota-tional speeds and tuning a controller’s gain; in particular,the gain of the controller for the position in the x coordi-nate axis of the spindle, at the left bearing x1. The hardwareused to deal with this task is the Agilent 35670A spectralanalyzer [1]. This device allows to precisely measure andcombine frequency response functions at execution time.Taking into account that the peak gain at each rotatingspeed must be neglected in consideration of the systemstability, Fig. [5] shows a maximum peak near to 12dB.So, according to the Table 1 the machine is operating inthe B/C zone in which it is possible to work with. Also,the same figure demonstrates that it is feasible to performthe peak gain measure in non-rotating state because thereis no change in the sensitivity function when it is rotatingsufficiently far from the critical speed.Hence, measuring experimentally the sensitivity peaks

for several gains of the controller in non-rotating state, theappropriate gain value for the controller can be achieved.

Table 1. Peak sensitivity at zone limits

Peak sensitivity

Level Factor Zone

9.5dB 3 A/B12dB 4 B/C14dB 5 C/D

100

101

102

103

5

0

0

5

10

15

20

25

30

Frequency (Hz)

Mag

nitu

de (

dB)

Sensitivity function for x1 at several rotating speeds

0 Hz

50 Hz

150 Hz

250 Hz

350 Hz

Rotation frequencies

Fig. 5. Sensitivity function for several rotating speeds.


FromFig. [6] it is clear that any change from thenominalgain with the controller (3.2) implies a worse behavior. Ifthe gain is augmented, the low frequency sensitivity func-tion peak grows up and, on the contrary, if it is decreased,the high frequency one becomes larger. So, the tuned gainis the best achievable in the neighboring of the chosenpoint and the experimentally implemented controller usesthis value.

H∞ based In option: the AMB based devices it iscommon to have some uncertainties which are difficultto measure and modeling, such as digital implementationand amplifiers delays, or sensors offsets. To deal with thischaracteristic and to get a robust control, an H∞ basedcontrol is proposed, only for stabilization purposes. Thiscontrol technique is based on the H∞-norm minimiza-tion using the Riccati equations [25, 4], which for largesystems, as FEM obtained MIMO systems, can lead tonumerical problems as ill-conditioning. This is the caseof the model presented in this article where the 2-normcondition number is 3.2 · 1010.

102

0

5

10

15

Frequency (Hz)

Mag

nitu

de (

dB)

Sensitivity peak change with controller gain

K=1K=0.9K=1.1

Fig. 6. Sensitivity function changing the controller gain.

To avoid the stated computing problems, the decom-position of the system into 4 SISO systems is performedfollowing the outline described in the section 2. In thisway, two designs are carried out for each coordinate axis,one with As + Bs and the other with As − Bs.

The proposed H∞ setup is depicted in Fig. [7]. Thisscheme, which is the same for each coordinate axis, iscomposed by G which is As + Bs or As − Bs, and theamplifiers and sensors blocks (Amp+δAmp, Sens+δSens).The equations which represent these blocks are,

Vo = 104Volt/m · x + δsensvolt |δsens| ≤ 1 (3.3)

ii = 0.25

1+ 2.2 · 10−4s·[1+ 1

20.1T(1+ δA)s

]Vi |δA| ≤ 1

(3.4)

where δsens and δA are the sensors and amplifiers uncer-tainties and T is the sampling period utilized in the digitalimplementation. It is supposed amaximumdelay of a 10%of the sampling period in the amplifier.The delay uncertainty is introduced into the amplifiers

dynamics (3.4), but it can include both the amplifiers andthe digital control implementation delays.On the other hand, the weighting functions WS , WT

and Wn are included to increase the importance of somefrequencies when minimizing the H∞-norm. In this case,they are chosen as follows,

Ws = 2 · 104s + 1.257 · 108s + 1257

(3.5)

WT = 10−6 (3.6)

Wn = 2.4s + 60

s + 6000(3.7)

Note that the equations used to model the uncertaintiesin sensors and amplifiers, and the weighting functions, arethe same in both H∞ problems.It is important to state that some other errors and per-

turbations could be also taken into account, such as model

S

Sens

WT

Wn

Z T

n

G

Amp

Amp

WS

ControllerV i i i V cx

rSens

y

Fig. 7. Block diagram of generalized plant.


0 6000 12000 18000 24000 30000−2

−1

0

1

2x 10

−5

Ω (rpm)

posi

tion

(m)

PID control

0 6000 12000 18000 24000 30000−2

−1

0

1

2x 10

−5

Ω (rpm)

posi

tion

(m)

H∞ control

Fig. 8. Comparison of the simulations of the speed frequency rotationbehavior of the system levitated with the PID or the H∞ controller.

errors or gyroscopic effects [12]. This coupling effect isnot notorious in the experimental testbed utilized in thispaper and it has not been included in the calculation ofthe controller. However, depending on the application, itcould be relevant.With the uncertainties and weights presented in (3.5)-

(3.7), two controllers P = Cs + Ds and Q = Cs − Ds arecalculated bymeans of a standard robust controlH∞ pack-age. Their linear combination leads to Cs andDs, reachingthe controller utilized to stabilize the shaft robustly.Although these controllers imply better simulation

results, even counteracting the disturbances (Figure 82),they are very large and difficult to implement experimen-tally without an order reduction (the achieved result is a4x4 MIMO controller with 12 states). Hence, the experi-mental results have been achieved with the experimentallytuned PID option.

3.3. Adaptive Feedforward Block for VibrationMinimization

The main purpose of this block is to counter the vibra-tion generated with the rotation as has been mentioned insection 2, in order to obtain the better positioning possible.The shaft rotates around the z-axis which defines the

horizontal plane together with the x-axis. Then, the y-axistakes into account the displacements along the verticaldirection. The vector r(t) = [

rxL ,rxR,ryL ,ryR]T is com-

posed by the reference signals corresponding to the rotor

2 The stability range does not fit with the one of figures 3 and 4. Thisis because the noise included in the position sensors in these lattersimulations is larger than the introduced to achieve Figure 8.

end coordinates as is shown in Fig. [1] and in Fig. [2] (inthe discrete domain).The control objective is, generally, to maintain the shaft

in an equilibrium position at any rotational speed. Such anequilibrium position is defined by the ends’ coordinateswhen the shaft is levitated without rotation. In this sense,the reference signal vector defines the desired displace-ments along x and y directions of the shaft ends respect tothe equilibrium position coordinates.In this article, the vector r(t) will be chosen zero for

designing the adaptive vibration controller and the sig-nal d(t) Fig. [1] represents the disturbance caused bythe unbalanced masses. This perturbation affects to thedisplacements along x and y coordinates of both endsof the shaft, which are compactly denoted by means ofx(t) = [

xL ,xR,yL,yR]T . These displacements are mea-

sured by the Hall effect sensors placed at the ends ofthe shaft. The rotor angular speed �(t) and the outputsignals measured from the sensors (related to the dis-placement of the corresponding coordinate respect to areference position) are the input signals of the adaptiveblock of the controller. Such a block provides a signalvector v(t) = [

vxL ,vxR,vyL ,vyR]T which tries to compen-

sate the vibrations produced by the unbalance masses. Forinstance, let xL(t) be the x-coordinate of the left end of theshaft. The output vxL(t) of the adaptive vibration controlleris a sinusoidal wave given by:

vxL(t) = A1xLsin�t + A2xLcos�t (3.8)

where A1xL and A2xL are time-varying parameters, whichare equivalent to the amplitude and the phase of thewave, updated by an adaptation algorithm in order tominimize the effect of the unbalanced masses upon xL .Analogous output signals are provided from the adaptivevibration block for yL , xR and yR (all of these signalsare generated experimentally to minimize the vibration insection 4).The adaptation process is in discrete-time and all the

adaptation parameters can be grouped as:

θc(k) = [A1xL(k),A2xL(k),A1xR(k),A2xR(k), (3.9)

A1yL(k),A2yL(k),A1yR(k),A2yR(k)]T

The adaptive control objective is to adjust the time-varying vector θc(k) so that the effect of the unbalancemasses d(t) upon the displacements x(t) is attenuated. Thesequences of values of such signals at sampling instantskT , where k being a non-negative integer and T denotingthe sampling period, can be expressed as:

x(k) = GD(q−1)d(k)+ GCL(q−1)v(k) (3.10)

where q−1 is the shift operator, d(k) and v(k) are, respec-tively, the value sequences of the signals d(t) and v(t) at


Fig. 9. Block diagram of the plant with filtered-x adaptive feedforward controller.

the sampling instants, and GD(q−1) and GCL(q1−) denotethe discrete-timematrix transfer function from x(k) tod(k)

and v(k), respectively.The aim is to design the adaptive vibration controller

so that the signals v(t) compensate the perturbation effectupon x(t). An appropriate design for this adaptive con-troller will result in a measure of the shaft position withoutvibrations or with very small amplitude vibrations aroundthe equilibrium position.The signals v(t) provided by the adaptive controller are

continuous-time but the parameters θc(k) associated withthe amplitude and phase of such signals are updated atdiscrete time instants by an estimation algorithmpresentedbelow.However, the adaptive signals (and the estimation algo-

rithm) depend on the knowledge of the frequency�. Noiseand variations in the measured speed can lead to loweffectiveness in the vibration cancellation and stabilityproblems, [17]. So, a FLL has been introduced for thesynchronization of the sinusoidal signals used in the con-trol structure with themeasured disturbance in the rotatingshaft [2].Similar devices can be utilized to achieve a frequency

estimation to improve the decoupling block, but, since thisblock is tried to be designed for a full frequency operationrange, the transitory of the estimator has to be very shortand an important design effort is already required.The estimation algorithm updates the time-varying

parameters θc(k) so that the vibrations upon the ends’coordinates is minimized. The ideal solution would beto calculate θc(k) to obtain x(k) equal to zero in (3.10).However, such a solution is not realistic since the pertur-bation d(k) is unknown. Then, a more realistic objectiveis to compute adaptively θc(k) in order to minimize thesquare of x(k) in (3.10), which can be written as,

x(k) = GD(q−1)d(k)+ GCL(q−1)ϕ(k)θc(k) (3.11)

where

ϕ(k) =

⎡⎢⎢⎣

s1 c1 0 0 0 0 0 00 0 s1 c1 0 0 0 00 0 0 0 s1 c1 0 00 0 0 0 0 0 s1 c1

⎤⎥⎥⎦ (3.12)

with s1 = sin�kT and c1 = cos�kT .A normalized least-squares algorithm with con-

stant covariance matrix is used to solve the proposedoptimization problem, [6]. Such an algorithm isdefined by:

θc(k) = θc(k − 1)+ Pϕ(k)T x(k)

1+ γ tr[ϕ(k)Pϕ(k)T ] (3.13)

where tr denotes the trace of a matrix, P ∈ R8x8 is the

covariance matrix and γ a real constant. P is chosen con-stant in order to reduce the computation effort. Since thesystemneeds a real time calculation of the stabilization andspeed controllers, the FLL and the vibration minimizationalgorithm, this choice could be compulsory to achieve anadequate sampling time.On the other hand, both γ and P must be chosen such

that the estimation process is much slower than that theclosed-loopdynamics describedbyGCL(q−1), [22]. In thisway, as it has been stated in the beginning of this section,the dynamics of the adaptive vibration blockwill not inter-fere with that of the stabilization block and therefore,the destabilization that could be caused by the dynam-ics coupling is avoided, allowing the 2-DOF control. Thisestimation algorithm ensures the boundedness and con-vergence of the adaptation parameters to constant values,which are crucial properties in adaptive systems.The experimental implementation of this scheme is pre-

sented in Fig. [9], where it is shown a “filtered-x” namedconfiguration [24]. This scheme type is performed to avoidthe noise of the plant (GCL(q−1)) in the input of the estima-tion algorithm. Concretely, the plant can be easilymodeled


Fig. 10. Experimental setup.

in the stationary state by means of a constant dependenton the rotational speed.

4. Experimental Results

In order to show the effectiveness of the proposed con-troller, it has been implemented in theDS1003 of dSPACEand positioning experiments have been carried out, at dif-ferent rotating speeds. The experimental setup is shown inFig. [10]. TheDSP based controller (DS1003 of dSPACE),on the left side of the picture, is connected with the PCformonitoring and configuration tasks, while the AD/DAspanel is in themiddle. Finally, theAMBbased system , theMBC500 Rotor Dynamics, is on the right of such figure,showing the input and output connections. This setup isschematically represented in Fig. [1].The two blocks of the final 2-DOF controller contain,

on the one hand, the PID experimentally tuned follow-ing the statements of the ISO 14839-3 and, on the otherhand, the feedforward adaptive controller. The first one isdesigned for maintain the stability of the shaft positioning,while the adaptive controller is designed for minimizingthe vibrations at rotating operation.The chosen parameters of the experiment are T =

1/12000seg, � = 2100rpm and γ = 5000. The period hasbeen chosen to obtain a stable behavior, while it is guaran-teed an implementable solution in this particular hardwaresetup at the chosen rotating speeds. Finally, the parame-ter γ allows a real 2-DOF behavior without implement adecoupling block.The experimental results are presented in Figs. [11, 12]

(similar results are achieved in the other bearing). In thefirst figure it is shown how a ∼ 80% of the unbalance pro-voked disturbance is rejected with the proposed 2-DOFcontroller. In consequence, the spindle is rotating veryclose to its rotation axis, and a more precise positioning is

0 0.005 0.01 0.015

0

2

4

6x 10

3

x axis position (mm)

yax

is p

ositi

on (

mm

)

Precise positioning of the spindle rotating at 2100rpm with and without the active control

Fig. 11. Achieved 2-DOF vibration reduction in one of the bearings viathe designed controller.

0 100 200 300 400 500 600 700

0

0.02

0.04

0.06

time (s)

Feedforward adaptive parameters

A1xL

A2xL

A1yL

A2yL

Fig. 12. Evolution of the feedforward adaptive parameters.

obtained. On the other hand, in the second figure Fig. [12],the temporal evolution of the adaptive parameters is pre-sented, showing its convergence. This convergence of theparameters implies a stabilization of the system in thenew enhanced positioning around the operation point. Theadaptation can be performed even faster reducing the valueof the parameter γ but the convergence of the adaptiveparameters is not ensured, so it is convenient to maintainthe adaptation “slow” to reduce the risk of destabilization.This fact is derived from the influence of the dynamicsintroduced for each control loop and corroborates the dis-cussion about the importance of the dynamics decoupling,which ensures a real 2-DOF control scheme.Another experimental measures have been performed

at lower rotational speeds with similar results.


0 2 4 6 8 10

0

1

2

3

4x 10

6

time (s)

posi

tion

(m)

Vibration reduction simulation in left bearing at 2100rpm

x

L

yL

Fig. 13. Simulation of achieved 2-DOF vibration reduction in one of thebearings via the designed controller.

0 2 4 6 8 10

8

6

4

2

0

0.02

0.04

0.06

0.08

0.1

time (s)

Feedforward adaptive parameters simulation

A

1xL

A2xL

A1y

L

A2yL

Fig. 14. Simulation of the evolution of the feedforward adaptive param-eters.

However, experiments with rotation velocity up to2400 rpm may end in system destabilization. To studyand explain this behavior, complete simulations have beendone Figs. [13, 14] at 2100 rpm and higher rotation speeds.These simulations do not reveal any malfunction at higherspeeds than 2100rpm, while the dynamics observed is sim-ilar to the experimental results at 2100 rpm. Therefore,the signals provided by the control hardware (DS1003 ofdSPACE) have been also tested in order to determine theorigin of the destabilization at high rotation speeds. It hasbeen corroborated that the generation of the sinusoid tocounter the vibration is not good enough for high rota-tional speeds and, thus, the system can became unstable.This problem is due to limited hardware calculation power,

which does not allow to generate a sinusoid with the nec-essary quality, starting from a particular rotating speed,since higher sampling rates are required. Better results areexpected with faster hardware such as FPGA based one.Although the results presented in this paper are general-

izable and applicable to general case, it must be remarkedthat the MBC500 Rotor Dynamics is more complex thana typical industrial device so the results obtained in thesection 4 are not directly comparable with those publishedin the industrial field. The differences came since thisdevice has a very flexible spindle being the resonances atlow frequencies. Also it has to be taken into account thatthe device is not oversized (particularly themagnetic bear-ings) and there is a limit in the force, both in magnitudeand time precision. Moreover, the frequencymeasurementsensor is not very precise. This fact enlarges the size ofthe controller because it implies the inclusion of an FLL.However, those characteristics make the control design

more challenging, but enable the study of systems pro-vided of lower resources in term of AMB forces. This lackof AMB resources, conveniently compensated by bettercontrol schemes, may result inmore economical solutions.It allows the improvement of the cost of themachinewhichis one of the main drawbacks of AMB based technology.With a better cost, the advantages ofAMBtechnologyget alot of impulse in a large number of industrial applications.

5. Conclusions

In this article, a precise positioningof a spindle levitated byAMBs is performed. To deal with this challenge, a 2-DOFactive controller is implemented. Concretely, it is com-posed by two specific purpose blocks, one for stabilizationand the other one for vibration rejection.Two of the most relevant characteristics of the systems

based on AMBs are their inherent instability and the cen-trifugal forces generated when the spindle is rotating dueto the mass unbalance. Therefore, the task of the con-troller is, on one hand, to maintain the stability and, onthe other hand, to reject the disturbance. Those actionshas been accomplished through the stabilization block andthe vibration minimization block, respectively. These twoblocks are combined with an optional decoupling blockto complete the mentioned 2-DOF controller. A generaldiscussion stating the effects of this block is performed.Also, considering that the decoupling block is based on aNotch filter, the relevance of the speed rotation measure-ment is studied. It is concluded that a bad measurementreduce the rotational speed stability range of the system.In particular, in the system under study, this decouplingblock implementation is not convenient since it destabi-lizes the plant in a large range of operating frequencies.


Hence, the controller to be implemented has been obtainedconsidering only the other two blocks.The process followed to obtain the stabilization block

begins with the application of a MIMO to SISO symme-try based decoupling. Then, each resulting SISO controlproblem is solved independently. Concretely, two controlstrategies are applied and compared, utilizing H∞ androot locus techniques, respectively. With the second tech-nique, a PID controller is achieved. This controller, hasbeen tuned experimentally using the ISO 14839-3 whichis specific for AMBbased devices. The election of a PID isbecause the derivative action is compulsory in this type ofsystems and because the integral part permits a good cen-tering of the spindle. This fact is beneficial for augmentingthe stability range of the system.On the other hand, to counter the vibration, an adaptive

feedforward adaptive controller is chosen. This block isin charge to generate a signal equal to the vibration but incounterphase, in order to reject it. With this controller, a∼ 80% of vibration reduction at 2100rpm rotational speedis achieved in a laboratory set-up based on the MBC500Rotor Dynamics. It has be taken into account that theadaptive algorithm does not take care of the correct mea-surement of the frequency. To deal with the frequencymeasurement noise and mean error, a FLL is includedin the design, allowing the good experimental resultspresented in the paper.Finally, it is remarkable that the proposed solution is

valid in a hardware with limitations in the magnetic forceor/and in the speed measurement sensor. In this way thecost of the elements which are needed to build the device isless and the industrial application of AMB based systemsis favored.The practical values of the presented results are derived

from the improvement of the position of the spindle. Morecomplex pieces can be designed with the advantages ofnon-contact technology: less mechanical wear, no need oflubrication, low maintenance, high-speed support,...

Acknowledgments

The authors are very grateful to the CICYT and theBasque Country Government for the support of this work,through projects DPI2002-04155-C02-01 and GIC07/49-208, respectively.

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6. Brief Discussion About Symmetry

Considering a symmetric plant G =[

A BB A

]and a

symmetric controller Gc =[

C DD C

]the closed-loop

characteristic equation results:

det

(1+ CA + DB CB + DA

CB + DA 1+ CA + DB

)= 0

⇒ det

(1+ A′ B′

B′ 1+ A′)

= 0 ⇒⇒ 1+ A′2 − B′2 + 2A′ = 0 ⇒⇒ (

1+ (A′ + B′))(1+ (A′ − B′)

) = 0 ⇒⇒ (1+ (C + D)(A + B))(1+ (C − D)(A − B)) = 0,

(6.1)

So, the MIMO system can be split into two differentcontrol design SISO problems, simplifying the overallcontrol procedure

1+ C1(A + B) = 0 (6.2)

1+ C2(A − B) = 0, (6.3)

and leading to the MIMO system controller

C + D = C1C − D = C2

⇒C1 + C2

2= C

C1 − C2

2= D.

(6.4)

Physically, this mode separation can be interpreted asthe superposition of the translational and conical modeswhich can describe any motion of the shaft.

Documents

2-DOF Controller Design for Precise Positioning a Spindle Levitated with Active Magnetic Bearings