Discrete-time flatness-based control of an electromagnetically levitatedrotating shaft

J. von Lowis, J. Rudolph, F. WoittennekInstitut fur Regelungs- und Steuerungstheorie

Technische Universitat Dresden, Mommsenstr. 13D-01062 Dresden, Germany

loewis,rudolph,woittenn @erss11.et.tu-dresden.de

Keywords: Differential flatness, discrete-time, trajectorytracking, time-varying observer.

Abstract

Differential flatness of the mathematical model of an electro-magnetically levitated shaft is exploited for trajectory track-ing. Emphasis is put on the flatness-based implementationof a discrete-time control law. The control law is completedby a time-varying observer. Both the control by coil currentsand voltages is illustrated in simulation.

1 Introduction

Electromagnetic bearings have a number of important advan-tages over conventional bearings. For example, they do notrequire a lubricant and have lower friction at high speeds.Furthermore, magnetic bearings can be used to suppress shaftvibration and position the rotor to follow some prescribed tra-jectory. These last two features cannot be readily achievedwith conventional bearings.

Magnetic bearings are unstable systems, as can be seenby the following reasoning. Consider a ferromagnetic bodyplaced between two electromagnets pulling into opposite di-rections. At the equilibrium position the resultant force, i. e.,the sum of the electromagnetic forces and the gravitationalforce, is zero. The attracting electromagnetic force decreaseswith increasing distance of the body from the magnet, and itincreases with decreasing distance. Therefore, if the body ismoved away from the equilibrium it is attracted by the elec-tromagnet pulling into the direction of the deviation. As aconsequence, feedback is required for the operation of mag-netic bearings.

The dynamics of the electromagnetically levitated rotat-ing shaft is nonlinear both due to the rigid body dynamic-s and due to the relations between coil currents and forces.However, the model isdifferentially flat[1, 2]. This proper-ty allows a relatively simple design of continuous-time con-trollers most appropriate for the trajectory tracking. Thisapproach has been discussed in some detail in [3]. Forthe implementation it is important that discrete-time control

laws can be directly derived from the continuous-time designwithout discretizing the model (this approach is related to[4]). This discrete-time design is discussed in the present pa-per. We describe control laws designed for a rotating spindlebuilt at the German company AXOMAT in Berggießhubel(Saxony). In view of their general applicability, our resultshave a certain theoretical importance.

We first discuss the control through the coil currents. Thisis the usual mode of operation. The extension to the con-trol through the coil voltages is also included. In both cases,time-varying observers can be used to estimate unmeasuredvelocities as well as disturbances. We propose a complexvariable notation for a simplified design of these observers.

The paper is structured as follows. The mathematical mod-el is given in Section 2 and the flatness-based controller forthe current controlled system is designed in continuous timein Section 3. The corresponding discrete-time controller isdiscussed in Section 4. Section 5 is dedicated to observerdesign, Section 6 treats the voltage controlled system, andsome conclusions are drawn in Section 7.

2 The mathematical model

The device considered is equipped with five pairs of electro-magnets which support a rotating shaft in its radial and theaxial directions — see Fig. 1.

A model of the shaft is

m X = Fx,p − Fx,n︸ ︷︷ ︸

Fx

+mgx (1a)

m Y = Fv,y,p − Fv,y,n︸ ︷︷ ︸

Fv,y

+ Fh,y,p − Fh,y,n︸ ︷︷ ︸

Fh,y

+mgy (1b)

mZ = Fv,z,p − Fv,z,n︸ ︷︷ ︸

Fv,z

+Fh,z,p − Fh,z,n︸ ︷︷ ︸

Fh,z

+mgz (1c)

Θ2 ψ = −(lf,v −X)Fv,z + (lf,h + X)Fh,z −Θ1 φ θ(1d)

Θ2 θ = (lf,v −X)Fv,y − (lf,h + X)Fh,y + Θ1 φ ψ(1e)

Θ1 φ = Dφ. (1f)

Figure 1: The shaft with its radial magnetic bearings.

HereX, Y , andZ are the coordinates of the center of massGof the shaft in a frame (with axesx, y, andz) fixed in space,at a point being considered as the “center” of the device. Theanglesφ, ψ, andθ describe the angular position of the axesof a body-fixed frame. The coil forces are denoted byF•, themotor torque asDφ. (Here and in the sequel bullets are tobe replaced by appropriate indices.) The shaft has massmand moments of inertiaΘ1 andΘ2, lf,v andlf,h are the dis-tances between the symmetry planes of the bearings (wherethe forcesF• are produced) and the pointG. The forces arerelated with the currents by

F• = k•i2•

(σ• − s•)2(2)

where the displacements from the magnetic centers are givenby

sv,y,• = ±(Y + lf,vθ) sh,y,• = ±(Y − lf,hθ)sv,z,• = ±(Z − lf,vψ) sh,z,• = ±(Z + lf,hψ)

sx,p = −sx,n = X.(3)

The system has eleven inputs: the currents through thecoils of the electromagnets and the torque produced by themotor (not shown in Fig. 1). However, only one force iscreated by each pair of coils, and considering the resultingforce as the input reduces the number of controls to six:Fx, Fv,y, Fh,y, Fv,z, Fh,z, Dφ. In the controller the ten coilcurrents can be computed from the five forces. This can bedone in such a way that at each time instant only the currentcreating a force in the direction of the resulting force is set d-ifferent from zero [3]. In contrast to this, “bias currents” arerequired in classical linear approaches. These lead to largeJoule effect losses, which can be avoided with the flatness-based nonlinear control.

In two measurement planes (perpendicular to thex-direction) atx = lm,v andx = −lm,h the deviationYm,v,

Ym,h, Zm,v, andZm,h from center position is measured iny- andz-direction:

Ym,v = Y + lm,vθ, Ym,h = Y − lm,hθ

Zm,v = Z − lm,vψ, Zm,h = Z + lm,hψ.(4)

These equations can be solved for the generalized coordi-natesY , Z, θ, andψ. A fifth sensor measures the axial po-sition X. Finally, the angular velocityω = φ is measured,too.

3 Flatness-based control

Considering the equations of motion (1) auxiliary (accelera-tion) variables can be introduced asax, ay, az, αψ, αθ, αφ:

m ax = Fx + mgx

may = Fv,y + Fh,y + m gy

m az = Fv,z + Fh,z + mgz

Θ2 αψ = −(lf,v −X)Fv,z + (lf,h + X)Fh,z −Θ1 φ θ

Θ2 αθ = (lf,v −X)Fv,y − (lf,h + X)Fh,y + Θ1 φ ψ

Θ1 αφ = Dφ.(5)

With this

X = ax, Y = ay, Z = az, ψ = αψ, θ = αθ, φ = αφ ,

and a stabilizing controller can easily be designed for each ofthe decoupled subsystems. For example for the positionX:

ax = Xref − k1ex − k0ex, (6)

whereXref is a twice differentiable reference trajectory andcontroller gainsk0, k1 > 0. The tracking errorex := X −Xref now satisfies:

ex + k1ex + k0ex = 0. (7)

Using (6), we compute the auxiliary accelerationsax, . . . , αφ. From these accelerations the forces andthe torqueDφ are obtained by solving the inhomogeneouslinear system (5). Finally, for a positive forceF• the controlcurrenti•,p in the corresponding winding is obtained from

i•,p =

√

F•k•,p

(σ•,p − s•,p), (8)

while the currenti•,n (generating a negative force) is keptzero. (For negative forceF• the currentsi•,p = 0 andi•,n =√

−F•/k•,n(σ•,n − s•,n) are used.) This approach avoidsbias currents required in classical linear approaches: at anyinstant there is a current in only one coil of each pair [3].The above design is based on thedifferential flatnessof themodel. The coordinatesX,Y, Z, φ, θ, ψ form a flat output ofthe mechanical subsystem [1].

The trajectory planning is simplified by the flatness prop-erty. For example, if we want to transfer the center of massof the shaft within timet∗ from the center position onto anelliptic trajectory we choose

Yref(t) = ry(t) cos ΩtZref(t) = rz(t) sinΩt

Xref(t) = ψref(t) = θref(t) = 0

φref = ω0 t + φ0 (ωref = ω0 = const.)

(9)

for the components of the flat output(X, Y, Z, φ, ψ, θ) and

ry(t) = r∗y p(t), rz(t) = r∗z p(t)

with

p(t) = 10(

tt∗

)3

− 15(

tt∗

)4

+ 6(

tt∗

)5

.

4 Discrete-time control

Since the procedure described in this section can be used todiscretize control laws for arbitrary flat systems with linearerror dynamics, we start with a general description beforedetailing it for the axial controller. The radial controllers aredesigned similarly. We useUk to denote constant inputs ap-plied over the sampling period[tk, tk+1). The correspondingcontinuous-time input will be denotedu(t).

Perhaps the simplest way to implement a continuous-timecontrol law on a microprocessor is to computeu(tk) frommeasurements and estimates, and to apply this value duringthe sampling interval[tk, tk+1), i. e., chooseUk = u(tk).However, in fast processes — like the spindle consideredhere — depending on the sampling timeTa, this might notachieve the desired performance. We refer to this method asquasi-continuouscontrol.

The flatness-based control approach allows us to proposea different method of discretization. Attk−1 + τ1 (τ1 <Ta, tk = tk−1 + Ta) both the measurements and the controlsUk−1 applied on[tk−1, tk) are known. An observer providesan estimate of the system state at timetk. It uses measure-ments attk−1 and the controlsUk−1.

Flatness is exploited in the computation of the control tobe applied on[tk, tk+1) through the use of the linear errordynamics assigned to the system by a continuous-time con-troller. To this end, one aims at computing a “good” approx-imationUk of the controlu(tk + τ), τ ∈ [0, Ta). Following[4], the mean value can be used

Uk =1Ta

∫ Ta

0u(tk + τ)dτ,

which can be approximated by

Uk ≈ 1n

n∑

i=1

u(tk + τi). (10)

with 0 ≤ τ1 < · · · < τn ≤ Ta.

Regarding the controller for the axial positionX, ix,p andix,n are the inputs. We now detail, howix,p(tk + τi) andix,n(tk+τi) corresponding tou(tk+τi) in (10) are computedin order to get the constant approximations

(Ikx,p, I

kx,n) :=

1n

n∑

i=1

(ix,p(tk + τi), ix,n(tk + τi)). (11)

In the sequel, we refer to this method asmean-value-discretization.

The tracking error and its time derivative are given by thedifference of the observer state and the reference trajectory

(e(tk), e(tk)) = (X(tk)−Xref(tk), ˙X(tk)− Xref(tk)).

The evolution of the tracking error in case of continuous con-trol is determined by the error dynamics (7)(

ex(tk + τ)ex(tk + τ)

)

= exp[(

0 1−k0 −k1

)

τ] (

ex(tk)ex(tk)

)

. (12)

When implementing the control law in real-time on a digitalcomputer, one will use ordinary exponentials, sines etc. toexpress the solution of (7). The matrix exponential in (12) isused only for notational convenience.

With (12) and (6)ax(tk + τ) can be predicted and withex(tk + τ) and the referenceXref(tk + τ) we determine theposition1 as

X(tk + τ) = ex(tk + τ) + Xref(tk + τ),

which is used in (8) to compute the input currents. For ex-ample, assumingax(tk + τ) > 0 we get

ix,p(tk + τ) =

√

ax(tk + τ) mkx

(σx −X(tk + τ))

ix,n(tk + τ) = 0.

Note, that since the sign ofax(tk + τi) is not necessarilythe same for allτi ∈ [0, Ta), the mean value ofboth in-put currents may be nonzero even though for eachτi eitherix,p(tk + τi) or ix,n(tk + τi) equals zero.

This can be seen on the simulation results shown in Fig. 2where transitions in they-direction from−100µm to100µm,and vice versa, are performed over5ms = 5Ta. Dur-ing the first 40 ms the controller is run with mean-value-discretization. The interval[0, Ta) is divided into 10 subin-tervals, i. e.,n = 11 in (10). For t ∈ [40ms, 100ms] thecontroller is run with quasi-continuous discretization, i. e.,the inputs are determined according to (6) and (8) with thesystem state and the reference evaluated attk.

Obviously, the tracking performance is much better whenthe controller with mean-value-discretization is used, whilethe currents are smaller. Notice the slight delay betweenYref andY at the beginning of the transitions with the quasi-continuous controller. This delay of approximatelyTa does

1The velocityX(tk + τ) can be computed similarly, but for this partic-ular system onlyX(tk + τ) is needed to determine the inputs.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−150

−100

−50

0

50

100

150

t in sec

Y r

ef ,

Y i

n µm

Y ref

Y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

t in sec

iv,

y,p ,

i v,y,

n in

A

iv,y,p

iv,y,n

Figure 2: Vertical positionY and referenceYref for tran-sitions within five sampling periods (Ta = 1ms) and cor-responding currents in one bearing. Beforet = 40 mswith mean-value-discretization, aftert = 40 ms with quasi-continuous discretization.

not appear with mean-value-discretization. This is not sur-prising since the discretization takes into account the refer-ence trajectory on the sampling interval for which the inputis computed.

The results of another simulation experiment are shownin Fig. 3. The center of mass of the rotating shaft is trans-ferred onto an elliptic path described by (9) whereω0 =Ω = 500 rad/s, i. e., the elliptic trajectory is synchronizedwith the angleφ of rotation about thex-axis. The trans-fer is performed withint∗ = 40 ms, once with mean-value-discretization and once with quasi-continuous discretization.Notice a rather large error with the quasi-continuous ap-proach. In all simulations, a dicrete-time observer as de-scribed in Section 5.3 is used.

−100 −50 0 50 100−100

−80

−60

−40

−20

0

20

40

60

80

100

Z ref

, Zm,v

in µm

Y r

ef, Y

m,v

in

µm

( Z ref

,Y ref

) ( Z

m,v,Y

m,v )

−100 −50 0 50 100−100

−80

−60

−40

−20

0

20

40

60

80

100

Z ref

, Zm,v

in µm

Y r

ef, Y

m,v

in

µm

( Z ref

,Y ref

) ( Z

m,v,Y

m,v )

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

t in sec

iv,

y,p ,

i v,y,

n in

A

iv,y,p

iv,y,n

Figure 3: Transition of the center of mass onto an elliptictrajectory according to (9), witht∗ = 40 ms, r∗y = 50 µm,andr∗z = 100 µm. Top figure: position of the shaft in thev-measurement plane (mean-value-discretization); middle fig-ure: same as top, but with quasi-continuous discretization;bottom figure: currents in the coils generatingFv,y,p andFv,y,n in case of mean-value-discretization.

5 Observer design

5.1 Complex variable notation

Introducing complex variables

χ := ψ + jθ and p := Y + jZ (13)

allows us to write the system equations as well as the equa-tions of the controller and the observers in a compact form.The acceleration of the center of mass (due to bearing forcesand gravity) in radial directions then reads

ap := ay + j az (14)

with ay andaz as in (5). The angular acceleration due tobearing forces can be written as

aχ := aψ + j aθ (15)

with

aψ =1

Θ2(−(lf,v −X)Fv,z + (lf,h + X)Fh,z)

aθ =1

Θ2((lf,v −X)Fv,y − (lf,h + X)Fh,y) .

(16)

The key point with the above definitions ofap andaχ is thatthey are functions of the measured generalized coordinatesand the input currents. This is essential for the observer de-sign since it allows to cancel these terms in the observer errordynamics.

With ap, aχ, p, andχ the system equations read

p = ap (17)

χ = jω(t)Θ1

Θ2χ + aχ. (18)

Equation (18) is a linear differential equation with a time-varying coefficient, while (17) is time invariant. The equationfor thex-direction is even simpler than (17) (sinceax andXare not complex) and it is omitted.

5.2 Estimating velocities and constant accelerations

Due to uncertainties on the model parameters steady stateposition errors are to be expected. These uncertainties havethe same effect as constant forces and torques acting on therigid body. To account for these perturbations the model canbe extended by differential equations for constant accelera-tions2 β

pand β

χ. With this extension equations (17) and

(18) can be written in state space form

xp = Ap xp + b ap (19)

xχ = Aχ xχ + b aχ. (20)

2For an approach to the estimation of harmonic perturbations due to im-balance see [6].

Here, the vectorsxp, xχ, andb are defined as

xp =

ppβ

p

, xχ =

χχβ

χ

, b =

010

, (21)

and the matricesAp andAχ are

Ap =

0 1 00 0 10 0 0

, Aχ =

0 1 00 ϕ 10 0 0

. (22)

Moreover,ϕ(t) := jω(t)Θ1Θ2

is a known function of time,ω(t) being measured.

The observer is composed of a simulation of the model anda correction by error injection

˙xp = Ap xp + b ap + lp (p− p) (23)

˙xχ = Aχ xχ + b aχ + lχ (χ− χ). (24)

Using (4), the (complex) positionp and the angleχ can becomputed from measurements. Therefore,p andχ are con-sidered as measured variables for the purpose of observer de-sign.

The interesting system is (24), because it is time-varying.The observer error dynamics are obtained as the differencebetween (24) and (20):

˙xχ = Aχ xχ − lχ χ =

−lχ,1 1 0−lχ,2 ϕ(t) 1−lχ,3 0 0

︸ ︷︷ ︸

=:Aclχ

xχ, (25)

with the observer errorxχ := xχ − xχ.Whenω varies slowly, it is legitimate to assumeAχ to be

constant, depending on the “parameter”ω. Then it makessense to compute the observer gainslχ,1, lχ,2, lχ,3 (depend-ing on ω) such that the characteristic polynomial ofAcl

χ isindependent ofω (see [3]). Extending this approach we aimat a time-invariant observer error dynamics.

For the reader’s convenience we rewrite (25) in terms of(x1, x2, x3) := (χ, ˙χ, β

χ):

x1 = −lχ,1 x1 + x2

x2 = −lχ,2 x1 + ϕx2 + x3

x3 = −lχ,3 x1.

(26)

By the change of coordinates

z1 := x1, z2 := x2 − ϕx1, z3 := x3

equations (26) become

z1 = −lχ,1 x1 + x2

= (−lχ,1 + ϕ) z1 + z2

z2 = x2 − ϕ z1 − ϕ z1

= −lχ,2 z1 + ϕx2 + z3 − ϕ z1 − ϕ(−lχ,1 z1 + x2)

= (−lχ,2 − ϕ + ϕ lχ,1) z1 + z3

z3 = −lχ,3 z1.

Substituting back,(χ, ξ, βχ) = (z1, z2, z3), we obtain the

error equations

˙χ˙ξ˙βχ

=

−lχ,1 + ϕ 1 0−lχ,2 − ϕ + ϕ lχ,1 0 1

−lχ,3 0 0

χξ

βχ

, (27)

which can be made time-invariant with the characteristicpolynomial3

c0 + c1λ + c2λ2 + λ3

by choosing the time-varying gains

lχ,1(t) = c0 − ϕ

lχ,2(t) = c1 − ϕ + ϕ (c0 − ϕ)

lχ,3(t) = c2.

(28)

The observer equations are given here for completeness:

˙χ = χ (c0 − ϕ) − ϕ χ + ξ˙ξ = χ (c1 − ϕ + ϕ (c0 − ϕ))− ϕ χ + β

χ+ aχ

˙βχ

= χ c2.

(29)

The angular velocity to be estimated follows immediatelyfrom the definition ofξ := χ− ϕ χ as ˙χ = ξ + ϕ χ.

5.3 Discrete-time observer

For the implementation on a digital computer the observerhas to be discretized. We restrict ourselves to the subsystem(24) for the angular coordinatesχ.

For constant or slowly varyingω the discretization of thelinear time-invariant system (24) is rather simple in case ofconstant inputs. The only obstacle is to find a “good” con-stant approximation foraχ. (For constant input currents theaccelerationaχ is usually not constant over the sampling pe-riod since the positionX, p, χ is changing and the general-ized forces depend on currentsandposition.)

In each sampling period, the currentsIk• were computed as

to achieve a prescribed error dynamics (considering as goodas possible the non-constant position). The currents corre-spond to the average accelerations

νkχ =

1n

n∑

i=1

aχ(tk + τi), (30)

computed in a similar fashion asIk• in (11). It seems reason-

able to useνkχ as input for the discrete-time observer

xk+1χ = exp(AχTa) xk

χ + bd νkχ + lχ,d (χk − χk),

with bd =∫ Ta

0 exp (Aχ (Ta − τ)) bdτ and the observer

gainslχ,d =(

lχ,d,1 lχ,d,2 lχ,d,3)T

such that the charac-teristic polynomial of

exp(Aχ Ta)− lχ,d(

1 0 0)

has its roots inside the unit circle of the complex plane.3Since the system is complex, the coefficientsc0, c1, c2 need not be

real. Hence, the eigenvalues of (27) need not be complex conjugates.

6 Voltage controlled system

Sometimes it is useful to control the system through the coilvoltages instead of the currents. Then, the model is extendedby the equations

Ψ• = V• −R• i•, (31)

whereR• are the resistances of the coils andV• are the coilvoltages. The relation between the flux linkageΨ• and thecoil currenti• is given by

Ψ• =2 k•

σ• − s•i•. (32)

Thus, equation (31) can be written as

Ψ• = V• −R•Ψ•σ• − s•

2 k•. (33)

The air gap lengthss• depend on the position of the shaftaccording to (3).

The controller may be designed using the idea that currenttracking essentially means force tracking (which is what weneed to stabilize the motion of the mechanical subsystem).In Section 4 we computed constant currents as “good” ap-proximations of the currents required by a continuous-timecontroller. In a similar fashion, the controller now generatesvoltages (constant over a sampling period) which cause cur-rents approximating those of the continuous-time controller.This will be discussed now.

From the coil currentsi•(tk−1) (assumed to be measured)we can compute the flux linkagesΨ•(tk−1) using (32). Withthe voltagesV k−1

• (which are known since they are the in-puts currently applied) we can integrate (33) on[tk−1, tk] toobtain estimatesΨ•(tk) of the flux linkages (or equivalentlythe currents) at timetk.

Equation (33) is time-varying, because the air-gap lengthss• depend on the positions. Since generalized positions aremeasured and generalized velocities are estimated, good es-timates ofs•(t) can be obtained on[tk−1, tk]. Alternatively,assuming the air gaps vary only slightly within a samplingperiod, one could simply chooses•(tk−1).

In order to compute the voltagesV k• to be applied over the

next sampling period, the currentsi•(tk+1) are determinedusing the error dynamics — similar to Section 4. In con-trast to Section 4, now we compute the currents requiredatthe endof the next sampling period, rather than average cur-rents. The voltages are then computed such as to reach thoserequired currents.

For constantV k• the solution of (33) is

Ψ•(tk+1) = e−R Ta0

R• (σ•−s•)2 k• dτΨ•(tk) + V k

• Ta. (34)

The desired currentsi•(tk+1) are computed as in Section 4,equations (32) then yield the flux linkagesΨ•(tk+1). UsingΨ•(tk+1) and the estimatesΨ•(tk) we can solve (34) for

V k• =

Ψ•(tk+1)− e−R Ta0

R• (σ•−s•)2 k• dτ Ψ•(tk)

Ta. (35)

The currents (resp. flux linkages, resp. forces) obtained withthese voltages,V k

• , are approximately the same as would re-sult from the continuous-time feedback at timetk+1, i. e., atthe end of the next sampling period.

Fig. 4 shows simulation results obtained with the proposedcontrol scheme. A transition inx-direction from center po-sition to X = 100 µm within 6 sampling periods (3ms) isperformed. The initial positions, velocities, and flux linkagesare all zero. To avoid singularities in the voltage controlledsystem (see [3]), which are related to the fact that the right-hand-side of (8) is not differentiable atF• = 0, the right-hand-side of (8) is replaced by a differentiable term causingnon-zero bias currents at small forces (see [3, 5] for a de-tailed discussion of this issue). For large forces again onlythe coil current corresponding to the direction of the force isnon-zero. As a consequence, now in order to retain the shaftin the center position non-zero currents are necesseary. Thiscan be seen in Fig. 4, where the first voltage peak is neededto reach the bias currents. The transition to100 µm is per-formed on[4ms, 7ms].

7 Conclusion

The differential flatness of the mathematical model of anelectromagnetically levitated shaft has been exploited todesign discrete-time control laws for trajectory tracking.The proposed discrete-time controllers compute constant in-puts as to achieve good approximations of correspondingcontinuous-time controllers with linear tracking error dy-namics. By this approach discrete-time control with ratherslow sampling can be designed for the differentially flatcontinuous-time system without discretizing the mathemat-ical model.

Acknowledgments

This work is part of a research project with AXOMATG.m.b.H., Berggießhubel, Germany, with financial supportby the European Union (EFRE) and the Free State of Saxony(P-No. 5051).

References

[1] M. Fliess, J. Levine, Ph. Martin, and P. Rouchon. Flat-ness and defect of non-linear systems: introductory the-ory and examples.Int. J. Control, 61(6):1327–1361,1995.

[2] M. Fliess, J. Levine, Ph. Martin, and P. Rouchon. ALie-Backlund approach to equivalence and flatness ofnonlinear systems.IEEE Trans. on Automatic Control,AC-44(5):922–937, 1999.

[3] J. Levine, J. Lottin, and J.-Ch. Ponsart. A nonlinearapproach to the control of magnetic bearings.IEEETrans. on Control Systems Technology, 4(5):524–544,1996.

0 0.005 0.01 0.0150

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t in sec

X r

ef, X

in

µm

X ref

X

0 0.005 0.01 0.0150

1

2

3

4

5

6

7

t in sec

ix,

p, ix,

n in

A

ix,p

ix,n

0 0.005 0.01 0.015−300

−200

−100

0

100

200

300

t in sec

Vx,

p, Vx,

n in

Vol

t

Vx,p

Vx,n

Figure 4: Voltage controlled system: PositionX and refer-enceXref for a transition within3ms (top figure). Currentsix,p andix,n in the positve and negative direction coils (mid-dle figure). VoltagesVx,p andVx,n across the coils (bottomfigure). Sampling periodTa = 0.5ms, voltages computedaccording to (35).

[4] Ph. Martin and P. Rouchon. Flatness and sampling con-trol of induction motors. InProceedings of the 13th I-FAC triennial World Congress, San Francisco, Califor-nia, 1996.

[5] J.-Ch. Ponsart.Asservissements numeriques de paliersmagnetiques. Application aux pompesa vide. These deDoctorat, Universite de Savoie, Annecy, France, 1996.

[6] J. Rudolph, F. Woittennek, and J. von Lowis. ZurRegelung einer elektromagnetisch gelagerten Spindel.at—Automatisierungstechnik, 48(3):132–139, 2000.