FACTA UNIVERSITATIS (NIS)

SER.: ELEC. ENERG. vol. 22, no. 2, August 2009, 183-195

Forced Dynamics Control Of An Actuator With LinearPMSM

Jan Vittek, Vladimir Vavr us, Jozef Buday, and Jozef Kuchta

Abstract: The paper presents design and verification of Forced Dynamics Control ofan actuator with linear permanent magnet synchronous motor. This control method isa relatively new one and offers an accurate realisation of a dynamic speed response,which can be selected for given application by the user. In addition to this, the an-gle between stator current vector and moving part flux vectoris maintained mutuallyperpendicular as it is under conventional vector control. To achieve prescribed speedresponse derived control law requires estimation of an external force, which is ob-tained from the set of observers. The first observer works in pseudo-sliding mode andobserves speed of moving part while the second one has filtering effect for eliminationof the previous one chattering. The overall control system is verified by simulationsand experimentally. Preliminary experiments confirmed that the moving part speedresponse follows the prescribed one fairly closely.

Keywords: Feedback linearization, Linear permanent magnet synchronous motor,Nonlinear system control, Observers,.

1 Introduction

TWO major categories of linear motors with low and high acceleration have beendeveloped. Low-acceleration linear motors are suitable mainly for ground-

based transportation applications while high acceleration motors are designed toaccelerate an object up to a very high speed. This paper develops a low accel-eration speed control of linear permanent magnet synchronous motor (LPMSM).

Manuscript received on June 14, 2009.J. Vittek and V. Vavrus are with University ofZilina, Faculty of Electrical Engineering, Univerz-

itna 1, 010 26Zilina, Slovak Republic (e-mail:jan.vittek@fel.uniza.sk). J. Buday, and J.Kuchta are with Electrotechnical Research Institute, Trencianska 19, 018 51 Nova Dubnica, SlovakRepublic (e-mail:jozef.buday@evpu.sk).

183

184 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

Control system exploits principles of feedback linearization [1] and forces movingpart speed to follow demanded speed with prescribed closed-loop dynamics.

Presented speed control system for LPMSM is based on Forced Dynamics Con-trol (FDC) [2] while respecting also principles of vector control [3, 4]. FDC methodenables to design such control algorithms, which have precisely defined the accel-eration profile and therefore it can contribute to the travelcomfort of the movingobjects. Derived control algorithms respect mutual orthogonality of the force pro-ducing primary part current vector and moving part magneticflux vector as it is inconventional vector controlled rotational drives with PMSM.

To achieve prescribed settling time during speed transients the informationabout external force is needed. The estimate of external force together with theestimate of translation speed is obtained from the set of twoobservers. The firstobserver is based on sliding-mode principles and to avoid chattering of the esti-mated variable this observer is completed with another one,which has filteringeffect. Disturbances estimation gives FDC a certain degreeof robustness not onlywith respect to external disturbances but also to plant parameter variations since theplant parameter variations are equivalent to load forces applied to the unperturbedplant model. The original contribution of this paper is a preliminary experimentalverification of the control algorithms presented in [5] capable to control speed ofmoving part with prescribed closed loop dynamics.

2/3

transf.

e-jJ

ia_d

ia

^Fext

^vmp

ExternalforceFext

Udc

meas.

ia

ib

ua , ia

ejJ

Inverter

3/2

transf.

ib -ic

ib_d

ic_d

ia_d

Position

sensor

ib_d

id_d

iq_d

Forced DynamicsControl Algorithm

( ) úû

ùêë

é+-

Y=

=

extmpdvPM

d_q

d_d

FvvT

M

c

1i

0i

Observers

vmp

J=smp/r

vmp

ud , id

uq , iqd

dt

vdType of

dynamics

moving

part PM

smpsmpJ=smp/r

LinearPMSM

Ffric

Slave Control LoopMaster Control Loop

ub , ib

Udc

Fig. 1. Overall LPMSM control system block diagram.

The overall speed control system of LPMSM, which is shown in Fig. 1, hasa nested structure comprising an outer master control loop and inner slave controlloop. Outer control loop computes such demanded primary part currents, which re-alizes the closed loop prescribed dynamic behavior of the drive. This way selectedoperational modes with defined acceleration and speed profile can be prescribed.The inner slave control loop forces real three-phase currents of primary part to fol-

Forced Dynamics Control of an Actuator With Linear PMSM 185

low their computed demands from master algorithm with negligible lag via controlof inverter power electronic switches.

2 Control Algorithm Design

FDC algorithm based on feedback linearisation and vector control principles isdeveloped in two steps. Firstly, the model of LPMSM is formulated in the dqco-ordinate system, which is coupled to the moving part of LPMSM:

dsmp

dt= v′mp (1)

dvmp

dt=

1M

[c(Ψdiq−Ψqid)−Fext] (2)

ddt

[

idiq

]

=

−Rs

Ld

PvmpLq

rLd

−PvmpLd

rLq−

Rs

Lq

[

idiq

]

−Pvmp

rLq

[

0ΨPM

]

+

1Ld

0

01Lq

[

ud

uq

]

(3)

where,[id, iq]T and [ud,uq]T are, respectively, column vectors of the LPMSM pri-

mary part current and voltage components,[Ψd,Ψd]T is column vector of moving

part magnetic flux with componentsΨd = Ldid +ΨPM andΨq = Lqiq as direct andquadrature magnetic fluxes in the moving part coupled transformation frame. Posi-tion and velocity of the moving part aresmp andvmp. Motor constant isc = 3P/2r,whereP is number of pole-pairs andr = Pτp/π is a constant parameter of LPMSMhaving the dimensions of length, andτp = 82.5 mm is pole pitch of the motor. Theexternal force acting on the moving part isFext, Rs is the phase resistance,Ld andLq are the direct and quadrature phase inductances,ΨPM is the permanent magnetlinkage flux andM is the mass of the linear motor moving part plus the equivalentmass of the driven mechanism.

2.1 Development of speed control algorithm

Secondly, the linearising function for translation speed of the moving part, whichforces this speed to obey specified closed-loop linear differential equation with aprescribedtime constant, Tv is developed:

amp =dvmp

dt=

1Tv

(vd −vmp). (4)

Linearisation function is obtained by equating the right hand side of (4) with theright hand side of the corresponding motor equation (2). This forcesthe non-linear

186 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

differential equation (2) to have the same response as the linear equation (4). Thus:

1M

[c(Ψdiq−Ψqid)−Fext] =1Tv

(vd −vmp). (5)

The following part of the control law is formulated on the basis of vector con-trol, which requires mutual orthogonality of moving part magnetic flux and primarypart current vector. Following conventional approach [3] to achieve maximum fluxup to nominal speed the current demand in direct axis is set tozero:

id = 0. (6)

Setting id = 0 in (5) on the assumption that the real current follows its de-mand,id = id d direct axis flux can be replaced by permanent magnet linkage fluxonly, Ψd = ΨPM. Solving equation (5) for force producing current component, iq d

yields the complete master control algorithm formulated for primary part currentdemands in thed- andq-axis:

id d = 0 (7a)

id q =1

cΨPM

[MTv

(vd −vmp)+ Fext

]

=Mamp+ Fext

cΨPM. (7b)

The numerator of the derived algorithm (7b) consists of two parts. The first onecontains the demanded output acceleration and creates dynamic force during tran-sients. The second part covers the external force, which needs to be estimated. Bychanging the prescribed acceleration,amp, in (7b) various operational modes canbe realised.

a) Linear first order speed response, (the case already described), where spec-ified settling time,Ts , is defined as the time taken for the error betweendemanded and real speed to fall and to stay bellow 5% (Ts = 3Tv):

ad =1Tv

(vd −vmp) =3Ts

(vd −vmp). (8)

The derived master control law (5) yields a moving part speedresponse withlinear, first order dynamics and unity dc gain defined as:

Fs(p) =vmp(p)

vd(p)=

11+ pTv

(9)

Forced Dynamics Control of an Actuator With Linear PMSM 187

b) Direct acceleration control with constant acceleration(ramp), where the mov-ing part produces an acceleration following a demanded constant one withnegligible dynamic lag. The demanded acceleration is determined by a de-manded of velocity,vd and the demanded acceleration time,Ts as:

ad =vd

Tssgn(vd −vmp). (10)

Via proper definition of smooth mathematical functions for drive accelerationas shown in [2] different operational modes can be achieved.Prescribed ideal accel-eration and speed responses to described modes forM = 1 kg are shown in Fig. 2:

0 0.5 1 1.50

20

40

60

80

100

120

[ms–1

, ms-2

]

[s]Ts

vmp

0,95vd

0,2ad

0 0.5 1 1.50

20

40

60

80

100

120

[ms–1

, ms–2

]

[s]

0.2ad

Ts

vmpvd

a) b)

Fig. 2. Prescribed profiles of speed and accelerations for a)first order dynamics and b) constantacceleration - ramp.

2.2 Representation of the load

Representation of the driven mechanical load, which is shown in Fig. 3, is anotherimportant feature of the designed control system.

This way only the mass of motor moving part,Mm is included in the forwardpath as a rigid body, movingwithout external force and friction[6]. Both, externalforce and friction are modelled in the feedback path. The justification for this isthat the not completely known friction forces can be for someapplications domi-nated, [7].

It is also important to note that it is unnecessary to providean accurate model ofthe inverse load dynamics. Since the error in the value of moving mass,Mm can berepresented as a dynamic load force,Medv/dt, and this together with any externalforce acts at the same point asFext, then the estimate,Fext from the observer willinclude both these forces and so both of them will be compensated.

188 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

Net

accelerating

force, Fa

motor

velocity,

vmp

Electrical

force, Fel

Driven dynamics

pMm

1

External

force Fext

pMe

p

1motor

position,

smp

Fig. 3. Representation of the driven mechanical load.

3 State Estimation And Filtering

The speed of moving part and external force, which are inputsfor master controlalgorithm are produced by the following set of observers. The first observer, whichis based on primary part current equation, works in pseudo-sliding-mode, [8] andgenerates an unfiltered estimate of translation speed. Thisobserver is completedwith filtering one, which provides filtered speed estimate together with estimate ofexternal force.

3.1 Pseudo-sliding-mode observer

The pseudo-sliding-mode observer for estimation of translation speed is shown inFig. 4a. Its real time model is based on a primary part current, iq eq., (3b), in whicharepurposely used only the terms without vmp. Term containingvmp are in (11)replaced in the model by the correction input, vq. Thus the current component,i∗qmodel is:

di∗qdt

= −Rs

Lqi∗q +

1Lq

uq−vq (11)

wherei∗q is estimate ofiq as in a conventional observer. In sliding-mode the estimateof vq is produced as a fast switching ofUmax, (12a). However, the useful observeroutput is the continuous equivalent value of the rapidly switchingvq.

Equation (12a) therefore cannot directly generateveqq. Instead, apseudo-sliding-modeobserver is formed, [8] by replacing signum function with high gain and sat-uration limits (12b):

vq = −Umaxsgn(iq− i∗q)

vqeq = −Ksm(iq− i∗q)(12)

where the gain,Ksm, is made as high as possible [9] within the stability limit. ForlargeKsm, the error between real and fictitious observer current is driven almost to

Forced Dynamics Control of an Actuator With Linear PMSM 189

zero, resulting in:

vq =−pvmp

rLqΨPM (13)

and this may be manipulated to yield an unfiltered velocity estimate,v∗mp. Thus:

v∗mp =−rLqvqeq

pΨPM. (14)

3.2 Filtering observer

Chattering of the pseudo-sliding mode observer due to its high gain can be limitedby ‘filtering’ function of the following observer which provides the velocity esti-mate and external force estimate [10]. Real time model of this observer is basedon the motor force equation (2), which is augmented by the second state equation,dFext/dt = 0, with assumption of the piecewise constant external force, Fext. Theobserver correction loops are actuated by the error betweenthe unfiltered speedestimate of pseudo-sliding-mode observer,v∗mp as input and the filtered translationspeed estimate, ˆvmp, as observer output. The observer is shown in Fig. 4b andcorresponding real time model is:

ev =v∗mp− vmp, (15)

dvmp

dt=

1M

[c(Ψdiq−Ψqid)− Fext]+kvev, (16)

dFext

dt=0+kFve. (17)

iq

-

rLq

pvmp

YPM

+Umax

-Umax

Ksm

Lq

1

Rs

p

1 -

Lq

1

Rs

--

rLq

pvmp

YPMvq=

uq

i*q

uq

p

1

Motor model Observer model

-

-

v*mp

kF

-Fext

p

1

M

1

p

1

kv

vmp

qPMiM

cY

a) b)

Fig. 4. Block diagrams of a) pseudo-sliding mode observer and b) filtering observer.

This is a second order linear observer with a correction loopcharacteristic poly-nomial, which may be chosen via the gains,kv andkF to yield the desired balanceof filtering between the noise from the measurements ofid andiq and the noise from

190 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

the pseudo-sliding-mode velocity estimate. The transfer function of the observer,shown in Fig. 4b, is:

F(p) =vmp(p)

v∗mp(p=

pkv + kFM

p2 + pkv + kFM

. (18)

The observer gains may be chosen to yield a non-oscillatory state estimationerror transient with prescribed settling time,Ts. Based on settling time formula,Ts = 1.5(1+ n)/ω0, (where n is order of the system andω0 are multiple poles ofthe desired polynomial), the observer poles can be placed atω0 = 9/2Ts yieldingfollowing polynomial:

(p+9

2Ts)2 = p2 +

9Ts

p+81

4T2s

, (19)

which if compared with (18) yields observer gains:

kv =9Ts

(20a)

kF =81M4T2

s. (20b)

Although the external force is assumed constant in the formulation of the ob-server real time model, the estimate ofFex will follow a time varying external forceand will do so more faithfully asTs is reduced, but at the expense of sensitivity toany noise contaminating the speed estimate.

4 Derived Control Algorithms Verification

Verifications of the designed speed control algorithms of LPMSM are carried outin two steps. The simulations were followed by the experiments on the test benchin laboratory shown in Fig. 5. Parameters of the LPMSM used insimulations andfor control are listed in Appendix. A sampling frequency of 12.5 kHz achievedduring experiments was used also for the power electronics switches of inverter insimulation.

Two simulations shown on Fig. 6 and Fig. 7 illustrate the designed controlsystem response to a step speed demand ofvdem= 1 ms−1 applied for time intervalt ∈ (0.51.0]s, which is followed by another step speed demand ofvdem= −1 ms−1

applied for time intervalt ∈ [1.01.5) s as shown in subplot a). Prescribed settlingtime or time of ramp isTr = Ts = 0.2s . External force due to significant frictionin the loading mechanism was modelled as a sum of friction force,Ff r = 50sign(v)

Forced Dynamics Control of an Actuator With Linear PMSM 191

Fig. 5. Experimental setup for LPMSM speed control.

and external force, which is applied att = 0.8s, and drops to zero att = 1s . Nega-tive external force is applied again att = 1.3s when speed is also negative. Subplot(b) shows primary part both current components for entire measured interval.

Simulations of filtering observer operation show estimatedexternal forceFext asa sum of friction force and applied external force for whole time interval,t ∈ (0.41.5)s are shown in subplot c).

0.4 0.6 0.8 1 1.2 1.4 1.6-1.5

-1

-0.5

0

0.5

1

1.5

vd , vmp [ms-1

]

vmp

vd

t [s]

0.4 0.6 0.8 1 1.2 1.4 1.6-10

-5

0

5

10

15

id , iq [A]

id

iq

t [s]

0.4 0.6 0.8 1 1.2 1.4 1.6-150

-100

-50

0

50

100

150

F^ext [N]

F^ext

t [s]

Fig. 6. Simulation results for the first order dynamics.

Simulation results for ramp speed demands are shown in Fig. 8and arrangedin the same pattern as for the first order dynamics. To eliminate control chatteringin steady state the original algorithm, (10) with signum function, was forvmp >0.95vdem replaced with the first order dynamics algorithm, (7) in which short timeconstant was selected.

During simulations and experiments the high gain of pseudo-sliding mode ob-server was set toKSM = 5e3 and settling time of the filtering observer was chosenasTso = 20ms. From the time function of estimated translation speedand externalforce it is possible to observe correct function of both observers including settlingtime of filtering observer. As can be seen from both simulations presented the pre-scribed settling time,Ts = 3Tv and ramp time,Tr including prescribed dynamics ofthe speed response were reached and good agreement with theoretical predictionswas achieved.

Experimental results for control of LPMSM with the first order dynamics and

192 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

0.4 0.6 0.8 1 1.2 1.4 1.6-1.5

-1

-0.5

0

0.5

1

1.5

vd , vmp [ms-1

]

t [s]

vmp

vd

0.4 0.6 0.8 1 1.2 1.4 1.6-8

-6

-4

-2

0

2

4

6

8id , iq [A]

id

iq

t [s]

0.4 0.6 0.8 1 1.2 1.4 1.6-150

-100

-50

0

50

100

150

F^ext [N]

F^ext

t [s]

Fig. 7. Simulation results for the ramp speed demand.

ramp speed demand are shown in Fig. 8 and Fig. 9, respectivelyand arranged intothree subplots which correspond to simulations of Fig. 6.

As can be seen from Fig. 8 the prescribed first order dynamic was achieved(95%of the demanded speed is achieved at prescribed settling time). If comparedwith simulation differences can be seen mainly in the external force estimates due tosudden changes of friction sign at the moment of speed reverse. This has reflectionin iq current component demand of sub-plot b), which arise significantly and realmotor current is not capable to follow such high demand.

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

vd , vmp [ms-1

]

vmp

vd

t [s]

0 0.5 1 1.5 2 2.5 3 3.5 4-20

-15

-10

-5

0

5

10

15

20

id d , id , iq d , iq [A]

t [s]

iq d , iq

id d , id

0 0.5 1 1.5 2 2.5 3 3.5 4-400

-300

-200

-100

0

100

200

300

400

F^ext [N]

F^ext

t [s]

Fig. 8. Experimental results for the first order dynamics.

Proper work of both observers (pseudo-sliding mode and filtering) demon-strates subplot c) showing the estimate of significant friction in the loading mech-anism, which consists of rotational loading motor and the set of pulleys and wire.Further external force was not applied during experiments and therefore presentedexperimental results are marked as ‘preliminary’.

Possibility to control LPMSM with the first order dynamics isvery importantfrom the position control point of view, which will follow asthe next step, becauseit simplifies design of the position control system. Thanks to this fact the overallspeed control system can be replaced with the first order delay as defined by (9),then completed for the kinematic integrator and whole position control system canbe designed as the second order system.

Forced Dynamics Control of an Actuator With Linear PMSM 193

Fig. 9 shows the measured results for ramp speed demand. As can be seen fromthis figure the prescribed ramp response was achieved including prescribed ramptime.

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

vd , vmp [ms-1

]

vd

vmp

t [s]

0 0.5 1 1.5 2 2.5 3 3.5 4-20

-15

-10

-5

0

5

10

15

20

id d, id, iq d, iq [A]

t [s]

iq d, iq

id d, id

0 0.5 1 1.5 2 2.5 3 3.5 4-250

-200

-150

-100

-50

0

50

100

150

200

250

F^ext [N]

F^ext

t [s]

Fig. 9. Experimental results for ramp speed demand.

To avoid control chattering of the ramp speed response in steady state due tosignum function in control law, this control law was replaced for speed vmp¿0,95vdem, with the first order dynamics law (14), in which short time constant, Tv wasprescribed (the same way as for simulations).

Preliminary experimental results of the proposed FDC method for an electricactuator with LPMSM also confirmed good agreement with the theoretical predic-tions made during control algorithms development. The actuator follows prescribeddynamics fairly closely during transients and in steady-state.

5 Conclusions and Recommendatios for Further Work

Presented preliminary experimental results indicate thatthe designed control sys-tem based on FDC for the actuator with LPMSM operates properly. It can beobserved from Fig. 8 for the first order dynamics and from Fig.9 for ramp speeddemand, respectively. Based on these results can be concluded that prescribed dy-namics were achieved as it was intended.

The designed pseudo-sliding mode observer and filtering observer, which pro-vide necessary estimates of moving part translation speed and external load forceincluding friction operate properly, have non-oscillatory character and estimate thestate variables necessary for the derived control algorithms.

The obtained preliminary experimental results of speed control of LPMSM areencouraging and sufficiently promising to warrant further investigations of LPMSMposition control with or without position sensor. The experimental verification ofthe position control system with LPMSM actuator based on thefirst order speeddynamic is sought as a continuation of this work.

194 J. Vittek, V. Vavrus, J. Buday, and J. Kuchta:

Appendix

LPMSM parameters:PN = 3.2 kW, number of pole-pairsP = 4, r = 0.105m,Rs =9.52Ω, Ld = 0.8H, Lq = 0.8H andΨPM = 0.467Vs. DC bus voltageUDC = 566V,nominal currentIN = 4.4A, total load massM = 57.5kg and nominal forceFN =200N.

Acknowledgment

The authors wish to thank the Slovak Grant Agency - VEGA and APVV for fundingthe project No.4087/07 ‘Servosystems with Rotational and Linear Motors withoutPosition Sensor’ and Project No. APVV 99 031 205.

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Forced Dynamics Control of an Actuator With Linear PMSM 195

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