Ján Vittek, Vladimir Vavrúš, Peter Briš, Lukáš Gorel

Forced Dynamics Control of the Elastic Joint Drive with SingleRotor Position Sensor

DOIUDKIFAC

10.7305/automatika.54-3.160621.532.8.03.015.4:621.313.13(621.828)2.1.4; 5.5.4

Original scientific paper

Position control system of moderate precision based on ‘forced dynamics control’ for the drive with significantvibration modes is described. To exploit the only position sensor on the motor side, all necessary control variablesare estimated in observers based on motor position and stator current measurements. The designed controller is ofa cascade structure, comprising an inner speed control loop, which respects vector control principles and an outerposition control loop, which is designed to control load angle with prescribed dynamics in the presence of flexiblemodes. Simulations of the overall control system indicate that the proposed control system exhibits the desiredrobustness and therefore warrants further development and experimental investigation.

Key words: Forced Dynamics Control, Observers, Position Control, Sensorless Control

Prisilno upravljanje dinamikom pogona elasticnog zgloba s jednim senzorom pozicije rotora.U raduje opisano upravljanje sustavom pozicioniranja srednje preciznosti s izraženim vibracijskim modovima korišten-jem metodologije prisilnog upravljanja dinamikom. Kako bi se iskoristio senzor pozicije na strani motora, svepotrebne varijable stanja estimiraju se na temelju mjerenja pozicije motora i statorskih struja. Projektirani regulatorje kaskadne strukture, s unutarnjom petljom po brzini vrtnje koja se temelji na principima vektorskog upravljanja,i vanjskom petljom po poziciji za upravljanje kutom tereta s definiranom dinamikom u prisustvu slabo prigušenihmodova. Simulacijski rezultati cjelokupnog sustava upravljanja potvr�uju da predloženi sustav upravljanja posje-duje željenu robusnost i time opravdava buduci razvoj i eksperimentalna istraživanja.

Klju cne rijeci: prisilno upravljanje dinamikom, obzerveri, upravljanje pozicijom, bezsenzorsko upravljanje

1 INTRODUCTION

To reduce the number of sensors for position control ofthe drive with flexible coupling, a control system based on‘Forced Dynamics Control’ (FDC) with measurement ofrotor position and stator current torque component is de-veloped. An overall control system is therefore completedwith observation of all necessary control variables. Posi-tion control algorithm of the drive with torsion vibrationsis developed in two steps.

Firstly, an inner speed control loop is formed for thePMSM rotor using feedback linearisation principles, [1].This control algorithm is formulated in the rotor fixed d_qframe, respecting mutual orthogonality of the stator currentvector and rotor magnetic flux vector, to achieve maximumtorque under vector control [2-3]. Assuming a known loadtorque, FDC forces the speed control system to respondwith a prescribed linear first order dynamic, which has aspecified time constant,Tω. This prescribed behavior ofthe speed control loop then enables to replace it with a first

order delay which substantially simplifies the design of theouter position control loop. Presence of the load torque es-timate in the speed control algorithm automatically coun-teracts the load torque on the motor shaft by producing anearly equal and opposite control torque component. Itsestimate is provided by the motor torque observer. Thisgives FDC a certain degree of robustness not only with re-spect to external disturbances, but also with respect to plantparameter variations, since such variations are equivalentto load torques applied to the unperturbed plant model.

The second step is the design of the position controlloop, which is also based on FDC and therefore complieswith the prescribed closed-loop dynamics for load anglecontrol, in spite of the presence of vibration modes andexternal load torque [4-5]. This approach achieves non-oscillatory position control with a settling time,Tss, whichcan be described as a function of the natural frequency oftwo mass system.

Linearisation of the speed control algorithm and the de-sign of the FDC based position control algorithm, operat-

Online ISSN 1848-3380, Print ISSN 0005-1144ATKAFF 54(3), 337–347(2013)

AUTOMATIKA 54(2013) 3, 337–347 337

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

ing with the motor position sensor only, require estimationof state variables including load torques. To achieve thesetasks three observers complete the overall control system.State dependence of the flexible load variables is exploitedto design their observer as a state observer. The second ob-server, which produces the first and second derivatives ofthe load torque is Luenberger’s type and the reason for itsseparation from the state observer is to decrease order ofthe first one. The third observer is based on similar princi-ples for estimation of load torque acting on the motor shaftfor correct operation of FDC speed control loop.

Systematic analysis of speed control of the drive withflexible coupling with PI and PID regulators designed bypole-placement method is described in [6-7]. Analysis re-sults in an important observation that the different pole-assignment patterns are necessary for the different inertiaratios between load and motor. If for an over damped sys-tem with a higher moment of load inertia PI regulator withspecified damping coefficients satisfies non-oscillatory op-eration, then for highly under damped systems with loadinertia lower than inertia of motor PID regulator is requiredto improve the control system performance.

An optimal speed controller design for a two inertiasystem stabilization based on ‘The Integral of Time multi-ply by Absolute Error’ criterion described in [8] results inanalysis of four different controllers. Improvement of con-troller tracking performance was achieved with pseudo-derivative feedback and for better disturbance rejection afeed-forward controller is proposed.

A servo-system, which exploits the model of two-inertia system and Kalman filter based observer to predictone step ahead all state variables as well as disturbances,is described in [9]. Capability of the proposed algorithm toestimate and control the speed and position of the drive isverified by simulation with good results.

Possibility to exploit FDC for control of the drive withflexible coupling was already verified in [10-11]. Simu-lation results including preliminary experimental verifica-tion, confirm the effectives of the proposed control method.These works have also confirmed that FDC is capable tocontrol vibration modes while offering precisely defineddynamic response to the reference position.

FDC based control system designed in [12] requirestwo position sensors for rotor and load position measure-ments as the inputs of FDC position control law. Prelimi-nary experiments confirmed the possibility to control rotorand load angle with prescribed dynamics.

Further study based on simulations, verifies the abilityto control the drive with flexible coupling with rotor po-sition and current torque component measurements only.The control algorithm fed by observed state variables in-cluding torque on the motor shaft and load torque oper-

ates in agreement with theoretical assumptions made dur-ing its development. Results of simulations confirmed thatthe designed control system is capable to eliminate the in-fluence of flexible coupling while controlling the load po-sition with moderate precision.

2 CONTROL LAW DEVELOPMENT

Development of the position control system is made intwo steps. At first the speed control system is linearised toachieve the first order response to the step angular speeddemand [13]. Secondly, FDC position control system withprescribed closed-loop dynamics is designed. If comparedwith previous works the designed observers require mea-surements of rotor position and torque component of statorcurrent. This way elimination of position sensor on theside of load was achieved.

2.1 Description of PMSM and Flexible Load

As a driving motor PMSM is supposed and its descrip-tion in the synchronously rotating d_q co-ordinate systemfixed to the rotor is as follows:

dθRdt

= ωR, (1)

dωR

dt=

1

JR[c (Ψdiq +Ψqid)− ΓLs] , (2)

diddt

=−Rs

Ldid + pωR

Lq

Ldiq +

1

Ldud, (3)

diqdt

=−Rs

Lqiq − pωR

Ld

Lqid −

pωR

LqΨPM +

1

Lquq, (4)

whereid, iq andud, uq are, respectively, the stator currentand voltage components,Ld, Lq are, respectively, the in-ductances in direct and quadrature axisRs is resistance ofstator phase,θR andωR are the rotor position and angularvelocity respectively andΓLs is the external motor torque,p is the number of pole pairs andc = 3p/2 andJR is therotor moment of inertia.

Flexible coupling between motor and load for its de-scription is shown in Figure 1.

Electro-magnetic

control torque ΓΓΓΓel

Inertial datum

Shaft spring

constant Ks

External load torque, Γ Γ Γ ΓLe , acting

on load mass Rotor moment of

inertia JR

Load mass

moment of

inertia JL

Shaft spring torque

External load torque,

ΓΓΓΓLR , acting on

( )LRSLs K θ−θ=Γ

θL

θR

Fig. 1. Flexible coupling between motor and load

338 AUTOMATIKA 54(2013) 3, 337–347

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

The driven mechanism is a balanced mass with mo-ment of inertiaJL, coupled to the motor shaft via a torsionspring representing the flexible coupling with spring con-stant,KS . The electrical torque developed by the motor isΓel. ΓLe andΓLs are load torques externally applied, re-spectively, to the load mass and the rotor. The torque,ΓLs

is direct proportional to the position displacement. Blockdiagram for flexible coupling representation is shown inFigure 2.

s

1

s

1

sJ

1

R

LLθJ &&

RRθJ &&

Lθ&

Lθ

RθRθ&

( )LRsLs θθKΓ −=

sJ

1

L

Fig. 2. Block diagram of flexible coupling

Mathematical description of flexible coupling betweenrotor and load is as follows:

θR = ωR, (5)

θR =1

JR[Γel − ΓLs] , whereΓLs = Ks (θR − θL) (6)

θL = ωL, (7)

θL =1

JL[ΓLs − ΓLe] . (8)

Using Mason’s formula the transfer function betweenthe electrical torque and the rotor angle can be derived di-rectly from Figure 2 as:

F (s) =θR (s)

Γel (s)=

s2 + Ks

JL

s2JR

(s2 + Ks

JR+ Ks

JL

)

=1

s2JR

s2 + v2ns2 + ω2

n

(9)

where:

vn =

√Ks

JLandωn =

√Ks

JR+

Ks

JL= vn

√1 +

JLJR

(10)Here, the ‘encastre natural frequency’, vn, is the frequencyof the oscillations of the spring and load with the rotorheld inertially fixed and withΓLe = 0. The ‘free natu-ral frequency’,ωn , is the frequency of the oscillations ofthe combined rotor, spring and load withΓLe = 0 andΓel = 0.

2.2 FDC of Motor Speed

FDC law for rotor speed is based on the feedback lin-earisation that yields the first order linear dynamics, whereTω is the prescribed time constant andθRdem is the de-manded rotor speed

θR =1

Tω

(θRdem − θR

). (11)

Linearisation is achieved through comparison of equa-tion for prescribed dynamics, (11) with equation for rotorspeed, (2). Settingid = 0 up to nominal speed for vectorcontrol of the PMSM, [2] and equating the RHS of (2) and(11) yields the following FDC law for speed control loop:

id dem = 0,

iq dem = 1cΨPM

[Jr

Tω

(θR dem − θR

)+ ΓLs

] (12)

henceid = id dem and iq = iq dem are regarded as thecontrol variables. A current controlled inverter is used tovary the stator voltage components,ud anduq, in such away that stator componentsid andiq follow their respec-tive demands,id dem andiq dem , with nearly zero dynamiclag.

Since the motor load torque,ΓLs appears on the righthand side of the demanded current,iq dem, (12) it is nec-essary to design an observer for estimation of the net loadtorque on the shaft of the motor (see section 3.3). Derivedstator current demands (12) are used for FDC of PMSMrotor speed with the first order dynamics and prescribedsettling time,Tω as it requires prescribed linearising func-tion (11).

2.3 Design of Load Position FDC

To design the prescribed response for a control sys-tem with n closed loop poles having equal real parts,−1/Tc = ω0, and with a specified settling time,Ts, thenthe Dodds settling time formula, [14] applies. If zero over-shoot of the step response is required, then:

y (s)

ydem (s)=

[1

1 + sTs/1.5 (1 + n)

]n(13)

wherey(s) is the controlled output andydem(s) is the ref-erence input. This formula is valid for the design of FDCspeed control algorithms as well as for the design of loadangle control algorithm of the drive with flexible coupling.

Plant for load position control is formed by the firstorder transfer function block representing FDC of rotorspeed completed with kinematics integrator and integratingthese blocks with the model of load. The resulting plant isshown in Figure 3. Load torque acting on the rotor shaft

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Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

s

1sJ

1

Ls

1ω

sT11

+Rθ&

RθLθ

LLθJ &&Lθ&

LθdemRθ&

Fig. 3. Block diagram for load angle control

does not appear because it has been cancelled in the speedFDC loop.

Successive differentiation of (8) combined with (6)gives:

...θL =

1

JL

[Ks

(θR − θL

)− ΓLe

], (14)

θIVL =1

JL

[Ks

(θR − θL

)− ΓLe

]. (15)

Substituting in (14) forθR from (11) and forθL from(8) yields for load angle control following 4th order equa-tion:

θIVL =Ks

JLTω

(θRdem − θR

)− K2

s

J2L

(θR − θL)

+Ks

J2L

ΓLe −1

JLΓLe.

(16)

Using (13) forn = 4, the following 4th order systemwill yield a specified settling time,Tsθ :

θL (s)

θL dem (s)=

(15

2Tsθ

)4

A(s),

A(s) = s4+4

(15

2Tsθ

)s3 + 6

(15

2Tsθ

)2

s2

+4

(15

2Tsθ

)3

s+

(15

2Tsθ

)4

.

(17)

The comparison of the 4th derivative (16) of load posi-tion, θL, with prescribed position (17) results in:

Ks

JLTω

(θRdem − θR

)− K2

s

J2L

(θR − θL) +Ks

J2L

ΓLe

− 1

JLΓLe +

4

b

...θL +

6

b2θL +

4

b3θL =

1

b4(θLdem − θL) ,

(18)

where θR dem is treated as the control variable andb = 2Tsθ/15. By manipulation of (18) the following FDC

control law for load angle control is derived:

θRdem = Tω

{JL

b4Ks(θLdem − θL) +

1

TωθR

− 4

b

(θR − θL

)−

[6

b2− Ks

JL

](θR − θL)− 4JL

b3KsθL

−[

1

JL− 6

b2Ks

]ΓLe +

4

bKsΓLe +

1

KsΓLe

}.

(19)

This control law also satisfies ideal transfer function:

F (s) =θL id (s)

θL dem (s)=

1

b4s4 + 4b3s3 + 6b2s2 + 4bs+ 1.

(20)

The derived position control algorithm, (19) is in theform of a state feedback control law with the gains alreadydetermined as functions of the plant parameters and thedesired closed loop system parameters (17). But in thiscase, the first and second derivatives of the external loadtorque also appear on the right hand side. This is a generalfeature of FDC when external disturbances are includedinto the plant model used for the control system synthesis.The designed overall control system is shown in Figure 4.

3 OBSERVATION OF CONTROL VARIABLES

All the state variables for the designed control law aregenerated by a state observer, which needs rotor positionand stator current torque component as the inputs. Forthe estimation of external load torque derivatives an ob-server with filtering effect is designed. Load torque actingon the shaft of the motor is estimated in the motor torqueobserver.

3.1 Observer of State Variables

Due to state dependence of the deflection torque,ΓLs

the ‘state variablesobserver’ is based on a real time modelof the two-mass system, whereΓLe is an external loadtorque. In this case the external torque is treated as if itis constant provided that its change over a period equal tothe observer correction loop settling timeTsO is negligible.System state equations are:

sθL = ωL, (21)

sθR = ωR, (22)

sωL = −Ks

JLθL +

Ks

JLθR − 1

JLΓLe , (23)

sωR =Ks

JRθL − Ks

JRθR +

1

JRΓel , (24)

sΓLe = 0. (25)

340 AUTOMATIKA 54(2013) 3, 337–347

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

s

1sJ

1

L K s

s

1PMSM

ωs1

1+

R θ&R θ

L θΓL

L θ&L θFDC

speed

law

FDC

position

control

de

R θ&de

L θ

Motor

torque

observer

P

cΨ*

L Γ qi

Load

torque

observer

State

observer

qP

e

icΨΓ ⋅=

R θ

L

Γ

L θ Rθ

L ω RωL

Γ

L

Γ ˆ&

L

Γ &&

Fig. 4. Overall position control system block diagram

Rewritten in matrix form:

pLpRωL

ωR

ΓLe

=

0 0 1 0 00 0 0 1 0

−a1 a1 0 0 −a2a3 −a3 0 0 00 0 0 0 0

θLθRωL

ωR

ΓLe

+[0 0 0 a4 0

]TΓel,

(26)

where constants are defined,a1 = Ks/JL, a2 = 1/JL,a3 = Ks/JR anda4 = 1/JR.Observer’s equation in matrix form is:

ˆθLˆθRˆωL

ˆωR

ˆΓLe

=

0 0 1 0 00 0 0 1 0

−a1 a1 0 0 −a2a3 −a3 0 0 00 0 0 0 0

θLθRωL

ωR

ΓLe

+

000a40

Γel +

kθ1

kθ2

kω1

kω2

kΓ1

(θR − θR

).

(27)

Equation of the ‘dynamical error system’ is obtainedby subtracting observer equations (27) from its real time

model (26):

εθLεθRεωL

εωR

εΓLe

=

0 −kθ1 1 0 00 −kθ2 0 1 0

−a1 a1 − kω1 0 0 −a2a3 −a3 − kω2 0 0 00 kΓ1 0 0 0

εθLεθRεωL

εωR

εΓLe

(28)

To ensure convergence of the state estimates toward thereal states the gains of observer,kθ1 , kθ2 , kω1

, kω2and

kΓ1must be chosen in such a way that dynamical error

system satisfies condition fort → ∞ εi(t) → 0. Suchconvergence is guaranteed if the eigenvalues of the systemmatrix have negative real parts.

det

λ kθ1 −1 0 00 λ+ kθ2 0 −1 0a1 kω1 − a1 λ 0 a2−a3 kω2 + a3 0 λ 00 −kΓ1 0 0 λ

=

=λ5 + λ4kθ2 + λ3 (a3 + kω2)+λ2a3kθ1

+λ (a1kω2 + a3kω1)+a2a3kΓ1.

(29)

Under assumption of collocations of all five error sys-tem eigenvalues atλ = −ω0 (the observers settling timecan be determined by formula,(13), which forn = 5 re-sults inTsO = 9/ω0), the desired characteristic equationhas form:

(s+ ω0)5= s5+5ω0s

4+10ω20s

3+10ω30s

2+5ω40s+ω5

0 .(30)

Comparing the coefficients of the same degree in (29)

AUTOMATIKA 54(2013) 3, 337–347 341

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

and (30) yields the required values of observer gains:

kΓ1 =ω50/a2a3kθ2 = 5ω0kω2 = 10ω2

0 − a1 − a3kθ1

= (10ω30 − a1kθ2)/a1kω1 =

(5ω4

0 − a1kω2

)/a3.

(31)

Although the load torque is assumed constant in theformulation of the observer real time model, its estimateΓLe, will follow a time varying load torque and will do somore faithfully asω0 is enlarged with respect of the com-putational step. Block diagram of load torque observer 1 isshown in Figure 5.

s

1

s

1 Lθ

s

1

s

1

RJ

1

LJ

1

s

1

Rθ

Rθ

Lω

Rω

LeΓ

Rθ

Fig. 5. Block diagram of state variables observer

Correct function of observer was verified for estimationof uncontrolled flexible coupling state variables when con-stant torque,Γel = 2 Nm was applied att = 0.1 s whichwas followed by equivalent load torqueΓLe = 2 Nm att = 0.5 s. Settling time of the observer was chosen asTsO = 50 ms. Simulation results are shown in Figure 6.

0 0.2 0.4 0.6 0.8 1-20

0

20

40

60

80

100

ω^L

ω^R

eω=10(ω^L - ωL)

ω^R, ω^

L [rads-1]

t [s]

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

2.5

Γel, ΓLe, Γ^Le, [Nm]

Γ^Le

ΓLeΓel

t [s]

Fig. 6. Simulation results of state variables observer

3.2 Load Torque Derivatives Observer

To produce the first and second derivative of loadtorque required by the derived control law (19), a similarobserver to the previous one was designed. In this case thesecond derivative of external torque load is treated as if itisconstant provided that its change over a period equal to the

observer correction loop settling time Tso, is negligible.System state equations in matrix form are as follows:

sΓLe

sΓLe

sΓLe

=

0 1 00 0 10 0 0

ΓLe

ΓLe

ΓLe

. (32)

If the error between load torque as the output of statevariables observer and load torque estimated in the deriva-tives observer defined ase = ΓLe − ΓLe is added withproper gain into every observer correction loop, then thetorque derivatives observer equations are as follows:

sΓLe

sˆΓLe

sˆΓLe

=

0 1 00 0 10 0 0

ΓLe

ˆΓLe

ˆΓLe

+

k1k2k3

(ΓLe − ΓLe

).

(33)

Subtraction of (33) from (32) gives dynamical errorsystem, which has the form

ε1ε2ε3

=

−k1 1 0−k2 0 1−k3 0 0

ε1ε2ε3

. (34)

Convergence of dynamical error system fort → ∞εi(t) → 0 is guaranteed for the eigenvalues of the systemmatrix with negative real parts. Under assumption of allthree eigenvalues collocations atλ = −ω0 (using(13) forn = 3 results in observer’s settling timeTso = 6/ω0).Comparing the desired third order characteristic equationwith equation of system matrix eigenvalues results in thefollowing gains of observer correction loops:

s3 + 3ω0s2 + 3ω2

0s+ ω30 = λ3 + λ2k1 + λk2 + k3,

(35)

k3 = ω30 , k2 = 3ω2

0 , k1 = 3ω0. (36)

Load torque observer block diagram of is shown inFig. 7.

Figure 8 illustrates correct function of the observerwhen the numerically computed first and second deriva-tives of exponential load torque are compared with thederivatives gained from the load torque observer.

342 AUTOMATIKA 54(2013) 3, 337–347

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

s

1

LeΓ

s

1 LeΓ

LeΓ&LeΓ&&s

1

Fig. 7. Load torque observer

0 0.2 0.4 0.6 0.8 1-10

0

10

20

30

40

50

dΓLe/dt , dΓ^Le/dt , [Nms-1]

dΓ^Le/dt

t [s]

dΓLe/dt

-

(a) computed and estimated firstderivatives

0 0.2 0.4 0.6 0.8 1-1000

0

1000

2000

d2ΓLe/dt , dΓLe/dt [Nms

-2, Nms

-1]

3000

4000

t [s]

d2ΓLe/dt

d2Γ^Le/dt

(b) computed and estimated sec-ond derivatives

Fig. 8. Computed and estimated the first and secondderivative

3.3 Motor Load Torque Observer

Load torque on the motor shaft needed for FDC speedcontrol is estimated in ‘motor load observer’. Due to thefact that the form ofΓLs(t) is unknown, its differentialequations cannot be formed. Motor load torqueΓLs istherefore treated as state variable, which is constant pro-vided that its change over a period equal to the observercorrection loop settling timeTsu is negligible. Thus, theobserver real time model is based on (1) and (2) augmentedby a new state equation,− dΓLs/dt = 0.

dθ∗Rdt

= ω∗R + kθe

∗θ, (37)

dω∗R

dt=

1

JR[cΨPM iq − Γ∗

Ls] + kωe∗θ, (38)

−dΓ∗Ls

dt= 0 + kΓe

∗θ. (39)

Mathematical description of the observer is available in[13] therefore to save some space only the final compari-son of characteristic polynomial of the observer’s transferfunction with desired characteristic equation is given:

s3+kθs2+kωs+

kΓJR

= s3+s218

Tsu+s

108

T 2su

+216

T 3su

. (40)

Observer’s gainskθ, kω andkΓ for a specified correc-tion loop settling time, Tso are as follows:

kθ = 3ω0, kω = 3ω20 , kΓ = JRω

30 . (41)

So with sufficiently small settling time of observerTsu,the observer, which is shown in Figure 9, produces a netload torque estimate,Γ∗

LR(t) able to track real load torque,ΓLs(t) with very small and defined dynamic lag.

s

1*Rθ

Rθ

s

1

*Rω

*LsΓ

s

1-

RJ

1

PMR

Ψ2J

3p

qi

Fig. 9. Block diagram of motor load torque observer

Correct function of the motor load torque observer il-lustrates Figure 10. Subplot a) shows the real and esti-mated rotor speed including the difference between themmagnified 5x. Applied and estimated load torque on theshaft of the motor is shown in subplot b) including the er-ror between them. In spite of dynamic lag of the estimatedvariable, which can be seen in this subplot, the observerproduces a correct estimate of load torque on the shaft ofthe motor demanded by FDC of rotor speed.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-60

-40

-20

0

20

40

60

80

100

ωR , ω^R [rads

-1]

ωR

ω^R

eω=5(ωR - ω^R)

t [s]

(a) real and estimated rotor speedincl. error between them

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-10

-5

0

5

10

15

20

ΓLs , Γ^Ls [Nm]

Γ^Ls

ΓLs

e=(ΓLs - Γ^Ls)

t [s]

(b) real and estimated motortorque incl. error

Fig. 10. Real and estimated rotor speed and load torque

4 OVERALL CONTROL SYSTEM VERIFICA-TION

Simulations of the proposed FDC system followed byverification of the designed observers via experimentaldata collected for control of flexible coupling with two po-sition sensors described in [12] were carried out to verifyoverall performances of the proposed control of flexiblecoupling with single position sensor on motor side.

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Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

Simulation results of FDC of the load position are pre-sented in Figure 11 and Figure 12. Verification of the ob-servers’ operation from measured data is shown in Fig-ure 13.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2

0

2

4

6

8

10

12

eθ=10(θid - θL)

θL

θid

t [s]

θid , θL [rad]

(a) ideal and real load positionand difference magnified (x10)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2

0

2

4

6

8

10

12

θL , θR [rad]

t [s]

eθ=10(θL - θ^L)

θR

θL

(b) rotor and load position andload position error magn. (10x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-20

0

20

40

60

80

100

ωR, ωL [rads-1]

t [s]

ωR

eω=10(ωR - ω^

R )

ωL

(c) rotor and load speed and rotorspeed error magn. (10x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-20

0

20

40

60

80

100

ω^R

t [s]

ω^L

eωL=10(ω^L - ω L )

ω^

R, ω^L [rads-1]

(d) est. rotor and load speedand load speed error (10x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-200

0

200

400

600

800

1000

ωR dem , ωR [rads-1

]

ωR

ωR dem

t [s]

(e) demanded and real rotor angularspeed

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-20

-10

0

10

20

30

40

50

60

id , iq [A]

id

iq

t [s]

(f) d_q components of stator cur-rent

Fig. 11. Simulation results for position control of the drivewith flexible coupling

The computational step of simulations ish = 1·10−4s,which corresponds to the sampling frequency achievedduring a previous implementation of the algorithm for FDCof load position. All the simulations presented are carriedout with zero initial state variables. A step load angle de-mandθL dem = 10 rad with settling timeTsθ = 0.2 s wasapplied to investigate response of the FDC based controlsystem. An external load torque with exponential increasegiven asΓLe = 5(1 − e−t/0.05) is applied att = 0.5 s,being zero for the time intervalt < 0.5 s. The settlingtime of the state and load torque observer were chosen asTsO = Tso = 12.5 ms, while settling time of motor torqueobserver is set asTsu = 1.5 ms respectively.

Subplot (a) of Fig. 11 shows the ideal response and re-sponse of the control system to the step load position de-mand,θL dem = 10 rad. including magnified difference

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2

0

2

4

6

8

10

12

θid , θL., eθ [rad]

eθ =10(θid ,- θL)

θL.

θid

t [s]

(a) ideal and real load positionand difference magn. (x10)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

50

100

150

200

eω=10(ωR-ω^R)

ω^R

ω^L

ω^L , ω

^R , eω [rads

-1]

t [s]

(b) estimated rotor and loadspeed and error in rotor speed(10x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2

0

2

4

6

8

10

12

θid , θL., eθ [rad]

� [��

eθ �10�θid - θL.�

θL.

θid

(c) ideal and real load position anddifference magnified (x10)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-20

0

20

40

60

80

100

120

140

ω^L , ω

^R , eω [rads

-1]

eω= 10(ωL - ω^L)

ω^R

t [s]

ω^L

(d) est. rotor and load speed anderror in load speed (10x)

Fig. 12. Simulation results for position control of the drivewith various moment of inertia

between them (10x). The realisation of the prescribed po-sition dynamics is very accurate, which is clear from themagnified difference. Also the prescribed settling timeis precisely kept because the load position,θL(t) passesthrough9.5 radian at a time very close to0.2s. Subplotb) shows load position and rotor position together with theerror between real and estimated load position (magnified10x). From this subplot can be seen also that in the steadystate, the motor electrical torque is transmitted via the tor-sion spring to counteract the external load torque appliedto the load mass. This entails a constant torsional deflec-tion of the spring, which is evident from the constant dif-ference betweenθR and θL occurring just after the loadtorque achieves its steady state.

Subplot (c) shows the angular velocities of the rotorand load as functions of time together with the differencebetween real and estimated rotor speed (magnified 10x).This subplot illustrate that the acceleration period is fol-lowed by the deceleration one, as expected. Correct func-tion of ‘states observer’ is shown in subplot (d) where theestimated rotor and load speeds are shown including thedifference between real and estimated load speed (magni-fied 10x). It can be seen that the errors in estimates occurduring transients only.

Demanded rotor speed as an input to FDC speed al-gorithm together with real response of the motor speed isshown in subplot (e). The PMSMiq-current torque com-ponent is shown in subplot (f) while theid-current compo-nent is kept at zero value up to the nominal motor speedsatisfying condition for vector control of PMSM.

344 AUTOMATIKA 54(2013) 3, 337–347

Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

Figure 12 shows simulation results of robustness inves-tigation when two different moments of load inertia wereapplied (JL1 = 0.75 · 10−3 kgm2 andJL2 = 12 · 10−3

kgm2) while the moment of motor inertia (JR = 3 · 10−3

kgm2) was constant. The corresponding encastre naturalfrequency (v1 = 109.5 rads−1 and v2 = 27.4 rads−1)and free natural frequency (w1 = 122.5 rads−1 andw2 =61.2 rads−1) were determined using (10) for spring con-stantKs = 9 Nmrad−1. To achieve satisfactory results theprescribed settling time for the drive with higher free nat-ural frequency was set toTsθ = 0.1 s and for lower freenatural frequency was set toTsθ = 0.2 s.

Simulation results for both chosen moments of inertiaare compressed into two subplots. Subplots (a) and (b)show the ideal response and response of the control systemto the step load position demand,θL dem = 10 rad withprescribed settling time ofTsθ1 = 0.1 s andTsθ2 = 0.2 sincluding difference between them (magnified 10x). Po-sition response of the drive including prescribed settlingtime is accurate, which is clear from the magnified differ-ence between ideal and real position response. Subplots (c)and (d) show the estimated velocities of the rotor and loadas functions of time together with the difference betweenreal and estimated rotor speed (magnified 10x). These re-sults also confirm correct function of the state observer un-der various load conditions.

Correct operation of the designed observers was inves-tigated using data collected during control of flexible cou-pling with two position sensors. Results are shown in Fig-ure 13. Experiments were carried out for load angle de-mandθL dem = 2π rad with settling timeTsθ = 0.2 sand with exponential increase of external load torque andspring constantKs = 24 Nmrad−1.

Subplot (a) of Figure 13 shows measured rotor posi-tion for whole data collecting interval. Measured compo-nents of stator current are shown in subplot (b). Properfunction of state observer, which exploits both measureddata as the inputs, shows subplot (c) where measured andobserved load position are shown including error betweenthem (magnified 10x). Subplot (d) shows estimated rotorand load angular speed. Estimated load position togetherwith both estimated speeds serve as the inputs of FDC al-gorithm to control load position.

One of the output of state variables observer is esti-mated load torque, which is utilized in FDC of load po-sition, as well as creates input for load torque derivativesobserver. Estimation of the first and second derivative ofload torque shows subplot (e). This way proper function ofload torque derivates observer was confirmed.

Presented simulation results and preliminary investiga-tions of correct observers operation confirmed possibilityto control load angle with single position sensor on the

0 0.5 1 1.5 2-1

0

1

2

3

4

5

6

7θR [rad]

t [s]

θR

(a) measured rotor positionas an input of state-variablesobserver

0 0.5 1 1.5 2-20

-15

-10

-5

0

5

10

15

20

25

id, iq [A]

t [s]

iq

id

(b) measured d_q componentsof stator current as an inputs

t [s]

0 0.5 1 1.5 2-6

-4

-2

0

2

4

6

8

θL, θ^

L [rad] θL θ

^L

eθ=10(θL-θ^L)

-2

-1

1

2

3

4

5

6

7

(c) real and est. load position in-cluding difference magn. (10x)

0 0.5 1 1.5 2-20

-10

0

10

20

30

40

50

60

70

t [s]

ω^R, ω

^L [rads

-1]

ω^L

ω^R

(d) estimated rotor and load an-gular speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

5

ΓL, Γ^Le [Nm]

ΓLs

Γ^Le

t [s]

(e) applied and estimated exter-nal load torque

2

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3000

-2000

-1000

0

1000

2000

3000

4000

5000

dΓ^Le/dt, d2Γ^

Le/dt2 [Nms-1, Nms-2]

t [s]

d2Γ

^Le/dt

2

10×dΓ^Le/dt

(f) estimated the first (10x) andsecond derivative of load torque

Fig. 13. Comparison of measured and estimated controlvariables by the designed observers

motor side using FDC. An important observation based onsimulation results is that the mechanical oscillations arecompletely damped and the peak of transient error if com-pared with ideal transfer function doesn’t exceed 0.5 radallowing position control of the load with moderate accu-racy.

5 CONCLUSION

A position control system based on the principles of‘Forced Dynamics Control’ for electric drives with flexiblecouplings has been presented and verified by simulations.Simulation results confirmed that the proposed positioncontrol system can be made to follow the prescribed idealclosed-loop dynamics with moderate precision in spite ofthe presence of flexible modes and external torque. Sim-ulation results also confirm accomplishment of the vectorcontrol conditions by keeping direct axis current to smallproportion.

Implementation of three observers enables control ofload position with fair precision and to eliminate the loadposition sensor. Preliminary tests of observers based onpreviously measured data has shown their capability to

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Forced Dynamics Control of the Elastic Joint Drive with Single Rotor Position Sensor J. Vittek, V. Vavrúš P. Briš, L. Gorel

provide estimates of all state variables for control algo-rithm including rotor torsion torque,ΓLs, and external loadtorque,ΓLe respectively. The same is valid about estima-tion of load torque the first and second derivatives requiredby the position control algorithm.

Simulation results indicate that the designed positioncontrol system exhibits the desired robustness and there-fore warrants further development and experimental inves-tigation.

ACKNOWLEDGMENT

The authors wish to thank for support to Slovak GrantAgency VEGA and R&D operational program Centre ofexcellence of power electronics systems and materialsfor their components II. No. OPVaV-2009/2.1/02-SORO,ITMS 26220120046 funded by European regional devel-opment fund (ERDF).

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[1] A. Isidori, Nonlinear Control Systems, London,Springer-Verlag, UK, 1995.

[2] I. Boldea, S. A. Nasar,Vector Control of AC Drives.2nd ed., Boca Raton, FL., CRC Press, 1992.

[3] D. W. Novotnı, T. A. Lipo, Vector Control and Dy-namics of AC drives, New York, Oxford UniversityPress, 1996.

[4] J. Vittek, S. J. Dodds, R. Perryman, M. Rapšík,“Forced Dynamics Control of Electric Drives em-ploying PMSM with Flexible Coupling” AustralasianUniversities Power Engineering Conference AUPEC2007, Perth, Australia, dec. 2007, CD-Rom.

[5] J. Vittek, P. Briš, P. Makyš, M. Štulrajter, “Control ofFlexible Drive with Permanent Magnet SynchronousMotor employing Forced Dynamics”, Proceedings ofEPE-PEMC 2008 conf., Sept. 2008, Poznan, Poland,CD-Rom, pp. 2242-2249.

[6] G. Zhang, J. Furusho, “Speed control of two-inertiasystem by PI/PID control”, IEEE Trans. on IndustrialElectronics, vol. 47, no. 3, 2000, pp. 603-609.

[7] S. Thomsen, N. Hoffmann, F. W. Fuchs, “PI control,PI-based state space control, and model-based pre-dictive control for drive systems with elastically cou-pled loads-A comparative study”, IEEE Transactionson Industrial Electronics Vol. 58 (8) , 2011, pp. 3647-3657.

[8] B. U. Nam, H. S Kim., H. J. Lee, , D. H. Kim,“Optimal Speed Controller Design of the Two-InertiaStabilization System”,International Journal of Com-puter and Information Engineering2009, pp. 41-46.

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[10] S. J. Dodds, K. Szabat, “Forced Dynamic Control ofElectric Drives with Vibration Modes in the Mechan-ical Load” , in Proceedings of the International conf.EPE-PEMC‘06, Portorož, Slovenia, 2006, CD-Rom.

[11] K. Szabat, T. Orlowska-Kowalska, “Damping of theTorsional Vibration in Two-Mass Drive System Us-ing Forced Dynamic Control”, . The InternationalConference on Computer as a Tool, EUROCON2007, pp. 1712 – 1717.

[12] J. Vittek, P. Bris, P. Makys, M. Stulrajter, “ForcedDynamics Control of PMSM Drives with Torsion Os-cillations”, COMPEL -The International Journal forComputation and Mathematics in Electrical and Elec-tronic EngineeringVol. 29, No. 1, 2010, pp.187-204.

[13] J. Vittek, S. J. Dodds,Forced Dynamics Con-trol of Electric Drives, Zilina, SK, EDIS,http://www.kves.uniza.sk, (E-learning), 2003.

[14] S. J. Dodds, “Settling Time Formulae for the Designof Control Systems with Linear Closed Loop Dy-namics”, International conf. AC&T‘07 - Advances inComputing and Technology, University of East Lon-don, UK, 2007.

[15] W. Leonhard, “Control of Electrical Drives”, 2nd ed.,Springer, Berlin 2001.

[16] K. Bose, “Power Electronics and Variable FrequencyDrives – Technology and Applications”,Instituteof Electrical and Electronics Engineers, New York1997.

[17] K. Ohnishi, M. Morisawa, “Motion control taking en-vironmental information into account”, in Proceed-ings of the EPE PEMC’02 International conf., Cavtat,Croatia, 2002, CD-Rom.

[18] O. Aguilar, A. G. Loukianov, J. M. Canedo,“Observer-based Sliding Mode Control of Syn-chronous Motor:”, in Proceedings of the intern.IFAC’02 congress on Automatic Control, Guadala-jara, Mexico, CD-Rom, 2002.

[19] S. Brock, J. Deskur, K. Zawirski, “Robust Speed andPosition Control of PMSM using Modified SlidingMode Method”, in Proceedings of the Internationalconf. EPE-PEMC 2000, Kosice, Slovakia, vol. 6,pp. 6-29–6-34.

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[20] V. Comnat., M. Cernat., F. Moldoveanu, R. Ungar,“Variable Structure Control of Surface PermanentMagnet Synchronous Machine”, International conf.PCIM 1999 Nurnberg, Germany, pp. 351-356, CD-Rom.

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Ján Vittek received his Ph.D. degree from theTechnical University of Transport Zilina in 1979.After two years service in CZ Railroads he joinedFaculty of Electrical Engineering of the Univer-sity of Transport and Communications in Zilina.In 1997 he became a professor for ‘Electric Trac-tion and Electric Drives’. He is responsible forthe development of Electric Drives division in thedepartment of Power Electrical Systems. Duringperiod 1998-2008 he was several times appointedto the position of Visiting Professor of the School

of Computing and Technology, University of East London, UK asthe re-flection of long-term research co-operation. His research interests arecontrol of electric drives, electronic energy conversion,electric traction(including locomotive drives and traction power supply) andapplied ro-bust control.

Vladimir Vavrúš received the MSc. degree inElectrical and Electronic Engineering from Uni-versity of Zilina in 2005 and thereafter he beganPhD study at the same place. He received hisPhD degree in 2009 and he is currently workingas a researcher at the Faculty of Electrical Engi-neering, Zilina University. His research interestscover control of electric drives with special in-terest in strategies for control of linear motors.He cooperates on the development of electronichardware for different industrial applications us-

ing digital signal processors and power PC as well.

Peter Briš received the MSc. degree in Electricaland Electronic Engineering from University ofZilina in 2007 and thereafter he began PhD studyat the same place. He received his PhD degree in2011 and he is currently working as a researcherat Railway Repairs and Machineries, Vrútky. Hisresearch interests cover control of electric driveswith special interest in energy saving control al-gorithms and robust control methods for driveswith flexible coupling.

Lukáš Gorel received his master degree in powerelectrical engineering from faculty of electricalengineering, University of Zilina in 2011. Cur-rently he is continuing his study as a doctoral stu-dent in the field of ‘electric drives’ at the samefaculty. His research interests are control of elec-tric drives, position sensors sensitivity and con-trol of electric drives with flexible coupling.

AUTHORS’ ADDRESSESJán Vittek, Prof.Vladimir Vavrúš, Ph.D.Peter Briš, Ph.D.Lukáš Gorel, M.Sc.Department of Power Electrical Systems,Faculty of Electrical Engineering,University of Zilina,Univezitna 1, Velky diel, 010 26 Zilina, SlovakRepublice-mail: jan.vittek@fel.uniza.skvladimir.vavrus@kves.uniza.skpeter.bris@gmail.comlukas.gorel@kves.uniza.sk

Received: 2012-01-09Accepted: 2012-11-21

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