FORCE RIPPLE COMPENSATOR FOR A VECTOR

CONTROLLED PM LINEAR SYNCHRONOUS MOTOR

Markus Hirvonen and Heikki Handroos

Institute of Mechatronics and Virtual Engineering, Lappeenranta University of Technology, Lappeenranta, Finland

Olli Pyrhönen

Department of Electrical Engineering, Lappeenranta University of Technology, Lappeenranta, Finland

Keywords: Cogging, Disturbance Observation, Linear Motor, Velocity Control, Vibration Suppression.

Abstract: A dynamic model including non-idealities for a permanent magnet linear synchronous motor (PMLSM) is

postulated and verified. The non-idealities acting on the physical linear motor are measured and analyzed.

These experimental results are utilized in the model. The verified simulation model is used in developing a

force disturbance compensator for the velocity controller of the motor. The force non-idealities, such as the

cogging force, friction and load force variation, are estimated using a disturbance observer. The acceleration

signal in the observer is derived through the use of a low-acceleration estimator. The significant effects of

the disturbance compensator on the simulated and measured dynamics of the motor are shown.

1 INTRODUCTION

The linear motor is an old invention but it is only

recently that, as a result of the development of

permanent magnets and their decreased costs,

permanent magnet linear motors have become a

viable alternative to rotating motors fitted with linear

transmissions. In machine automation, linear

movement has traditionally been transmitted from a

rotary actuator by means of a ball screw, rack and

pinion or belt. The linear motor simplifies the

mechanical structure, eliminating the contact-type

nonlinearities caused by backlash, friction, and

compliance. In addition, the main benefits of a linear

motor include its high-power density, reliability and

efficiency.

Nowadays, the controllers commercially

available, mainly PID algorithms with fixed gains,

are unable to compensate for the undesirable

phenomena that reduce the precision of motion such

as backlash, static friction, load variations etc. Large

controller gains are needed in order to maintain the

stiff control required when suppressing load

disturbances that tend to reduce the stability of a

system. Therefore, extended methods for the

compensation of disturbance have become an

important topic of research. By compensating for an

unknown time-varying force based on the estimation

of such a force, faster speed responses and smaller

speed ripples can be achieved.

In disturbance compensation, the compensation

technique itself is a very simple feed-forward

control, but the difference arises from the different

disturbance estimation algorithms. In (Castillo-

Castaneda et al., 2001), the friction compensation

has been studied using model-based estimation. One

disadvantage of this technique is that it is suitable

for tracking only, since the desired velocity must be

known in advance. Kim et al. (2002) and Tan et al.

(2002) have studied sliding mode estimators in

compensation feedback, while Godler et al. (1999),

Deur et al. (2000), Bassi et al. (1999) and Hong et

al. (1998) have studied the disturbance observer of a

more general algorithm. Godler et al. (1999)

compared load disturbance compensation with an

acceleration control loop inside a speed loop. They

have found that control implemented using an

acceleration control loop can better tolerate

parameter variation as well as disturbance in

comparison to robust control with a disturbance

observer. On the other hand, Deur et al. (2000)

suggested the use of a disturbance observer in

industrial applications due to its simple

177

Hirvonen M., Handroos H. and Pyrhönen O. (2004).

FORCE RIPPLE COMPENSATOR FOR A VECTOR CONTROLLED PM LINEAR SYNCHRONOUS MOTOR.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 177-184

DOI: 10.5220/0001128001770184

 SciTePress

implementation as well as simple design which has

no robustness constraints.

The motor model discussed in the above-

mentioned papers is a simplified model. The

development of computers and software has made it

feasible to simulate the more detailed dynamical

behavior of machine systems. This paper discusses

the use of a more detailed non-linear dynamic model

for the analysis of a linear transmission system. The

motor studied in this paper is a commercial three-

phase linear synchronous motor application. The

moving part (the mover) consists of a slotted

armature and a three phase windings, while the

surface permanent magnets (the SPMs) are mounted

along the whole length of the path (the stator).

Figure 1: The structure of the studied linear motor.

First of all, the equations for modelling the

vector-controlled motor drive and non-idealities are

derived. Then, the simulation model is verified with

the measurements from physical linear motor

applications, and the comparison of the responses is

shown in the study. Finally, a disturbance observer

based on (Hong et al., 1998), (Godler et al., 1999),

(Deur et al., 2000), and (Bassi et al., 1999) is

implemented in the physical motor system after

being tested in the simulation model, and the results

and conclusions are presented.

2 SIMULATION MODEL

2.1 Model of LSM

The modeling of the dynamics of the linear

synchronous motor examined in this paper is based

on the space-vector theory. The time-varying

parameters are eliminated and all the variables

expressed on orthogonal or mutually decoupled

direct and quadrature axes, which move at a

synchronous speed of

. The d- and q-axes

equivalent to the circuit of the PMLSM are shown in

figure 2, and the corresponding equations are (1) and

(2), respectively.

The voltage equations for the synchronous

machines are

uRi

dad q

=+−, (1)

uRi

qaq sd

=++ , (2)

where u

and u

are the d- and q-axis components of

the terminal voltage, i

and i

the d- and q-axis

components of the armature current, R is the

armature winding resistance and

are the d-

and q-axis flux linkage components of the armature

windings. The synchronous speed can be expressed

=πv

, where v

is the linear synchronous

velocity and

the pole pitch. Although the physical

system does not contain a damper, which in PMLSM

usually takes the form of an aluminum cover on the

PMs, virtual damping must be included in the

model, due to eddy currents. The voltage equations

of the short-circuited damper winding are

=+, (3)

=+, (4)

where R

and R

are the d- and q-axis components

of the damper winding resistance and i

and i

the d-

and q-axis components of the damper winding

current. The armature and damper winding flux

linkages in the above equations are

Figure 2: Two axial models of the linear

nchronous motor.

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178

Li L i

dadadmdDpm

ψψ

=++, (5)

Li L i

qaqaqmqQ

=+, (6)

Li Li

DmdadDDpm

ψψ

=++, (7)

Li Li

QmqaqQQ

=+, (8)

where

and L

are the d- and q-axis components

of the armature self-inductance,

and L

the d- and

q-axis components of the damper winding

inductance,

and L

the d- and q-axis

components of the magnetizing inductance and

is the flux linkage per phase of the permanent

magnet. By solving the flux linkage differential

equations from (1) to (4) and substituting the current

equations from (5) to (8) into these equations, the

equations for the simulation model of the linear

motor can be derived. The electromagnetic thrust of

a PMLSM is (Morimoto

et al., 1997)

()

Fii

dx d aq q ad

ψψ

== − , (9)

where

is the electrical power.

2.2 Non-idealities of PMLSM

The force ripple of the PMLSM is larger than that of

rotary motors because of the finite length of the

stator or mover and the wide slot opening. In the

PMLSM, the thrust ripple is caused mainly by the

detent force generated between the PMs and the

armature. This type of force can be divided into two

components, for tooth- and core-type detent force. In

the tooth detent force, the force component is

generated between the PMs and the primary teeth,

while the core-type detent force component is

generated between PMs and the primary core. The

wavelength of the core component is usually one

pole pitch, while that of the teeth component is one

slot pitch. The core-type detent force can be

efficiently reduced by optimizing the length of the

moving part or smooth-forming the edges of the

mover, and the tooth-type detent force can be

reduced by skewing the magnets and chamfering the

edges of the teeth (Jung

et al., 2002), (Hyun et al.,

1999), (Inoue

et al., 2000), (Zhu et al., 1997), (Hor

et al., 1998).

The detent force effect tends to move the mover

to a position in which the energy of the magnetic

circuit is at its minimum. This phenomenon attempts

to stall the mover at the stator pole positions and is

always present, even when no current is flowing

through the motor coils (Otten

et al., 1997). The

ripple of the detent force produces both vibrations

and noise and reduces controllability (Chun

et al.,

2000). The force ripple is dominant at low velocities

and accelerations. At higher velocities, the cogging

force is relatively small and the influence of

dynamic effects (acceleration and deceleration) is

more dominant (Otten

et al., 2000). In this study, the

detent force was measured in the reference system.

The force ripple can be described by sinusoidal

functions of the load position,

x, with a period of

and an amplitude of

, i.e.,

sin( ) sin( )

11 12 2

AxAA x

ripple r r r

ϕϕ

⎡

⎤

⎣

⎦

. (10)

In figure 3, the results of the simulation are

compared with those measured in the reference

system.

Figure 3: Comparison of measured and simulated

detent forces.

The reluctance force is another phenomenon that

occurs in linear synchronous motors. A lot of

research has been carried out into the reluctance in

linear induction machines, in which the phenomenon

depends on velocity. The reluctance force in

PMLSMs has been studied to a lesser extent. The

reluctance force is due to the variations in the self-

inductance of the windings with respect to the

relative position between the mover and the magnets

(Tan

et al., 2002). The reluctance force was

observed to be relatively small in the reference

system and, therefore, has not been included in the

model.

The model also takes into account the effect of

FORCE RIPPLE COMPENSATOR FOR A VECTOR CONTROLLED PM LINEAR SYNCHRONOUS MOTOR

179

friction. Friction is very important for control

engineering, for example, in the design of drive

systems, high-precision servomechanisms, robots,

pneumatic and hydraulic systems and anti-lock

brakes for cars. Friction is highly nonlinear and may

result in steady-state errors, limit cycles and poor

performance (Olsson

et al., 1998). Friction was

modeled using the simple gradient method in which

the linear motor system was set in a tilted position

and the moving part allowed to slide down freely.

Friction was measured at several tilt angles and the

results obtained were used to plot the friction

function as a function of speed. The friction model

took into account the Coulomb (static) and viscous

(dynamic) components

()

()FsignvF absvF

coulomb viscous

⎡⎤

⎣⎦

, (11)

where

v is the velocity of the motor. The friction

function was incorporated into the simulation model

in such a way that it acts between the stator and the

mover. In the simulation, the smoothing of the

friction function was used to obtain a numerically

efficient model in order to improve simulation rate.

The effect of load variation was also taken into

consideration. This was carried out using a forced

vibrating non-homogenous two-mass model, where

the load force variation

∆F

is calculated using a

spring-mass equation, which is the sum of the spring

force,

, and the damping force, F

. The equation of

motion for such a system is

()()

12 12 22

FF kxx cxx mx

lsd

∆= + = − + − =

&& &&

,(12)

where

k is the spring constant, c the damping

coefficient, and

x ,

and

are the displacement,

velocity and acceleration of masses

, respectively.

The total disturbance force equation can be

described using the equations of detent force,

friction force and load force variation; i.e., the

disturbance force,

dist

, is

FFF

dist ripple l

=++∆ (13)

This resultant disturbance force component is

added to the electromotive force to influence the

dynamical behavior of the linear motor system.

2.3 Current Controller of the Linear

Motor

The current control of the system is implemented in

the form of vector control. Vector control is based

on the space vector theory of electrical machines

and, therefore, can be easily implemented in the

motor model that is also based on the space vector

theory. Vector control is suitable for the force

(torque) control of both induction and synchronous

motors. Generally in the vector control theory, the

force and flux components are analyzed separately

from the motor currents using the mathematical

model of the machine, and control algorithms

control these components separately. In the vector

control used in this study, the direct axis current,

is set to zero (

=0) assuming that it does not

influence the generation of force; i.e. equation (9)

transforms to

()

dx d aq

. (14)

This means that angle

, between the armature

current and q-axis, always remains at 0

° and that the

thrust is proportional to the armature current,

The drawback of vector control is its low robustness

for changes in the machine parameters. The

resistance values change considerably due to

temperature variations, and the inductances rapidly

reach their saturation levels. However, a vector

controller is appropriate for applications in which

good dynamics and/or accurate velocity control is

needed.

In literature, vector control is presented in many

ways. In this study, we have used a simulation

principle in which the incoming thrust command,

F*, is converted to the i

current component by

dividing the force value by the force constant of the

motor,

. The current control algorithms are

executed in the rotor flux coordinates and the

outputs of the controllers are transformed back into

the stator reference frame, and these values,

, u

and

, are the inputs of the control inverter. In the

simulation model, the modulation technique used is

sinusoidal pulse width modulation (SPWM) with

ideal switches.

2.4 Verification of the Simulation

Model

The simulation model was implemented and

analyzed in the MatLab/Simulink

® software using

the previously mentioned equations. The PWM

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180

inverter is modeled as an ideal voltage source and

common Simulink blocks are used for the model.

The time step of the integrator in the analysis was 10

µs, except for the velocity controller, which had a

time step of 1 ms. The parameters used in the

simulation are introduced in table I in the appendix.

Figure 4 shows the Simulink model of the system.

The simulation results were compared with those

measured in the reference system. The motor studied

in this paper is a commercial three-phase linear

synchronous motor application with a rated force of

675 N. The moving part is set up on an aluminum

base with four recirculating roller bearing blocks on

steel rails. The position of the linear motor was

measured using an optical linear encoder with a

resolution of approximately one micrometer. The

parameters of the linear motor are given in table II in

the appendix.

The spring-mass mechanism was built on a tool

base in order to act as a flexible tool (for example, a

picker that increases the level of excitation). The

mechanism consists of a moving mass, which can be

altered in order to change the natural frequency of

the mechanism and a break spring, which is

connected to the moving mass on the guide. The

purpose of the mechanism is to increase the level of

excitation when the motor’s vibrational frequency is

equal to the mechanism’s natural frequency, which

was calculated at being 9.1 Hz for a mass of 4 kg.

The motion of the moving mass was measured using

an accelerometer.

The physical linear motor application was driven

in such a way that the PI velocity controller was

implemented in Simulink to gain the desired velocity

reference. The derived algorithm was transferred to

C code for dSpace’s digital signal processor (DSP)

to use in real-time. The force command, F*, was fed

into the drive of the linear motor using a DS1103

I/O card. The computational time step for the

velocity controller was 1 ms, while the current

controller cycle was 31.25

µs. The measured and

simulated velocity responses and force generating

quadrature currents are compared in figure 5. Sine

and step functions were used as the reference

velocity.

(a)

(b)

Figure 5: A comparison of the measured and

simulated quadrature currents and velocity responses in

the case of (a) a sine velocity reference and (b) a step

velocity reference.

ure 4: The Simulink model of the motor s

stem.

FORCE RIPPLE COMPENSATOR FOR A VECTOR CONTROLLED PM LINEAR SYNCHRONOUS MOTOR

181

3 DISTURBANCE

COMPENSATION

Disturbance compensation is applied to the motor

model to reduce a detrimental force ripple. Force

ripple compensation improves the speed response

and robustness of the system. The force ripple, i.e.

the detent force, friction, and load variation, is

estimated using a disturbance observer, or in other

words, a load force observer. Figure 6 shows the

construction of the compensator.

Figure 6: Disturbance compensation scheme utilizing

load force estimation.

The concept of the observer is based on the

comparison of the actual input force with the ideal

one. This gives rise to an error, which after proper

filtering, is used to produce the compensation

current,

comp

. The filter implemented in this study is

a second-order Butterworth digital low-pass filter

with a cut off frequency of 50 Hz. The main function

of the filter is to reduce high-frequency noise due to

input signal derivation but also break the algebraic

loop in the simulation model between the currents

ref

and i

comp

. Unfortunately, the time delay of the

filter also limits the performance of robust control,

since it delays the estimated force disturbance

(Godler

et al., 1999); therefore, the cut-off

frequency should be as high as possible. Hong et al.

(1998) suggested that an artificial delay be used in

the filtering bath of the observer in order to improve

dynamic behavior.

The limitations of this method are highlighted by

the fact that acceleration, which is needed in the

disturbance observer, is generally not available.

Usually, acceleration is calculated as the time

derivative of the output of the pulse encoder,

although the signal becomes easily erroneous due to

the high noise ratio in the encoder signal, and the

filtering of this kind of signal increases the

undesirable time delay, which leads to an unstable

response. In this study, acceleration is estimated

using an acceleration estimator, which is based on

the construction introduced in (Lee

et al., 2001).

This so-called low-acceleration estimator is based on

the fact that the displacement signal from the

encoder is accurate and numerical integration

provides more stable and accurate results than does

numerical differentiation. Figure 7 shows the

structure of the accelerated estimator.

Figure 7: Structure of low-acceleration estimator.

The estimation of acceleration,

, is calculated

from the displacement signal,

x, of the encoder using

a double integrator. The estimator takes the form of

a PD controller, in which the estimated

displacement,

, is set to follow the actual

displacement,

x; i.e. the acceleration estimate is

()

aKxx K

=−−

(15)

and the transfer function from

x to x

222

sKsK s s

ωω

++ + +

, (16)

where

represents the bandwidth of the

acceleration estimator and

is the damping ratio.

Gains

and K

from the PD controller can be

determined from the required bandwidth of the

estimator. Lee

et al. (2001) propose that a good

guideline for the damping ratio is 0.707, which

corresponds to critical damping.

The proposed disturbance regulator has been

tested in the simulation model introduced earlier and

implemented in the physical application. The

algorithm of the controller was run in real-time with

a frequency of 1 kHz. The damping ratio of the

acceleration estimator,

, was 0.707 and the

bandwidth

1000 Hz, i.e. the gains are K

=10e

and

=1414. Figure 8 shows a comparison of the

velocity errors between the reference and actual

velocities in compensated and non-compensated

systems, when the amplitude of the reference

velocity sine signal was 0.1 m/s.

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Figure 8: The error between the actual and reference

velocities in the compensated and non-compensated

systems, when the reference velocity signal is sine with an

amplitude of 0.1 m/s.

4 CONCLUSION

This paper presented and verified a dynamic model

for a PMLSM including non-idealities. The model

appeared to be an effective tool for designing the

controller of such a system. A disturbance estimator,

which included a low-acceleration estimator, was

proposed and successfully implemented in the

control of the motor. By means of an accurate

simulation model, it was possible to design and test

the controller without fear of physical damage. The

implementation of the proposed controller was

easily carried out by using a DSP system that

supported the used simulation software. Preliminary

parameter tuning was performed by using the

simulation model and final tuning was carried out in

the physical linear motor application.

It was observed that mechanical non-idealities

have important effect on the dynamics of the motor

system. This effect can be reduced by constructional

modifications and/or a suitable control algorithm. As

mentioned before, the acceleration signal for

disturbance estimation or another control algorithm

is not usually available. The double derivation of the

encoder signal produces a very noisy signal, and

filtering this kind of signal leads to undesirable

stability problems. With the proposed method,

acceleration can easily be estimated from the

position signal.

APPENDIX

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