Design of control system for forced dynamics control of an electric drive employing linear permanent magnet synchronous motor

Abstract

The paper presents forced dynamics control of an electric drive with linear permanent magnet synchronous motor. This control method offers an accurate realisation of a dynamic speed response, which can be selected for given application by the user. In addition to this, the angle between stator current vector and moving part flux vector are maintained mutually perpendicular as it is in conventional vector control. To achieve prescribed speed response derived control law requires an external force information, which is obtained from the set of observers. The first set of observers is based on sliding-mode, while the second observer is classical full-state observer and exploits information of the position sensor. Simulations of the overall control system together with preliminary experimental results for the drive with rotational synchronous motor predict intended performance of the drive

Full text

Page 1

Design of Control System for Forced Dynamics Control of an Electric Drive
Employing Linear Permanent Magnet Synchronous Motor
Ján VITTEK, Ján MICHALÍK, Vladimír VAVRÚŠ, Vladimír HORVÁTH
University of Zilina, Faculty of Electrical Engineering
Univerzitná 1, 010 26 Zilina, Slovak Republic
jan.vittek@fel.utc.sk
Abstract - The paper presents forced dynamics control of
an electric drive with linear permanent magnet synchronous
motor. This control method offers an accurate realisation of a
dynamic speed response, which can be selected for given
application by the user. In addition to this, the angle between
stator current vector and moving part flux vector are
maintained mutually perpendicular as it is in conventional
vector control. To achieve prescribed speed response derived
control law requires an external force information, which is
obtained from the set of observers. The first set of observers is
based on sliding-mode, while the second observer is classical
full-state observer and exploits information of the position
sensor. Simulations of the overall control system together with
preliminary experimental results for the drive with rotational
synchronous motor predict intended performance of the drive.
I. INTRODUCTION
The exploitation of linear AC motors for industrial and
transportation applications is increasing. In this paper, speed
control system for linear permanent magnet synchronous
motor (LPMSM) is developed. This system enables to
prescribe speed profile and settling time of the drive or
vehicle and such way substantially contribute to the gentle
load handling or to the travel comfort of passengers during
speed transients. The conventional vector control methods
can also prescribe shape of acceleration for the drive, but the
settling time of the overall control system during speed
transients depends on external disturbances.
Developed speed control system for LPMSM is based on
forced dynamics control principles and principles of vector
control. The forced dynamics control method is a new
method for control of a.c. drives based on principle of
feedback linearisation [1], [2]. The main reason for utilising
of this method is that it enables to design control algorithm
for the drive, which has precisely defined the profile of the
acceleration and speed including settling time. The
prescribed acceleration can therefore sufficiently exploits the
limits determined by the maximum electrical force of the
linear motor.
Mutual orthogonality of the force producing linear stator
current vector and moving part magnetic flux vector is
maintained, as it is usual for conventional vector controlled
rotational drives with PMSM [3], [4].
To achieve prescribed settling time during speed transients
the information about external force is needed. The estimate
of external force together with the estimate of translation
speed is obtained from the observers. Two various sets of
observers are developed and compared. The first set is based
on sliding-mode and to avoid chattering of the estimated
variable, the observer is completed with another one, which
has filtering effect. The second observer is designed as third
order state observer, which exploits the measured position of
the drive or vehicle.
2/3
transf.
e-jϑ
iα_d
ia
a
^Fext
^vmp
External
force
Fext
Udc
meas.
ia
ib
uα, iα
ejϑ
Inverter
3/2
transf.
ib
a
-ic
ib_d
ic_d
ia_d
Position
sensor
iβ_d
id_d
iq_d
Forced Dynamics
Control Algorithm
0
(
v
)






+−
Ψ
c
=
=
ext mpd
v PM
d_q
d_d
Fˆv
T
M1
i
i
Observers
vmp
ϑ=smp/r
vmp
ud, id
uq, iq
d
dt
vd
Type of
dynamics
moving
part PM
smp
smp
ϑ=smp/r
Linear
PMSM
Fext
Slave Inner Control Loop
Master Outer
Control Loop
uβ, iβ
Udc
Fig. 1 Overall LPMSM control system block diagram
0-7803-9419-4/05/$20.00 ©2005 IEEE

Page 2

The overall control system for LPMSM has a nested
structure shown in Fig. 1, comprising an outer master control
loop and inner slave control loop. Outer control loop
computes such demanded stator currents, which realises the
closed loop prescribed dynamic behaviour of the drive. This
way selected operational modes with defined acceleration
and speed profile can be prescribed. The inner slave control
loop forces real three-phase stator currents to follow their
computed demands from master algorithm with negligible
lag via control of power electronic switches.
II.
CONTROL ALGORITHM DESIGN
Forced dynamics control of the LPMSM exploits
principles of feedback linearisation and principles of vector
control. Firstly, the model of LPMSM is formulated in the
d_q co-ordinate system, which is coupled to the moving part:
ds
=
,
mp
mp
dt
dv
v
(1a)
()
[]
extdqqd
mp
dt

i
Fiic
M

−
1
−Ψ−Ψ=
,
(1b)








/ 1
+




Ψ
−
−









−
−
=



q
d
q
d
PM
q
mp
L
q
d
qsqd mp
dq
/
mpds
L
q
d
u
u
L / 10
0L
0
r
pv
i
i
LR rL/pv
rL/L pvL/R
i
dt
d
,
(1c)
where, [ id, iq ]T and [ ud, uq ]T are, respectively, column
vectors of the LPMSM stator current and stator voltage
components, smp and vmp are the position and velocity of the
moving part, c=3p/2r where p is number of motor pole-pairs
and r is a constant parameter of LPMSM, which depends on
the linear motor structure having the dimensions of length,
Fext is the external force, Rs is the phase resistance, Ld and
Lq are the direct and quadrature phase inductances, ΨPM is
the permanent magnet linkage flux and M is the mass of the
linear motor moving part plus the equivalent mass of the
driven mechanism.
A2. Development of Control Algorithm
Secondly the principles of feedback linearistion enable to
formulate the linearising function for speed of the moving
part, which forces this translation speed to obey specified
closed-loop differential equation. This equation is assumed
linear, first order with a prescribed time constant, Tv :
(
mpd
v
Tdt
The computation technique for feedback linearisation is to
equate the right hand side of (2) with the right hand side of
the corresponding motor equation (1b). This forces the non-
linear differential equation (1b) to have the same response as
the linear equation (2). Thus:
(
v
TM
The following part of the control law is formulated on the
basis of vector control, which requires mutual orthogonality
)
mp
mp
vv
1
dv
a
−==
.
(2)
)
[]
()
mpdextdqqd
vv
1
Fiic
1
−=−Ψ−Ψ
.
(3)
between the rotor magnetic flux and stator current vectors.
Following conventional approach [3] to achieve maximum
magnetic flux up to nominal speed the current demand for
the magnetic flux component in direct axis is set to zero:
0id=
Setting id=0 in (3) on the assumption that the real current
follows its demand, id=id_d and solving (3) for force producing
component, iq_d yields the following master control algorithm
formulated for stator current demands in the d- and q-axis:
0i
=
.
(4)
()
PM
ext mp
c
Ψ
ext mpd
v PM
d_q
d_d
Fˆ Ma
Fˆvv
T
M
c
1
i
+
=






+−
Ψ
=
.
(5)
This way the derived master control law (5) yields a moving
part speed response with linear, first order dynamics and
unity dc gain:
( )
( )
vd
pT1pv
+
The numerator of the derived control algorithm (5) consists
of two parts. The first one contains the demanded output
acceleration and creates dynamic force during transients. The
second part covers the external force, which needs to be
estimated. By changing the prescribed acceleration, amp ,
various operational modes of the drive can be realised.
( )
p
mp
S
1
pv
F
==
.
(6)
B2. Operational Control Modes
Via proper definition of smooth mathematical functions
for drive acceleration in (5), various operational modes can
be achieved. Following modes were chosen as the examples:
a) Direct acceleration control with constant acceleration
(ramp), where the drive produces acceleration of a moving
part following a demanded constant acceleration with
negligible dynamic lag. In this case, demanded
acceleration is determined by a constant demand of
velocity, vd and the demanded acceleration time, Ts :
(
mpd
s
T
b) Direct acceleration control with linearly dependent
acceleration (S-curve), where the drive produces linearly
increasing and decreasing acceleration (constant jerk)
during speed transients. The corresponding equations for
the demanded acceleration can be easily derived, [5].
c) Linear first order speed response, which is the case
already described during master control law development
(Ts=3Tv), therefore briefly:
(
d
sv
TT
d) Second order speed response (fluent change of
acceleration), where the drive acceleration is prescribed
such way that the closed-loop system has a response given
by the second order characteristic equation. In this case the
derivative of demanded acceleration is done by (9) and to
gain desired acceleration (9) needs to be integrated.
da
ω+ξω−===ε
&& &
)
d
d
vv sgn
v
a
−=
.
(7)
)()
mpmpdd
vv
3
vv
1
a
−=−=
.
(8)
()
mpd
2
n mpn mp
d
vvv2v
dt
−
.
(9)

Page 3

If damping factor ξ=1 for critically damped system is
chosen and the poles of the characteristic equation are
purposely designed as coincident the settling time formula
(10), where n is order of the system, may be used to
determine ωn , which will fit chosen settling time:
()
n
2
ω
Ideal acceleration and speed responses based on described
operational control modes are shown in Fig. 2.
s
1
n1
3
T
+=
.
(10)
00.51 1.5
0
20
40
60
80
100
120
[ms–1, ms–2]
vd
[s]
0.2ad
Ts
vmp
0 0.51 1.5
0
20
40
60
80
100
120
[ms–1, ms-2]
vd
[s]
Ts
vm p
0.2ad
a) constant acceleration, rampb) linear acceleration, S-curve
00.511.5
0
20
40
60
80
100
120
[ms–1, ms-2]
[s]
Ts
vmp
0,95vd
0,2ad
00.511.5
0
20
40
60
80
100
120
[ms–1, ms-2]
[s]
Ts
0,95vd
0,2ad
vmp
c) first order dynamic responsed) second order dynamic response
Fig. 2 Prescribed profiles of speed and accelerations
C2. Representation of the Load
An important feature of the designed control system is the
representation of the driven mechanical load, which is shown
in Fig. 3. Thus only the mass of motor moving parts, Mm
used is included in the forward path as a rigid body, moving
without external force and friction [6]. Both, external force
and friction are modelled in the feedback path, as it is shown.
The justification for this is that the not completely known
friction forces can be for some applications dominated, [7].
Net
accelerating
force, Fa
motor
velocity,
vmp
Electrical
force, Fel
Driven dynamics
1
pMm
External
force Fext
pMe
p
1
motor
position,
smp
Fig. 3 Representation of the driven mechanical load
It is also important to note that it is unnecessary to provide
an accurate model of the inverse load dynamics. Since the
error in the value of moving mass, Mm used can be
represented as a dynamic load force, Medv/dt, and this
together with any external force acts at the same point as
Fext , then the estimate, F^
both these forces and so both of them will be compensated.
ext from the observer will include
III. STATE ESTIMATION AND FILTERING
The speed of moving part and external force, which are
inputs for the master control algorithm are produced by the
following set of observers. The first observer, which is
based on stator current equation, works in pseudo-sliding-
mode, [8], [9] and generates an unfiltered estimate of
translation speed. This observer is completed with filtering
observer, which provides filtered speed estimate together
with estimate of external force. The second observer, which
exploits the measurement of motor position is full-state
observer and provides the estimates of translation speed
together with external force.
A3. The pseudo-sliding-mode observer
For the translation speed estimation pseudo-sliding-mode
observer shown in Fig. 4 is used. The observer is based on a
stator current iq equation, (1c) as a real time model but
purposely using only the terms without vmp :
(
dd
q
q
rLLdt
The terms containing vmp in (11) are in observer model,
(12) replaced by the correction input, vq. Thus equation of
the model i*
*
q
vu
LLdt
where i*
sliding-mode the estimate of vq is produced as fast switching
of Umax , (13a). However, the useful observer output is the
continuous equivalent value of the rapidly switching vq .
Equation (13a) cannot directly generate veq q. Instead, a
pseudo-sliding-mode observer, [8] may be formed by
replacing (13a) with (13b):
(
qqmax q
iisgn Uv
−−=
eq q
v
where the gain, Ksm, is made as high as possible [10], [11]
within the stability limit. For large Ksm, the error between
real current and fictitious observer current is driven almost to
zero, resulting in (14):
()
PMdd
q
rL
and this may be manipulated to yield an unfiltered velocity
estimate, v*
vrL
v
Ψ+
)
q
q
PM
mp
q
s
q
u
L
1
iL
pv
i
R
di
+Ψ+−−=
.
(11)
q component is:
qq
q
*
q
q
s
1
i
R
di
−+−=
,
(12)
q is estimate of iq as in a conventional observer. In
)
*
,
()
*
qq sm
iiK
−−=
,
(13a,b)
mp
eqq
iL
pv
v
Ψ+
−
=
(14)
mp . Thus:
()
PMdd
eq_qq
*
mp
iLp
−
=
.
(15)
iq
-
-
rLq
pvmp (Ldid + ΨPM)
+Umax
-Umax
Ksm
Lq
1
Rs
p
1
-
-
Lq
1
Rs
rLq
pvmp
(Ldid+ΨPM)
vq=
uq
i*
q
uq
p
1
Motor
model
Observer
model
Fig.4 Block diagram of pseudo-sliding-mode observer

Page 4

B3. Filtering Observer
Filtering function of the velocity estimate, (15) is
performed by filtering observer, which provides also the
external force estimate, [12], [13]. The real time model of
this observer is based on the motor force equation (1b), which
is augmented by the second state differential equation,
dFext/dt=0, for assumption of the piecewise constant external
force, Fext . The observer correction loops are actuated by the
error between the unfiltered speed estimate, v*
pseudo-sliding-mode observer and the filtered translation
speed estimate, v^
filtering observer equations are:
mp , from the
mp , from the real time model output. The
mp
(
Ψ
*
mpv
v ˆ
v ˆ
[
c
ve
−=
,
(16a)
)
]
vv extdqqd
mp
dt
Fˆ
ekFˆii
M
1
d
+−Ψ−=
,
(16b)
eF
ext
dt
vk0
d
+=
.
(16c)
This is a conventional second order linear observer with a
correction loop characteristic polynomial, which may be
chosen via the gains, kv and kF to yield the desired balance of
filtering between the noise from the measurements of id and iq
and the noise from the pseudo-sliding-mode velocity estimate.
Transfer function of the observer shown in Fig. 5 is:
( )
( )
kpkppv
Fvmp
++
( )
pF
M/
M/kpk
pv ˆ
2
Fv
*
mp
+
==
.
(17)
-
v*mp
kF
-Fext
p
1
M
1
p
1
(
Ψ
)
dqqd
ii
M
c
Ψ−
kv
vmp
Fig.5 Block diagram of filtering observer
The observer gains may be chosen to yield a non-oscillatory
state estimation error transient with prescribed settling
time, Ts. Based on settling time formula, (10) the observer
poles can be placed at ω0=9/2Ts yielding desired polynomial:
()
81T/9pT2/9p
ss
++=+
)T4 /(
2
s
2
2
,
(18)
which yields observer gains:
sv
T/9k =
,
2
sF
T4/M 81k
=
.
(19a,b)
Although the external force is assumed constant in the
formulation of the observer real time model, the estimate of
Fˆ
will follow a time varying external force and will do so
more faithfully as Ts is reduced, but at the expense of
sensitivity to any noise contaminating the speed estimate.
ext
C3. State Observer Based on Position Measurement
Since a direct measurement of the external force is
assumed unavailable, its estimate for the master control law
can be provided by a standard full-states observer, which
exploits the position measurement. Real time model of the
observer is based on the motor position, (1a) and velocity,
(1b) equations augmented again by the third state equation
for piecewise constant external force. The error between real
position and estimated position, es=smp–s^
corresponding gain into every observer correction loop:
mp is added with
ssmp
mp
dt
v ˆd
eKv ˆ
s ˆ d
+=
,
(20a)
()
[]
svextdqqd
mp
dt
Fˆd
eKFˆiic
M
1
+−Ψ−Ψ=
,
(20b)
sF
ext
dt
eK0
+=
.
(20c)
Block diagram of observer for estimates of external force
and translation speed of the moving part is shown in Fig. 6.
-
smp
KF
-Fext
p
1
M
1
p
1
p
1
(
Ψ
)
dqqd
ii
M
c
Ψ−
Kv
Ks
smp
vmp
vmp
Fig. 6 Moving part speed and external force observer
The observer transfer function is given as:
( )
( )
p
ps
mp
+
If the observer poles are chosen as coincident and placed at
p=-6/Tso with settling time, Tso , then the desired polynomial is:
3
108
p
T
T


Comparing characteristic polynomials (21) and (22) yields:
18
K =
,
2
so
T
The force estimate, F^
varying external force and will do it more closely as Tso is
reduced.
( )
p
M/K pKKp
M/K
+
pK
+
Kp
p s ˆ
F
Fvs
23
Fvs
2
mp
O
++
==
.
(21)
3
so
2
so
2
so
3
so
T
216
p
T
18
p
6
p
+++=




+
.
(22)
so
s
T
v
108
K =
and
3
so
F
T
M216
K =
.
(23)
ext will again follow arbitrary time
IV. DESIGNED CONTROL SYSTEM VERIFICATION
Verifications of the overall designed control system were
performed in two steps. The first one was the verification by
simulation and the second step was the preliminary
experiments with rotational PMSM.
The simulations of the design control system were
performed with the LPMSM having parameters: Pn=800 W,

Page 5

p=3, r=0.156 m, Rs=0,59 Ω, Ld=3,7 mH, Lq=3.5 mH and
ΨPM=0,3 Vs. The total load mass and external force are
M=5 kg and Fext=250 N. Settling time for both types of
observers was chosen as Tso=10 ms. A sampling frequency
of 10 kHz achieved during experiments with rotational
PMSM was assumed also for the power electronics switches
of inverter in simulation.
All simulations show the designed system response to a
step speed demand of vdem=10 ms-1 applied for time interval
t∈(0, 0,25] s with zero initial states of all state variables and
prescribed settling time Ts=0,1 s followed immediately by
step speed demand of vdem=-10 ms-1 applied for time interval
t∈[0,25 0,6) s with twice longer settling time. The external
force Fext=250 N is applied at t=0,15 s, which drops to zero at
t=0,25 s . Negative external force is applied again at t=0,5 s
when speed is also negative. The simulations were
performed for both types of the designed observers.
Simulation results for described dynamics show demanded
speed, vd , real speed, vmp , estimated speed from filtering
observer or full-state observer, v^
Fext and estimated external force F^
Fig. 7 shows control system ideal response to prescribed
first order dynamics, when pseudo-sliding-mode observer
completed with filtering observer were employed (translation
speed is multiplied by factor 10 and external force is divided
by factor 5 in all simulation results presented further).
mp , together with applied
ext as a functions of time.
00.10.20.30.40.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
vd
vm p
v^m p
F^ext
Fext
Fig. 7 Simulation results for the first order speed dynamics
Ideal control system response to the prescribed first order
dynamic for full-state observer are shown in Fig. 8.
00.10.20.30.40.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
vd
vmp
v^mp
F^ext
Fext
Fig. 8 Simulation results for the first order speed dynamics
As can be seen from both figures the prescribed settling
time and dynamics of the system were achieved. From the
time function of estimated external force it is possible to
observe correct design of observer poles, when 95% of
estimated force is achieved at settling time Ts=0,01 s.
Fig. 9a shows both stator current components and Fig. 9b
shows estimated speed of pseudo-sliding-mode observer.
00.10.20.30.40.50.6
-30
-20
-10
0
10
20
30
00.10.20.30.40.50.6
-10
-5
0
5
10
15
a) id and iq =f(t)
Fig. 9 Stator current components and estimated speed of pseudo-
sliding-mode observer
b) v*
mp =f(t)
To save some space, simulation results for ramp speed
demands are shown for full-state observer and simulations of
second order dynamics are shown for pseudo-sliding-mode
observer only. To get more realistic simulations the 12 bits
AD converters (final length of digital word) for sensing of
moving part position and for sensing both stator current
components were included into simulations.
Fig. 10a shows control system response to the prescribed
ramp speed demand and control system response to the
second order dynamics is shown in Fig. 10b. As can be seen
from both figures the prescribed system dynamics were
achieved (demanded speed or 95% of the demanded speed is
achieved at prescribed settling time).
00.10.20.30.40.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
vd
vmp
v^mp
F^ext
Fext
00.10.20.30.40.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
vd
vmp
v^
mp
F^ext
Fext
a) ramp speed demand
Fig. 10a,b Simulation results for ramp speed demand
and second order dynamic
b) second order dynamic
Current components id and iq for ramp speed demand and
second order dynamic are shown in Fig. 11a,b. The ratio
among the peak currents of the first order dynamic, Fig. 9a,
the second order dynamic, Fig. 11b and ramp speed demand
Fig. 11a is 3 : 1,5 : 1 as it is clear also from acceleration
peaks of Fig. 2.
00.10.20.30.40.50.6
-15
-10
-5
0
5
10
15
00.10.20.30.40.50.6
-20
-15
-10
-5
0
5
10
15
20
Fig. 11 Stator current components as a function of time
Due to the absence of experimental bench with LPMSM
the preliminary experiments were carried out with rotational
PMSM Dutym Ax-DS having the following parameters:
Pn=375 W at ωn=314,16 rad/s, p=3, Rs=36,5 Ω,
Ld=Lq=50 mH, ΨPM=0,312 Vs and Jr=0,032 kgm2.