chapter 2 modelling and parameter estimation...
TRANSCRIPT
17
CHAPTER 2
MODELLING AND PARAMETER ESTIMATION OF
BLDC SERVO SYSTEM
2.1 INTRODUCTION
Physical systems are modelled in order to analyse their behaviour under different operating conditions and employ them for various control
applications. Modelling of physical systems involves estimation of their parameters. Parameter estimation plays a vital role in perfect tuning of
controllers. This is essential to fulfill the desired performance specifications from the system.
Over the years, a great deal of research has been carried out in the computation of system parameters using genetic algorithm, fuzzy logic and
neural networks. Moment of Inertia and friction coefficient of motor alone are determined but that of load are not considered, even though various
optimization techniques, including artificial intelligence techniques and adaptive control methods are employed (Al-Qassar & Othman 2008; Babau et
al 2007; Despalatovic et al 2005; Hadef et al 2007; Kapun et al 2008). Load parameters are obtained using genetic algorithm but friction coefficient of
motor is not considered by Zhang & Bai (2008). Viscous friction coefficient of motor is determined while that of load is not considered for precise position control tasks (Campa et al 2008). The importance of estimation of
load parameters is emphasized by Lin et al (2010) but strategies for determining moment of inertia and friction coefficient at different load
conditions are not highlighted.
18
Ther
efor
e, a
sim
ple
met
hod
is p
ropo
sed
in t
his
thes
is f
or t
he
dete
rmin
atio
n of
mec
hani
cal p
aram
eter
s, vi
z. m
omen
t of
iner
tia a
nd f
rictio
n
of a
BLD
C d
rive
at d
iffer
ent l
oads
. Thi
s ch
apte
r dea
ls w
ith th
e m
odel
ling
and
com
puta
tion
of v
ario
us p
aram
eter
s of
BLD
C s
ervo
sys
tem
. The
sig
nific
ance
of
dete
rmin
atio
n of
th
e m
echa
nica
l pa
ram
eter
s at
di
ffer
ent
load
s is
emph
asis
ed. DC
pos
ition
con
trol s
yste
m e
mpl
oyin
g a
trape
zoid
al B
LDC
mot
or
was
con
side
red
in th
is th
esis
for
the
inve
stig
atio
n. T
he e
ffec
t of
load
on
the
para
met
er v
aria
tion
was
em
phas
ised
by
anal
ysin
g th
e tra
nsie
nt r
espo
nse
of a
clos
ed lo
op B
LDC
driv
e fe
d po
sitio
n co
ntro
l sys
tem
at d
iffer
ent l
oads
. In
this
chap
ter,
an a
ttem
pt is
mad
e to
ana
lyse
the
influ
ence
of p
aram
eter
var
iatio
n on
the
PID
con
trolle
r tun
ing
for t
he d
ynam
ic lo
ad v
aria
tion.
2.2
MO
DE
LL
ING
OF
BL
DC
DR
IVE
SY
STE
M
The
mat
hem
atic
al m
odel
ling
of B
LDC
driv
e sy
stem
is
desi
gned
(Kup
erm
an
&
Rab
inov
ici
2005
; Pa
rk
et
al
2003
) w
ith
the
follo
win
g
assu
mpt
ions
.
1.A
ll th
e st
ator
pha
se w
indi
ngs
have
equ
al re
sista
nce
per p
hase
and
cons
tant
self
and
mut
ual i
nduc
tanc
es.
2.Po
wer
sem
icon
duct
or d
evic
es a
re id
eal.
3.Iro
n lo
sses
are
neg
ligib
le.
4.Th
e m
otor
is u
nsat
urat
ed.
The
sche
mat
ic d
iagr
am o
f a B
LDC
driv
e sy
stem
is s
how
n in
Fig
ure
2.1.
The
thr
ee-p
hase
inpu
t vol
tage
s ar
e ex
pres
sed
as f
ollo
ws
with
the
abov
e
assu
mpt
ions
.
19
aedta
diL
aRi
av
(2.
1)
bedtbdi
Lb
Ribv
(
2.2)
cedtcdi
Lc
Ricv
(
2.3)
The
elec
trom
agne
tic to
rque
is e
xpre
ssed
as
)(
1ci ce
bi beai ae
eT
(2.
4)
Figu
re 2
.1 S
chem
atic
dia
gram
of a
BL
DC
dri
ve sy
stem
20
where va, vb and vc are the stator input voltages of phase a, b and c,
respectively, ea, eb and ec are the back emfs of phase a, b and c, respectively,
ia, ib and ic are the phase currents of phase a, b and c, respectively, R and L are
per phase resistance and inductance of each stator winding, TL is the load
torque, J is moment of inertia, is angular speed, B is friction coefficient, Kb
is back emf constant, KT is torque constant and is the angular position.
cebeaeE (2.6)
cibiaiI (2.7)
The block diagram of a typical position control system employing
BLDC motor is shown in Figure 2.2.
Figure 2.2 Block diagram of DC position control system
Desired angular position is fed as an input to the DC position
control system. Actual angular position (s) which is the output of the system
is fed as the feedback signal to the system. The difference between these two
positions actuates the per phase armature voltage Ea(s) to the BLDC motor
whose mathematical model is shown as an inner loop to the position control
system. Transfer function of BLDC motor is found out from its mathematical
model as
21
)])(([)()(
aaTb
T
a sLRBJsKKsK
sEs
(2.8)
where KT, Kb, Ra, La, J and B are specified in the name plate details of BLDC
motor.
2.3 ESTIMATION OF PARAMETERS OF DC POSITION
CONTROL SYSTEM
Parameters KT, Kb, Ra, La, J, B of DC position control system are
significant in assessing the transient and steady performance of DC position
control system. KT, Kb, Ra and La do not vary with respect to load. However,
J and B are found to vary appreciably with respect to load. Therefore, they
have to be estimated at different loads.
2.3.1 Estimation of Friction Coefficient B
The torque equation of the BLDC motor with load arrangement is
given by
BdtdJTe (2.9)
where J is moment of inertia of BLDC motor and load and B is friction
coefficient of motor and load.
When the speed is constant, the torque equation becomes
BTe (2.10)
60N2BiKT aTe
22
aTiKB (2.11)
where ia is the per phase armature current measured at steady state for the
given load current. B is determined for the given load current using Equation
(2.11).
2.3.2 Estimation of Moment of Inertia J
When the supply to the motor is switched off, motor speed reduces
to zero from its steady speed. Hence, the torque equation becomes
0BdtdJ
The solution for this equation obtained using the steady state speed
as the initial value of speed is expressed by
t)J/B(e eBT
(2.12)
When t = =J/B, mechanical time constant of the BLDC motor and load, the
motor speed reduces from steady state speed to 36.8% of steady state speed.
From the time constant, the moment of inertia of the motor and load is given
by,
BJ (2.13)
J is determined for the given load current by substituting the values
of B and time constant in the Equation (2.13).
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2.3.3 Estimation of J and B of BLDC Drive System
Figure 2.3 shows SIMULINK model of a BLDC motor fed by a
six- step inverter used at different loads. The specifications of BLDC motor
used for the proposed work are given in Table 2.1.
Figure 2.3 BLDC motor fed by a six-step inverter
24
Table 2.1 Specifications of BLDC motor
Parameters Values
Rated voltage 24 V dc
Number of poles 8
Rated speed 4000 rpm
Rated torque 0.125 Nm
Torque constant 0.036 Nm/A
Moment of inertia 48 10-7 kg-m2
Friction coefficient 1 10-5 Nm-sec/rad
Armature resistance 1.08 per phase
Armature inductance 1.8 mH per phase
24V DC supply was given to the BLDC motor through a switch
controlled by a timer. The gates of the respective switch in the six-step
inverter were controlled based on hall sensor signals. The three-phase outputs
of the inverter were applied to stator windings of BLDC motor. The
specifications of BLDC motor from Table 2.1 were chosen in block
parameters of permanent magnet synchronous machine (PMSM). Load torque
(TL) was given as step input of size corresponding to the desired load setting.
The simulated electromagnetic torque (Te) and the speed responses of the
motor were observed. Mechanical time constant of motor and load ( ) was
obtained from the speed response.
The electromagnetic torque and speed responses obtained at no load
are shown in Figures 2.4 and 2.5, respectively. The steady state speed and the
mechanical time constant are found from Figure 2.5 as 5994 rpm and 0.462
25
sec, respectively. The electromagnetic torque at steady state speed of 5994
rpm is determined as 5.47 10-3 Nm from Figure 2.4. Using Equations (2.11)
and (2.13), friction coefficient and moment of inertia of BLDC motor with
loading arrangement at no load are found out as 8.72 10-6 Nm-sec/rad and
4.03 10-6 Kg-m2, respectively. The mechanical parameter values J and B
computed are in agreement with the specifications of the machine as it is
available in the name plate (BM=1 10-5 Nm-sec/rad and JM=4.8 10-6 Kg-m2).
This procedure to compute moment of inertia and friction coefficient was
repeated from no load to full load in steps of 10% and the results are tabulated
in Table 2.2.
Figure 2.4 Electromagnetic torque response
26
Figure 2.5 Speed response
Table 2.2 Estimation of J and B at different loads
TL
(Nm)
% of full load (rad/sec)
Te
(Nm) (sec)B
(Nm-sec/rad)J
(Kg-m2)
0 0 627.5 0.00547 0.462 8.72 10-6 4.03 10-6
0.0125 10 567.8 0.014 0.1051 2.47 10-5 2.59 10-6
0.025 20 517 0.0278 0.055 5.38 10-5 2.96 10-6
0.0375 30 472.7 0.0449 0.0353 9.50 10-5 3.35 10-6
0.05 40 433.5 0.0629 0.025 1.45 10-4 3.63 10-6
0.0625 50 398.7 0.0637 0.0187 1.54 10-4 2.94 10-6
0.075 60 366.3 0.08 0.0146 2.18 10-4 3.18 10-6
0.0875 70 338 0.0834 0.0117 2.47 10-4 2.89 10-6
0.1 80 311.2 0.0974 0.0096 3.13 10-4 3.00 10-6
0.1125 90 287.7 0.1336 0.0081 4.64 10-4 3.76 10-6
0.125 100 265 0.1218 0.0068 4.54 10-4 3.13 10-6
27
2.3.4 Experimental Determination of J and B of BLDC Drive System
The BLDC motor was fed with 24V DC supply. The motor was
loaded in steps using slotted weights up to 40% of full load. The steady state
load current was found to be 0.6A from Figure 2.7. Then the supply to the
motor was switched off and the speed and current responses were obtained as
shown in Figures 2.6 and 2.7, respectively.
Figure 2.6 Speed response at 40% of full load
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Figure 2.7 Stator current (load current) response
Mechanical time constant ( ) was determined as 0.1 sec
corresponding to time taken for the speed to drop from 4409 rpm to 1623 rpm
(36.8% of initial speed 4409 rpm). B and J were found to be 4.68 10-5 Nm-
sec/rad and 4.68 10-6 Kg-m2, respectively, using Equations (2.11) and (2.13).
Similarly, B and J were determined at different load currents and are tabulated
in Table 2.3. B and J as estimated experimentally were used to develop
transfer function model of the position control system that is used for the
design of proposed compensator in chapter 5.
Table 2.3 Determination of mechanical parameters J and B
ia(A) % of full load (rad/sec) (sec) B(Nm-sec/rad) J(Kg-m2)
0.32 20 482.97 0.2 2.39 10-5 4.78 10-6
0.6 40 461.71 0.1 4.68 10-5 4.68 10-6
1.2 60 387.46 0.045 1.11 10-4 4.88 10-6
1.6 80 374.89 0.032 1.54 10-4 4.93 10-6
29
The mechanical parameters, viz. moment of inertia and friction
coefficient of BLDC drive are found to vary with respect to load from Table
2.3 and therefore, they will certainly have an adverse effect on the
performance of the position control system. Hence, there is a need for the
determination of these parameters at different loading conditions.
2.4 EFFECT OF LOAD ON SYSTEM PERFORMANCE
The transfer function model given in Equation (2.8) is further
simplified as Equation (2.14) and Equation (2.15) for no load and full load,
respectively, using the experimentally estimated J and B parameter values.
9 3 -6 2a
(s) 0.036E (s) 7.254 10 s 4.368 10 s 0.001305s (2.14)
9 3 -6 2a
(s) 0.036E (s) 5.634 10 s 4.198 10 s 0.001786s (2.15)
The unit step response of the closed loop uncontrolled transfer
function model and the experimental hardware were obtained, as shown in
Figure 2.8 and Figure 2.9, respectively, for no load condition. Figure 2.10 and
Figure 2.11 indicate the simulated and hardware step response of the system
under full load condition respectively.
30
Figure 2.8 Simulated step response of position control system at no load
Figure 2.9 Real time step response of position control system at no load
31
Figure 2.10 Simulated step response of the system at full load
Figure 2.11 Real time step response of position control system at full load
32
The rise time and settling time obtained from the response
characteristics revealed that the time domain behaviour of the BLDC drive
based position control system depended on the load.
Unit step response of closed loop Parr (Parr 1989) tuned PID-
controlled BLDC drive based position control system at full load condition is
shown in Figure 2.12. PID controller parameters Kp, Td and Ti were obtained
as 18.55, 0.0022 sec and 0.0111 sec respectively using Equation (2.15). Time
domain specifications, viz. rise time, peak overshoot and settling time were
obtained as 2.87 msec, 17.6% and 29.3 msec, respectively from Figure 2.12.
Kp, Td and Ti were deduced as 11.2, 0.0029 sec and 0.0146 sec, respectively,
using no load transfer function of Equation (2.14). Unit step response of
closed loop Parr-tuned PID-controlled BLDC drive based position control
system at full load condition was obtained using these controller parameters at
no load and shown in Figure 2.13. Rise time, peak overshoot and settling time
were obtained as 4.52 msec, 15.8% and 43.1 msec, respectively, from
Figure 2.13.
Figure 2.12 Step response of PID-controlled system at full load
33
Figure 2.13 Step response of PID-controlled system at full load using
no-load model
These results are sluggish and not in agreement with the results of
Figure 2.12 since the no load model was used for tuning PID controller at full
load condition. It is observed from Figure 2.12 and Figure 2.13 that PID
controller parameters of the position control system have to be tuned for the
mechanical parameter variation in order to achieve better results under
dynamic load variation.
2.5 CONCLUSION
In this chapter, a simple method to compute the mechanical
parameters of a BLDC drive was developed for dynamic load variation. The
importance of estimation of mechanical parameters was emphasised. The
important observations made using the proposed approach are listed as
follows:
34
The mechanical parameters, i.e., moment of inertia and friction
coefficient vary with respect to the loading conditions and,
therefore, they have an adverse effect on the performance of
BLDC drive system.
The controller parameters for the closed loop position control
system are found to vary dynamically since the mechanical
parameters vary with respect to load.
It is essential to tune the controller parameters with respect to
load in order to achieve better position control since the load
influences the system dynamics.
There is a need to employ suitable tuning algorithm for PID
controller that can adjust itself its parameters in order to achieve
optimum results in the proposed closed loop BLDC drive fed
position control system under dynamic load variation.