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Modeling of Brushless DC Motor Using
Adaptive Control
N. Veeramuthulingam
(&)
, A. Ezhilarasi, M. Ramaswamy,
and P. Muthukumar
Department of Electrical Engineering, Annamalai University, Annamalai Nagar,
Chidambaram, Tamil Nadu, India
sethukumark@gmail.com, jee.ezhiljodhi@yahoo.co.in,
aupowerstaff@gmail.com
Abstract. The paper proposes the introduction of an adaptive controller for
enhancing the performance of a brushless DC motor (BLDC). It involves the use
of the state space model for the motor in an effort to articulate the theory of
model reference adaptive control and follow the principles of sensorless feed-
back mechanism. While the measure of the back EMF reflects the repository
speed and enables the computation of the speed error, the reference frame fosters
the estimate of the ripple in the torque of the motor. The two stage converter
interfaces attach support to assuage the most intriguing of the corrective action
through changes in the duty cycle and the modulation index, respectively. The
methodology forges its adaptability to various speed ranges and remains
immune to source and load distubances without affecting the accuracy of the
results. The MATLAB based response illustrates the merits of the algorithm in
terms of its speed regulating capability and minimizing the torque ripple to
allow the motor find a place in the utility industry.
Keywords: Modelling BLDC motor Model reference adaptive control
1 Introduction
The modern world continues to promote the resurgence of solid state drives and invite a
thrust towards enhancing its performance. The applications among the many include
the brushless dc engines, electrical vehicles, HVAC industry, military application and
medical equipments. The advantages rely on the higher power capacity, better effec-
tiveness and lower maintenance.
The brushless dc motor (BLDC) motor driven by direct current, operates on
electronic commutation rather the mechanical commutation system based on brushes.
The motor torque, current, voltage and speed experience a linear relationship with each
other and the significance owes to the removal of the brushes, which swing sparking
and brush maintenance. It evokes considerable interest in the industry and finds a wide
scope for its use. The modeling and simulation form part of the design of the BLDC
motor and the tuning of the parameters of the controller evince a paradigm shift to
manifest fresh utility areas.
©Springer Nature Singapore Pte Ltd. 2018
I. Zelinka et al. (Eds.): ICSCS 2018, CCIS 837, pp. 764–775, 2018.
https://doi.org/10.1007/978-981-13-1936-5_78
The control of the BLDC motor emphasizes the location of the magnitude of the back
EMF through its rotor speed operations. At low speed operating range it becomes difficult
to measure the speed because of inverter and nonlinearities in the parameter. Though the
normal PID based speed control solutions show to be simple, stable, and highly reliable,
still several modern control results continue to materialize [1]. However currently many
degrees of nonlinear approaches occupy prominence in controlling the motor [2].
A number of closed loop control algorithms have been recommended to design an
optimal converter [3] and a torque ripple minimization strategy discussed in [4]to
address the trapezoidal shape of back EMF and in addition enliven an easy active
method to design an angular speed controller and build a torque controller. The model
reference adaptive controller for the motor has been handled by [5] and seen to depend
on the correct parameters and the sensor position [6,8]. The experimental methods
have been to shown to involve a PLL [7] and arrive at driving nearly constant speed.
However the trend augurs exploring further new control methodologies to enunciate
improved performance for the drive motor.
2 Modeling of Brushless DC Motor
The transfer function offers powerful and simple design techniques although it suffers
from certain problems in the sense the model appears to be defined under zero initial
conditions. Besides it behoves its applicability to linear time-invariant systems and
remains limited to single input single output systems. The other restriction relates to the
fact that it provides the system output only for a given input and does not reveal any
data regarding the internal state of the system.
The approach involves the following assumptions in the modeling of the BLDC motor
•Star wound type stator.
•Symmetric in nature of inductance, resistance and mutual inductance.
•No change in rotor reluctance angle.
•Stator winding aligned with proper orientation.
•Free from saturation effects.
•Identical back- EMF shape in all three phase.
•Inverter remains ideal.
•Negligible hysteresis, eddy current effect and iron losses.
It comprises of both mechanical and electrical equations
The electrical equations:
Va¼RiaþLdia
dt þemfað1Þ
Vb¼RibþLdib
dt þemfbð2Þ
Vc¼RicþLdic
dt þemfcð3Þ
Modeling of Brushless DC Motor Using Adaptive Control 765
emfa¼KewmFðheÞ
emfb¼KewmFðhe2p=3Þ
emfc¼KewmFðheþ2p=3Þ
Te¼½emfaiaþemfbibþemfcic=wm
The mechanical equation relates as is Eq. (4)
Te¼BwmþJdwm
dt þTlð4Þ
V
k
–Phase voltage, I
k
–Phase Current, emf
k
–Phase back emf
R, L –Phase resistance and inductance of the stator winding
T
e
–Total Electromagnetic torque and T
l
–load at motor
h
e
–Electrical angle of the rotor
h
m
–Mechanical angle of the rotor
W
m
–angular speed of rotor
P–number of poles on rotor
K
e
–motor back emf current (V/rad/Sec)
J–moment of inertia (kg-m
2
)
B–Dynamic Frictional Torque constant (NM/rad/Sec)
Mechanical and Electrical angle of the rotor can be related as in Eq. (5)
he¼p
2hmð5Þ
F(h
e
) follows the back emf reference which bears a trapezoidal shape and mag-
nitude in in Eq. (6)
FðheÞ¼
10he2p=3
16
pðhe2p=3Þ2p=3hep
1phe5p=3
1þ6
pðhe5p=3Þ5p=3he2p
9
>
>
=
>
>
;
8
>
>
<
>
>
:
ð6Þ
On solving the equation, the complete state-space model can expressed as in Eq. (7)
x¼Ax þBu
dia
dt
dib
dt
dic
dt
dwm
dt
dhm
dt
2
6
6
6
6
6
4
3
7
7
7
7
7
5
¼
R
L100kp
L1fahr
ðÞ 0
0R
L10kp
L1fbhr
ðÞ 0
00R
L1kp
L1fchr
ðÞ 0
kp
Jfchr
ðÞ kp
Jfchr
ðÞ kp
Jfchr
ðÞ
B
j0
000 p
20
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
ia
ib
ic
xm
hr
2
6
6
6
6
4
3
7
7
7
7
5
þ
1
L1000
01
L10
001
L10
0000
0000
2
6
6
6
6
4
3
7
7
7
7
5
Va
Vb
Vc
T1
2
6
6
43
7
7
5
ð7Þ
766 N. Veeramuthulingam et al.
The Eq. (7) is a state space representation of complete linear modeling in which,
linear analysis and control methods can be applied to it.
3 Proposed Methodology
The primary focus owes to elicit the theory of adaptive control through the modeling
equations and assuage measures for regulating the speed of the BLDC motor and
minimizing the ripple in the torque. The principle of an adaptive control system gen-
erates an error from two models using which it allows computing the unknown
parameter. The measured quantity from the output of the adaptive system serves as the
feedback and enables the closed loop stability through the Popov’s Hyper stability
criterion [15].
Depending on the quantity (i.e. the functional candidate), it formulates the error
signal and develops the model reference adaptive system (MRAS) with d-q compo-
nents of flux [11]. It relies on the stator resistance variation and experiences from the
integrator problems like saturation and drift. The MRAS with on-line stator resistance
evaluation reported in [12], reactive power-based MRAS [13,14] and neural network
(NN) based MRAS outline the efforts to attenuate the solutions.
The instability of drive parameters can be fulfilled using MRAC and gainschedling
operation [9], Self-turning [14]. The MRAC with parameter transformation [9] involves
proportional and integral parts of the algorithm and requires iteration for the opti-
mization of tuning of the adaptation algorithm. The most important merit of the MRAS
with signal adaptation is that it does not limit integral parts and essentially turning of
controller parameters used for changing plant parameters [10].
The error between the estimated quantities obtained from the models leaves way to
drive a fit adaptive mechanism for creating the estimated rotor speed [1] and finds the
error and adaptation controller parameters by MIT Rules as seen from Figs. 1and 2.
Fig. 1. Adaptive control system
Modeling of Brushless DC Motor Using Adaptive Control 767
The MIT Rule
The MIT rule methods aims to reduce the squared model cost function and owing to the
error being minimum it forms an accurate tracking between the actual output and the
reference output.
Designing Steps
The adaptation error computed as the difference between the parameter output and the
model output as in Eq. (8)
e¼ypðtÞymðtÞð8Þ
Where, Y
m
,Y
p
is the output of the model, plant
The Eq. (9) shows the cost function J
J¼1
2e2ðtÞð9Þ
The procedure evolves to adjust the parameter hin order that the objective function
can be minimized to zero and as a consequence necessitates hto be in the direction of
the negative gradient of J as seen from the Eq. (10)
dh
dt ¼dJ
dt ¼ce @e
@hð10Þ
The procedure enables the choice of the second order transfer function Eq. (11)
GmðsÞ¼ bm1þbm0
S2þam1sþam0
ð11Þ
The formulations tracks the error in a manner as specified through Eqs. 12 and 13
Fig. 2. Model reference adaptive control simulation block diagram of BLDC
768 N. Veeramuthulingam et al.
e¼rypð12Þ
de
dt ¼dyp
dt ð13Þ
The Eq. (14) represents the control law of system for PI controller
uðtÞ¼kpeðtÞþkiZeðtÞdt ð14Þ
The Eq. (15) gives the Laplace Transform of Eq. (14)
UðsÞ¼KpEðsÞþ Ki
sEðsÞð15Þ
The closed loop transfer function of control law deduces as in Eq. (16)
yp¼GpKprþGpKir
s
1þGpKp
GpKi
s
ð16Þ
By solving for Y
p
in terms of r and substitute Y
p
in Eq. (8), the adaptation error can
be obtained from Eq. (17)
e¼ðGpKpsþGpKiÞr
Sð1þGpKpÞþGpKi
ymð17Þ
The Eq. (18) shows the adaption error view to MIT rules for K
p
,K
i
de
dkpki
¼dyp
dkpki
ð18Þ
The Eq. (20) can be obtained by rewriting the Eq. (17)
e¼GpKprþGpKir
S
1þGpKpþGpKi
S
ð19Þ
e¼GpKprþGpKir
S
1þGpKprþGpKprþGpKir
S
1
ð20Þ
Applying MIT rules for obtaining K
p
, the gradient obtains a form as in Eq. (21)
de
dKp
¼Gpr
1þGpKpþGpKi
S
GpGpKprþGpKir
S
1þGpKpþGpKi
S
2ð21Þ
Modeling of Brushless DC Motor Using Adaptive Control 769
Substituting Eqs. (16)in(21) to get the Eq. (22)
de
dKp
¼Gpr
1þGpKpþGpKi
S
GpYp
1þGpKpþGpKi
S
ð22Þ
The Eq. (23) can be derived by rearranging Eq. (22)
de
dKp
¼GpE
1þGpKpþGpKi
S
ð23Þ
Applying MIT rules for obtaining K
i
, it becomes as in Eq. (24)
de
dKi
¼
Gp
Sr
1þGpKpþGpKi
S
Gp
SGpKprþGpKir
S
1þGpKpþGpKi
S
2ð24Þ
Substituting Eqs. (16)in(24) it gets the form as in Eq. (25)
de
dKi
¼
Gp
Sr
1þGpKpþGpKi
S
Gp
SYp
1þGpKpþGpKi
S
ð25Þ
The Eq. (26) can be obtained by rearranging the above equation
de
dKi
¼
Gp
SE
1þGpKpþGpKi
S
ð26Þ
Under the usual approximations it follows that the parameters fedback relate to the
ideal value and the plant becomes the model reference.
den Gp
1þGpKpþGpKi
s
!
¼s2þam1þam0ð27Þ
Applying MIT rules for adjusting the parameters h
1
,h
2
, and equating it to K
P
and
K
i
. in order that the Eq. (10), gives the adjustment parameters
dh1
dt ¼dkp
dt ¼ke de
dkp
¼ kp
s
es
a0s2þam1sþam2
eð28Þ
dh2
dt ¼dki
dt ¼ke de
dki
¼ kp
s
e1
a0s2þam1sþam2
eð29Þ
Also the second order transfer function of the model reference is given by the
Eq. (30)
770 N. Veeramuthulingam et al.
HmðsÞ¼ 16
S2þ4Sþ16 ð30Þ
4 Simulation
The exercise compares the performance of BLDC motor speed control using MRAC
with the normal PI controller through MATLAB based simulation as reflected in Fig. 3.
The Table 1shows the parameters of BLDC motor.
Table 1. Simulation parameter
S. no Name of parameter Rating of parameter
1. Stator resistance (Rs) 2.8750 ohms
2. Stator inductance (Ls) 8.5e−3H
4. DC voltage (Vdc) 146.6077 volts
5. Rotor flux (k) 0.175
6. Moment of inertia (J) 0.0008 kgm
2
7. Friction (B) 0.001 Nm/rad
8. Poles (P) 4
9. Load Torque (TL) 1.4 NM/RPM
10. Speed (N) 1500 RPM
Fig. 3. Simulation model of BLDC-MRAC
Modeling of Brushless DC Motor Using Adaptive Control 771
The Fig. 4explain the steady state characteristics of the BLDC motor and shows
the ability of the control algorithm to reach the reference speed smoothly.
The Fig. 5depicts the variation of torque in the motor being subjected to sudden
changes in load intervals of 100 percent at 0.2 sec, 200 percent at 0.4 sec, 100 percent
at 0.82 respectively and in each case brings out the feature to settle at the new steady
state operating values.
The Fig. 6exhibit the nature of the control mechanism to enable the speed to
restore back to the reference speed for each of the three sudden changes in load at the
chosen time intervals.
Fig. 4. Speed characteristics of BLDC motor
Fig. 5. Comparison ref speed and actual speed
Fig. 6. Torque characteristics
772 N. Veeramuthulingam et al.
The Fig. 7displays the Speed characteristics of BLDC motor and brings out the
ability to different load condition. Besides the response relates to rejecting the sudden
change in torque of 200 percent and establishes the ability of control algorithm is being
able to settle at the chosen reference speed even at the new operating point.
Figures 8and 9displays the inverter output voltage characteristics and the EMF
characteristics of BLDC motor has respectively at the chosen operating load.
Fig. 7. Stator current characteristics
Modeling of Brushless DC Motor Using Adaptive Control 773
5 Conclusion
The model of the BLDC motor has been developed in the state space framework and
allowed to operate with the usual approximation for responding to operational changes.
The theory of MIT rule has been introduced to linearise the model and enables it to
adapt the system for envisaging the benefits of a MRAC mechanism. The performance
has evaluated using simulation to project the capability of the methodology for
rejecting servo and regulatory disturbances. The speed regulating feature of the scheme
has been the highlight and showcases the strength to replace the drive system is modern
utilities.
Acknowledgement. This publication is an outcome of the R&D work undertaken project under
the Visvesvaraya Ph.D Scheme of Ministry of Electronics & Information Technology,
Government of India, being implemented by Digital India Corporation.
Fig. 8. Inverter output voltage
Fig. 9. Back EMF
774 N. Veeramuthulingam et al.
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