With the novel improvement of using DC motor
model-based in Forward and Inverse Model, the new
IMC system for BLDC speed control is much less amount
of calculation than the classical IMC model based on
BLDC motor. The results show that all the advanced
features of the traditional IMC model are considerably
remained in the developed IMC system such as
robustness with many parameters and coefficients (J, R, L,
Ke
, Kt, β, VDC, TL) changing in wide ranges. In detail, the
ranges of parameter J, R, β can be from a haft to double
of their model values, while the inductance L may vary
within +/- 50% of its model value. The back-EMF and
torque constants can be 20% different from their model
values. Moreover, robustness of the improved IMC
system is also tested under some bad operating conditions
such as 20% decrease of DC voltage and 20% increase of
load torque. These wide ranges are well satisfied the
reasonable changing of practical parameter values when
the motor is operating. These unavoidable changes may
be caused from effect of temperature, electromagnetic
field, measurement errors, etc. Additionally, the DC
model-based IMC system also achieves benefits of IMC
principle such as easily controlling output response by
setting IMC filter coefficient, and simply designing steps.
Notice that, the designing process is even much easier
since using the simpler DC model.
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DC Model-based IMC Method for Brushless DC
Motor Speed Control
Hon H. Trinh and Thinh D. Le
Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Vietnam
Email: {trinhhoanghon09, thinh1113}@gmail.com
Abstract—This paper focuses on a novel idea of using DC
motor model-based to build Forward and Inverse Model of
internal model control (IMC) method for brushless DC
motor (BLDC) speed control. This contribution reduces
large amount of calculation compared to using BLDC motor
model-based but it still achieves all positive characteristics
of IMC method. In detail, the robustness of IMC method for
BLDC speed control is compared between the traditional
BLDC model-based and the novel DC model-based on
MatLAB Simulink environment.
Index Terms—brushless DC (BLDC), internal model control
(IMC), speed control, direct current (DC) motor model-
based, robustness control
I. INTRODUCTION
Nowadays, Brushless DC motors are used in very large
range of application such as industrial automation,
aerospace, automotive, instrumentation and appliance. As
the name described, BLDC motors have no brush like
DC’s, it used electronic commutation instead. Moreover,
there is no slip in BLDC motor because it is a kind of
permanent magnet synchronous motor. With these
advantages, it leads to several other benefits of BLDC
motors such as long life and noiseless operation, high
dynamic response and efficiency, large speed range and
so on. Because of electronic commutation, the input
voltage supply is relative to the rotor position which can
be achieved by 2 common methods. Those are using Hall
sensors [1]-[3] and using back electromotive force (back-
EMF) [4], [5] methods. Since the latter has no sensors, it
is also called sensorless method. About back-EMF
signals, there are some different experiments on its
waveform as sinusoidal [5], [6] and trapezoidal [2], [7],
[8].
There are many BLDC motors applications which
require the angular speed to be stable such as computer’s
hard disk drivers and helicopter robots, etc. However, the
motor behaviors may be changed with time for example,
load torque increase or motor parameters change during
operation like higher resistance when higher temperature.
Furthermore, there are usually some incomplete
information of motors. Therefore, the motor output speed
is needed to be stable under those uncertainties, which
Manuscript received October 7, 2014; revised May 15, 2015.
leads to the concept of robustness control [9]. There were
some researches about this issue such as using model
reference adaptive backstepping approach [10], auto
tunning algorithm [11], and neural net-based [12]. The
internal model control (IMC) which was introduced by
Garcia and Morari in 1980’s [13] and widely applied in
chemical process control is another typical robustness
control method. IMC method is very easily designed
based on system model, well working with linear system,
controlling output response through a simple low-pass
filter, and very robust with wide ranges changing of
parameters. However, it suffers some drawbacks as
problems with non-linear and multi-input multi-output
(MIMO) systems, especially large amount of calculation
[14, 15]. There are researches which simplify the IMC
with phase-lock loop assisting for BLDC speed control
[16, 17].
However, it seems very rare report about solving the
major weakness of IMC method which is large amount
calculating. Hence, this paper describes a solution to
reduce IMC method calculation for BLDC speed control
by using DC model-based instead of BLDC’s. To
compare between the two model-based IMC systems, a
BLDC motor with 3 Hall sensors and sinusoidal back-
EMF is worked with 3-phase sinusoidal inputs voltage. In
addition, the BLDC motor’s parameters and coefficients
are copied from reference [18].
The remaining parts of this paper are arranged as
follows. First, the modeling of BLDC motor is based on
its mathematical equations is expressed in Section II.
Then, in Section III, the detail construction of IMC
method for controlling speed of BLDC motor based on
BLDC model is clearly presented. After that, the Section
IV describes the construction of DC model-based IMC
method for BLDC speed control. Next, the comparison of
robustness between those 2 IMC model-based systems is
shown in Section V. Finally, some conclusions are
claimed in Section VI.
II. MODELING BLDC MOTOR
A. Mathematical Equations of BLDC Motor in Laplace
Domain [8]
The 3 phase currents ia, ib and ic are calculated as
follows:
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doi: 10.12720/joace.4.2.104-110
a a a
a
b b b
b
c c c
c
V E Ri
i
Ls
V E Ri
i
Ls
V E Ri
i
Ls
(1)
where
Va, Vb, Vc stator phase voltage;
R stator winding phase resistance;
L stator winding phase effective inductance;
Ea, Eb, Ec phase back electromotive force (back-EMFs);
s Laplace factor.
Additionally, the back-EMFs are given as:
( )
2
( )
3
4
( )
3
a e m e e m a
b e m e e m b
c e m e e m c
E K F K F
E K F K F
E K F K F
(2)
where
Ke back-EMF constant, voltage constant;
ωm rotor mechanical angular speed;
θe rotor electrical position;
Fa, Fb, Fc commutation functions of electrical rotor
position.
Since the back-EMFs are sinusoidal, the commutation
functions Fa, Fb, Fc are also sinusoidal (see Fig. 1), and
these 3 functions are 1200 out phase of each other.
Figure 1. Commutation functions relative to rotor position.
Next, the electrical torque (Te) is the sum of each phase
torque:
e a b cT T T T (3)
With Kt is torque constant, each phase torque
component Ta ,Tb ,Tc are defined as:
a t a a
b t b b
c t c c
T K i F
T K i F
T K i F
(4)
And the relationship between electrical torque, load
torque, and angular speed is indicated in (5):
e L m mT T Js (5)
where
J motor moment inertia;
β friction coefficient;
ωm rotor angular speed;
TL load torque.
And the mechanical angular speed (ωm) is calculates
from the mechanical angular distance (θm):
m ms (6)
where the mechanical angular distance (θm) is relative to
the electrical angular distance (θe) by number pair of
poles (PP):
.e mPP (7)
B. Modeling BLDC Motor on MatLAB Simulink
The detail construction of a BLDC motor is shown in
Fig. 2 with the Current I_abc block is built from (1).
Similarly, the Torque Te, Omega, Theta_e and Back
EMF E_abc blocks are based on (3), (6), (7), and (2)
relatively. About the F_abc block, it is based on its
definition as mentioned in Fig. 1.
Figure 2. BLDC motor model construction in simulink.
III. MODELING BLDC MODEL-BASEL IMC SYSTEM
FOR BLDC MOTOR SPEED CONTROL
To control the rotor speed of BLDC motor, an IMC
system based on BLDC model is structured as described
in Fig. 3 where the BLDC motor is built in Section II.
Importantly, the Forward ( Pˆ ) and Inverse (Q) are based
on BLDC model. In detail, the Forward Model is kept the
same as the motor model except 3 things. First,
parameters and coefficients are added a subscript M
which is stands for Model to distinguish from the
practical one’s. Second and third, load and friction are
eliminated (TLM=0, βM=0). Similar to the Forward Model,
the Inverse Model also has those 3 different things.
Furthermore, in the Inverse Model, there are some
differential components which may cause extremely high
overshoot at initial. This phenomenon is known as kick-
starter which can be prevented by using a low-pass filter
(with very small TdM coefficient) at each differential
component as presented in (8).
1dM
d s
dt T s
(8)
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Now, the 3-phase voltage and electrical torque are
calculated as (9), (10) relatively, and these 2 equations
are used in Inverse Model.
1
1
1
1
1
1
a M a M a a
dM
b M b M b b
dM
c M c M c c
dM
V R i L si E
T s
V R i L si E
T s
V R i L si E
T s
(9)
1
1
e M m
dM
T J s
T s
(10)
Moreover, inside the Inverse Model block, the
component phase currents are calculated from each phase
torque. The difficulty is calculating 3 torque components
Ta, Tb, Tc from total torque Te via just (3). Therefore, the
phase currents are assumed to be ideal as sinusoidal with
Io amplitude as indicated in (11) since input voltage and
back-EMF are both sinusoidal.
a o a
b o b
c o c
i I F
i I F
i I F
(11)
Hence, combined (11) with (3) and (4) the electrical
torque now becomes:
2 2 2sin ( ) sin ( 2 / 3) sin ( 4 / 3)e t o e e eT K I
1.5e t oT K I
(12)
Equation (12) proves that the BLDC motor with
sinusoidal back-EMF can be equivalent to a DC motor
which has 1.5 times of torque constant (Kt). This is an
important conclusion for using DC model-based instead
of BLDC’s in IMC method, which is clearly described in
Section IV.
In addition, the inverter block (in Fig. 3) can generate
sinusoidal 3-phase voltage supply to the motor depending
on the rotor position. Since input voltage and back-EMF
are both sinusoidal and concern with rotor position, they
are almost the same as indicated in Fig. 4. Fig. 3 also
illustrates the IMC filter which is used to control the
system output response and overshoot. In this project, a
first order low-pass filter is used as IMC filter (see Fig. 5).
There are some experiments with several values of IMC
filter coefficient (Tf), these experiments are done with
the motor parameters in the Table I. The Fig. 6
presents 2 output responses when Tf =0.005 (upper
of Fig. 6(a)) and when Tf =0.5 (upper of Fig. 6(b)).
Comparing these results with the IMC filter response
(lower graphs of Fig. 6), the IMC filter is concluded that
it can be used to control the output response. In other
words, the response of the IMC filter is the expected
output of the system. This is a huge advantage of IMC
method since the filter is very simple. For other
experiments, a mid-value 0.05 is set to be the standard of
Tf.
TABLE I. PARAMETERS AND COEFFICIENTS OF MOTOR IN
SIMULATION [18]
NO. Parameters and Coefficients (sign) Value (unit)
1 IMC filter (Tf) 0.05
2 Differential filter (TdM) 0.001
3 Load torque (TL) 0.03 (Nm)
4 Reference angular speed (𝜔ref) 1400 (RPM)
5 Moment inertia (J) 6.5 × 10-5(Kg.m2)
6 Phase resistance (R) 0.1(Ω)
7 Phase inductance (L) 0.5 (mH)
8 Phase back-EMF constant (Ke) 0.03 (V/(rad/s))
9 Phase torque constant (Kt) 0.03 (Nm/A)
10 DC voltage (VDC) 24 (V)
11 Friction (β) 5 × 10-6(Nm/(rad/s))
12 Simulation time (Tsim) 3 (s)
13 Pairs of pole (PP) 2
Figure 3. BLDC model-based IMC system for BLDC motor
speed control.
(a) 3-phase input voltage
(b) 3-phase back-EMF
Figure 4. 3-phase voltage and back-EMF at transient state.
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Figure 5. IMC filter block in MatLAB Simulink.
(a) Output response with Tf=0.005
(b) Output response with Tf=0.5
Figure 6. Experiment with several Tf values and BLDC model-based,
IMC filter response (lower graphs).
Similarly to the IMC filter, the differential low-pass
filter which is mentioned from (8) to (10) is tested with
some coefficient values of TdM. Comparing the results
when TdM=0.0001 and TdM=0.01 with the expected output
which are shown in Fig. 7, it is seen that the differential
filter does not affect much on the output system.
However, the simulation becomes failed on MatLAB
Simulink if the filter coefficient TdM is zero. Therefore,
standard value of TdM is set to be 0.001 for later
experiments.
(a) Output response with TdM=0.0001
(b) Output response with TdM=0.01
Figure 7. Experiment with several TdM values and BLDC model-based,
IMC filter response (lower graphs).
IV. MODELING DC MODEL-BASEL IMC SYSTEM FOR
BLDC MOTOR SPEED CONTROL
The DC motor model is described by just only 4
equations, which are much less than BLDC’s. Since the
DC model is used in the Forward and Inverse Model (see
Fig. 8), all parameters and coefficients are also with
subscript M and load, friction are simplified. The DC
current is:
a M
M
V E R i
i
L s
(13)
With the armature back-EMF Ea:
a eM mE K (14)
As mentioned in the 12, Section III. The torque is
calculated as:
1.5e tDC tMT K i K i (15)
And rotor angular speed (ωm) is:
e
m
M
T
J s
(16)
The Forward Model based on DC motor is built from those 4
equations above. Similar to the BLDC model-based, the Inverse
Model also needs differential filter as calculating of voltage (17)
and torque (18):
1
1
M M a
M
V R i L si E
T s
(17)
1
1
e M m
M
T J s
T s
(18)
Figure 8. DC model-based IMC system for BLDC motor
speed control.
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It is clear that the number of equations is much less
than so the calculating amount is much reduced by using
DC model-based instead of BLDC’s in Forward and
Inverse Model. With the same motor parameters as in
Section III, two experiments with IMC filter coefficient
Tf are done and the results are presented in Fig. 9.
Comparing this figure with Fig. 6, it can be easily seen
that the IMC filter in the DC model-based system plays
the same role as in BLDC model-based system. Hence,
the output response is simply controlled by the IMC filter
coefficient in both model-based IMC systems. In addition,
the experiments with differential filter coefficient TdM
(see Fig. 10) with DC model-based system give the same
results as the IMC system based on BLDC (see Fig. 7).
That means, the TdM value should be considerable small
but not zero, otherwise it causes failed of the simulation
on MatLAB Simulink environment.
(a) Output response with Tf =0.005
(b) Output response with Tf=0.5
Figure 9. Experiment with several Tf values and DC model-based,
IMC filter response (lower graphs).
(a) Output response with TdM=0.0001
(b) Output response with TdM=0.01
Figure 10. Experiment with several TdM values and DC model-based,
IMC filter response (lower graphs).
V. ROBUSTNESS EXPERIMENT OF BLDC AND DC
MODEL-BASED IMC SYSTEM
To know how good the performance of the new
improved system, the classical IMC for BLDC speed
control with BLDC model-based is compared with the
novel system based on DC motor model in this Section V.
Robust control can keep the system remain stable of
differences between practical and model parameter values.
Let denoted all model parameters with the subscript M.
Hence, those robustness experiments are made as
expressed in Table II with the external disturbance as
load change and reference output response as shown in
Fig. 11. Under those situations, in this project, the system
is considered stable if its output is remained within +/-
5% (upper and lower dash line in the lower graph of Fig.
11) of the expected output value which is 1400 round per
minutes (RPM). In addition, the experiment results are
from Fig. 12 to Fig. 19.
TABLE II. PARAMETERS AND COEFFICIENTS RANGES OF
ROBUSTNESS EXPERIMENTS
NO.
Parameters and Coefficients
(sign)
Upper
Limit
Lower
Limit
1 Moment inertia (J) 2JM 0.5JM
2 Phase resistance (R) 2RM 0.5RM
3 Phase inductance (L) 1.5LM 0.5LM
4 Phase back-EMF constant (Ke) 1.2KeM 0.8KeM
5 Phase torque constant (Kt) 1.2KtM 0.8KtM
6 Friction (β) 2β 0.5β
7 DC voltage (VDC) -20%
8 Load Torque (TL) +20%
According to the Table II, the order of robustness
experiments and comparison between 2 IMC model-
based systems are as follows. First, the output responses
of those 2 systems when practical and model parameters
are matched are described in Fig. 12. Second, the
comparison of robustness with J when practical moment
inertia are double and a half of its model value in
presented in Fig. 13. Third, the Fig. 14 illustrates the
results of 2 model-based systems output response when
R=2RM and R=0.5RM. Next, the sensitive parameters L is
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supposed to be increased 50% and decrease 50% around
its model value, the 2 systems output responses are
clearly indicated in Fig. 15. After that, the 2 systems
robustness are compared when back-EMF constant is
20% higher and lower than its model value, this leads to
the involving of torque constant Kt in the experiment
because the two constant are equal, and their results are
seen in Fig. 16.
Figure 11. External load disturbance (upper graph) and reference
output response (lower graph).
(a) BLDC model-based system (b) DC model-based system
Figure 12. Output response (upper graphs) of 2 model-based IMC
systems when practical and model parameters are equal.
(a) BLDC model-based system (b) DC model-based system
Figure 13. Output response when J=2JM and J=0.5JM.
(a) BLDC model-based system (b) DC model-based system
Figure 14. Output response when R=2RM and R=0.5RM.
(a) BLDC model-based system (b) DC model-based system
Figure 15. Output response when L=1.5LM and L=0.5LM.
(a) BLDC model-based system (b) DC model-based system
Figure 16. Output response when Ke=1.2KeM, Kt=1.2KtM and
Ke=0.8KeM, Kt=0.8KtM.
Practically, friction is very hard to determine and
easily changed depending on operating environment.
Therefore, the 2 model-based IMC systems are assumed
to work with double and a half friction for robustness test.
The test results are shown in Fig. 17. In reality, a BLDC
motor may work under bad conditions such as the battery
is running out of power (suppose 20% DC voltage lower)
or the load torque is higher than the nominal load
(suppose 20% load increase). The results of 2 IMC
systems robustness under these 2 bad situations are given
in Fig. 18. Finally, let all parameters and coefficients in
Table II are at their upper and lower values combined
with the 2 bad working conditions to make the total
robustness experiments of 2 IMC model-based systems,
the results of output speed response are described in Fig.
19.
(a) BLDC model-based system (b) DC model-based system
Figure 17. Output response when double and half friction.
(a) BLDC model-based system (b) DC model-based system
Figure 18. Output response when 20% DC voltage decrease
and 20% load torque increase.
(a) BLDC model-based system (b) DC model-based system
Figure 19. Output response when 20% DC voltage decrease,
20% load torque increase and parameters are at upper and
lower limits.
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From those output angular speed results of 2 IMC
systems from Fig. 12 to Fig. 19, it is clearly seen that the
outputs always stay within +/- 5% criteria around the
reference value under the disturbance external load even
though many parameters are mismatched with their
model values in very wide ranges. That means, the 2 IMC
model based systems are equally well robust with those
many parameters and working conditions.
VI. CONCLUSION
With the novel improvement of using DC motor
model-based in Forward and Inverse Model, the new
IMC system for BLDC speed control is much less amount
of calculation than the classical IMC model based on
BLDC motor. The results show that all the advanced
features of the traditional IMC model are considerably
remained in the developed IMC system such as
robustness with many parameters and coefficients (J, R, L,
Ke, Kt, β, VDC, TL) changing in wide ranges. In detail, the
ranges of parameter J, R, β can be from a haft to double
of their model values, while the inductance L may vary
within +/- 50% of its model value. The back-EMF and
torque constants can be 20% different from their model
values. Moreover, robustness of the improved IMC
system is also tested under some bad operating conditions
such as 20% decrease of DC voltage and 20% increase of
load torque. These wide ranges are well satisfied the
reasonable changing of practical parameter values when
the motor is operating. These unavoidable changes may
be caused from effect of temperature, electromagnetic
field, measurement errors, etc. Additionally, the DC
model-based IMC system also achieves benefits of IMC
principle such as easily controlling output response by
setting IMC filter coefficient, and simply designing steps.
Notice that, the designing process is even much easier
since using the simpler DC model.
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Hon H. Trinh was born in Dong Nai, Vietnam, in
1973. He received B.E and M.E degrees from
Electrical Electronic Engineering of Ho Chi Minh
city University of Technology, Vietnam, in 1997
and 2002 respectively. He received Ph.D. degree
in computer vision area at Electrical Engineering
Department from University of Ulsan, Korea in
2008. He is currently a lecturer in Faculty of
Electrical and Electronic Engineering, Ho Chi
Minh city University of Technology, Vietnam. His research interests
include computer vision, pattern recognition, understanding and
reconstructing outdoor scenes, designing the outdoor mobile robot for
civil and military applications, electrical machinery, controlling
electrical machinery, robotics, Human Computer Interaction (HCI),
pattern recognition.
Thinh D. Le was born in Hue city, Vietnam, in
1991. He is currently an undergraduate in Faculty
of Electrical and Electronic Engineering, Ho Chi
Minh city University of Technology, Vietnam.
His research interests include power electronic,
computer vision, and modern control.
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