Development of a robust scalar control system for an induction
squirrel-cage motor based on a linearized vector model
TSELIGOROV N.A., OZERSKY A.I., CHUBUKIN A.V., TSELIGOROVA E.N.
Department of Computing Systems and Information Security
Don State Technical University
1, Gagarin square, Rostov-on-Don, 344000
RUSSIA
Abstract: The paper considers the problem of developing a digital system for an induction motor speed control which has
a sensor and a speed regulator to increase accuracy of speed control. Speed control is carried out by a scalar method due to
consistent change in the stator frequency and voltage. To obtain the uniformity of the motor overload capability in a given
range the control mode is used associated with maintaining uniformity of flux linkage of the motor stator. Induction motor
scalar models do not possess high accuracy and their parameters and their parameters can vary over a wide range, which
complicates the controller design and achievement of robustness of the speed control system. To eliminate these
disadvantages, it is proposed to use a vector model in a rotating coordinate system having subjected it to linearization at
different points of the operating mode with the account of the adopted law of frequency control, to ensure robust absolute
stability of the system on the basis of application of a graphical method for constructing a modified amplitude-phase
characteristic.
Key-Words: stability, robust absolute stability, interval polynomial, induction motor; V/f control; PID controller
Received: June 25, 2021. Revised: October 31, 2021. Accepted: November 15, 2021. Published: January 3, 2022.
1 Introduction
A modern frequency controlled electric drive of
common application consists of an induction
squirrel-cage electric motor and a static frequency-
converter (SFC) with a DC link. The frequency
converter generates voltage variable in frequency and
amplitude from the constant voltage of the DC link.
A change in the frequency of the voltage and its
amplitude results in a change of rotating frequency of
the stator magnetic field and, as a consequence, to a
change in the shaft rotational speed of the electric
motor. At present, the following laws of frequency
control of an electric drive are known:
– scalar control;
– field-oriented control (FOC);
– direct torque control (DTC).
2 Problem Formulation
The following papers are known that make it
possible to perform frequency control of the electric
drive using scalar control [1-11]. As it is known,
scalar models do not possess high accuracy and their
parameters can change over a wide range which
complicates the controller design and achievement of
robustness of the speed control system. Field oriented
frequency control (FOC) of the induction motor [12-
23] is the most promising and often used in modern
industry. Diagrams of automatic control for an
alternating current electric drive with discontinuous
control that got the name “systems with direct torque
control” (DTC) [24-26] are also widely used. The
best static and dynamic characteristics within this
method belong to a direct torque control using
controllers on the basis of artificial neuron networks
(ANNDTC) [27-29].
3 Problem Solution
Usually, modern frequency converters make it
possible to implement several laws of electric motor
control, for this purpose, a software switching of the
known laws is factored in them. Despite the success
in the field of creating highly dynamic electric drives
on the basis of the field-oriented control (FOC) and
DTC, scalar control systems have not lost their
importance due to simplicity of implementation and
adjustment [30, 31]. Scalar control systems do not
always require identification of accurate parameters
of the induction motor substitute diagram. A scalar
control diagram is based, as a rule, on the consistent
control of frequency and voltage of the induction
motor stator.
The generation of the required static and dynamic
properties of the induction frequency-controlled
electric drive is only possible in a closed control
system of its coordinates. The functional diagram of
the speed control system with maintaining uniformity of
flux linkage of the stator in a steady state is presented in
fig.1.
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Tseligorov N. A., Ozersky A. I., Chubukin A. V., Tseligorova E. N.
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K
Fig.1 Functional diagram of the automatic control
system (ACS) of the induction motor speed with the
stator uniform flux linkage
The induction squirrel-cage motor M is powered
from the power network through an uncontrolled
rectifier with a capacitive filter and a frequency
converter (FC) controlled with the help of pulse-
width modulator (PWM). Signals at the input of
PWM: UН controls the amplitude of the phase
voltage, and Uf controls its frequency. The reference-
input signal UЗС is given through the power-up sensor
(PUS) to the comparison element, the second input
receives the signal UОС from the speed sensor (SS),
at the output of the speed controller there appears
voltage that is proportional to the rotor slip frequency
UРС and coming to the input of the functional
converter FC which implements the required
dependency of the voltage amplitude at the stator
windings of the induction motor on the current slip
frequency of the IM rotor.
3.1 Calculation of the vector model
parameters of an induction motor
Let us make a vector model of the induction motor
and determine its parameters on the basis of the
substitution diagram of its phase. The vector model
of the IM in the rotating synchronous system of
coordinates is presented by the following system of
equations:
(1)
To calculate the parameters of the vector model
for the induction motor with a squirrel-cage rotor,
АИР90 type, we will use the rated data and
parameters of the substitution diagram:
Рн=2.2 kW – the rated power, nн=1420 rev/min – the
rated rotational speed, ηн=0.81 – the rated efficiency,
соsφн= 0.83 – the rated cosine φ, sн = 0.053 – the
rated slip, sк = 0.321 – critical slip, R1 = 2.852 Ohm
– resistance of the stator phase winding, R2 = 2.785
Ohm – resistance of the rotor phase winding, Lσ1 –
leakage inductance of the stator phase winding, Lσ2 -
leakage inductance of the rotor phase winding, Lm –
magnetizing inductance of phase windings of the
stator and the rotor.
L1= Lσ1 + Lm, L2= Lσ2 + Lm are inductances of the
stator and rotor phase windings.
The calculated parameters of the IM vector model
are equal to:
,
As the vector model is made for the rotating
synchronous system of coordinates, for the purpose
of its simplification we assume U1β = 0, а U1αα=U1m.
To maintain uniformity of flux linkage of the stator,
it is necessary to adopt the law of frequency control
М
SS
FC
PUS
Upus
(+)
PWМ
USС
SС
(–)
UFC
UОС
Uf
FC
2
1 1 1 0 2
1 1 1
2
1 1 0 1 1 2
11
1
2 1 2 2
22
1
2 1 2 2
22
1
2 1 1 2
2
0
11
1
1
1
3
2
1
l
l
p
p
дп
дc
plп
k
p U ;
T T T
k
p U ;
TT
k
p;
TT
k
p;
TT
k
M p ( );
L
p ( M M );
Jp
p.
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in the form of the following dependence of the
voltage amplitude at the stator U1m = αω0д+βωр. The
angular frequency of the electrical power supply
voltage can be expressed in terms of the motor speed
and frequency of the rotor EMF as ω0д=рпω+ωр.
Having substituted the expressions obtained for
U1m and U1β into the equations of the IM vector model
and having carried out the necessary transformations,
we will get a vector model of the IM speed control
system where ω0l represents an input signal and ω is
an output value. The vector model obtained in this
way can be written in a vector-matrix form, but it is
non-linear due to the availability of the product of
variables. Therefore, it is inapplicable in the obtained
form for the speed controller design and needs
linearization which can be carried out by the way of
going to increments of values relative to their initial
values at different points of the operation modes.
ψ1α= ψ1αn + Δψ1α; ψ1β= ψ1βn + Δψ1β; ψ2α= ψ2αn + Δψ2α;
ψ2β= ψ2βn + Δψ2β; ω=ωn+Δω; ; ωр=ωрn+Δωр.
After the corresponding transformations we
obtain the linearized matrix equation of the form:
A=
11
1 0 2 1 1
11
0 1 2 1 1
11
1 2 2
0
0
00
0
lпn
lпn
рn
T k T p
T k T p
k T T
( ) ( )
( ) ( )
( ) ( )
11
1 2 2
2 2 1 1
0
0
рn
м n м n м n nм
k T T ( ) (
k / J k / J k / J k / J
)
x= [Δψ1αΔψ1βΔψ2αΔψ2βΔω]T;
B=[ ψ1βn +α+β - ψ1αnψ2βn - ψ2αn 0];
C=[0 0 0 0 1].
The frequency control law for an induction motor
used in an electric drive has the following expression
U1m = αω0l+βωр, где α=0.902, а β=1.05. Let us
assume the range of motor speed control equal to 10.
We assign the voltage frequency ω0'ln=31, 130 and
314 1/s and the rotor frequency ωр=0 and 50 rad/s. To
determine the initial values of flux linkages of the
stator and the rotor for the selected nominal design
points, it is necessary to solve the matrix equation for
the static mode obtained from the considered
matrices at р=0: x= -A-14*4B4*1U1m, where A4*4 is a
square matrix of the first four columns and four rows
highlighted in the matrix А, В4*1=[1 0 0 0]T.
3.2 Investigation of a continuous model
of an induction motor
Given the voltage frequency at the stator of an
induction motor and the slip frequency within the
specified ranges, initial values of flux linkages of the
stator and the rotor are calculated.
Using the values of flux linkages in the matrix А5*5
and taking into account the vectors B, C, we find a
transfer function of the linearized model of the
induction motor in the form of zpk.
For example, for the given ω0'ln=250 1/s and ωр=0 1/s,
the found transfer function has the following form:
р
р
Taking into consideration that ωр= ω0'l - рпω, we can
develop a linearized model of the induction motor, the
input of which is ω0'l , and the output is ω. The structural
diagram of this model is presented in fig.2.
Fig.2. Structural diagram of the system with
input ω0'l and output ω
Accordingly, the transfer function of such a
system is determined by the equation
3.3 Investigation of a discrete model of
an induction motor
With the purpose of further design of the digital
speed controller, we convert the obtained continuous
transfer function into a discrete form taking into
account the availability of a zero-order extrapolator
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and the discreteness period of Т=0.01s. As a result,
we obtain the following discrete transfer function:
р
We will carry out a discrete PID-controller design
with the help of the pidtool program, while in order
to achieve robustness we ensure the maximum
possible phase margin in an open digital system. The
synthesis is based on the transfer function (4) as
among the obtained linearized transfer functions it
possesses the highest gain factor. As a result of the
controller design, we will determine the controller
transfer function of the following form:
Let us find a discrete transfer function of the open
speed control system
The transfer function in the calculated linearized
contour is presented in fig.3.
Fig.3. Transfer function of the closed linearized
contour of speed
As it is presented in figt.3, we can see that the
transition process of the induction motor discrete
model has been carried out without over-shoot in a
time of 0.15s.
3.4 Investigation of robust absolute
stability of control system of AC drives
A fair number of publications are devoted to the
investigation of robustness of control system of AC
drives. In the papers [32 - 34] a robustness solution
on the basis of model reference adaptive system
(MRAS) was proposed for simultaneous
consideration of stator resistance (Rs) and rotor
rotation speed (ω), that do not affect the drive
characteristics. In [35-37] to solve this problem the
MRAC approach was used making it possible to
measure the rotor speed without sensors. The
robustness of vector control for the double fed
induction motor (DFIM) mode was investigated in
[38]. The application Lyapunov functions made it
possible to implement stability of the robust
nonlinear feedback control. In the papers [39,40] a
sensorless control of induction motor speed was
applied. Due to the use of genetic algorithm in a
speed controller with fuzzy logic, the problem of
robustness was solved in [41]. The use of neuron
network was proposed in [42]. In [43] and [44] this
problem is investigated for changing time constant of
the rotor and for load disturbances. With regard to
changes in the parameters of the system, an
investigation was carried out in [45].
Let us consider the system under study as a
nonlinear pulse control system (NPCS).
3.5 Mathematical model for analysing
absolute stability of NPCS
To investigate NPCS, we will use a criterion of
absolute stability that is written in the following way
[46]
, (7)
where ν is a pseudo-frequency.
After the transformation, (7) can be presented in
the form
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),()()( xhBxAxP
(8)
where A(x), B(x) are polynomials, h=ck, x=
2
.
Taking into consideration that the transfer
function of the system under study has interval real
coefficients, then the equation (7) can be represented
as real interval polynomials of the form [22]
,
, (9)
making it possible, with their help at the variation of
h parameter, to construct a root locus [47, 48] with
interval coefficients.
Usually, when using interval polynomials,
Kharitonov’s strong theorem is applied [49], in which
it is proved that necessary and sufficient condition for
robust absolute stability of interval polynomials (8)
is that four Kharitonov’s polynomials are Hurwitz
polynomials.
3.6 Simulation of a discrete control
system of an induction motor
To estimate robust absolute stability of
the system it is necessary to perform w-
transformation of the initial transfer function
represented in z-form (6) into a transfer function
represented in a w-form with nominal
coefficients, which after the w-transformation is
written in the form
(10)
We transform this transfer function into a
transfer function with interval coefficients, the
values of which differ from the coefficients of
the transfer function with nominal values (10) by
10%. After the transformation (10) will have
the following form
. (11)
Obtaining an interval function (11) is done
under the assumption that endogenous and
exogenous changes in the parameters of the
system under study are within the range of
±10%, which is reflected on the values of the
transfer function coefficients [47].
Applying the method of root locus, we will
construct a locus for the transfer function (10),
presented in fig.4.
Fig. 4. Root locus of the transfer function of the
induction drive with nominal coefficients
It can be seen from the presented figure that
the trajectories of branches of the root locus do
not fall on positive real axis and therefore, the
system under study is absolutely stable.
Let us carry out an investigation of the control
system using a modified method of the root locus
[50], which allows us to graphically represent
and estimate the robust absolute stability. The
root locus, constructed for the transfer function
(11), is presented in fig.5.
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Fig. 5. Modified root locus of the transfer function
of the induction drive with interval coefficients
It can be seen from fig.5 that the blurred branches
of the root locus do not fall on the real positive axis
and therefore, the system under study is robustly
absolutely stable.
4 Conclusion
The proposed approach to the implementation of
the control system has shown that the possibility of
applying a vector model of an induction motor,
obtaining PID-controller ensuring robust absolute
stability of the discrete system with the parameters
spread of ± 10% has been demonstrated.
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DOI: 10.37394/23205.2022.21.1
Tseligorov N. A., Ozersky A. I., Chubukin A. V., Tseligorova E. N.
E-ISSN: 2224-2872
8
Volume 21, 2022
Simulation, Saint Petersburg, Russia, 2014, pp.
75-80.
[49] Kharitonov V. L. Asymptotic stability of
an equilibrium position of a family of systems of
linear differential equations. Differential
Equations. 1978.Vol. 14, pp. 2086-2088.
[50] Tseligorov N.A. Modeling complex
"Sustainability" for the study of robust absolute
stability of nonlinear impulsive control systems.
In Proceeding of the Computer Modeling and
Simulation, Saint Petersburg, Russia, 2012, pp.
9-14.
Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Author Contributions:
Tseligorov N.A. carried out investigation of the
control system of the induction drive for stability
Chubukin A.V. developed a linearized model of
the induction drive
Tseligorova E.N., Ozersky A.I. performed
simulation.
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DOI: 10.37394/23205.2022.21.1
Tseligorov N. A., Ozersky A. I., Chubukin A. V., Tseligorova E. N.
E-ISSN: 2224-2872
9
Volume 21, 2022
