1. ISSN: 2277 – 9043
International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
Direct torque control of induction motor using
discrete events of a hybrid system
B. M. Manjunath, A. vinay Kumar
 comparators, a flux and torque estimator and a voltage
Abstract— In this paper, the direct torque control (DTC) is vector selection table. The torque and flux are controlled
employed for fast and slow torque and flux control of induction simultaneously by applying suitable voltage vectors, and
motor coupled to an inverter (Inv-IM). This paper describes a
combination of direct torque control (DTC) and space vector by limiting these quantities within their hysteresis bands,
modulation (SVM) for an adjustable speed sensor-less induction de-coupled control of torque and flux can be achieved.
motor (IM) drive. The motor drive is supplied by a two-level
inverter. The inverter reference voltage is obtained based on
However, as with other hysteresis bases systems, DTC
input-output feedback control, using the IM model in the stator drives utilizing hysteresis comparators suffer from high
– axes reference frame with stator current and flux vectors torque ripple and variable switching frequency. . The
components as state variables. We first model the DTC of torque and flux are controlled simultaneously by applying
Inv-IM as a hybrid system (HS). Then, we abstract the suitable voltage vectors, and by limiting these quantities
continuous dynamics of the HS in terms of discrete events. We within their hysteresis bands, de-coupled control of
thus obtain a discrete event model of the HS. And finally, we use torque and flux can be achieved. However, as with other
Supervisory Control Theory of discrete event system (DES) to hysteresis bases systems, DTC drives utilizing hysteresis
drive Inv-IM. comparators suffer from high torque ripple and variable
switching frequency. The most common solution to this
Index terms- direct torque control (DTC), discrete event
problem is to use the space vector modulation depends on
system (DES), inverter coupled induction motor
(INV-IM), supervisory control theory (SCT). the reference torque and flux. The main advantages of
SVM-DTC are minimal torque response time and the
absence of coordinate-transform, voltage modulator
I. INTRODUCTION block, controllers such as PID for flux and torque for
these advantages, DTC is the control method adopted in
The main advantage of Induction motors (IM) is that no this paper.
electrical connection is required between the stator and the We propose a three-step method to model the DTC of
rotor, they have low weight and inertia, high efficiency and a Inv-IM. In a first step, we model the DTC of Inv-IM as a
high overload capacity [1]. There exist several approaches to hybrid system (HS) with a discrete event dynamics
drive an IM. Induction motor control methods can be defined by the voltage vectors used to control IM; and a
broadly classified into scalar control and vector control. continuous dynamics defined by continuous equations on
In scalar control, V/F control is the important control the stator flux vector(Φs) and the electromagnetic
technique, it is the most widespread, reaching torque(Г).
approximately 90% of the industrial applications. The Hybrid system in the sense that it consist of discrete
structure is very simple maintaining a constant relation component (inverter) and the continuous component
between voltage and frequency and it is normally used (induction motor).
without speed feedback, hence this control does not In a second step, we abstract the continuous dynamics of
achieve a good accuracy in both speed and torque the HS in terms of discrete events. Some events are used
responses mainly due to the fact that the stator flux and to represent the entrance and exit of the torque Γ and the
the torque are not directly controlled. Vector control is a amplitude Φs of in and from a working point region.
technique that can reach a good accuracy, but its main
And some other events are used to represent the passage
disadvantage is the necessity of a huge computational
of the vector Φs between different zones. By this
capability and of a good Identification motor parameters.
abstraction, the continuous dynamics of the system IM
The method of Field acceleration overcomes the
is described as a discrete event system (DES).
computational problem of vector controllers by achieving
In a third step, we use Supervisory Control Theory
some computational reductions [2][4]. And the technique of
(SCT) to drive Inv-IM.
Direct Torque Control (DTC) been developed by Takahashi
[5][6][7][8] permits to control directly the stator flux and the
II.INVERTER AND ITS DISCTERT EVENT
torque by using an appropriate voltage vector selected in a
MODEL
look-up table. And the technique of Direct Torque Control
(DTC) been developed by Takahashi [5][6][7][8] permits to
The inverter (Fig.1) is supplied by a voltage Uo and
control directly the stator flux and the torque by using an
contains three pair of switches for i = a, b, c.
appropriate voltage vector selected in a look-up table. The
The input of the inverter is a three-bit value (Sa Sb Sc)
conventional DTC drive contains a pair of hysteresis
where each Si can be set to 0 or 1. A value 0 of Si sets
to (close, open), and a value 1 sets it to (open,
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2. ISSN: 2277 – 9043
International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
A. Model of torque and flux
close). The output of the inverter is a voltage vector Vs
that drives IM. The inverter vector s is determined by With DTC, the voltage vector VS generated by the
Equation (1), and thus, depends uniquely on Uo and (Sa inverter is applied to the IM to control the flux Φs and
Sb Sc). the torque Γ. Let us first see how and Φs, Γ can be
expressed. In a Stationary reference frame, the flux
s = ]… vector Φs is governed by the differential Eq. (3), where
(1) Rs is the stator resistance and Is is the stator current
The inverter can produce eight vectors k = 0, 1,…7) vector. Under the assumption that , is negligible
corresponding respectively to the eight possible w.r.t VS (realistic if the amplitude of Φs is sufficiently
values(0=000 to 7=111) of (Sa Sb Sc).from equation high); we obtain Eq. (4) which approximates the
(1),we obtain easily equation (2) that computes the eight evolution of Φs from Φso after a delay t.
vectors , k=1,2,..7. Note that v0 and v7 are null.
Figure (2) represents the six non-null voltage vectors in
the D-Q axes which represent the stationary reference
frame fixed to the stator.
B. evolution of flux and torque
Eq. (4) implies that the application of a vector voltage
generates a move of the end of Øs in the direction
of . Note that consists of a radial vector
(parallel to ) and a tangential vector
(orthogonal to ). Increases (resp. decreases) the
flux Øs (i.e., the amplitude of ) if it has the same
(resp. opposite) direction of . rotates
clockwise (resp. counterclockwise) if the angle from
to is +π/2 (resp. –π/2). From Equation (5). We
deduce that increases (resp. decreases) the torque Г
The inverter can be modeled by a 8-state automaton if the angle from to is π/2 (resp. –π/2).
who’s each state qk (k = 1... 7) means: “Vk is the current
voltage vector”. To adopt the terminology of hybrid
systems, the term mode will be used as a synonym of
state. The transition from any mode q⋆ to a mode qk
occurs by an event Vk which means “starting to apply
Vk”.
III. INDUCTION MOTOR AND ITS CONTINUOUS
MODEL
The induction motor is a continuous system because its
behavior is modeled by algebraic and differential
equatons on two continuous variables, the stator flux
(Φs) and the electromagnetic torque(Γ).
Figure 3 illustrates the evolution of when have
the same direction as and the angle from to is
+90 degrees. Therefore, in this example both the flux Øs
and the torque Г increase. As proposed in [5] to divide
the possible global locus of into the six zones Z1,
Z2... Z6 of Fig. 4. Table I shows how the flux
magnitude Øs and the torque Г evolve when is in Zi
(i = 1,.6) under the control of each of the eight vectors
(k = 0, 7, i − 2 · · · i+3), where indices are defined
modulo 6 (and not modulo 8). Symbols ↑, ↓ and = mean
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3. ISSN: 2277 – 9043
International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
“increases”, “decreases” and “is constant”, respectively. and under the control of i-2, i-1, i+1, i+2, 0
We see that under the control of i-2, i-1, i+1, i+2, and 7, the evolution of Øs and Г is thus the one already
0 and 7, the evolution of Øs and Г is known. But indicated in Table I or Zi. Table II shows the evolution
vectors i and i+3 are problematic because they can of Øs and Г in zone Zi,j under the control of i and i+3.
both increase and decrease the torque Г in the same
zone Zi, depending if Øs is in the first or the second 30
degrees of Zi. This problem will be called
non-determinism of the six-zone division.
IV. MODELING OF IM AS DES BY ABSTRACTING
ITS CONTINUOUS DYNAMICS
Let us show how the continuous dynamics of IM
presented in Sect. III is abstracted in terms of discrete
events. The first abstraction consists in translating by
events the entrance and exit of (ØS, Г) in and from a
working point region. The second abstraction consists in
translating by events the passage of the vector
C. solving the non-determinism of six-zone division between orientation zones.
In this two approaches were proposed to solve the A. Abstracting the entrance and exit of (Øs, Г)
non-determinism of the 6-zone division. The first
approach is based on the observation that the Let Øwp and Гwp the flux magnitude and the torque
non-determinism occurs when is in a zone Zi while defining the targeted working point. That is, the aim of
control will be to drive IM as close as possible to (Øwp,
one of the control vector i or i+3 is applied A solution
Гwp). We define a flux interval [Øwp-, Øwp+] centered in
is to leave non-determinism as soon as it appears, by
Øwp, and a torque interval [Гwp, Гwp] centered in Гwp. We
applying a control vector different from i and i+3.
partition the space of (Øs, Г) into sixteen regions Ru,v
We suggest to select the control vector to be applied
for u, v = 1, 2, 3, 4, as shown in Fig. 6. The objective of
among the four control vectors i-2, i-1, i+1, i+2 the control will be to drive IM into the set of regions
because these four vectors permit to obtain all the {Ru,v : u, v = 2, 3} (shaded in Fig. 6) and to force it to
combinations of the evolution of (Øs, Г) (see Table I). remain into this set. We define the event that
represents a transition from Ru,v to Ru′,v for any v, and the
event that represents a transition from Ru,v to Ru,v′
for any u. Since only transitions between adjacent
regions are possible, the unique possible events are the
following: if u<4, if u>1, if v<4,
if v>1. With the above abstraction, the evolution of (Øs,
Г) can be described by a 16-state automaton, whose
states are noted (u, v) and correspond to the sixteen
regions Ru,v, u, v = 1, 2, 3, 4. The transitions between
states occur with the events defined
above , , , .
B. Abstracting the passage of between orientation
zones
A second approach to solve the non-determinism is to
use twelve zones by dividing each of the six zones Zi In Sec. III-B and III-C, we have shown how to partition
into two zones Zi,1 and Zi,2 comprising the first and the
the global locus of into six or twelve zones (Figs. 4
second 30 degrees, respectively [13], [1]. Figure 5
and 5). This partitioning is very relevant, because we
represents the twelve-zone division. In each zone Zi,j
have seen that from the knowledge of the current zone
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4. ISSN: 2277 – 9043
International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
occupied by . interpreting Table I, we determine the transitions of Mk
We can determine the control vector to be applied as follows, where Vk is the control vector currently
applied by the inverter. From state (u, v, i)k of Mk :
for obtaining a given evolution of (Øs, Г) (Tables I and
i′ • The event can occur when u < 4 and ØS
II). With the 6-zone partition, we define the event Zi
increases, i.e., when k is equal to one of the following
that represents a transition from Zi to Zi’. Since only values: i−1, i+1, i, 0 if i is odd, 7 if i is even. This event
transitions between adjacent zones are possible, the
leads from (u, v, i)k to (u+1, v, i)k.
unique possible events are the following: , ,
where i-1and i+1 are defined modulo 6. We can thus • The event can occur when u > 1 and ØS
decreases, i.e., when k is equal to one of the following
abstract the evolution of by a 6-state automaton,
whose states are noted (i) and correspond to the zones values: i−2, i+2, i+3,0 if i is odd. This event leads
Zi, i = 1…6. The transitions between states occur with from (u, v, i)k to (u−1, v, i)k.
the events defined above: , , We can use the • The event can occur when v < 4 and Γ increases,
same approach with the 12-zone partition, by defining i.e., when k is equal to one of the following values: i+1,
the event that represents a transition from Zi,j to i +2, i, i+3. This event leads from (u, v, i)k to (u,
Zi′,j′ . Since only transitions between adjacent zones are v+1, i)k.
possible, the unique possible events are the • The event can occur when and v > 1 and Γ
following: , , , , where i−1 and i+1 decreases, i.e., when k is equal to one of the following
are defined modulo 6. We can thus abstract the values: i−2, i−1, i, i+3. This event leads from (u, v,
evolution of by a 12- state automaton, whose states i)k to (u, v−1, i)k.
correspond to the zones Zi,j , i = 1 ···12 and j = 1, 2. • The event can occur when rotates clockwise,
i.e., when Γ increases, i.e., when k is equal to one of
C. modeling IM as a DES i+1the following values: i+1, i+2, i, i+3. This event
leads from (u, v, i)k to (u, v, i+1)k.
In Sec. IV-A, we have shown how to abstract the
evolution of , Г) by a 16-state automaton. In Sec. • The event can occur when rotates counter
IV-B, we have shown how to abstract the evolution clock-wise, i.e., when Γ decreases, i.e., when k is equal
to one of the following values: i−2, i−1, i, i+3. This
of by a 6-state or 12- state automaton. In the sequel,
we consider uniquely the 6- state automaton because it event leads from (u, v, i)k to (u, v, i−1)k.
reduces the state space explosion which is inherent to Due to the non-determinism of Sect. III-B, the events
the use of automata. As we have seen in Sect. III-C, the depending on the evolution of Γ ( , , ,
6-zone partition necessitates to apply a control vector ) are potential but not certain when k is equal to i or
different from Vi and Vi+3 when Øs is in Zi. We will i+3.
explain in Sect. V how this requirement can be
guaranteed by supervisory control of DES. V. USE OF SCT TO DRIVE IM
Let us see how the two automata (16-state and 6-state) A. introduction to SCT
are combined into an automaton Mk that abstracts the
behavior of IM when a given control vector Vk is In supervisory control, a supervisor Sup interacts with a
applied by the inverter. DES (called plant) and restricts its behavior so that it
respects a specification. Sup observes the evolution of P
(i.e., the events executed by the plant) and permits only
the event sequences accepted by S. To achieve its task,
Sup will disable (i.e., prevent) and force events. The
concept of controllable event has thus been introduced,
meaning that when an event e is possible, then Sup can
disable it if and only if e is controllable; e is said
uncontrollable if it is not controllable [9]. We will also
use the notion of forcible event, meaning that when an
event e is possible, then Sup can force e to preempt (i.e.,
to occur before) any other possible event, if and only if e
is forcible; e is said unforcible if it is not forcible [14]. A
method has been proposed to synthesize Sup
automatically from P, S and the controllability and
forcibility of every event [9].
B. The Plant Inv-IM Modeled as a DES
A State of Mk is noted (u, v, i)k since it is a combination
of a state (u, v) (corresponding to Ru,v) and a state i The plant to be controlled is the system Inv-IM (i.e.,
(corresponding to Zi ). Mk can therefore have at most 6 inverter with IM). In Section II, we have modeled the
× 16 = 96 states (u, v, i)k, (u, v = 1…4, i = 1…6). By inverter by an automaton A with 8 states qk (k = 0…7)
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International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
Corresponding to the 8 control vectors Vk, respectively. added an event Null in the model of IM. The interface G
And in Section IV-C, when a given VK is applied by the generates these events.
inverter to the IM, we have modeled the evolution of IM Sup observes the events Z, Г, Ø, Null. Since these
by an automaton MK that can have at most 6 × 16 = 96 events are generated by IM through G, Sup has no
states (u, v, I)k, (u, v =1,…4,i=1,…6).Therefore, the control on them. Hence, these events are uncontrollable
system Inv-IM can be modeled by replacing in A each and unforcible. Sup generates the events VK, and thus,
mode qk by the automaton Mk . The transition from any has all control on them. Hence, these events are
state (u, v, i)* to a state (u, v, i)k occurs by an event Vk . controllable and forcible.
The obtained automaton, noted P , can therefore have at
most 8 × 96 = 768 states. The initial state is (1; 1; 1)0, VI. CONCLUSION
that is, initially: the flux and the torque are in Region
R1,1, the flux vector is in zone Z1 and the null control In this control method, a much better behavior of the
vector Vo is applied. The set of marked states is { (u; v; DTC-SVM performance is presented , achieving one of
i)k : u, v = 2, 3},because the objective of the control is to the main objectives of the present work, which was to
drive Inv-IM into the set of regions {Ru,v : u, v = 2, 3} control the torque by reduce the torque ripple and
(i.e., the set of states { u; v; i k : u, v = 2, 3}), and then to consequently improve the motor performance compared
force it to remain into this set. For the purpose of to classical DTC. This control method shows better
control, we define an undesirable event Null meaning results in high as well as in low speed also as shown in
that the flux or the torque has decreased to zero, and a results for low speed. As a future work, we intend to
state E reached with the occurrence of Null .We will see improve our control method by using a hierarchical
later how Null and E are necessary. Therefore, the control and a modular control, which are very suitable
automaton P has actually at most 769 (768+ the state ), to take advantage of the fact that the event based model
and its alphabet Σ is: of the plant has been constructed hierarchically and
modularly.
C. control architecture
We propose the control architecture illustrated in Figure
7. The interaction between the plant and the supervisor
is realized through two interfaces A and G.
A: In Sect. II, we have modeled the inverter by an
automaton executing the events Vk, where k = 0,.…7.
The interface A translates every event Vk generated by
the supervisor into (Sa Sb Sc), which is the 3-bit code of VII. SIMULATION AND RESULTS
k. And the inverter translates (Sa Sb Sc) into the control
vector Vk, which is applied to IM.
G: In Sect. IV, we have modeled IM by an automaton
executing the events Z, Г, Ø. and in Sect. V-B, we have
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International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
Fig: 8. simulation diagram for DTC of IM
Fig: 9. Simulation results for Torque, current IABC,
speed and flux in DQ axis.
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International Journal of Advanced Research in Computer Science and Electronics Engineering
Volume 1, Issue 5, July 2012
AUTHORS
B. M. Manjunath received his B.Tech (Electrical and Electronics
Engineering) degree from the Jawaharlal Nehru Technological
University, and M.Tech (Power Electronics) from the same university.
He is currently an Asst. Professor of the Dept. Electrical and
Electronics Engineering, Rajeev Gandhi Memorial College of Engg.
& Tech, Nandyal. His field of interest includes renewable energy
sources and Power electronics & Drives. (E-mail:
manjumtech003@gmail.com). Nandyal , Andhra Pradesh, India.
A. Vinay Kumar received his B.Tech (Electrical and Electronics
Engineering) degree from the Jawaharlal Nehru Technological
University in 2009 and M.Tech (Power Electronics) pursuing from the
same university. His field of interest includes power systems and
fig.10: flux on XY plot power electronics. (E-mail: coolvinay207@gmai.com). Nandyal,
Andhra Pradesh, India.
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