IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40,NO. 1, FEBRUARY 1993
23
Sliding Mode Control Design Principles and
Applications to Electric Drives
Vadim I. Utkin
Abstract-The paper deals with the basic concepts, mathematics, and design aspects of variable structure systems as well as
sliding modes as a principle operation mode. The main arguments in favor of sliding mode control are order reduction,
decoupling design procedure, disturbance rejection, insensitivity
to parameter variations, and simple implementation by means of
power converters. The control algorithms and data processing
used in variable structure systems are analyzed. The potential of
sliding mode control methodology is demonstrated for versatility
of electric drives and functional goals of control.
I. INTRODUCTION
Al
converters. This reason predetermined both the high efficiency of sliding mode control for electric drives and the
author choice of the application selection topic in this
paper.
MODESIN VSS
11. SLIDING
Variable structure systems consist of a set of continuous subsystems with a proper switching logic and, as a
result, control actions are discontinuous functions of system state, disturbances (if they are accessible for measurement), and reference inputs. In the course of the
entire history of control theory, intensity of discontinuous
control systems investigation has been maintained at a
high enough level. In particular, at the first stage, on-off
or bang-bang regulators are ranked highly due to ease of
implementation and efficiency of control hardware.
Futhermore, we shall deal with the variable structure
systems governed by
high level of scientific and publication activity, an
nremitting interest in variable structure control enhanced by effective applications to automation problems
most diverse in their physical nature, and functional purposes are a cogent argument to consider this class of
nonlinear systems as a prospective area for study and
applications.
The term “variable structure system” (VSS) first made
X = f ( x , t , u ) , x E Rn,U E R”
its appearance in the late 1950’s. Since that time, the first
expectations of such systems have naturally been reevaluu + ( x , t ) if s ( x ) > 0 (for each component)
ated, their real potential has been revealed, new research
u - ( x , t ) if s ( x ) < 0
directions have been originated due to the appearance of
(1)
new classes of control problems, new mathematical methods, recent advances in switching circuitry, and (as a
consequence) new control principles.
The paper is oriented to base-stone ideas of VSS design
methods and selected set of applications rather than the The VSS (1) with continuous functions f,s, U + , U - consurvey information or a historical sequence of the events sists of 2” subsystems and its structure varies on m
accompanying VSS development since at its different surfaces at the state space. From the point of view of our
stages survey papers on theory [1]-[4] and applications [51, later treatment, it is worth quoting the elementary exam161 have been published. In addition, monographs [7]-[12] ple of a second-order system with bang-bang control and
summarize the results of these stages.
sliding mode:
Furthermore, it will be shown that the dominant role in
VSS theory is played by sliding modes, and the core idea
x + a2X + a , x = U ,
of designing VSS control algorithms consists of enforcing
this type of motion in some manifolds in system state
U = - M signs
spaces. Implementation of sliding mode control implies
s = cx + X,M , c , a , , a 2 - const ( 2 )
high-frequency switching. It does not cause any difficulties
when electric drives are controlled since the “on-off”
operation mode is the only admissible one for power which was considered by Andronov et al. [131 in connection with his study of autopilot dynamics. It follows from
analysis of the (X,x) state plane (Fig. 1) that, in the
Manuscript received June 6, 1992. A revised version of the paper was
presented at the IEEE VARSCON ’91 Workshop, Reno, NV, June 6, neighborhood of segment mn on the switching line s = 0,
1991.
the trajectories run in opposite directions, which leads to
The author is with the Institute of Control Sciences, 117 806, Moscow,
the appearance of a sliding mode along this line. The
Russia.
switching line equation s = 0 may be treated as a motion
IEEE Log Number 9204071.
0278-0046/93$03.00
0 1993 IEEE
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24
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 1, FEBRUARY 1993
Fig. 2. State planes of linear structures.
Fig. 1. Sliding mode in a second-order relay system.
one
cx+x=o
(3)
with solutions depending only on the slope gain c and
invariant to plant parameters and disturbance (should the
plant be subjected to).
The sliding mode domain is bounded in the above
example, but if the amplitude of discontinuous control is
made state dependent ( M = klxl, k = const), it may coincide with the whole switching line. The system consists of
two linear structures (U = krc and U = -la)
shown in Fig.
2 for a, = 0, a2 < 0. Due to the sliding mode (Fig. 3) after
the state reaches s = 0, it decays exponentially in accordance with (3).
The systems with discontinuous control (1) are known
to generate sliding modes with state trajectories running
in discontinuity surfaces as well. Similar to the above
examples, state velocity vectors may be directed toward
one of the surfaces and sliding mode occurs along it (arcs
ab and cb in Fig. 4). It may arise also along their intersection (arc bd). Fig. 5 illustrates the sliding mode at the
intersection even if it does not exist at each of them taken
separately.
Let us discuss major reasons why sliding modes were
and are of exceptional significance in VSS control
methodology. First, in sliding mode the input of the element implementing discontinuous control is close to zero
while its output (exactly speaking its average value U,)
takes finite values (Fig. 6). Hence the element implements
high (theoretically infinite) gain, that is the conventional
mean to suppress influence of disturbances and uncertainties in system behavior. Unlike systems with continuous
control, the invariance is attained using finite control
actions. Second, since sliding mode trajectories belong to
some manifold of a dimension lower than that of the
system, the order of a motion equation is reduced as well.
This enables simplification and decoupling design procedure. Both order reduction and invariance to plant dynamics are transparent for the above second-order example. And, finally, a pure technological aspect of using
sliding mode control should be mentioned. To improve
the performance, inertialess power thyristor and transistor
converters are increasingly used as actuators in control
systems. Even if continuous algorithms are employed, the
control is shaped as a high-frequency discontinuous signal
with an average value equal to the desired continuous
control since a switching mode is the only operation one
\
i
x
Fig. 3. Sliding mode in a variable structure system.
Fig. 4. Sliding mode in discontinuity surfaces and their intersection.
Fig. 5. Sliding mode in the intersection of discontinuity surfaces.
f
Fig. 6.
Implementation of high-gain control via the sliding mode.
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UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
1
M a t h c
25
S L X D X N a
MODE
CONTROL
I
1
-
Applications
m a t Ic a 1
Methods
1
,
M
E
P
0
x
T?
t
0
0
I.
s
t
n
c
I
-
C
-
e
a
a
n
c
=
9
U
V
a
t
i
c
0
m
n
0
0
&
t
i
0
n
i
n
n
I
.
0
d
U
=
t
I rl
Fig. 7. Scope of sliding mode control theory.
for the converters. It seems more natural to employ the
algorithms oriented toward discontinuous control actions.
The study of sliding modes is a multifacet problem that
embrances mathematical, control theoretical, and application aspects. The chapters and sections of this study are
shown in Fig. 7.
111. MATHEMATICAL
METHODS
To justify strictly the arguments in favor of employing
multidimensional sliding modes we need mathematical
methods of describing sliding modes in the intersection of
discontinuity surfaces s = 0 and the conditions for this
motion to exist.
The first problem arises due to discontinuity of control
since the relevant differential equations do not satisfy
conventional theorems on existence uniqueness solutions.
We confine ourselves to a second-order example to
demonstrate that discontinuous control systems may need
subtle treatment. Let the discontinuous control in the
system
X I = o.3X2 -k uxI
+
i , = - 0 . 7 ~ ~ 4u3x,,U = -sign xis, s
= x1
+ x2
be implemented by a limiter and then by a hysteresis relay
element so that A-the width of the limiter linear zone
and the hysteresis loop-is small enough when compared
with the magnitude of control. The experiment with A =
0.01 shows that in spite of closeness of the controls, the
motion along the switching line is unstable in the first
case and asymptotically stable in the second one (Fig. 8).
Fig. 8. Ambiguity of sliding mode equations.
In cases when conventional methods are not applicable,
the usual approach is to employ the regularization approach or replacing the initial problem by a closely similar
one, for which familiar methods can be used. In particular, taking into account delay or hysteresis of a switching
element, small time constants neglected in an ideal model,
replacing a discontinuous function by continuous approximation, are the examples of regularization since discontinuity points (if they exist) are isolated.
In our opinion, the universal approach to regularization
consists of introducing a boundary layer llsll < A around
manifold s = 0 where an ideal discontinuous control is
replaced by a real one such that the state trajectories of
system (1) are not confined to this manifold but run
arbitrarily inside the layer (Fig. 9). The only assumption
for this motion is that the solution exists in ordinary
sense. If, with the width of the boundary layer tending to
zero, the limit of this solution exists, it is taken as a
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26
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 1 , FEBRUARY 1993
Fig. 9. Boundary layer regularization.
solution to the system with ideal sliding modes. Otherwise
we have to recognize that the equations beyond discontinuity surfaces do not derive unambiguously the motion
equation in their intersection.
Boundary-layer regularization enables substantiation of
the so-called equicalent control method [ 141 intended for
deriving a sliding mode equation in the systems depending
on linear control:
X =f(x,t)
+ B(x,t)u
(4)
where B ( x , t) is an n x m matrix. In accordance with the
method control, U should be replaced by the equivalent
control, which is the solution to
S = Cf + GBu,,
=
0, G
=
(as/ax)
For det GB # 0 (ue! = - ( G B ) - ' G f ) , the sliding mode
equation in the manifold s = 0 is
i
=
[I
-
B(GB)-'G]f.
(5)
Since s ( x ) = 0 in sliding mode m components of the state
vector x may be found as a function of the rest ( n - m)
ones: x 2 = so(x,), x 2 , s o E R", x , E R"-'" and, correspondingly, the order of the sliding motion equation may
be reduced by m :
The idea of the equivalent control method may be easily
explained with the help of geometric consideration. Sliding mode trajectories lie in the manifold s = 0 and the
equivalent control ueq being a solution to the equation
S = 0 implies replacing discontinuous control by such a
continuous one that the state velocity vector lies in the
tangential manifold.
Uniqueness of sliding mode equations explains why the
study of control systems with linear dependence on control has turned out the main (if not the only) stream in
VSS theory. Note that in the above second-order example,
nonlinear dependence of the motion equation on control
resulted in the ambiguous sliding mode equations.
The second mathematical problem relates to conditions
for a sliding mode to exist. They are equivalent to conditions of state trajectory convergence to the intersection of
discontinuity surfaces s = 0. Therefore, the existence conditions may be formulated in terms of stability of the
origin in m-dimensional space s, or subspace of the distances to discontinuity surfaces. To derive existence conditions in analytical form, the equation of the projection
of overall motion on subspace s
S
=
Gf
+ GBu
(7)
should be analyzed for example by designing a Lyapunov
function. The simplest case is for GB being an identity
matrix. Then for U = - M signs ((signs)T = (signsl;..,
signs,)) with M exceeding the upper estimates of vector
Gf elements, the functions S, and s, ( i = 1;",m) have
different signs. It means that the sliding mode will occur
in each discontinuity surface.
The most interesting fact is that the Lyapunov function
testifying to convergence to the manifold s = 0 is a finite
function of time. It vanishes after a finite time interval. AS
a result, sliding mode arises in a finite time instant in
contrast to continuous systems with only asymptotic tending to any manifold consisting of system trajectories [111.
IV. DESIGNPROCEDURE
The discussed methods pertaining to sliding mode
equations and existence conditions constitute the back-
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UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
27
ground for the variety of design procedures in VSS. Decoupling or invariance or both are inherent in any of
them. High dimension and uncertainties in system behavior are known to be serious obstacles in applying efficient
control algorithms and using both analytical and computational methods.
In connection with control of high-dimensional plants,
the design methods permitting decoupling of the overall
motion into independent partial components are of great
interest. Decoupling in discontinuous control systems (4)
is easily feasible. The sliding mode equation (6) is of a
reduced order, does not depend on control, and does
depend on the discontinuity surface equation. The design
procedure consists of two stages. At stage 1, the function
s,, is handled as a control in (6) and designed in accordance with some performance criterion-a standard control task. In stage 2 selection of discontinuous control
follows to switching logic in (1) to enforce the sliding
mode, which is equivalent to stability task in s space ( 7 ) of
a reduced order as well. It should be noted that the last
problem is not very difficult since its dimension and that
of control coincide. As a result, the control design is
decoupled into two independent tasks of lower dimensions: ( n - m)th order at the first stage and mth order at
the second. In a thus-designed system starting from some
finite time instant, the motion with the prescribed properties will arise.
Time interval preceding the sliding mode decreases
with the growth of control magnitude ill], and if it is
small enough it is the very sliding equations that predetermine control system properties.
What shall we expect of sliding modes in systems operating under uncertainty conditions? Suppose that in the
system equation
i =f(x,t)
+ B ( x , t ) u + h(x,t)
(8)
vector h(x, t ) represents all the factors whose influence
on the control process should be eliminated. If for each x
and t
h
E
range { B )
(9)
which means that disturbances act in control space, then
there exists control U , such that Bu, = -h and hence
the system is invariant to h(x,t). But control U,, would
hardly be implementable since the disturbances may be
inaccessible for measurement.
As we had established the sliding mode equation in any
manifold does not depend on control. Similarly, via the
equivalent control method, it can be shown that sliding
mode is independent on h(x,t) as well, therefore condition (9) is the invariance condition for sliding mode control. It is important that for the design of an invariant
system there is no need to measure vector h. To ensure
sliding mode existence, only an upper estimate of h (a
number or function) is needed.
v. SLIDING MODECONTROL 1N LINEARSYSTEMS
Consider conventional control tasks for linear plants
i =Rx + B u , x
Rn,U E R"
(10)
( A , B are constant matrices, rank B = m ) to demonstrate
the sliding mode design procedure based on the decoupling principle.
The system (10) may be transformed to the regular
form [151
i ,= A , , x ,+A,?X,
i2= A 2 , x , + A,?xz + B 2 u
(11)
E
where A , , (i, j = 1,2), B, are constant matrices of relevant dimensions, x, E R"-"', x 2 E R'", det B , # 0. Assuming that control vector components have discontinuities on linear surfaces,
s =
cx,+I,, s E R"'
( 12)
we find that, when the sliding mode appears on manifold
s = 0 (i.e., x2 = -Cx,),the system behavior is governed
by the ( n - m)th-order equation
i , = ( A , ,- A & > x , .
(13)
One of the possible ways of obtaining the required
dynamic properties of the control systems is assigning
eigenvalues of a closed-loop system with linear feedback.
However, whereas in the context of initial system (10) we
are concerned with a task of full dimensionality, introducing a sliding mode reduces it, since the order of the sliding
equation (13) is decreased by an amount equal to the
control dimension. For controllable systems (lo), there
always exists a matrix C, ensuring the desired eigenvalues
of the system (13) [16]. The matrix C being a solution to
the (n-m)th-order eigenvalue assigning task determines
the equation of discontinuity surfaces (12). The second
stage of the design procedure is choice of discontinuous
control such that the sliding mode always arises at the
manifold s = 0, which is equivalent to stability of the
origin in m-dimensional space s. The motion projection
on the s space is described by an equation similar to (7):
S = Rx + B , u ,
fi = ( C A , , + A , I ) X , + ( C A , , + A 2 2 ) X 2 .
The discontinuous control
U =
-alxlB,'
sign s, a
-
const
(1x1 is the sum of vector x component moduli) leads to
S = Rx - alxlsigns.
(14)
There exists such positive value of a that the functions S,
and s, ( i = l;.., m) have different signs. It means that the
sliding mode will occur in each discontinuity surface.
Within the same framework a discontinuous control
may be designed in accordance with a mean square criterion
I
=
['kTQxdt, Q
2
0.
If we are concerned with optimization of sliding motions,
then the sliding manifold should be linear while coeffi-
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28
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40,NO. 1, FEBRUARY 1993
cients of its equation are found from the Riccati equation
of a reduced order similar to the above eigenvalue assignment [ll]. The control (14) fits to generate a sliding mode
in manifold (12) with both constant and time-varying
matrices C.
Invariance to disturbances and plant parameter variation is one of the main problems of control theory. As
mentioned in Section IV generating sliding modes results
in invariance to all factors to be rejected acting in control
subspace. For linear systems
i = A+ Bu
-,
EL>DT,
Fig. 10. Unmodeled dynamics of actuator and sensor.
+ D f ( t ) ,f ( t ) E R '
this condition was formulated in terms of system and
input matrices in [17]. The sliding mode in manifold (12)
is invariant to the disturbance f ( t ) and the variations of
the parameters A A ( t ) ( A = A , , + AA(t), A,, = const) if
D E range { B } ,A A ( t ) E range { B } ,correspondingly. Similarly, the decoupling conditions in the interconnected
systems may be found: interconnection matrices in each
of the subsystems should belong to corresponding control
subspace.
Fig. 1 1 . System with continuous control.
t
,
L(t)
+>
-1141n.
moa-
Fig. 12. Chattering in system with discontinuous control.
VI. THECHA~TERING
PROBLEM
The subject of this section is of great importance whenever we intend to establish a bridge between the recommendations of the theory and applications. Bearing in
mind that the control has a high-frequency component,
we should analyze the robustness or the problem of correspondence between an ideal sliding mode and real-life
processes at the presence of unmodeled dynamics. Neglected small time constants ( p l and p2 in Fig. 10) in
plant models, sensors, and actuators leads to dynamic
discrepancy (zl and z2are the unmodeled-dynamics state
vectors).
In accordance with the singular perturbation theory [ 181
in systems with continuous control, a fast component of
the motion decays rapidly and a slow one depends on the
small time constants continuously (Fig. 11).
In continuous control systems the solution depends on
the small parameters continuously as well. But unlike
continuous systems, the switchings in control excite the
unmodeled dynamics, which leads to oscillations in the
state vector at a finite frequency (Fig. 12). The oscillations, usually referred to as chattering, are known to
result in low control accuracy, high heat losses in electrical power circuits, and high wear of moving mechanical
parts. These phenomena have been considered as serious
obstacles for applications of sliding mode control in many
papers and discussions. A recent study [19] and practical
experience showed that the chattering caused by unmodeled dynamics may be eliminated in systems with asymptotic observers (Fig. 13). In spite of the presence of
unmodeled dynamics, ideal sliding arises, it is described by
a singularly perturbed differential equation with solutions
free from a high-frequency component and close to those
of the ideal system (Fig. 14). As shown in Fig. 13 an
asymptotic observer serves as a bypass for a high-frequency
I
I
Aslmptotic observer
component, therefore the unmodeled dynamics is not
excited. Preservation of sliding modes in systems with
asymptotic observers predetermined successful applications of the sliding mode control [61.
The alternative approach to handling dynamic discrepancies is a continuous approximation of discontinuous
control [20]-[22]. It should be noted that too high a slope
in the middle part of the approximation functions (Fig. 15)
may result in excitation of the unmodeled dynamics as
well, and the trade-off between accuracy and robustness
must be achieved. In addition to that, continuous approximation is nonadmissible for many applications where
on-off operation is the "way of life" for actuators (e.g.,
thyristor or transistor conversions).
VII. CONTROL OF ELECTRIC
DRIVES
The experience that has been gained in the applications
of sliding mode algorithms testifies to their efficiency and
versatility [6]. Control of electric drives is one of the most
challenging applications due to wide use of electric servomechanisms in control systems, the advances of highspeed switching circuitry, and insufficient linear control
methodology for internally nonlinear high-order plants
such as ac motors. Implementation of sliding modes by
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~
UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
29
TABLE I
CONTROL
OF E L K IRIC MO I ORS
“)R
APPLIED STATE VARIABLB
observers
brectld
mCNoBs
uex
1. Position a
a6
i
n, dn/dt
2. Speed n
n6
1
dn/dt
__
IgWCTIOB HOlVR
LPosition and flux
a,
it,^^
n, dn/dt
2, Speed and flux
LR
1. Position
Fig. 14. Sliding mode in system with observer.
a,iffil
n,dn/dt
tions to pulse-width modulation seems unjustified, it is
reasonable to turn to the algorithms with discontinuous
control actions. Introduction of discontinuities is dictated
by the very nature of converter elements. The results of
sliding mode control applications to dc, induction and
synchronous motors [6], [23]-[33] are summarized in
Table I.
All the systems have much in common: enforcing sliding modes leads to low sensitivity of disturbances (load
torques) and plant parameters variations; independently
of a plant operator motion equations are of a reduced
order and linear. The wide range of functional goals of
control should be noted: angle position, rotation speed,
magnetic flux, and optimization in accordance with mechanical, power, and efficiency criteria. Commonly used
transducers of angle, speed, and current are installed in
the systems. The rest of the state variables (flux, time
derivatives of its components, angle acceleration) are restored with the help of linear or nonlinear observers.
VIII. DC MOTORS
Fig. 15. Continuous approximation of discontinuous control.
means of the most common electric components has
turned out to be simple enough. The commercially available electronic converters enable one to handle powers of
several dozen kilowatts at frequencies of the same order.
When using converters of this type confining their func-
From the point of controllability a dc motor with constant excitation is the simplest. Its motion is governed by
the second-order equation with respect to shaft angle
speed n and current i with voltage U and load torque M L
as a control and a disturbance:
Ldi/dt
=
Jdn/dt
-ir
=
-
k,,i
ken
-
+U
M,
where L , J, r , k , , and k,, are constant parameters.
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(15)
( 16)
IEEE TRANSACTIONS ON INDUSTRIALELECTRONICS, VOL. 40, NO. 1, FEBRUARY 1993
30
Let n,(t) be a reference input, then the second-order
motion equation with respect to the error x , = n,(t) - n
is of form
where a,, a 2 , and b are positive parameters, f(t)-a time
function depending on M,(t), n & t ) and their time derivatives.
For discontinuous control
u
= U()
sign s, s
=
cxl
+ x2
ul,,c
-
const
(18)
the error x , decays exponentially should the sliding mode
occur on the line s = 0 since its equation
cxl
=
+
dt
with a control error An = no - n. Let us write down
formally the equation for An making L be equal to zero.
Then, substitution of the solution of (15’) with L = 0
+ il= 0
cx2 - a l x l - a 2 x ,
+f ( t )
ke
(n, - An)
r
I =
is linear and does not depend on f ( t ) ,
As follows from
S
motion components may be separated by rates and then
the fast one is neglected. A similar situation may happen
when controlling a dc motor with a mechanical motion
being much slower than an electromagnetic one. Formally
it means that L a J in (15), (16), which may be presented
as
di
L - = -ir - k,(n,, - A n ) u
(15‘)
dt
d An
J-= -k,i
+ M , + Jiz,
-
--
1
+ -Ur
into (16‘) yields
bu, sign s
in the system (17), (18) with
> 1~x2- ~ 1 x 1- ~ 2 x 2+ f ( t > I
(19)
the values of s and S have opposite signs and the state
reaches the sliding line s = 0 after a finite time interval.
Inequality (19) determines the voltage needed for enforcing the sliding mode, as a result, the control error is
steered to zero.
For implementation of control (181, angle acceleration
is needed ( x 2 = ii, - iz). Under the assumptions that the
angle speed n and the current i can be measrued directly
and the load torque varies slowly or
dM,/dt
=
0
(20)
Equation (23) is taken as a reduced-order model of a dc
motor. Within the framework of the model (23), discontinuous control U = uOsign A n , depending only on the control error (in contrast to (18), depending on its time
derivative) for high enough U , , provides the sliding mode
in “manifold” An = 0; and, as a result ideal tracking the
reference input n,,(t) by the shaft rotation speed. However, as it was discussed in Section VI, the unmodeled
dynamics (15‘) may excite nonadmissible chattering. Following the recommendation of Section VI, chattering may
be eliminated by using asymptotic observers.
Bearing in mind that M , = 0, let us design an asymptotic observer to estimate An (23) and M,:
a conventional Luenberger reduced-order observer may
be designed:
dM/dt
=
1/J( -1M
+ l’n + lk,i)
d An
Jdt
=
as an estimate of M = M , + In. Acwith 1 - const,
cording to (161, (2!), and (21) the equation for the mismatch M = M - M is of the form
By a proper choic? of gain 1, the desired convergence rate
of
to zero or M - In to M , may be provided. It means
that the load torque is known and the time derivative
d n / d t may be found from (16).
Similarly, the sliding mode control may be designed for
position and torque control with or without measurement
of the motor current. In addition to control of mechanical
coordinates, optimization in accordance with a power
consumption criterion may be provided for dc motors with
a controlled excitation current [6].
Section VI was dedicated to sliding mode control in the
systems with unmodeled dynamics in which the partial
-
An)
krn
-
-U
r
+ M, + .hio+ l,(An - A n )
(21)
1dM/dt= --M.
J
k,,k,
r
-(no
dML
~
dt
=
/,(An
-
(24)
An)
where A n and kLare estimates for An and M,, li and 1,
are constant. Control is a discontinuous function of the
error estimate
u
The values of
ent signs if
U,
> k,(n,
-
=
u0 sign A n .
(26)
A, and its time derivative (24) have differAn)
r
,.
Jr
1,r
k
km
+ -MI> + -iz, + - ( A n
krn
A
-
An) .
Origination of the sliding mode means that the discyntinuous control (26), (27) reduces the error estimate An to
zero. To derive a sliding mode equation in accordance
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UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
31
with the equivalent control method (Sect. III), the solution to dii/dt = 0 (24) with respect to control
should be substituted into system (15'), (16'), (25) for
d An
Jdt
=
+ M L + Jli,
-k,i
U:
(29)
dkL
( 30)
-12An.
-=
dt
Equations (281, (29), (3?), and ML = 0 describe the sliding
mode in the manifold An = 0. According to the theory of
singularly perturbed systems [18] for L 4 J , the fast motion of the linear system may be neglected by zeroing the
parameter L. Substitution of the solution to algebraic
equation (28) ( L = 0) with respect to i j n t o (29) results in
a motion equation for A n and
= ML - MI.:
JAri=
-
aL
a - 1, An
kekm
-An
r
ML =
-
-1, An
Apparently the eigenvalues of the homogeneous system
may be assigned at our will by a proper choice of 1, and 1,
and the desired rate of the control error An decay may be
guaranteed.
The principal advantage of the reduced-order-based
method is that the angle acceleration is not needed for
designing sliding mode control.
IX. INDUCTION
MOTORS
The most simple, reliable and economic of all
motors-maintenance-free
induction motors-supersede
dc motors in today's technology, although, in terms of
controllability, induction motors seem the most complicated. Their behavior is described by a nonlinear highorder system of differential equations:
whtre n is a rotor angle velocity, and two-dimensional
vectors 4T = (4a,4p);iT = ( i , , i p ) , uT = ( u a , u p )are rotor flux, stator curent, and voltage in the fixed coordinate
system ( a , p), respectively; M and M L are a torque
developed by a motor and a load torque, u R ,U , , U,-phase
voltages, which may be made equal either to uo or -uo;
e R ,e,, e , are unit vectors of phase windings R , S, T ; and
J , x H ,x s , x R , rR,rs are motor parameters.
The control goal is to make one of the mechanical
coordinates, for example, an angle speed n(t),be equal to
a reference input n o ( t ) and the magnitude of the rotor
flux Il+(t>ll be equal to its scalar reference input 4,(t>.
The deviations from the desired motions are described by
the functions
and c,, c2 are const positive values.
The static inverter forms three independent controls
u R , u s , u T , so one degree of freedom can be used to
satisfy some additional criterion. Let the voltages
uR,u s , uT constitute a three-phase balanced system, which
means that the equality
should hold for any t .
If all three functions s,, s2,s3 are equal to zero then, in
addition to balanced system condition (32), the speed and
flux mismatches decay exponentially since s, = 0, s, = 0
with c1,c2> 0 are first-order differential equations. This
means that the design task is reduced to enforcing the
sliding mode in the manifold s = 0, sT = (sl, s,, s3) in the
system (31) with control uT = ( u R ,u s , U , ) . Projection of
the system motion on subspace s can be written as
ds
dn
1
_ - - ( M - M L ) ,M
dt
J
'4a
- -
dt
'R
- -4,
-
xR
' H
.
=
XR
-==++U
dt
ip4,)
where vector F = (f l , f 2 , 0) and matrix D do not depend
on control and are continuous functions of the motor
state and inputs. Matrix D is of form
x H .
n4b + rR-z,
XR
I':[
D=
di, _
dt
dip
_ -
'R
x,xR
- X;
i-'H
'4,
x R dt
(33)
- r,i,
+ U,
- rsi,
+ up
D,
=
and
d
=
(1,1, l ) , k
-
const,det D f 0.
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32
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 1, FEBRUARY 1993
Discontinuous control will be designed using the Lyapunov function
4, +,,
As
foliows from (31) the estimation error $, =
4, = 4, - 4, should satisfy differential equations
0 . 5 2~ 0.~ ~
U =
~
Find its time derivative on the state trajectories of system
(33):
du/dt
=
s T (F
d$,
dt
d p
dt
(34)
U =
sign s * , s*
U = -Ug
=
-
-
'R
n4,
+ n$,.
-4,
(37)
XR
The time derivative of Lyapunov function
Substitution of control
(s*f
-
-4,
xR
- - - -
+ Du).
-
'R
=
-
=
DTs
OS($:
+ $;)
>0
on the solutions of (37)
(s;,s;,s;)
into (34) yields
du/dt
=
is negative, which testifies tc expon:ntial convergence of U
to zero and the estimates 4, and 4, to the real values of
4, and 4,. The known values
i, and i, enable
one to find the time derivatives d+,/dt and d+,/dt from
the estimator equation (36) and then dll4ll/dt needed for
designing the discontinuity surface s2 = 0.
The equation of the discontinuity surface s, depends on
acceleration dn/dt. Since the values of 4, and 4, have
been found and the currents i , and i, are measured
directly, the motor torque may be calculated:
( s * ) ~ F *- uoIs*I
+,, +,,
where
The conditions
Ug
> Ifi*l, i
=
1,2,3
(35)
provide negativeness of du/dt and hence the origin in the
space s* (and by virtue of det D f 0 in the space s as
well) is asymptotically stable. Hence sliding mode arises in
the manifold s = 0, which enables one to steer the variables under control to the desired values. Note that the
existence condition (35) are inequalities, therefore the
only range of parameter variations and a load torque
should be known for the choice of necessary values of
phase voltages.
The equation of discontinuity surfaces s* = 0 depend
on an angle acceleration, rotor flux, and its time derivative. These values may be found using asymptotic observers under the assumption that an angle speed n and
stator currents iR,is, i , are measured directly. Bearing in
mind that
= (xH/xR)(icr4,
Under the assumption that the load torque M L varies
slowly the value of dn/dt may be found using the observer (21) designed for a dc motor.
The maintenance of an electric drive would be simplified considerably if it can be designed with no transducers
of mechanical coordinates. A rotor flux and angle speed
may be found simultaneously with the help of a nonlinear
observer with discontinuous parameters and stator currents and voltages as its inputs [30]:
di,
-_
dt
4,)
as an
design an observer with the state vector (J,,
estimate of rotor flux components +a and 4, and with i,
and i, as inputs:
d $a
dt
-
rR
-4,
XR
A
-
n4,
xH.
+ rR-ia
-
-
i-xH
xR
x,xR - x i
d6a
xR dt
-
&
r,i,
+U,
where J,, J,, L, and
are estimates of the current and
flux components. The estimate of the angle speed ii and
auxiliary parameter CL. are discontinuous functions of the
current estimate errors
XR
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UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
33
For large enough no and po,a sliding mode aJises in the
:n
:urfaces of s, = 0 and sp = 0, resulting in i, = i, and
.
.
= I , . Functions fi and p in (28) should be replaced by
h, and p,,-solutions of the system S, = 0, S, = 0 with
respect to A and p. The analysis of the system (38)
dynamics in sliding mode shows that fie, = n , peq- 0 is
an asymptotically stable equilibrium point and the observer enables restoration of both an angle speed and
stator flux. The value of he, may be found using a
low-pass filter.
Efficient speed (position) control algorithms imply decoupling the overall motion into two components, depending on the orientation of the motor flux, and then correFig. 16. “Torque-speed” diagram of induction motor.
spondingly design of control components providing desired values of the motor flux and torque. On one hand,
In the framework of the model (39) and (40), the sliding
the field-oriented control design methods need information on the current values of flux components, obtained mode control is a discontinuous function of the control
error
with the help of sensors, and, on the other hand, nonlinear state-dependent coordinate transformations. These
s = s,, sgn [ n o (t ) - n ] .
(41)
reasons may hinder implementation of induction motor
control systems, in particular for low-power electric drives For M,, > IM, + Jriol the values (T = no - n and b have
when application of complex control algorithms may prove different signs, therefore after a finite time interval the
to be unjustified.
sliding mode occurs and the motor shaft rotation speed is
Similarly, to dc motors application of reduced-order equal to the reference input identically.
models enables simplification of control algorithms. The
The second approach to the design of the sliding mode
dynamic processes in induction motors may consist of control algorithm is based on the assumption that the
partial motions of different rates. The rate of varying of a time constant related to the motor flux is considerably
magnetizing current may be much faster than that of greater than that of the leakage flux.
mechanical rotation; the time constant associated with
The angle cp between the vectors U and i,, (magnetizstator and rotor currents is much less than a magnetizing ing component of the stator current) is handled as a
one. As follows from the theory of singularly perturbed control action. Within the framework of the reduced-order
systems [18] the existence of rate-separated motions en- model, jumpwise increment of cp by n- leads to a jumpwise
ables order reduction of the system and, as a result, change of the stator current while the flux remains continsimplfication of the design procedure.
uous in time. The motor torque being a vector product of
We shall consider two versions of induction motor current and flux undergoes discontinuity, hence the rightcontrol systems based on reduced-order models-of the hand side in the equation of the mechanical motion may
first and of the third orders 1331. In the first case, the change sign, which results in acceleration or deceleration
electromagnetic dynamics is neglected and in the second of the motor shaft rotation. Inversion of the voltage phase
the processes associated with leakage fluxes. The motor is performed in correspondence with
slip and phase are handled as control actions and de77
signed as discontinuous functions of control error, which
- sgn a )
(T = n o ( t ) - n
(42)
is steered to zero due to enforcing sliding modes. The first a,(t) = a ( t ) - -(1
2
design method is oriented to induction motors with a high
inertia moment reduced to the rotor shaft.
where a ( t ) is continuous function depending on the conNeglecting the electromagnetic dynamics means that trol algorithm (in particular a ( t >= ~ , t o1
, = const corthe rotation speeds of the flux and voltage coincide and responds to voltage rotation at constant speed). Similarly,
to (41) the discontinuous control (42) makes the signs of U
w1 = n + s, where s is a motor slip. The above procedure
results in
and 6 opposite, and due to origination of sliding mode
the rotor speed tracks the reference input.
Jri = M ( s ) - ML
(39)
For the above control algorithms it is assumed that the
motor slip or the phase of the voltage are discontinuous
where M ( s ) is the well-known induction motor “torquefunctions of the control error. However, the only motor
slip” characteristics (Fig. 16).
slip control with constant voltage amplitude may result in
The maximum value of the motor torque M,, corretoo high magnitude of the current (the flux) at low rotor
sponds to the critical value of slip and for Is1 < s,,
speed, also the only phase control is unable to provide the
wide range of rotor speed control. Then the system combining both control methods will be needed since changing the voltage phase by n- means that inversion of its sign
I,
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34
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 1, FEBRUARY 1993
and switching at high frequency is equivalent to reducing
average value of voltage amplitude.
Let the value of the input voltage rotation speed be
formed in correspondence with the slip control algorithm.
w1 = n
+ s,,
magnets governed by differential equations:
did
xd- dt
= -
~
df
xd iq n
f
ud
sgn a
Taking into account the phase control algorithm (42) and
relationships
IT
CY, =
( w , dt
-
-(1 - sgn a )
2
dY _
-n
(44)
dt
obtain
CY, =
l ( n + s, sgn u ) d t
?r
- -(1
2
- sgn a ) .
Origination of the sliding mode means that the control
mismatch is steered to zero.
The small time constants having been neglected in the
reduced order models may result in an oscillatory component in system coordinates since switching in control
excites the unmodeled dynamics. The chattering phenomenon is eliminated in the sliding mode control systems with asymptotic observers (see Section VI). In practical applications, the observers were designed under the
assumption that the load torque varies much slower than
the motor state variables.
In the system with an observer, the estimate A of the
rotor speed is used in the switching function
u=no-A
where (d, q ) is an orthogonal coordinate system with the
d axis oriented along a rotor winding; i d , i,, u d ,U , are
stator current and voltage components; $ = const - rotor
flux;n - rotor speed, M and M L are the motor and load
torques; and r , xd, x,, J motor parameters,
u s , u T ,i,, is, i, are phase stator voltages and currents,
and y is the angle between phase R and the rotor. Let
the functions
U,,
while the observer is governed by equations
dh
Jdt
dkL
--
=
( p , sgn u - p , -~k L +
) l,(n
-
A)
(43)
- -l,(n
-
Sg =
[(U,
f Us
+ U,)dt
A)
dt
with pl,p , , I,, 1, constant.
The first equation of the observer is similar to motion
equation with respect to n in the reduced-order system of
the 3-d order, the second one implies that the load torque
varies slowly. The main advantage of using an estimate A
instead of the real value of n consists in the possibility of
generating an ideal sliding mode in spite of the presence
of unmodeled dynamics. Indeed, for high enough value of
p 1 the values of u and 6 have different signs and it does
not matter whether the full or reduced-order model is
dealt with. The above condition testifies to the existence
of the sliding mode with a = 0 or A = no.As follows from
(43) n = A = no in the steady-state mode, which is the
goal of control. It is known that in the systems with no
observers unmodeled dynamics results in chattering instead of sliding mode.
similar to an induction motor be deviations from the
desired motion. The third goal of control is to make the
component id be equal to a reference input io. It means
that the control uT = ((U,, u s , U , ) should enforce the
sliding mode in the manifold sT = (sl, s, s,) = 0, s2 = io
- id = 0. The equations of the system (44)motion projection on s subspace are derived by the differentiating
vector s
ds
-=F++u
dt
where F T = (f,,f,,f3),f3 = 0, scalar functions f,,f2 depend on the motor state and reference inputs, load torque,
and their time derivatives,
X. THESYNCHRONOUS
MOTOR
The sliding mode control design methods will be
demonstrated for synchronous motors with permanent
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dT
1
UTKIN: SLIDING MODE CONTROL DESIGN PRINCIPLES
x = -[(
35
hence d u / d t 5 -(2r/x,)u, which testifies to exponential
convergence of U to zero and the observer state to real
values of stator current components.
‘ d - x q ) / J x d ] Zq
Y = -(l/J x,)[(x,
-
xy)i, +
4.
XI. CONCLUSION
The discontinuous control is designed within the framework of the induction motor control design method discussed in Section IX:
U = -U,,
sign s*
s*
=
DTs
Only the ranges of plant parameters and disturbances
should be known to find necessary magnitudes of phase
voltages to generate the sliding mode in the manifold
s = 0 with desired dynamics.
The choice of the reference input i,, is usually dictated
by requirements for a motor static mode. For instance, a
motor torque is maximal if
Because in real-life conditions the stator current is always
bounded, for i, =f(i,) enforcing sliding mode in the
surface s2 =_O enables maximization of a motor torque. In
many cases q 2P 4(x, - x , ) i , and then i,, = 0. Similarly,
minimization of heat losses and maximization of motor
efficiency may be provided by a proper choice of the
reference input i,,.
The design method under discussion implies that angle
position and speed and phase currents are measured
directly while an acceleration d n / d t may be found with
the help of the above asymptotic observer. From the point
of practical applications it is of interest to design a ctntrol
system with no current transducers. Let i,, and i, be
estimates of stator currents components and satisfji the
observer equations
d;,
x
- = -rid
dt
+ x d i , n + U,,
and, corTespondingly, equations for mismatches
= i, - i, are of the form
iYd =
Ld -
i d , i,
did
xd-
dt
4
di
=
- r i d + x d i4 n
=
-n, - x , i , n .
dt
Calculate the time derivative of the Lyanpunov function
U = 0.5(ii
E:) > 0 taking into account inequality x,,>
+
Y
The paper has outlined the mathematical background
and sliding mode control design philosophy oriented to
high-dimensional nonlinear systems operating under uncertainty conditions and has demonstrated its applicability
to control of different types of electric motors. The electric drives with sliding mode control have already been
used in many applications: metal-cutting machine tools
(feed and spindle drives), robotics (tracking position and
speed control of manipulator links), transport (batterydriven cars and trams), and process control (fiber drafting
process) 131, [61.
An assessment of the scientific arsenal accumulated in
the sliding mode control theory is beyond the scope of the
paper therefore we confine ourselves to mentioning new
research areas initiated by scientific groups of many countries: geometric approach to design, control of infinitedimensional (including distributed and time-delay) plants,
sliding mode in discrete-time systems, Lyapunov function
based design methods, control of power electronic converters, aircraft, and combustion engines.
REFERENCES
A. Bakakin, M. Gritsenko, and N. Kostyleva, “Control algorithms
for variable structure systems,” Variable Structure Systems and Their
Uve in Flight Automation Problems. Moscow: Nauka, pp. 13-25 (in
Russian), 1968.
V. 1. Utkin, “Variable structure systems with sliding modes,” IEEE
Trans. Automat. Contr., vol. AC-22, no. 2, pp. 212-222, 1977.
-,
“Discontinuous control systems: State of the art in theory
and applications,” Preprints of the 10th IFAC Congress, Germany,
Munich, 1987, pp. 75-94.
R. A. DeCarlo, S. H. Zak, and G . P. Matthews, “Variable structure
control of nonlinear multivariable systems: A tutorial,” Proc. IEEE,
vol. 76, no. 3, pp. 212-232, March 1988.
V. 1. Utkin and N. E. Kostyleva, “Design principles for algorithms
of local automation systems with variable structure,” Izmerenie,
Control, Automatizatziya, no. 1, pp. 27-35, 1981, (in Russian).
V. I. Utkin, A. S. Vostrikov, S. A. Bondarev et al., “Applications of
sliding modes in automation of technological phnts,” Izmerenie,
Control, Automatizatziya, no. 1, pp. 74-84, 1985, (in Russian).
S. V. Emel’yanov, Variable Structure Automatic Control Systems.
Moscow: Nauka, 1967 (in Russian).
S. V. Emel’yanov, Ed., Theor)?of Variable Structure Systems. Moscow:
Nauka, 1970 (in Russian).
V. 1. Utkin, Sliding Modes and Their Applications in Variable Structure Syytems. Moscow: Nauka, 1974 (in Russian), and Moscow:
Mir-Publisher, 1978 (in English).
K. K. Zhil’tsov, Approximation Methods of Variable Structure System
Design. Moscow: Nauka, 1974 (in Russian).
V. I. Utkin, Sliding Modes in Optimization and Control. Moscow:
Nauka, 1981 (in Russian) and New York: Springer-Verlag, 1992 (in
English).
A. S. I. Zinober, Ed., Deterministic Control of Uncertain Systems.
London: Peter Pcregrinus Ltd., 1990.
A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theoiy of Oscd[alions. Moscow: Fizmztgiz, 1959 (in Russian).
V. 1. Utkin, “Equations of slipping regime in discontinuous systems. I, 11,” Aut. Remote Contr.,vol. 32, no. 12, pt. 1, pp. 1897-1907,
1971, vol. 33, no. 2, pt. I , IY71, pp. 211-219.
A. G. Luk’yanov and V. 1. Utkin, “Methods of reducing equations
of dynamic systems to a regular form,” Aut. Remote Contr., vol. 42,
no. 4, pt. I, 1981, pp. 413-420.
V. I. Utkin and K.-K.D. Young, “Methods for constructing discon-
Authorized licensed use limited to: SWINBURNE UNIV OF TECHNOLOGY. Downloaded on November 2, 2009 at 18:21 from IEEE Xplore. Restrictions apply.
36
[201
1211
[221
[231
IEEE TUNSACTICINS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 1, FEBRUARY 1993
tinuity planes in multidimensional structure variable systems,”
Aut. Remote Contr., vol. 39, no. 10, pt. 1, 1978, pp. 1466-1470.
B. Drazenovic, “The invariance conditions in variable structure
systems,” Automafica, vol. 5, no. 3, pp. 287-295, 1969.
P. V. Kokotovic, R. B. O’Malley, Jr., and P. Sannuti, “Singular
perturbation and order reduction in control theory,” Automafica,
vol. 12, pp. 123-132, 1976.
A. G. Bondarev, S. A. Bondarev, et al. ., “Sliding modes in systems
with asymptotic state observers,” Aut. Remofe Contr., vol. 46, no. 6,
pt. 1, pp. 679-684, 1985.
J. J. Slotine and S. S. Sastri, “Tracking control of nonlinear
systems using sliding surfaces, with applications to robot manipulator,” Inf. J. Contr., vol. 38, no. 2, pp. 465-492, 1983.
J. J. Slotine, “Sliding controller design for nonlinear systems,” Int.
J. Contr., vol. 40, no. 2, pp. 421-434, 1984.
J. A. Burton and A. S. I. Zinober, “Continuous approximation of
variable structure control,” Inc. J. System Sci., vol. 17, no. 6, pp.
875-885, 1986.
D. B. Izosimov, B. Matic, et al., “Using sliding modes in control of
electrical drives,” Dokl. A N S S S R , vol. 241, no. 4, pp. 769-772,
1978. (In Russian).
Y. Dote and R. C. Hoft, “Microprocessor based sliding mode
controller for dc motor control drives,” in IEEE U S Con$ Rec.,
Cincinnati, OH, 1980.
A. Sabanovic and D. Izosimov, “Application of sliding modes to
induction motor control,” IEEE Trans. Industiy Applications, vol.
LA-17, no. 1, pp. 41-49, 1981.
A. Sabanovic, D. Izosimov, 0. Music, and F. Bilalovic, “Sliding
modes in controlled motor drives,” in Proc. IFAC Conf. on Control
in Power Electronics and Electrical Drice~,Lausanne, Switzerland,
1983, pp. 133-138.
F. Harashima, H. Hashimoto, and S. Kondo, “MOSFET converterfed position servo system with sliding mode control,” IEEE Trans.
Ind. Elecrron., vol. IE-32, no. 3, pp. 238-244, 1986.
H. Hashimoto, H. Yamomoto, S. Yanagisava, and F. Harashima,
“Brushless servomotor control using variable structure approach,”
in ConJ Rec. 1986 IEEE Industry Application Society Annu. Meet.,
pt. 1, 1986, pp. 72-79.
G. P. Matthew, R. DeCarlo, and Lefebvre, “Towards feasible
variable structure control design for a synchronous machine connected to an infinite bus,” IEEE Trans. Automat. Contr., vol.
AC-31, no. 12, 1986.
[30] D. B. Izosimov, “Sliding-mode nonlinear state observer of an
induction motor,” in Control of Multiconnected Systems, M. V.
Meerov and N. A. Kuznetsov, Eds. Moscow: Nauka, pp. 133-139,
1983, (in Russian).
[31] Y. Dote, T. Manabe, and Murakami, “Microprocessor-based force
control for manipulator using variable structure with sliding mode,”
in Proc. IFAC Conf: on Control in Power Electronics and Electrical
DriL,es, Lausanne, Switzerland, 1983, pp. 145- 149.
[32] F. Harashima, H. Hashimoto, and K. Maruyama, “Practical robust
control of robot arm using variable structure control,” in Proc.
1986 IEEE Int. Conf: on Robotics and Automation, Apr. 7-10, San
Francisco, pp. 532-539, 1986.
[33] H. Hashimoto et al., “Sliding mode control of induction motors
based on reduced order models,” in Korean Automatic Control
Con$, vol. 2, pp. 1607-1610, 1991.
Vadim 1. Utkin was born in Moscow, USSR in
1937. He received the Dipl. Ing. degree from the
Moscow Power Institute in 1960 and the Candidate (Ph.D.) and Doctor degrees from the Institute of Control Sciences in 1964 and 1971, respectively.
Since 1960 he has been with the Institute of
Control Sciences, since 1973 he has been Head
of the Diwmtinuous Control Systems Laboratory He is d part-time Professor at Polytechnical Institute He was a Visiting Professor at the
University of Illinois, Urbana-Champaign, from 1975 to 1976 and the
University of Tokyo in 1991.
Dr. Utkin is Honorary Doctor at the University of Sarajevo, Yugoslavia (1978). Dr. Utkin was awarded the Lenin Prize in 1972. His
research interest$ are sliding mode controls, discontinuous dynamic
systems, infinite dimensional systems, and control of electric drivers,
vehicles, manipulators, and industrial processes He has published four
books and more than 170 technical papers.
Dr. Utkin is an Associate Editor of the IFAC JournalAutomatica and
the Intemational Journal of Control He was a Chairman of the International Program Committee of the l l th IFAC Congress in 1990.
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