Robust chattering-free (higher order) sliding mode control for a
vector-controlled induction machine
K.B. Goh, M.W. Dunnigan and B.W. Williams
Heriot-Watt University
School of Engineering and Physical Science,
Electrical, Electronic and Computing Engineering Department,
Riccarton, Edinburgh, EH14 4AS,
Scotland, U.K.
K.B.Goh@hw.ac.uk, M.W.Dunnigan@hw.ac.uk and B.W.Williams@hw.ac.uk
Abstract
The chattering behaviour that is inherent in standard
sliding mode control is often an obstacle for practical
application if neglected. In this paper, special attention is
given to a particular chattering-free control method: 2sliding “super twisting” control algorithm. Several
control algorithms, including the standard and chatteringfree control algorithms, are assessed via position and
speed control of a vector-controlled induction machine.
Attention has also been paid to the friction dynamics in
the induction machine system. Linear and non-linear
friction dynamics in the induction machine model are
considered.
Simulation results using an induction
machine model are first reported, followed by
implementation results. A pre-defined reference signal is
employed for tracking capability assessment as well as
robustness against a high inertia load (about double the
machine inertia) and DC motor load.
1
Introduction
The chattering effect is the main drawback in the standard
sliding mode control algorithm. The high frequency
switching behaviour of the control signal has often made it
impossible for practical application unless a solution is
considered. A common solution to this problem is to
attempt to smooth the signum function (i.e. sgn(s)=s/_s_) to
obtain a continuous approximation. The signum function
is seen to be relay-like in nature. The ideal relay
characteristic is often impossible to implement. One
possible answer is to replace the signum function by a
sigmoid-like function (i.e. f(s)=s/(_s_+G)) to induce pseudosliding instead of ideal sliding [1]. The variable, G is a
small positive scalar, and can be used to trade-off between
maintaining ideal performance and obtaining smooth
control action. In some cases, the control signal is thought
to comprise of ‘low’ and ‘high’ frequency components. A
natural solution is to pass the discontinuous control signal
through a low pass filter to eliminate the high frequency
component in the control signal. The tuning reaching law
is another possible solution to the chattering problem [2]
by introducing a linear term (i.e. f(s)=KLs+Usgn(s)) to the
existing discontinuous signum term. By selecting the gain
U to be small, the amplitude of the chatter will be reduced.
However, U cannot be chosen equal to zero because the
reaching time would become infinite. A large value for
KL increases the reaching rate when the state is not near
the switching surface. By this method, chattering can
practically be suppressed. Slotine proposed a technique to
change the dynamics near to the sliding surface in order to
avoid a real discontinuity and at the same time to preserve
the sliding mode properties [3]. This technique introduces
a boundary layer on both sides of the sliding surface in
order to avoid the chattering effect in the control signal.
The sliding variable will reach within the vicinity of the
sliding surface. However, the accuracy and robustness of
the sliding mode are partially lost.
Another approach proposed to eliminate the chattering
problem uses higher order sliding modes (HOSM). The
HOSM concept emerged in the 1980s with the motivation
of tackling the chattering problem. A number of
publications on the HOSM concept were from
[4][5][6][7]. The HOSM method generalises the basic
sliding mode idea by acting on the higher order time
derivatives of the sliding variable instead of influencing
the first time derivative as happens in the standard sliding
mode control. Keeping the main advantages of the
original approach, at the same time, HOSM totally
mitigates the chattering effect. HOSM control has the
potential to provide greater accuracy. It is known that in
practice, the classical sliding mode precision is
proportional to the time interval between the
measurements or to the switching delay. An r-sliding
mode realization may provide up to r-th order of sliding
precision with respect to the measurement interval. The
sliding order is also a measure of the degree of
smoothness of the sliding variable in the vicinity of the
sliding mode. The sliding order is defined as the number
of continuous total time derivatives of the sliding variable
[8].
This paper also pays attention to the friction dynamics in
the system under consideration. The common friction
dynamic terms in a machine are viscous, Coulomb,
windage, static and Stribeck friction. Among these
frictions, the Coulomb friction is the main problem, as far
as control system design is concerned, because of its
discontinuous nature (i.e. the signum term). Several
general survey of machine friction models can be found in
the survey paper [9] and books [10][11]. Dunnigan et al
have presented the viscous friction components in an
induction machine, using Slotine’s sliding mode control
method [3] to counter the linear friction dynamic terms.
The design has shown that superior performance can be
achieved via this control method [12]. Lee et al
considered the Coulomb friction in a nanometer cutting
machine model for a position control system design. A
sliding mode position controller, based on [3], was
designed. The simulation results suggested that this
technique would allow ultra-precision machining with
motions on the order of micrometers. Bartolini proposed
a second order sliding mode algorithm to simulate a
mechanical system with friction [13]. The simulation
showed that the control algorithm was efficient in
counteracting the discontinuous disturbances such as
Coulomb friction.
In this paper, the control scheme under consideration is
based on a particular 2-sliding (second order) supertwisting algorithm [5]. A sixth-order induction machine
model (i.e. electrical system) is derived from the induction
machine equivalent circuit in a stationary frame. For
completeness, a mechanical system model is derived
which has linear and non-linear friction dynamics
incorporated into it. The designed controller is firstly
simulated using the induction machine model.
The
controller is then implemented on a practical induction
machine system. The controller is tested at several
different conditions (i.e. nominal, high inertia load and
DC motor load) to test the tracking capability and
robustness against the external load disturbance. To
compare the controller performances, other control
algorithms include Slotine’s sliding mode (single
component), proportional-integral and signum (single
component) are tested and compared.
2
Mathematical model of induction machine
The induction machine equivalent circuits in a stationary
frame are show in Figure 1. Figure 1(a) shows the
equivalent circuit for the d-axis of a twin axis model,
while Figure 1(b) shows the q-axis [14].
ids
(a)
Rs
Lls
idr
Rr
+
-Zr\qr
idm
iqs
(b)
Vqs
Rs
Lls
_
iqr
Llr
Rr
Lm
_
Zr\dr
iqm
+
Figure 1: Induction Machine equaivalent circuit in the
stationary reference frame. (a) d-axis, (b) q-axis.
The general mathematical dynamic model of the induction
machine can be represented by six electrical equations and
two mechanical equations as follows:
Electrical system:
di
1
(Vds ids Rs dm Lm )
Lls
dt
didr
dt
di
1
(idr Rr dm Lm (iqm Lm iqr Llr )Z r )
Llr
dt
didm
dt
Rc
(ids idr idm )
Lm
diqs
diqm
1
(Vqs iqs Rs Lm )
Lls
dt
dt
diqr
dt
diqm
dt
diqm
1
(iqr Rr Lm (idr Llr idm Lm )Z r )
Llr
dt
Rc
(iqs iqr iqm )
Lm
(1)
Mechanical system:
Te
JZ r
3
Lm (iqm iqr )idr (idm idr )iqr
2
>
@
Te TL
(2)
where TL , comprises of a linear, FL and a non-linear,
FNL friction terms which are defined as:
TL
FL FNL
FL
FvZ r
FNL
2
FwZ r2 ( Fc Fs e Fst Z r ) sgn(Z r )
(3)
where Fv , Fc , Fw , Fs and FSt are viscous, coulomb,
windage, static, Stribeck friction constants respectively.
The signum function in (3) is approximated to the
following function to avoid simulation difficulty.
sgn Z r #
Llr
Lm
Vds
dids
dt
Zr
Z r G fric
(4)
where G fric is a small positive constant.
The viscous
friction is a linear relationship between force and velocity.
Windage loss normally arises in the machine due to fluid
drag on a rotating body, i.e. windage in air [15]. Coulomb
friction provides a constant retarding force and also
introduces a discontinuity force (i.e. when direction
changes) at zero velocity. Static friction occurs at the
initial phase of motion, i.e. moving from 0 rad/s. It is due
to friction resistance between two surfaces and the
threshold of motion is characterised by the coefficient of
static friction. Stribeck effect is a low velocity friction
effect, i.e. pre-sliding displacement, rising static friction
etc.
The induction machine model
saturation and the skin effect.
neglects
magnetic
Remark: Other losses in the induction machine are
neglected.
3
Higher order sliding mode control
The sliding order characterises the dynamic smoothness in
the vicinity of the sliding mode. If the control task is to
keep a constraint given by equality of a smooth function, s
to zero, the sliding order is the number of continuous total
derivatives of s in the vicinity of the sliding mode. Thus,
the rth order sliding mode is determined by the equalities
s
s ... s ( r 1)
(5)
0
This forms an r-dimensional condition on the state of the
dynamic system. The word “rth order sliding” is also
referred to as “r-sliding”. The discontinuity is seen to act
on s ( r ) 0 .
They are thus characterised by a
discontinuous control acting on the higher order time
derivatives of the sliding variable instead of acting on its
first time derivative ( s is discontinuous) only, as happens
in standard sliding modes. Thus, the standard sliding
modes are deemed to be the special case of HOSM
obtained when the sliding order is one. The HOSM is
considered as a generalisation of standard sliding modes.
The main problem in implementation of the HOSM is the
increasing information demand. Generally, any r-sliding
controller keeping s r 0 needs s, s,, s ( r 1) to be
available. One of the known exclusions is a so-called
“super twisting” 2-sliding controller, which needs only
measurement of s [5]. It is assumed that upper bounds on
the non-linear dynamics are known.
)
! 0,
*m
O2 t
4)*M (W ) )
,
*m3 (W ) )
0 U d 0.5
(8)
where W, U and O are variable controller parameters, ) is
positive norm bound on the smooth uncertain function I,
*m and *M are lower and upper positive bounds on the
The super-twisting
smooth uncertain function, J.
algorithm does not need any information on the time
derivative of the sliding variable. The choice of U 0.5
assures that sliding order 2 is achieved [5].
0
ds
s
W!
0
4
s
Statement of super-twisting algorithm
Consider that the induction machine system is of the form,
x(t )
f (t , x(t ), u (t ))
(6)
where x is the system state vecter, u is a bounded input
and t is the time variable. The trajectories of the supertwisting algorithm are characterised by twisting around
the origin on the phase portrait of the sliding variable.
Figure 2 shows the phase plot of the super-twisting
algorithm. The trajectories perform an infinite number of
rotations while converging in finite time to the origin.
The vibration magnitudes along the axes as well as the
rotation times decrease in as a geometric progression.
The super-twisting algorithm defines the control law, u (t )
as a combination of two terms. The first is defined in
terms of a discontinuous time derivative while the second
is a continuous function of the sliding variable. The
super-twisting algorithm is defined as follows:
where
u (t )
u1 (t ) u2 (t )
u1 (t )
­° u
®
°̄- W sgn( s )
u 2 (t)
­° O s U sgn( s ).
0
®
U
°̄ O s sgn( s ).
(7)
Figure 2: Phase plot of the super-twisting algorithm.
Remark: The system under consideration is assumed to
be bounded.
5
Control of an induction machine
A switching function, s, is defined in term of speed and
position errors.
s
Z error E hosmT error
where Z error
Z rmea Z rref and T error
s I (t , s ) J (t , s )u
u d1
I (t ) d ) ! 0
and sufficient conditions for finite time convergence are:
(10)
where I (t , x), J (t , x) are smooth uncertain functions and
are assumed in the range of operation
0 *m d J (t ) d *M
s d s0
T mea T ref . The
switching surface s (Z error ,T error ) is designed so that by
zeroing it, the control objective is achieved. It satisfies a
second order differential equation of the form:
u !1
s ! s0
(9)
(11)
The super-twisting algorithm in equation (7) can be
simplified as follows:
u (t )
u1
U
O s sgn( s ) u1
W sgn( s )
(12)
This control algorithm does not need any information on
the time derivatives of the sliding variable nor any explicit
knowledge of other system parameters. Effectively the
controller can be tuned via three parameters, U , O and
W.
6
Simulation and practical induction machine
system setup
The induction machine models in equations (1) and (2) are
written in an S-function for simulation in the
Matlab/SimulinkTM environment. The stator d-q voltages
are calculated through frame conversion and these
voltages are fed to the electrical model in equation (1).
The simulation system is setup in the vector control
environment as shown in Figure 4. For position and speed
tracking test, a set of position and speed reference signals
are used, as shown in Figure 5.
The speed reversal
(passing through zero rad/s) will provide information
about the friction effect on the induction machine system.
For the high inertia (about double the machine inertia)
load condition test, a flywheel weight is attached to one
end of the rotor shaft to increase the effective inertia of the
rotor shaft. A dc motor is also used to simulate a load
condition on the induction machine. The dc motor is
controlled
by
Gemdrive-micro2
micro-processor
controller. The magnitude of the dc motor torque is
directly proportional to the armature current. As a result,
the armature current is a measure of the degree of motor
loading. The armature current is set to 100% via the
micro-controller to give a ‘full-load torque’ in the test.
The specification of the induction machine system is
given in the Appendix. The system uses the standard
indirect vector-control technique for torque control. The
control technique configuration for the test is shown in
Figure 5.
Figure 3: Simulation system setup.
The data acquisition and control hardware systems used in
the test consist of a Texas InstrumentTM TMS320C6701
DSP board and a PC set. The C6701 DSP is a high
performance floating point digital signal processor. The
PC runs Texas Instruments Code Composer Studio (CCS)
integrated development environment [16]. The software
provides a compiler, assembler, and linker to allow
development of C or Assembly language code for the
DSP. A useful feature is the Real Time Data Exchange
(RTDX), which allows access to read or write data from
the DSP to the PC whilst the DSP is running in real time
without interrupting the operation. A FPGA AED-106
Multi-channel analog expension daughterboard by
Signalware [17] is employed to provide A/D interface,
PWM generation and a shaft encoder interface. The
generated PWM signals feed through a power inverter
which consists of a dc link, six gate drive circuits and a
three phase dc-ac IGBT inverter.
iderr
i*dse
+
-
idse
iqerr
PI torque
controller
+
i
*
-
qse
V*dse
PI flux
controller V*qse
Figure 5: Position and speed reference signals.
V*ds
synchronous
to stationary V*
qs
frame
sin(T), cos(T)
V*a
2/3 phase
conversion
3/2 phase
conversion
iqse
ids
position &
speed
reference
signal
V*b
Power
inverter
PWM
V*c
source
board
ia
ib
ic
Va Vb Vc
iqs
idse
designed
controllers
Tr
calculation
iqse stationary to
synchronous
frame
induction
machine
Zr
1/s
setup within DSP system and PC
Figure 4: Induction machine test setup.
hardware setup
7
‘Slotine’ and ‘PPI’ respectively in the following
description.
Simulation results
The friction and control parameters are shown in Table 1.
Table 1: Friction and control parameters.
2
Table 2: Control system parameter settings.
E sgn = 5.0
E slot = 10.0
KPP = 100.0
KPS = 6.0
J = 0.06 kgm
G fric = 0.0001
Fs = 0.0085
Fw = 6.25 x 10-5
ksgn = 9.0 (max)
KIS = 50.0
E hosm = 1.0
U = 0.5
Fv = 6.56 x 10-4
k slot = 9.0 (max)
I = 0.8
Fc = 0.0085
W = 8.0
Fst = 0.047
O = 0.1
Figure 6 and Figure 7 show the simulation results of the
position and speed errors respectively. The results with
and without the friction in the model are included. The
error amplitudes in the presence of the friction tend to be
larger. The control signal that is required to track the
reference signal is shown in Figure 8. The friction has an
impact on the control signal when the machine direction
changes (at about 0.65s) compared to the one without
friction dynamics.
Figure 9 shows the simulated
switching variable and Figure 10 gives the respective
simulated friction dynamic terms and the overall friction
dynamics. The friction constants are chosen in such a way
that the amplitudes of the respective friction dynamics are
distributed equally at about r 0.1 Nm.
8
Figure 6: Position error.
Implementation results and control system
performance comparison
To investigate the chattering effect on the control signal,
three additional control algorithms are considered. A
single component, signum term is employed in the test.
This component is commonly used in sliding mode
control design and is defined as
u
ksgn sgn ( s )
(13)
where ksgn is a positive constant and s
Z error E sgnT error
is the sliding variable.
A single component version of Slotine’s sliding mode
control [3] is included and is represented by the following
equation
u
k slot sat ( s / I )
sat ( s / I )
­°sgn( s )
®
°̄ s / I
Figure 7: Speed error.
(14)
s !I
s dI
where k slot is a positive constant, s Z error E slotT error is
the sliding variable, I defines the width of the boundary
layer, u is the controller output which is the torque
*
command reference current, iqse
. A fixed-gain control
system is also considered and comprises of nested position
and speed loops with proportional (KPP) and
proportional-integral (KPS, KIS) controllers, respectively.
The additional control system design parameters are
shown in Table 2. The signum term controller, Slotine’s
controller and fixed-gain controller are denoted as ‘sgn’,
Figure 8: Control signal.
Figure 9: Switching variable.
Figure 10: Simulated friction dynamic amplitude.
The first practical test is to investigate the U design
parameter in equation (12). The value of U is set to be
0.5. By testing U at different values, it became apparent
that for values less than 0.5, the tracking capability of the
controller degraded. Figure 11 shows the position error as
the U value is reduced.
Figure 11: Position error at different U values.
Figure 12, Figure 13 and Figure 14 show the results under
three different test conditions: nominal, high inertia load
and dc motor load respectively. Figure 12(a) and 12(b)
show the induction machine position and speed errors
respectively. The maximum position error for the fixedgain (PPI) is r0.12 rad. The HOSM, Slotine and ‘sgn’
controller have shown to have similar amplitudes of
maximum position error which is r0.025 rad. The
maximum speed errors for the HOSM, Slotine, ‘sgn’ and
PPI controllers are r0.40 rad/s, r0.40 rad/s, r1.0 rad/s and
r0.70 rad/s respectively. The results show that high
precision position tracking can be achieved using the
sliding mode controller, particularly the HOSM controller.
The ‘sgn’ controller generates the largest maximum speed
error. It is believed that the chattering effect is the cause.
The required control signals to maintain the reference
trajectories are shown in Figure 12(c) and the required
control signals are within r3.5A. As predicted the ‘sgn’
controller shows the most control activity, due to the
chattering phenomenon. The rest of the control signals
show no sign of chattering except at the moment the
induction machine changes its direction. Greater control
activity exists during the change of direction and this can
be explained by the friction dynamics that exist in the
system and also due to the inertia of the system. The
HOSM controller has shown to be a solution to the
chattering problem. These control signals replicate the
signals obtained via simulation (with friction dynamic
terms) in the earlier section. The plot in figure 12(d)
shows the sliding variable of both the HOSM, Slotine and
‘sgn’ controller. It is noticed that the sliding variable is
within the boundary limit of r0.8 and it is below the
theoretical tracking precision (i.e. I / E slot ) for the Slotine
controller.
In the presence of the high inertia load, the errors
produced by PPI controller are larger than the sliding
mode controllers. In fact, the peak position error (i.e.
r0.12 rad) for the PPI controller is about two times as
great as the peak errors of the HOSM and Slotine
controllers (i.e. r0.05 rad) with the high inertia load
incorporated. The HOSM has shown to have close control
performance to the Slotine controller. as shown in Figure
13(b). The PPI controller again has the largest peak error
and the ‘sgn’ controller is not much better for speed
tracking with a peak error of about the same as the PPI
controller. Similarly, the ‘sgn’ controller has shown to
produce chattering control. Other control signals show no
sign of chattering apart from when the machine is
changing its direction where more control activity is
observed. The control signals are approximately within
r5.0A.
With the dc motor load incorporated in the system, the
three sliding mode controllers again outperform the PPI
controller. The HOSM has shown to be similar to the
other two sliding mode controllers. It generates slightly
larger maximum position errors although the peak-to-peak
variation is about the same. It performs as well as the
Slotine controller in terms of the peak speed error. The
control signals for all the controllers unusually comprise
of high control activities throughout the tracking process
in the presence of the dc motor load. One explanation for
this is that it is because of the sampling rate (i.e. on the
feedback current) and the firing angle calculation of the
armature current which is controlled by a separate
industrial Gemdrive micro-controller [18]. This results in
a ‘ripple-like’ armature current being generated and hence
replicated in the load dynamics. The ‘sgn’ controller
again shows its inherent chattering problem in the control
signal.
9
Conclusion
In this paper, a chattering-free higher order (second order)
sliding mode control algorithm for the vector-controlled
induction machine has been described and implemented.
An induction machine model with linear and non-linear
friction dynamics has been proposed. The effect of the
friction dynamics has been shown in the implementation
results as well as the simulation results and close
replication of the results can be obtained.
The performance of the HOSM controller has been
investigated under three different test conditions together
with three additional control algorithms. The HOSM
controller has shown to be able to mitigate the chattering
effect completely in the tests. The performance results of
the HOSM controller have been shown to be comparable
with the Slotine’s controller. The implementation results
also highlight the robustness of the HOSM controller
against the external load disturbances and superior
tracking performance can be obtained.
10
References
[1] Edwards C. and Spurgeon S.K., Sliding mode
control: theory and applications, Taylor & Francis,
UK, 1998.
[2] Zinober, A.S.I., Deterministic control of uncertain
systems, Peter Peregrinus Ltd, London, 1990.
[3] Slotine, J.J.E., Sliding controller design for
nonlinear system, International Journal of Control,
vol.40, no.2, 1984.
[4] Emel’yanov, S.V., S.K. Korovin and Levant, A.,
High order sliding model in control systems,
Computational Mathematics and Modelling, Vol.7,
No.3, pp. 294-318, 1996.
[5] Levant, A., Sliding order and sliding accuracy in
sliding model control, International Journal of
Control, vol.58, no.6, pp.1247-1263, 1993.
[6] Levant, A., Higher order sliding: collection of
design tools, Proceeding of the European Control
Conference, Brussels, 1997.
[7] Bartolini, G., Ferrara, A. and Usai, E., Chattering
avoidance by second-order sliding mode control,
IEEE Transaction on Automatic Control, vol.43,
no.2, pp.241-246, 1998.
[8] Levant, A., Universal SISO sliding mode controllers
with finite-time convergence, IEEE Transactions on
Automatic Control, vol.46, no.9, pp.1447-1451,
2001.
[9] Armstrong-Helouvry, B., Dupont, P. and Canudas
De Wit, C., A survey of models, analysis tools and
compensation methods for the control of machines
with friction, Automatica, vol.30, no.7, pp.10831138, 1994.
[10] Armstrong-Helouvry, B., Control of machines with
friction, Kluwer, Boston, MA, 1991.
[11] Armstrong-Helouvry, B. and Canudas de Wit, C.,
The control handbook, chapter Friction modeling
and compoensation,” pp.1369-1382, CRC Press,
1996.
[12] Dunnigan, M.W., Wade, S., Williams, B.W. and Yu,
X., Position control of a vector controlled induction
machine using Slotine’s sliding mode control
approach, IEE Proc.-Electr. Power Appl., vol.145,
no.3, pp.231-238, 1998.
[13] Bartolini, G. and Punta, E., Chattering elimination
with second-order sliding modes robust to Coulomb
friction, Journal of Dynamic Systems, Measurement,
and Control, vol.122, pp.679-686, 2000.
[14] Bose, B.K., Power electronics and AC drives,
Prentice-Hall International, 1986.
[15] Etemad,
M.R.
and
Pullen
K.R.,
http://www.me.ic.ac.uk/department/review97/case/ca
serr12.html, 1997.
[16] Texas Instruments, Code Composer Studio Users
Guide, Lit. No. SPRU328, May, 1999.
[17] Signalware Corporation, Documentation Package for
AED-106
Multi-Channel
Analog
Expansion
Daughterboard, V.1., 2000.
[18] Gemdrive Micro 2, Gemdrive-micro 2 technical
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Controls), 1993.
11
Acknowledgements
This project is supported by EPSRC Grant Reference
GR/28447. Invaluable advice from Dr. Steve J. Finney
throughout this work is gratefully acknowledged.
12
Appendix
The definition of symbols are shown in Table 3.
Table 3: Symbol definition.
Symbol
Definition
Zr
rotor speed (rad/s)
magnetising inductances (H)
stator and rotor leakage inductance (H)
stator and rotor resistances (:)
stator, magnetising and rotor current (A)
d and q-axis stator voltages in the stationary
frame (V).
Lm
Lls , Llr
R s , Rr
is , im , ir
vds , vqs
The induction machine used in the experiments has the
specifications shown in Table 4 and Table 5. All
parameters are referred to the stator unless stated
otherwise
Table 4: Induction machine rating.
No. phases
3
Connection Star
No. poles
4
Power
3 HP (~2.24 kW)
230 V rms (line)
Line
l current 9 A rms
Line
Rated speed 1430 rpm
Rated
14.96 Nm (calculated at rated speed)
Rotor
Wound rotor with slip rings allowing
construction the connection of external impedances
(a)
Table 5: Induction machine parameters.
Stator resistance
Rs 0.55 :
Stator leakage inductance
Lls 5 mH
Magnetising inductance
Lm 63 mH
Rotor leakage inductance
Llr 5 mH
Rotor resistance
Rr 0.75 :
Core loss resistance
Rc 320 :
N 0.91 : 1
Stator:rotor winding ratio
External rotor resistance
RrEXT 1.0 :
(b)
(c)
(d)
Figure 12: Nominal condition - (a) position error, (b)
speed error, (c) control signal and (d) switching variable.
(a)
(a)
(b)
(b)
(c)
(c)
(d)
Figure 13: High inertia load condition - (a) position error,
(b) speed error, (c) control signal and (d) switching
variable.
(d)
Figure 14: DC motor load condition - (a) position error,
(b) speed error, (c) control signal and (d) switching
variable.