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Sliding mode field
oriented control of
3-phase induction motor
Presented by
M.M.V Prabhakar
07341D4207

Department of Electrical Engineering
GMR Institute of Technology
Rajam, Srikakulam(D.T)

Deliver The Promise Learning Social Responsibility Respect for Individual


ABSTRACT
• The design includes rotor speed estimation from measured
stator terminals of voltages and currents.
• The estimated speed is used as feedback in an IFOC
system achieving the speed control without the use of
‘shaft mounted transducers’.
• This paper presents a new sensor less vector control
consisting on the One hand of speed estimation algorithm
which overcomes the necessity of the speed sensor and on
the other hand of a variable structure control law with an
integral sliding surface, that compensates the uncerta
inities that are present in the system.



• Now a days, AC drives become popular in many
applications because of the advances in power electronics
and microelectronics technology

• It is well known that field-oriented control (FOC) is an
effective scheme for the variable speed control of IM
drives. However, difficulties arise from the modelling
uncertainties due to parameter variations, magnetic
saturation, load disturbances and unmodelled dynamics



Why FOC ?
• IM is superior to DC machine with respect to size,
weight, inertia, cost, speed

 DC motor is superior to IM with respect to ease
of control
– High performance with simple control due de-coupling
component of torque and flux

 FOC transforms the dynamics of IM to become
similar to the DC motor’s – decoupling the
torque and flux components



Basic Principles DC machine
By keeping flux constant,
torque can be controlled
by controlling armature
current
φa
Te = k If Ia

Current in
φf Current out



Basic Principles of IM
φs
a φr Stator current produce stator
flux
c’ b’
Stator flux induces rotor
current → produces rotor
flux

Interaction between stator
and rotor fluxes produces
b torque
c

Space angle between stator
and rotor fluxes varies with
load, and speed



Introduction to speed control

 Scalar control
• Magnitude variation of control variables

 Vector control
• Both the magnitude and phase alignment of vector variables

• Scalar control is somewhat simple to implement, but the inherent coupling
effect i.e. both the torque and flux are functions of V or I & f, gives sluggish
response & system is easily prone to instability because of a high order
system harmonics.

• Ex. If torque is increased by incrementing the slip (frequency), the flux
tends to decrease. This is then compensated by sluggish flux control loop
feeding in additional voltage. The temporary dipping of flux reduces the
torque sensitivity with slip & lengthen the response time.



DC DRIVE ANALOGY
Ia ψa
If Te = K t Ψf Ψa = K t' I a I f
If

Torque component
ψf
Decoupled Field component

(Neglecting armature reaction & field saturation)

I qs ^

Ids
Te = K t Ψ r I qs = K t' I qs I ds
*
I ds

ωe
*
I qs
∧

(Synchronously rotating frame) ψr



Principle of vector control

Control Machine

*
I ds I ds*
s *
Ia Ia s
I ds d s − s I ds
d −q
a −b − c q machine
d −q
e e s s
*
* Ib Ib to to
I qs to I qs*
s
to *
s
I qs I qs d e −q e
I Ic d s − qs de − e
q
d s −q s a −b −c
c
mod el
I qs

I ds

ωe
cos θ e sin θ e cos θ e sin θ e
∧
ψr

Inverse Machine model Transformation

transformation



. FOC is called as decouple, orthogonal and Trans vector control
. FOC technique dpcouples the 2 components of stator current. One
providing the airgap flux & another producing the torque. It provides
independent control of flux and torque , another control charecterstics
are liniarized.
. The stator currents are converted to synchronously rotating reference
frame aligned with the flux vector and transformed back to the stator
frame before feeding back to the machine.
. The 2 components of currents are d-axis Ids analogous to armature
current, q-axis Iqs analogous to field current.
. FOC offers more precauce control of A.C motor compare to vector
control. Therefore FOC is used in high performance drives like
Robotic actuators, centrifuges and servos.



FOC of IM drive
Torque equation : 3p
Te = ψ s × is
22
3 p Lm
Te = ψ r × is
In d-q axis : 2 2 Lr
3 p Lm
Te = (ψ rd i sq − ψ rq i sd )
2 2 Lr
Choose a frame such that:

ψ ψr
rd = ψr ψ ψr = 0
rq



FOC of IM drive

ψψ = ψr
rd
r
ψψ = 0
r
rq

qs
As seen by stator reference frame:

is
isq
Ψr
3 p Lm Ψrq
Te = (ψ rd isq − ψ rq isd )
2 2 Lr
isd Ψrd ds



FOC of IM drive

ψψ = ψr
rd
r
ψψ = 0
rq
r

qs
Rotating reference frame:
q Ψr

Te =
3 p Lm
(ψ rd i sq − ψ rq i sd )
is
d Ψr
2 2 Lr Ψr
Ψ
isqr Ψ
isdr
3 p Lm ψ
Te = ψ r isqr
2 2 Lr
ds



Direct vector control
Feedback

Indirect vector control

Feedforword



Direct vector control

iqs qe ^
Ψ = Ψr cos θe
s
dr

Ψqr
s ^
qs Ψ = Ψ sin θe
s
qrr

ids
− iqs θe Ψs
Ψr^ cos θe = dr
e Ψ ^
Ψ dr
d
s
r
Ψ s

sin θ
qr
e =
d s
Ψr
^

ˆ = Ψ s2 + Ψ s2
Ψr dr qr



Indirect Vector Control

qe

iqs ψ qr = 0

ψ qr
s

∧ qs
θ ids I s
s
iqs ∧
ids ψ dr = ψ r
θe θ sl
ωsl
θr de
ψ s
dr r
ωe
d
ωr
Rotor
s
d Axis



The stator voltage equations in d-q equivalent circuit is given as,

Lm d
v =
s
ds (Ψ dr ) + ( Rs + σ Ls S )ids
s s

Lr dt

Lm d
v =
s
qs (Ψ qr ) + ( Rs + σ Ls S )iqs
s s

Lr dt
where

L2
σ= − m
1
Lr Ls
Which is called the motor leakage coefficient



Salient Features of vector control :

The machine is essentially self controlled .

No fear of instability

The transient response will be fast

Speed control is possible in four quadrants



Sliding mode control

 A sliding mode control with a variable control structure is basically
an adaptive control that gives robust performance of a drive with
parameter variation load torque disturbance.

 The control is nonlinear and can be applied to a linear or nonlinear
plant.

 The drive response is forced to tract or slide along a predefined
trajectory or reference model in a phase plane by a switching control
algorithm, irrespective of the plant’s parameter variation.

 SMC is sensor less vector control, where the speed signal is
estimated from the machine terminal voltage without using any speed
sensor or any other type of secondary transducer.



Sliding mode field oriented control

* *
iqs iabc
ω *
r i*
qs
e(t) VSC
limiter
*
ids dq-abc
Current
Controller
ωr*
ωr controller

Ψ dr*
e θe pulses

1
ωr i *
ds s
Field
PWM
calculation
weakening
ωe
inverter
iabc
ωr Wr & We
vabc
estimator

IM



The mechanical equation of the induction motor is given by,
•
J wm + Bwm +TL =Te
•
ω +aω + f =bi
m m
e
qs

The speed tracking error is given as,

e(t ) = ω m (t ) − ω m (t )
*

• • •
*
e(t ) = ω m − ω m = − ae(t ) + u (t ) + d (t )

Where u (t) and d (t) are collected as,


•
u (t ) = bi e qs (t ) − aω* m (t ) − f (t ) −ωm (t )
*

d (t ) = −∆aωm (t ) − ∆f (t ) + ∆bi e qs (t )

Now we are defined the sliding variable s (t) with an integral component
as,

t
s (t ) = e(t ) − ∫ ( k − a )e(τ ) dτ
0

The sliding surface is defined as,

t
s (t ) = e(t ) − ∫ ( k − a )e(τ )dτ = 0
0



The variable structure speed controller is designed as,

u (t ) = ke(t ) − β sgn( s )

In order to obtained the speed trajectory two assumption are taken as,

1.The gain k must be chosen such that the term (k-a) is strictly
negative, so K<0
2. The gain β must be chosen so that β ≥ |d(t)| for all time.

if assumptions are verified, the control law leads the rotor mechanical
speed wm(t) so that the speed tracking error e (t) tends to zero as the
time tends to infinity.



The proof of this theorem will be carried out using the Lyapunov stability
theory.
1
v (t ) = s (t ) s (t )
2
Its time derivative is

.
V (t )
. .
= s (t ) s (t ) = s[e− (k − a )e]
= s[(− ae + u + d ) − (ke − ae)] = s[u + d − ke]
= s[ke − β sgn( s ) + d − ke] = s[d − β sgn( s)]
≤ −[ β − | d | s ] ≤ 0
When the sliding mode occurs on the sliding surface , and the dynamic
behavior of the tracking problem is equivalently governed by the
following equation



Current controller:

The block current controller consists of three hysteresis band current PWM
control. it is basically an instantaneous feedback current control method of
PWM where the actual current continuously tracks the current command
within a hysteresis band.

SPWM

i*
K2
K1 +
s PWM
i

φ



The control circuit generates the sin reference current wave of desired
magnitude and frequency and it is compared with the actual phase current
wave. As the current exceeds the prescribed hysteresis band, the upper
switch in the half bridge is turned off and lower is turned on. The output
voltage transitions from +0.5Vd to -0.5Vd. As the current crosses the lower
band limit the lower is turned off and upper is on.

Upper band HB
Hysterisis band 2HB

Sine reference wave
Lower band HB

Actual current

+0.5Vd

0 ωt

-0.5Vd



Estimation of rotor speed
The rotor flux equation in the stationary frame is
• Lm 1
Ψ dr = ids − ω r Ψ qr − Ψ dr
Tr Tr
• Lm 1
Ψqr = iqs + ωr Ψdr − Ψqr
Tr Tr
The angle between the rotor flux in relation to d axis of the stationary frame is
Ψ
θ =arctan(
e
qr
)
Ψdr
• •
• Ψ Ψ −Ψ Ψ
qr
θe =ω =
e
dr qr dr

Ψ +Ψ
2
dr
2
qr

Lm Ψ dr iqs − Ψ qr ids
ωe = ωr − ( )
Tr Ψ dr + Ψ qr
2 2

1 • • L
ω r = 2 [Ψ dr Ψ qr − Ψ qr Ψ dr + m (Ψ dr iqs − Ψ qr ids )]
Ψr Tr



Field weakening controller
The block field weakening gives the flux command based on the rotor
speed ,so that the PWM controller does not saturate.

if ω r < ω b , Ψ * = Ψ drRated
dr
ωb
ω r > ω b , Ψ = Ψ drRated ×
*
dr
ωr
With the proper mentioned field orientation, the dynamic of the rotor flux is given
as: e
dΨdr Rr e Lm
+ Ψdr − Rr ids − ωsl Ψqr = 0
e

dt Lr Lr
dΨ e
qr
For decoupling Ψ e = 0,
qr =0
dt
Lr dΨ e
dr
+Ψ = Lm ids
e
dr
Rr dt
Ψ = Lm ids
dr

In other words the rotor flux is directly proportional to the current ids in steady state


Results and discussion

Reference and estimated rotor speed signal (rad/sec)

-------ω m
*

------ -ω m



Plot of motor torque vs time Stator current isa (A)

80
60

60
50

40 40

30 20

current(A)
torque(N-M )

20 0

10 -20

0 -40

-10 -60
0 0.5 1 1.5 0 0.5 1 1.5
tim e(s) time(s)



Time (s)

stator voltage Vsa (V)



Stator current ids (A) Stator current iqs(A)

60 80

40 60

20 40

0 20

current(A)
current

-20 0

-40 -20

-60 -40

-80
0 0.5 1 1.5 -60
0 0.5 1 1.5
time(s)
time(s)
Time (s) Time (s)



Total rotor flux

200

180

160

140

120
total flux

100

80

60

40

20

0
0 0.5 1 1.5
time(s)



UNDER LOAD TORQUE VARIATION

Reference and estimated rotor speed
signal (rad/sec) Plot of motor torque vs time

200
6000

150
4000
100

2000
50
speed(rad/sec)

torque(N-M)
0 0

-50
-2000

-100
-4000
-150

-200 -6000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(S)
time(s)



Data for the sliding mode control
Electrical parameter Mechanical parameter
Sl. Name of the Numerical Sl Name of the Numerical
No Parameters value No Parameters value
1. J (Moment of 1.662 kg-
1. Pole 4 Nos
Inertia) m.sq
2. Rs (Stator 0.6 Ohms 2. B( Frictional 0.1 Nms
resistance) constant)
3. Rr (Rotor 0.412
resistance) Ohms Controlling parameter
4. Ls(Stator 1.9 mH Sl Name of the Numerical
inductance) No Parameters value
5. Lr (rotor 1.9 mH 1. K (Constant gain) -100
inductance)
6. Lm(Mutual 41.2 mH
2. Β (switching gain) 30
inductance)


Conclusion and future work
Sliding Mode Control (SMC) is a robust control scheme based on the concept of
changing the structure of the controller in response to the changing state of the
system in order to obtain a desired response. The biggest advantage of this
system is stabilizing properties are preserved, even in the presence of large
disturbance signals. The dynamic behavior of the system may be tailored by the
particular choice of switching function and the closed-loop response becomes
totally insensitive to a particular class of uncertainty. One of the problems
associated with implementation of SMC is Chattering which is essentially a high
frequency switching of the control. Chattering in torque & speed may large, but can
be minimized by small computation sampling time higher pwm frequency &
minimizing additional delay in feedback signal.

Scope of future work.

 Application of higher order sliding mode control to other non linear systems may
be attempted.

A higher order discrete sliding mode control law may be developed.



References
[1] Nihat Inanc “A new sliding mode flux and current observer for
direct field oriented induction motor drives” Electric Power
Systems Research 63 (2002) 113-118.

[2] Nihat Inanc “A robust sliding mode flux and speed observer for
speed sensorless control of an indirect field oriented induction
motor drives” Electric Power Systems Research
77(2007)1681-1688
[3] B.K. Bose, Modern Power Electronics and AC Drives, Prentice
Hall, New Jersey, 2001.

[4] P. Vas, Vector Control of AC Machines, Oxford Science
Publications, Oxford, 1994.



[5] Williams, B. W., Goodfellow, J. K. and Green, T. C. “Sensorless
speed measurement of inverter driven squirrel cage induction motors.”
in Proc.. IEEE 4th Int. Con$ Power Electron. Variable Speed Drives,
(1987).

[6] Benchaib, A. Edwards, C. “Nonlinear sliding mode control of an
induction motor.” Int. J. Adapt. Control Signal Process. Vol. 14, (2000):
pp. 201–221.


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