1) The document reviews an adaptive control method for induction motors with uncertain load torque and magnetic saturation. 2) A nonlinear adaptive controller is designed using a field oriented control approach. The controller estimates the unknown time-varying load torque through online identification. 3) Once the load torque is identified, the control decouples regulating rotor speed and flux amplitude to improve efficiency without affecting speed regulation. Adaptation laws are developed to track the load torque and regulate the motor's speed and flux.

1. REVIEW PAPER
International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010
Adaptive Control of Saturated Induction Motor
with Uncertain Load Torque
T.S.L.V.Ayyarao1, G.I.Kishore2, and M.Rambabu3
1
GMR Institute of Technology/EEE, Rajam, India
Email:ayyarao.tummala@gmail.com
2
GMR Institute of Technology/EEE, Rajam, India
Email: gi_kishore@yahoo.co.in
3
GMR Institute of Technology/EEE, Rajam, India
Email:m.rambabu@gmail.com
Abstract—The most popular method for high motor and then field oriented model. An adaptive control
performance control of induction motor is field of induction motor with saturation is developed which
oriented control. Variation of load torque cause input- estimates a time varying load torque.
output coupling, steady state tracking errors and
deteriorated transient response in this control. A II.INDUCTION MOTOR MODEL
nonlinear adaptive controller is designed for induction
motor with magnetic saturation which includes both A.Model of Induction Motor
electrical and mechanical dynamics. The control
An induction motor is made by three stator windings
algorithm contains a nonlinear identification scheme
and three rotor windings.
which asymptotically tracks the true value of the
unknown time varying load torque. Once this
dψ sb
parameter is identified, the two control goals of
regulating rotor speed and rotor flux amplitude are Rsisb + = usb
decoupled, so that power efficiency can be improved
without affecting speed regulation.
dt
Index Terms—Adaptive control, Induction motor with d ψ rd '
saturation, Matlab/Simulink R r i rd ' / + =0
dt
I. INTRODUCTION
dψ sa
Induction motor is known for constant speed
applications. It is maintenance free, simple in operation,
Rsisa + = usa
rugged and generally less expensive than either DC or dt
synchronous motors. On the other hand its model is more
complicated than other machines and because of this, it is
considered as ‘the benchmark problem in nonlinear
d ψ rq '
R r i rq ' + =0 (1)
control’. There are many approaches to induction motor dt
control. Field oriented control of induction motor was Where R, i, ψ, us denote resistance, current, flux linkage,
developed by Blaschke. Field oriented control achieves and stator voltage input to the machine; the subscripts s
input-output decoupling. Applications clearly indicate and r stand for stator and rotor, (a,b) denote the
that even though nonlinearities may be exactly modeled, components of a vector with respect to a fixed stator
the physical parameters involved are most often not reference frame, (d’,q’) denote the components of a
precisely known. This motivated further studies on vector with respect to a frame rotating
adaptive versions of feedback linearization and input- We now transform the vectors (ird’, irq’), (ψrd’, ψrq’) in the
output linearization [1], since cancellations of rotating frame (d’,q’) into vectors (ira, irb), (ψra, ψrb) in the
stationary frame (a,b) by
nonlinearities containing parameters are required. Load
torque and rotor resistance are estimated. The common
⎡ira ⎤ ⎡cosδ − sin δ ⎤ ⎡ird / ⎤
assumption made in these control laws is the linearity of
⎢i ⎥ = ⎢ sin δ ⎢ ⎥
cosδ ⎥ ⎢irq / ⎥
the magnetic circuit of the machine. In many variable ⎣ rb ⎦ ⎣ ⎦⎣ ⎦
torque applications, it is desirable to operate the machine
(2)
in the magnetic saturation region to allow the machine to
develop higher torque. The paper is organized as follows.
We shall first describe a fifth model [2] of induction
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DOI: 01.IJRTET.03.04.197

2. REVIEW PAPER
International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010
⎡ψ ra ⎤ ⎡cosδ − sin δ ⎤ ⎡ψ rd / ⎤ ⎛ψ d ⎞ ⎡ cos ρ sin ρ ⎤⎛ψ a ⎞
⎜ ⎟=⎢
⎢ψ ⎥ = ⎢ sin δ ⎢ ⎥ ⎜ψ q ⎟ − sin ρ cos ρ ⎥⎜ψ ⎟
⎜ ⎟
cosδ ⎥ ⎢ψ rq/ ⎥
(7)
⎣ rb ⎦ ⎣ ⎦⎣ ⎦ ⎝ ⎠ ⎣ ⎦⎝ b ⎠
(3) Since cos ρ = (ψ a / |ψ| ), sin ρ = (ψb/|ψ|),
The system (1) becomes with | ψ | = ψa2 + ψb2.
ψ a i a + ψ b ib
dw n p M id =
= (ψ a ib − ψ a i a ) − T L ψ
dt JL r J
dψ a R R ψ a ib − ψ b i a
= − r ψ a + r Mi a − n p ωψ b iq =
dt Lr Lr ψ
dψ b R R
= − r ψ b + r Mib + n p ωψ a ψ d = ψ a 2 +ψ b 2 = ψ
dt Lr Lr
ψ q = 0. (8)
We now interpret field oriented model as state feedback
di a MR r npM
= ψa + wψ b transformation (involving state space change of
dt σL s L r 2
σL s L r coordinates and nonlinear state feedback) into a control
system of simpler structure. Defining the state space
change of coordinates
⎛ M 2 Rr + Lr 2 Rs ⎞
−⎜ ⎟ia + 1 ua ω =ω
⎜ ⎟ σL
⎝ σLs Lr
2
⎠ s ψ d = ψ a 2 +ψ b 2
di b MR r npM ψb
= ψb − wψ a ρ = arctan
dt σLs Lr 2
σLs Lr ψa
ψ a i a + ψ b ib
id =
ψ
⎛ M 2 Rr + Lr 2 Rs ⎞
−⎜ ⎟ib + 1 ub ψ a ib − ψ b i a
⎜ σL L 2 ⎟ σL (4) iq = (9)
⎝ s r ⎠ s ψ
Substituting the above equations (10) in (9) the system
Where i, Ψ, us denote current, flux linkage and stator yields the field oriented model
voltage input to the machine; the subscripts s and r stand dω T
for stator and rotor; (a,b) denote the components of a = μψ d i q − L
vector with respect to a fixed stator reference frame and dt J
σ = 1-(M2/LsLr).
Let u = (ua,ub)T be the control vector. Let α = RrN/Lr, ψ
dd
β = (M/ σLsLr), γ = (M2RrN/ σLsLr)+(Rs/ σLs),
μ = (npM/JLr). =αψα d
− d+ M
System in (4) can be written in compact form as
x = f ( x ) + u a g a + u b g b + p1 f 1 + p 2 f 2 ( x )
&
dt
2
(5) did iq 1
= −γid + αβψ d + n pωiq + αM + ud
B. Field Oriented Model dt ψ d σLs
It involves the transformation of vectors (ia, ib), (ψ a, di q
ψ b) in the fixed stator frame (a, b) into vectors in a frame = − γ i q − β n p ωψ d
(d, q) which rotate along with the flux vector (ψ a, ψ b); dt
defining iq id 1
ρ = arctan (Ψb/Ψa) (6) − n p ω id − α M + uq
the transformations are
ψd σ Ls
⎛ id ⎞ ⎡ cos ρ sin ρ ⎤⎛ ia ⎞ dρ iq
⎜ ⎟=⎢ = n pω + αM .
⎜ iq ⎟ − sin ρ cos ρ ⎥⎜ i ⎟
(10)
⎜ ⎟ dt ψd
⎝ ⎠ ⎣ ⎦⎝ b ⎠
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3. REVIEW PAPER
International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010
ψ
III.INDUCTION MOTOR CONTROL
dd
A. Control Strategy
Let ω r (t) and ψ r (t) be the smooth bounded
=αψα dr
− d+Mi
reference signals for the output variables to be controlled
which are speed ω and flux modulus ψ . The control
dt
strategy is shown in Figure (1). The goal is to design dρ i qref
= n pω + αM .
control inputs u d and u q so that for any unknown time dt ψd
varying TL (t), we obtain (14)
lim
[ ω ( t ) − ω r ( t )] = 0
t → ∞ Now i qref and i dref are the inputs and these inputs are
lim
[ψ d ( t ) − ψ r ( t )] = 0 used to control ω and ψ . Different models are
t → ∞ (11)
available to incorporate flux saturation. Let’s use the
A standard approach for simplifying the dynamics for the model presented in [3] as given below.
dψd
design of the speed and flux loops is to use proportional
= M (−f −1(ψd ) +idref)
α
plus integral control loops to generate current command.
That is, choose the inputs as
t
u d = − k d 1 ( i d − i dref ) − k d 2 ∫ ( i d (τ ) − i dref ) d τ . dt
0
dρ i qref
(12) = n pω + αM . (15)
t
dt ψd
u q = − k q 1 ( i q − i qref ) − k q 2 ∫ ( i q (τ ) − i qref (τ )) d τ .
0 Where M and α are at their nominal values. In steady
(13) state ψ d = f ( i d ), which represents the saturation
Proper choice of the gain results tracking the reference curve for induction motor. In the absence of saturation
currents so that current dynamics can be ignored.
ψ = Mi d and f −1
(ψ d ) = ψ d / M . Therefore (11)
Therefore the system can then be replaced by the d
following reduced order. reduces to linear case.
Defining the error variables
~
ω = ω − ωr
~
ψ = ψ −ψ r (16)
Using the reduced motor equations and substituting
T L (t ) = c o + c1 (t − t o )
as in [4]. Where co and c1 are constant coefficients of the
first order polynomial used to estimate the time varying
parameter TL (t). We deduce the error equations.
~
& c o ( t ) c1 ( t )
ω = μψ d i q − − (t − t o ) − ω r
&
J J
~&
ψ = − M α ( f −1 (ψ d ) + i d ) − ψ r
&
(17)
Figure 1. Field Oriented Control of Induction motor.
Choosing the Lyapunov function
~ 1 ~ 2 ~2 ~2
dω V L= t ) (γ 1ω 2 + ψ d + γ 2 c o + γ 3 c1 )
T ( (18)
Mr.T.S.L.V.Ayyarao is an Assistant Professor = μψ d i qref − 2
at GMR Institute of Technology, Rajam. dt J
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4. REVIEW PAPER
International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010
Where γ 1 , γ 2 ,γ
3 are positive design parameters 350
~ ˆ ~
chosen by the designer. c o = c o − c o , c1 = c1 − c1 are
ˆ
300
ˆ ˆ
the coefficient estimation errors and c o , c 1 are
estimates of the coefficients. 250
reference speed (rad/s)
200
The desired currents inputs are chosen as follows
150
1 ~ cˆ ˆ
c
i qref = [ − k w ω 2 + o + (t − t o ) + ω r ]
&
μψ d
100
J J
(19)
1 50
i dref = [ Mαf −1
(ψ d ) + ψ r − kψ d ]
&
Mα 0
0 1 2 3 4 5 6 7 8
time(s)
Where kw and k ψ d are positive design parameters. Figure 2. Reference Speed
The adaptation laws are given by
1.4
1 ~
λ1ω
&
c0 =
ˆ (− ) 1.35
λ2 J 1.3
1 ~
λ 1ω ( t − t o )
&
c1 =
ˆ (− )
1.25
λ3
referenc e flux (w b)
J 1.2
(20) 1.15
1.1
With
& ~ ~ 2
V = −λ1 k wω 2 − kψdψ d
1.05
1
(21) 0.95
0.9
0 1 2 3 4 5 6 7 8
IV. SIMULATION time(s)
Figure 3. Reference Flux
The proposed algorithm has been simulated in
Matlab/Simulink for a 15 KW motor, with rated torque
70 Nm and rated speed 220 rad/s, whose data are listed in Simulation results for f
−1
(ψ d ) = ψ d / M are shown in
the Appendix. The simulation test involves the
[9]. Simulation results for f (ψ d ) = ψ d / M
−1 3
following operating sequences: the unloaded motor is
required to reach the rated speed and rated value of 1.3 (saturation) are shown in Figure.4 & 5. The controller
Wb for rotor flux amplitude. At t = 2s a load torque parameters are chosen as kw=12000, k ψ d =1000,
40Nm, which is unknown to the controller, is applied. At kd1=46000, kd2=88000, kq1=196000, kq2=188000. Figure.4
t = 5s., the speed shows the actual speed and it tracks the reference speed
is required to reach 300 rad/s., well above the nominal even though load torque changes as a function of time.
value, and rotor flux amplitude is weakened. The The flux response tracks the reference flux as shown in
reference signals for flux amplitude and speed consists of Figure.5.The simulation results show the superior
tep functions, smoothed by means of second order behavior of the control algorithm.
polynomials. A small time delay at the beginning of the
speed reference trajectory is introduced in order to avoid
overlapping between flux and speed transients.
Figure 2 & 3 shows the reference speed and flux.
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5. REVIEW PAPER
International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010
Speed vs time measurement of noise, simplifying modeling assumptions
350
and unmodeled dynamics. The authors can conclude on
the topic discussed and proposed. Future enhancement
300
can also be briefed here.
250
Appendix A
Induction motor Data
speed(rad/s)
200
Rs 0.18 Ω
150
Rr 0.15 Ω
ψr 1.3 Wb rated
100 ω 220 rad/sec rated
np 1
50 J 0.0586 Kgm2
M 0.068 H
Lr 0.0699 H
0
0 1 2 3 4 5 6 7 8 Ls 0.0699 H
time(s) TL 70 Nm
Figure 4. Actual Speed T Rated power 15 Kw
1.4
REFERENCES
1.35 [1] R.Marino, Peresada and Valigi, “Adaptive input-output
Linearizing control of Induction motor”, IEEE transactions
1.3
on Automatic control, vol. 38, pp. 209-222.
1.25 [2] P.C.Krause, ‘Analysis of Electric Machinery’, New York:
1.2
McGraw-Hill, 1986.
[3] G.Heinemann and W.Leonhard, “Self-tuning field oriented
flux(wb)
1.15 control of an induction motor Drive”, in Proc. Int. Power
1.1
Electro. Conf., Tokyo, Japan, April 1990.
[4] Ebrahim, Murphy, “Adaptive Control of induction motor
1.05
with magnetic saturation under time varying load torque
1 uncertainty”, IEEE transactions on systems theory, Macon,
March 2007.
0.95
[5] Alberto Isidori, Nonlinear Control, Springer-Verlag,
0.9
0 1 2 3 4 5 6 7 8
London:1990.
time(s) [6] J.J.E.Slotine and Weiping Li, Applied Nonlinear Control,
Figure 5. Actual Flux.
Prentice-Hall, 1991.
[7] W.Leonhard, Control of Electric Drives, Berlin:Springer-
Verlag, 1985.
[8] I.Kanellakopoulos, P.V.Kokotovic, and R.Marino, “An
CONCLUTION extended direct scheme for robust adaptive nonlinear
In this paper we propose a detailed nonlinear model control,” Automatica, vol. 27, no.2, pp. 247-255, 1981.
[9] T.S.L.V.Ayyarao, A.Apparao and P.Devendra, “Control of
of an induction motor, an adaptive input-output
Induction motor by adaptive input-output linearization,”
decoupling control which has some advantages over the NADCAP, Hyderabad, 2009.
classical scheme of field oriented control. With a
comparable complexity exact decoupling between speed
and flux regulation is achieved and critical parameter is
identified. The main drawback of the proposed control is
the requirement of flux measurements. Additional
research should analyze the influence of sampling error,
88
© 2010 ACEEE
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