adaptive control of saturated induction motor with uncertain load torque

REVIEW PAPER International J. of Recent Trends in Engineering and Technology, Vol. 3, No. 4, May 2010

84 © 2010 ACEEE DOI: 01.IJRTET.03.04.197

Adaptive Control of Saturated Induction Motor with Uncertain Load Torque

T.S.L.V.Ayyarao1, G.I.Kishore2, and M.Rambabu3

1GMR Institute of Technology/EEE, Rajam, India Email:[email protected]

2GMR Institute of Technology/EEE, Rajam, India Email: [email protected]

3GMR Institute of Technology/EEE, Rajam, India Email:[email protected]

Abstract—The most popular method for high performance control of induction motor is field oriented control. Variation of load torque cause input-output coupling, steady state tracking errors and deteriorated transient response in this control. A nonlinear adaptive controller is designed for induction motor with magnetic saturation which includes both electrical and mechanical dynamics. The control algorithm contains a nonlinear identification scheme which asymptotically tracks the true value of the unknown time varying load torque. Once this parameter is identified, the two control goals of regulating rotor speed and rotor flux amplitude are decoupled, so that power efficiency can be improved without affecting speed regulation. Index Terms—Adaptive control, Induction motor with saturation, Matlab/Simulink

I. INTRODUCTION

Induction motor is known for constant speed applications. It is maintenance free, simple in operation, rugged and generally less expensive than either DC or 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 control’. There are many approaches to induction motor control. Field oriented control of induction motor was developed by Blaschke. Field oriented control achieves input-output decoupling. Applications clearly indicate that even though nonlinearities may be exactly modeled, the physical parameters involved are most often not precisely known. This motivated further studies on adaptive versions of feedback linearization and input-output linearization [1], since cancellations of nonlinearities containing parameters are required. Load torque and rotor resistance are estimated. The common assumption made in these control laws is the linearity of the magnetic circuit of the machine. In many variable torque applications, it is desirable to operate the machine 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

motor and then field oriented model. An adaptive control of induction motor with saturation is developed which estimates a time varying load torque.

II.INDUCTION MOTOR MODEL

A.Model of Induction Motor An induction motor is made by three stator windings and three rotor windings.

sbsb

sbs udt

diR =+ψ

0''/

=+dt

diR rd

rdrψ

sasa

sas udt

diR =+ψ

0'' =+

dtd

iR rqrqr

ψ (1)

Where R, i, ψ, us denote resistance, current, flux linkage, and stator voltage input to the machine; the subscripts s and r stand for stator and rotor, (a,b) denote the components of a vector with respect to a fixed stator reference frame, (d’,q’) denote the components of a vector with respect to a frame rotating We now transform the vectors (ird’, irq’), (ψrd’, ψrq’) in the rotating frame (d’,q’) into vectors (ira, irb), (ψra, ψrb) in the stationary frame (a,b) by

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡/

/

cossinsincos

rq

rd

rb

ra

ii

ii

δδδδ

(2)


85 © 2010 ACEEE DOI: 01.IJRTET.03.04.197

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡/

/

cossinsincos

rq

rd

rb

ra

ψψ

δδδδ

ψψ

(3) The system (1) becomes

( )J

TiiJL

Mndtdw L

aabar

p −−= ψψ

bpar

ra

r

ra nMiLR

LR

dtd

ωψψψ

−+−=

apbr

rb

r

rb nMiLR

LR

dtd

ωψψψ

++−=

brs

pa

rs

ra wLLMn

LLMR

dtdi

ψσ

ψσ

+= 2

as

ars

srr uL

iLL

RLRMσσ1

2

22

+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

ars

pb

rs

rb wLLMn

LLMR

dtdi

ψσ

ψσ

−= 2

bs

brs

srr uL

iLL

RLRMσσ1

2

22

+⎟⎟⎠

⎞⎜⎜⎝

⎛ +− (4)

Where i, Ψ, us denote current, flux linkage and stator voltage input to the machine; the subscripts s and r stand for stator and rotor; (a,b) denote the components of a vector with respect to a fixed stator reference frame and σ = 1-(M2/LsLr). Let u = (ua,ub)T be the control vector. Let α = RrN/Lr, β = (M/ σLsLr), γ = (M2RrN/ σLsLr)+(Rs/ σLs),

μ = (npM/JLr). System in (4) can be written in compact form as

)()( 2211 xfpfpguguxfx bbaa ++++=& (5)

B. Field Oriented Model It involves the transformation of vectors (ia, ib), (ψ a, ψ b) in the fixed stator frame (a, b) into vectors in a frame (d, q) which rotate along with the flux vector (ψ a, ψ b); defining ρ = arctan (Ψb/Ψa) (6)

the transformations are

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

b

a

q

d

ii

ii

ρρρρ

cossinsincos

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

b

a

q

d

ψψ

ρρρρ

ψψ

cossinsincos

(7)

Since cos ρ = (ψ a / |ψ| ), sin ρ = (ψb/|ψ|), with | ψ | = ψa

2 + ψb2.

ψψψ bbaa

dii

i+

=

ψψψ abba ii

iq−

= ψψψψ =+= 22

bad ψ q = 0. (8) We now interpret field oriented model as state feedback transformation (involving state space change of coordinates and nonlinear state feedback) into a control system of simpler structure. Defining the state space change of coordinates

22bad ψψψ

ωω

+=

=

a

b

ψψ

ρ arctan=

ψψψ bbaa

dii

i+

=

ψψψ abba iiiq −

= (9)

Substituting the above equations (10) in (9) the system yields the field oriented model

JTi

dtd L

qd −= μψω

ddd M

dtd

ααψψ

+−=

qpddd ini

dtdi ωαβψγ ++−= d

sd

q uL

iM

σψα 1

2

++

dpqq ni

dtdi

ωψβγ −−=

qsd

dqdp u

Lii

Minσψ

αω 1+−−

.d

qp

iMn

dtd

ψαωρ

+= (10)


86 © 2010 ACEEE DOI: 01.IJRTET.03.04.197

Figure 1. Field Oriented Control of Induction motor.

III.INDUCTION MOTOR CONTROL

A. Control Strategy

Let ω r (t) and ψ r (t) be the smooth bounded reference signals for the output variables to be controlled which are speed ω and flux modulus ψ . The control strategy is shown in Figure (1). The goal is to design control inputs du and qu so that for any unknown time varying TL (t), we obtain

0)]()([lim

=−∞→

ttt rωω

0)]()([lim

=−∞→

ttt rd ψψ (11)

A standard approach for simplifying the dynamics for the design of the speed and flux loops is to use proportional plus integral control loops to generate current command. That is, choose the inputs as

∫ −−−−=t

drefdddrefddd diikiiku0

21 .))(()( ττ

(12)

∫ −−−−=t

qrefqqqrefqqq diikiiku0

21 .))()(()( τττ

(13) Proper choice of the gain results tracking the reference

currents so that current dynamics can be ignored. Therefore the system can then be replaced by the following reduced order.

JtTi

dtd L

qrefd)(

−= μψω

drdd Mi

dtd

ααψψ

+−=

.d

qrefp

iMn

dtd

ψαωρ

+=

(14)

Now qrefi and drefi are the inputs and these inputs are used to control ω and ψ . Different models are available to incorporate flux saturation. Let’s use the model presented in [3] as given below.

))(( 1drefd

d ifMdt

d+−= − ψα

ψ

.d

qrefp

iMn

dtd

ψαωρ

+= (15)

Where M andα are at their nominal values. In steady state ),( dd if=ψ which represents the saturation curve for induction motor. In the absence of saturation

dd Mi=ψ and ./)(1 Mf dd ψψ =− Therefore (11)

reduces to linear case.

Defining the error variables

r

r

ψψψωωω

−=−=

~~

(16) Using the reduced motor equations and substituting

)()( 1 ooL ttcctT −+=

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.

rdd

roo

qd

ifM

ttJ

tcJ

tci

ψψαψ

ωμψω

&&

&&

−+−=

−−−−=

− ))((~

)()()(~

1

1

(17)

Choosing the Lyapunov function

)~~~~(21 2

132

222

1 ccV od γγψωγ +++= (18)

Mr.T.S.L.V.Ayyarao is an Assistant Professor

at GMR Institute of Technology, Rajam.


87 © 2010 ACEEE DOI: 01.IJRTET.03.04.197

Where 1γ , 32 , γγ are positive design parameters chosen by the designer. 111 ˆ~,ˆ~ cccccc ooo −=−= are the coefficient estimation errors and oc , 1c are estimates of the coefficients.

The desired currents inputs are chosen as follows

])([1

])(ˆˆ~[1

1

2

drddref

roo

wd

qref

kfMM

i

ttJc

Jc

ki

ψψψαα

ωωμψ

−+=

+−++−=

− &

&

(19)

Where kw and dk ψ are positive design parameters. The adaptation laws are given by

)~

(1ˆ 1

20 J

c ωλλ

−=&

))(~

(1ˆ 1

31 J

ttc o−

−=ωλ

λ&

(20)

With 22

1~~

ddw kkV ψωλ ψ−−=& (21)

IV. SIMULATION

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

the Appendix. The simulation test involves the following operating sequences: the unloaded motor is required to reach the rated speed and rated value of 1.3 Wb for rotor flux amplitude. At t = 2s a load torque 40Nm, which is unknown to the controller, is applied. At t = 5s., the speed is required to reach 300 rad/s., well above the nominal value, and rotor flux amplitude is weakened. The reference signals for flux amplitude and speed consists of tep functions, smoothed by means of second order 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.

0 1 2 3 4 5 6 7 80

50

100

150

200

250

300

350

time(s)

refe

renc

e sp

eed

(rad

/s)

Figure 2. Reference Speed

0 1 2 3 4 5 6 7 80.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

time(s)

refe

renc

e flu

x(w

b)

Figure 3. Reference Flux

Simulation results for Mf dd /)(1 ψψ =−are shown in

[9]. Simulation results for Mf dd /)( 31 ψψ =−

(saturation) are shown in Figure.4 & 5. The controller parameters are chosen as kw=12000, dk ψ =1000, kd1=46000, kd2=88000, kq1=196000, kq2=188000. Figure.4 shows the actual speed and it tracks the reference speed even though load torque changes as a function of time. The flux response tracks the reference flux as shown in Figure.5.The simulation results show the superior behavior of the control algorithm.


88 © 2010 ACEEE DOI: 01.IJRTET.03.04.197

0 1 2 3 4 5 6 7 80

50

100

150

200

250

300

350

time(s)

spee

d(ra

d/s)

Speed vs time

Figure 4. Actual Speed

0 1 2 3 4 5 6 7 80.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

time(s)

flux(

wb)

Figure 5. Actual Flux.

CONCLUTION

In this paper we propose a detailed nonlinear model of an induction motor, an adaptive input-output decoupling control which has some advantages over the 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,

measurement of noise, simplifying modeling assumptions and unmodeled dynamics. The authors can conclude on the topic discussed and proposed. Future enhancement can also be briefed here.

Appendix A

Induction motor Data

Rs

Rr ψr ω np J M Lr

Ls TL T

0.18 Ω 0.15 Ω

1.3 Wb rated 220 rad/sec rated

1 0.0586 Kgm2

0.068 H 0.0699 H 0.0699 H 70 Nm

Rated power 15 Kw

REFERENCES [1] R.Marino, Peresada and Valigi, “Adaptive input-output

Linearizing control of Induction motor”, IEEE transactions on Automatic control, vol. 38, pp. 209-222.

[2] P.C.Krause, ‘Analysis of Electric Machinery’, New York: McGraw-Hill, 1986.

[3] G.Heinemann and W.Leonhard, “Self-tuning field oriented control of an induction motor Drive”, in Proc. Int. Power Electro. Conf., Tokyo, Japan, April 1990.

[4] Ebrahim, Murphy, “Adaptive Control of induction motor with magnetic saturation under time varying load torque uncertainty”, IEEE transactions on systems theory, Macon, March 2007.

[5] Alberto Isidori, Nonlinear Control, Springer-Verlag, London:1990.

[6] J.J.E.Slotine and Weiping Li, Applied Nonlinear Control, Prentice-Hall, 1991.

[7] W.Leonhard, Control of Electric Drives, Berlin:Springer-Verlag, 1985.

[8] I.Kanellakopoulos, P.V.Kokotovic, and R.Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, no.2, pp. 247-255, 1981.

[9] T.S.L.V.Ayyarao, A.Apparao and P.Devendra, “Control of Induction motor by adaptive input-output linearization,” NADCAP, Hyderabad, 2009.

adaptive control of saturated induction motor with uncertain load torque

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