Adaptive Back Stepping Control Of A Speed-sensorless Im

359

39th Southeastern Symposium on System Theory Mercer University Macon, GA, 31207, March 4-6, 2007

TB2.6

Adaptive Backstepping Control of a Speed-Sensorless Induction Motor Under Time-Varying Load Torque and Rotor Resistance Uncertainty Arbin Ebrahim and Gregory Murphy Abstract- A new global adaptive backstepping controller is designed for induction motor speed control based on measurements of stator current and estimation of rotor speed. The designed partial state feedback controller is singularity free and guarantees asymptotic tracking of smooth reference trajectories for the speed of the motor under time varying load torque and rotor resistance uncertainty. The new control algorithm generates estimates for unknown time varying load torque, rotor resistance and rotor speed, which asymptotically tracks and converges to their true values. The rotor flux modulus asymptotically tracks a desired reference signal which allows the motor to operate within its specifications. As in the field-oriented control scheme; the control algorithm generates references for the magnetizing flux component and for the speed component of the stator current. The control strategy yields decoupled rotor speed and rotor flux amplitude tracking control goals which allow the selection of an appropriate flux modulus for the rotor to maximize the efficiency. I. INTRODUCTION

Induction motor has grown in popularity for industrial applications due to its low cost and ruggedness. Advanced controllers are employed in applications which require robust, precise and fast system response. Typically these involve the use of speed sensors for rotor speed measurements which lead to high costs and unreliability in the system. Therefore in recent years the development of controllers based on elimination of speed sensors has gained attention which involves developing speed-sensorless control algorithms that guarantee reliable high performance control. In [6] a speed-sensorless controller is designed based on measurements of rotor position and stator currents. An asymptotic controller with an explicitly computed domain of attraction is designed in [5] for induction motor control with uncertainties. A global controller design for sensorless induction motors with known parameters can be found in [7]. In [8] a local asymptotic controller is designed for the induction motor based on current measurements under unknown constant load torque. The contribution of this paper is to design an adaptive backstepping controller for induction motors which is adaptive with time-varying load torque and uncertain rotor resistance conditions. The controller design is based only This work was supported by the Graduate Council Research/Creative Activity Fellowship, The University of Alabama Arbin Ebrahim is a Doctoral Student in the Department of Electrical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA [email protected] Gregory Murphy is the Chair, Electrical Engineering Department, Tuskegee University, Tuskegee, AL 36088, USA gvmur-

[email protected]

1-4244-1051-7/07/$25.00 ©2007 IEEE.

341

on measurements of stator currents. We use the polynomial approximation [10] to estimate the time-varying load torque. The main idea is to represent it as a polynomial in a finite interval of time. The accuracy of approximation depends on the order of polynomial and the width of interval. During each interval, the coefficients of the polynomial can be considered as constant and approximated by adaptive laws. The dymamic control algorithm generates estimates for load torque, rotor resistance and rotor speed that converge to their true values. II. INDUCTION MOTOR MODEL

Based on the assumptions of linear magnetic circuits, i.e, a proportional change in magnetic flux due to change in current, the dynamic model of a balanced induction motor in a fixed reference frame (a-b frame) is given by the fifth order model (see [1] for derivations, modelling assumptions and general machine theory)

dw dt d4a dt

d
dt dia dt dib dt

8(aib- bia

-T
-

)-L (t)

pb +Tta

-Tbb + lpa -a

+ TMib

+ TPba + npPW&b + -Ua

--ib + Tp
(1)

where s = L(1 _M2/L,Lr), T= RrILr, p = M'§Lr, p = npM/JLr. The state variables of the system are the rotor speed w, rotor fluxes ba, b and stator currents ta, ib. The known parameters of the motor are the moment of inertia J, stator resistance R5, stator self-inductance L,. rotor self-inductance Lr, mutual inductance M, and the number of pole pairs np. The control objective is to control the rotor speed w and the rotor flux modulus 4bO2 4b by using the stator voltages Ua, Ub as control inputs based on the measurements of ia, ib. The load torque TL (t) and rotor resistance Rr are uncertain parameters. The torque changes are typically a function of time while the resistance is usually a constant unknown with respect to time.

Introducing an angle nprTo(t) [2], the dynamics of which will be defined later, we can define a transformation of

360

by choosing woo(t) and control inputs ud, Uq so that for any unknown TL (t) and T we obtain

varibles CoS(npTio)

[Ud]

Sin(npTio)

[-sin(npTlo) cos(npo) [d cos(npro) Sin(nprTo)) tq -sin(npTlo) coS(nprTo)

uqj

ua

oq

liMt

'ib

cS(npTo7) sin(npTro) Oa -sin(npTlo) cos(npTlo) Ob

[id]

(2)

to obtain the rotor flux components 4bd, 4bq, stator current vectors td, iq and stator voltage vectors ud, Uq. These vectors represent varibles with respect to a time-varying d-q frame, rotating at a speed npwoo(t) and contributing to the angle npilO(t). The equations for the dynamics of motor in the arbitrary rotating d-q frame, are given by (see for example [11] for detailed d-q frame modeling)

dw dt

d
diq dt

did dt diq dt

dr,o dt

H(iqeq

-T
-Tq

d

-Wr(t)] [Od(t) r(t)l liM t--.C)c [Oq (t)]

lim [w(t)

Ubj

lia1

-

=

0

(4)

=

°

(5)

0

(6)

Equations (5) and (6) indicates that the flux vector rotates at speed npwo, which shows that field orientation is achieved. In otherwords the rotating (d,q) frame rotating at speed npr0o tends to have the d-axis coincident with the rotating flux vector as t goes to infinity IV. REPRESENTATION OF TIME-VARYING FUNCTIONS

f(t)A

_TL(t

np(wo0-/)q + TM'1d

f (t

f,(t)

./

(t)

-np(wo -W)d + TMiq

--id + npWOJq + -Ud + TP
-TMp'Jd --q-npWOidU +

Pq

T

T

rel="nofollow">'

)

npPWfd

41 tr

-TMpiq wo

III.

T <

<

(3)

CONTROL STRATEGY

t

t

+I

Fig. 2. Local approximation of a continuous function. Each fr (t) approximated by a polynomial in time.

can be

In order to represent a general time-varying function, we introduce the following lemma from [4] Lemma 1. Let I be an open interval in X, and f be a p-times continuously differentiable function of I into X; then, for any pair of points to, t in I f (t) = f (to) + It

l!

(t

+0

(p

tof()(to) + ... + )P

(t

to)P

J (1

1)!f

P

f(P-

1)(t) (7)

)<

where f(i) (.) denotes the ith derivative of the function f(.), c [to, t]. From Lemma 1, it follows that the time-varying function and its time derivative can be represented locally at to as polynomials of time with constant coefficients given by Fig. 1.

f (t)

Field-Oriented Control

co(to) + cl(to)(t -to) + +

-to)p

1

Qf (t, to)

(p- 1)

Let Wr(t) and /r(t) be the smooth bounded reference signals for the output variables to be controlled which are the speed and the rotor flux modulus /+ = /id + . the field-oriented control Adopting strategy [14] as shown in Figure 1, the goal is to design a compensator ro (t) = zo(t)

+ cp(to)(t

=

ci(to)(t -to)i +

5f

(t, to), t C [to, to + T)

i=O (p- 1)

/q

(t)

=

ic

i=l

342

(to)(t

-

to)i-t

+ 6f (t,

to)

(8)

361

where t C [to,to + T), c(to) (1/i!)f()(to), = 0, 1, ..., (p -1), f() (to)is the ith time derivative evaluated at t = to, and T is the window length chosen. Assuming that the window length is sufficiently small, 6f (t, to) is negligible. Suppose the pth derivative of f(t) is bounded, that is, SUpt f (P) (t) 1 < sp, then 6f (t, to) can be bounded by j

(9) - p! Therefore it is possible to approximate f(t) closely by choosing either a higher-order polynomial, that is p large, or a small interval T such that (t- to) < T or both. Figure 2 illustrates the idea of dividing the estimation time t into many small time intervals of size T. During each time interval the load torque TL(t)can be approximated by a local polynomial of time. At the beginning of each time interval to = tr (r = 0,1,2...), coefficient ci needs to be reset. The resetting condition ensures that the estimated timevarying parameter is continuous and the local polynomial in each interval is different. Let tr be the time when the r-th window begins, and tr±+ be the time when the r+lth window begins. T is the length of the window, and T = (tr+l- tr). Let [Co(tr+i)Cl(tr+l) ...Cp(tr+i)]T be the coefficient vector of the estimated parameter TL (t) during the time interval beginning at tr±l and [CO(tr)Cl(tr)...Cp(tr)]T be the coefficient vector during the time interval beginning at tr. As in [10], the resetting condition is given by (10), f (t,to) <

Ci

)

CO

(tr)

(tr+ )

Ci

(tr)

p x p,

and

bij

is

j

Tt

V i,j

O, 1, 2....p (I11)

1ddes

1qdes

M

k=

CD

=

Tor +

4r)

+ Cr)

(15) (TMiqdes) np r where k, is a positive design parameter. Defining the estimate parameters TL (t), w; the expressions in (15) are modified by substituting the estimates for the unknown parameters TL (t), +

T,

T,

1ddes

M

(Tr

WOdes

+r)

-kw(c

iqdes = 8

-

br) + J

.r)

+

(16)

(TMiqdes)

+

The adaptation law for the estimate will be designed using the adaptive backstepping design approach and will include a projection algorithm to ensure that that :t 0 V t > 0 singularity is avoided and (16) is well defined. so

errors

ed =

id -ddes

eq

iq

=

e, =

and speed observation

-tqdes

(17)

w-

where is the estimate of rotor speed. The dynamics of the rotor speed observer is chosen as given below Wi

II

Yd

'q

/)q'd)

TL

(18)

q-U -

-k(w -ew) + P(jdiq -

r

(12)

O=-d

by using the motor equations in the d-q rotating reference frame, we can deduce the error equations

II(Od'q Oq'd)-

+

where X is a parameter that will be suitably designed using the adaptive backstepping design technique. From (3), (13) and (16) we get

Od = Od oq = 'Oq

=

(-/Wiq

(k +

WOdes

V. CONTROLLER DESIGN

A. Non-Adaptive Design Defining the error variables

(14)

q)

T,

Cp (tr)

bj=j

d+

2+

By computing the time derivative Vo of Vo, we can choose the virtual control laws iddes, qdes, control law WOdes to make V0 negative semidefinite, assuming TL (t) are known

(10)

,i

=-

B. Adaptive Design Defining the current tracking error as below

=Bx

Cp (tr±+1 )

Vo

p!

where B is the transition matrix of order the element of B. CO(tr+ i

Choosing the Lyapunov function [12]

49q

+TMid

(13)

+ I/reqr

+ np(O -)q + T(Middes

T-bqq-np(wo +TMeq

g

O/d = -T/d + np(WO -W/)q Tr V)q = T-Tq -np(O)(O d + Or) + TMiq

T/d

bqid)

w>/) d +

)

TMiqdes +nlpe

+ TMed r

(19)

where T = T- and TL(t) = TL(t) -TL(t). Substituting TL(t) = co+c1 (t -to) from (8), where co and cl are constant coefficients of the first order polynomial used to estimate the time-varying parameter TL(t). Defining the estimates co, cl

343

362

and estimation errors (co = co -co, c= cl-cl), TL in (18) and TL(t) in (19) is replaced by co + c(t -to) and co + c,(t- to) respectively. The adaptation law for the estimates co, cl will be designed using the adaptive backstepping design procedure and the estimates are reset at the beginning of each interval as given in (10). Defining a new set of error variables ¢d

=

¢,q

ed + P
=

eq

+ p
(20) and choosing

a

T

2

(d +

+7~3CO

where 7i, 72, 73,

+

q

+

¢d

+

¢q

+Ye

+Y2 (21)

-74Cl)

pa-

rameters. The dynamics of ¢d, ¢,q by making use of (13), (20)

-)r)d+M&d<)d

(28)

T)

where Proj(,, T) is a smooth projection algorithm as given in [3]. These selections of the variables results in the derivative of the Lyapunov function to be

V>

-

(T + TMP)[/d + /q -kd(d

-kqq2

(29)

The rotor flux measurements which is unavailable is obtained by open-loop online integration from zero initial conditions; assuming nominal values for the motor inductances and stator resistance, using the equations given below 4'/sd

given as

*

modify the

we

-kew,i,,e2

suitably chosen positive design

are

Proj (-[(Middes

=

+M(,dQd1,

Lyapunov function given by

V

is

Inorder to ensure that T(t) :t 0 V t > 0, dynamics of (26) as

1

=

4bsq

¢d =-Uld+OWdl

Ud- Rids+no, d(O) Uq

-Rsiq'

pWo,4

q(0)

=

0

=

0

(30)

s

&q

= -Uq

(22)

+ ql

where the design variables Odl, Oql are suitably chosen using the adaptive backstepping design procedure as Odl

- Rs.id + npWO iq + TP)r -TMPJddes + npPWO4/q <)Jr +

<)r m

M

Oql =-

Rs. q

p02

<>d

oq Osd, Osq

M'

td -

LrLs

Lr

M

M

M id +

Mi- LrLs. M iq + Lr M Osq

m+ T2M _2T

TM

- npWoidlppWYrr-MPT-qdes-lppWOYd

-kw (Co-k -or) + f+ f(t - to) + Car]

rwC

+

WJr)

[r dtk (w

+

CO

Cl

ff+ jf

A

A

t

to)

x

is chosen

as

(24) X( =-kewew e-npO.fq where kew is a positive design parameter. The control inputs Ud, Uq selected are given by Ud uq

S(-kd(d -dl- TM
q

(25)

where kd and kq are positive design parameters suitably chosen. The adaptation laws are given by T

[(Mtddes

cl

=:

q

4bq,

Cf +Wr]

The design variable

(31)

where are the stator flux vectors. From (29) it follows that V < 0. Consequently from (21) and (29) we conclude that all the signals (eC, bd, (d, (q, T, Co, c1) in the closed loop system remain bounded, it follows that (eu, bd, bq, (d, (q) C L, from Lyapunov theory. Besides, due to the boundedness of all state variables and control inputs from (25) we can further guarantee (eS,Cd,Cq,¢,d,¢,q) C L.O. Hence (ev,bd,bq,(d,(q) are bounded and therefore (e,,, (d, (q) are uniformly continuous. We can also easily obtain f0° IV dt V(oc)V(0) < oc which in turn implies that (eC, 4bd, (d, (q) C L2. Therefore by Barbalat's lemma ([16],[17]) we can conclude that d

-

(1

Mtq q

bq,

4gr

d

MOsd

- br)d + M(d/d + M((qbq] (26)

1

TY4

(tto)1

J

(27)

t

liM

0

. C)c

liM

t--.C)c

[;F- (t) .o (t) .j (t)] .

.

0

(32)

From (32) and (12) we can conclude that e, Wr= ; and this can be used in (24) and (27) for implementing the controller. From (19) we can deduce that C tends asymptotically to zero. This shows that asymptotic speed and flux tracking is achieved. VI. SIMULATION Simulations are carried out in Matlab/Simulink to demonstrate the effectiveness of the above adaptive backstepping design. The parameters of the motor are RS=5.3 Q, Rr=3.3 Q, np=2, L,=.365 H, Lr=.375 H, M=.34 mH, J=.0075 kgm2 In the simulation, the desired speed of the motor WJr iS required to reach the rated speed of 100 rad/sec at t=0.5

344

363

sec at a constant rate of change starting from 0 rad/sec at t=0.3 sec, and maintained constant as shown in Figure 3. The desired rotor flux modulus
Load torque

E, 0

F--

Fig. 5. Load torque profile

Actual speed

120

T

100 _

a)

-a)a1) CD Q1

80 60 40

20

Speed reference

120

0

100 _

0.5

1

Time (sec)

Fig. 6.

1.5

Actual motor speed

a1) -a)

-a)Q1 (1

0

0.5

Fig. 3.

1

Time (sec)

1.5

2

C)-

-1

Et

-2

Z5

Desired reference speed

L1

-3

Flux modulus reference 1.2 _

Fig. 7. -M D

Motor speed observation

0.8 0.6 06--

Flux modulus

0.4 0.2

Fig. 4.

Desired flux reference

LL

VII. CONCLUSIONS AND FUTURE WORKS A. Conclusions In this paper, we have presented a new adaptive backstepping controller that achieves global asymptotic rotor speed tracking for the full-order nonlinear model of an 345

Fig. 8.

Actual flux modulus

error

2

364

Estimate of tau

d

-

axis Flux

-o

a1)

5:

Time (sec)

Fig. 9. Estimate of tau

Fig. 11.

Load torque estimate

E, 23

q

-

d-axis flux axis Flux

-

CD

CZ

0

2

-o

F--

Time (sec)

Fig. 10. Estimate of load torque Fig. 12.

induction motor despite the uncertainty in rotor resistance and time-varying load torque conditions based only on the measurements of stator currents. B. Future Works Future research will be directed towards designing a new adaptive backstepping controller for a speed-sensorless induction motor with flux observers under time-varying load torque and rotor resistance uncertainty. REFERENCES

[1] P.C Krause, Oleg Wasynczuk and Scott D.Sudhoff, Analysis of Electric Machinery and Drive Systems, John Wiley & Sons, NY; 2002. [2] David C. White and Herbert H. Woodson, Electronechanical Energy Conversion, John Wiley & Sons, NY; 1959. [3] Miroslav Krstic, loannis Kanellakopoulos and Petar Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons NY; 1995. [4] Kennan T. Smith, Primer of Modern Analysis, Springer-Verlag, NY; 1983. [5] Riccardo Marino, Sergei Peresada, and Patrizio Tomei, "Adaptive control of speed-sensorless induction motors with uncertain load torque and rotor resistance", Int. J. of Adap. Control and Signal Process., vol. 19, pp. 661-685, 2005. [6] P. Aquino, M. Feemster, and D.Dawson and A. Behal, "Adaptive Partial State Feedback Control of Induction Motor: Elimination of Rotor Flux and Rotor Velocity Measurements", Int. J. ofAdap. Control and Signal Process., vol. 141, pp. 83-108, 2000. ,

346

q-axis flux

[7] R. Marino, P. Tomei and C.M Verrelli, "A Global Tracking Control for Speed-Sensorless Induction Motors", Automatica, pp. 1071 -1077, 2004. [8] Marcello Montanari, Sergei Peresada, and Andrea Tilli, "Adaptive Backstepping Control of Induction Motor with Uncertainties", Automatica, 2006, pp. 1637-1650. [9] Hualin Tan, and Jie Chang, "Adaptive Backstepping Control of Induction Motor with Uncertainties", in Proc. American Contr Conf:, San Diego, California, 1999, pp. 1-5. [10] Arbin Ebrahim and Gregory Murphy, "Adaptive Backstepping Control of an Induction Motor Under Time-Varying Load Troque and Rotor Resistance Uncertainty", in IEEE Proc. SouthEast. Symp. Sys. Theory, Cookeville, Tennessee, 2006 [11] John Chiasson, Modeling and High-Performance Control of Electric Machines, Wiley- Interscience, NY; 2005. [12] J-J.E. Slotine and Weiping Li, Applied Nonlinear Control, PrenticeHall, NJ; 1991. [13] C.M Kwan, F.L Lewis, K.S Yeung, "Adaptive Control of Induction Motors without Flux Measurements", Automatica, vol. 32, pp.903908, 1996. [14] B.K Bose, Power Electronics and AC Drives, Prentice-Hall, NJ; 2002. [15] Ian T. Wallace, Donald W. Novotny, Robert D. Lorenz and Deepakraj M. Divan "Increasing The Dynamic Torque Per Ampere Capability of Induction Machine", IEEE Trans. Ind. Applicat., vol. 30, pp.146-153, 1994. [16] Kumpati S. Narendra and Anuradha M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, NJ; 1989. [17] Riccardo Marino and Patrizio Tomei, Nonlinear Control Design, Prentice-Hall, London; 1995.