Nonlinear sensorless indirect adaptive speed control of induction motor with unknown rotor resistance and load

Proceedings of the American Control Conference San Diego, California June 1999

Nonlinear Sensorless Indirect Adaptive Speed Control of

Induction Motor with Unknown Rotor Resistance and Load

Yu-Chao Lin

',

Li-Chen

Fu

2,'p

Chin-Yu Tsai

Department

of Electrical Engineering'

Department of Computer Science and Information Engineering

National Taiwan University

Taipei, Taiwan, Republic of China

Abstract

In this paper, a nonlinear indirect adaptive sensorless speed

controller for induction motors is proposed. In the con-

troller, only the stator currents are assumed to be measur- able. Flux observers and rotor speed estimator are designed

t o relax the need of flux and speed measurement. Besides,

two estimators are also designed t o overcome drifting prob-

lem of the rotor resistance and unknown load torque.

Nomenclature

{ Va,

&

1

are stator voltages; { la, Ib } are stator cur-

rents; { $ a , $6 } are rotor fluxes; wr is the mechanical

angular speed of the rotor;

R,

is the stator resistance; R,

is the rotor resistance; L, is the stator self-inductance; L,

is the rotor self-inductance; M is the mutual inductance; p

is the number of pole-pairs; J is the rotor inertia; D is the

damping coefficient; TL is the load torque; kT is the torque

constant

(=%I;

L, equals t o $ ( L ,

-

E);

p1

equals t o

&g.

,

pz

equals t o PL,; 0 3 equals t o

$

1

Introduction

The induction motor is a coupled system with highly non- linear dynamics. In the early years, all system states are assumed t o be measurable and parameters are assumed t o

be known. Under these assumptions, techniques such as

classical field-orientation [l] and input-output linearization

[2] are utilized t o design the controller. Then flux observers

are then designed t o relax the need of flux measurement

[3]. These flux observers are designed under the assump-

tion that the rotor resistance is known. Following these researches, further efforts were then t o design controllers and flux observers which are adaptive with respect to both

system parameters and/or the load [7] [9].

All the schemes above require speed measurement. How-

ever, the speed measuring device is rather costly relative

t o the price of a n induction motor in general. Besides,

the measured signals are usually noisy and difficult t o deal with. Therefore, controllers that does not require speed measurement are obviously preferable for practical imple- mentation. Many research results on sensorless vector con-

trol have been proposed [6], of which analyze are mainly

based on the steady state behavior and only rough proves

are supplied. O n the other hand, in [8], many researches

on sensorless control were discussed and compared, includ- ing vector control and other modem control theory, such

as robust and MRAS (Model Reference Adaptive System).

Here, we follow this trend t o design a full nonlinear adap-

tive sensorless speed controller for induction motors based

on flux observer.

In this paper, an introduction of induction motors and

related researches are discussed in section 1. In section 2,

the mathematical model will be presented

.

The main part

of this paper is section 3 in which observers and controller

are designed and proved in detail. Experimental results are presented in section 4. Finally, we will make some conclu- sions in section 5.

2

Mathmetical Model

In this section, we introduce the induction motor model.

If the induction motor never goes into the saturation re-

gion, and the air-gap MMF is sinusoidal, then it can be

characterized by the following dynamic equations:

Lofa =

-

& l a

+

%$a

+

b h $ b

+

P3Va Loib = -MRrIb

-

plrb

-

&&$a

+

Rr$b

+

p 3 &

Lr4a = -%$a -k M%la

-

&?wr$b

Lr4b = -Rr$b M&Jb

+

pZWr$a

Te = kT ($016

-

$ b r a ) (1)

where Lo,

PI,

Pz,

p 3 are constants defined in the nomen-

clature. The mathematical model listed above is referred

to the well-known stator fixed reference frame.Also, the dy- namics of the mechanical part can be derived.

JLjr

+

Dw,

+

TL = Te

where J

>

0 is the rotor inertia,

D

>

0 is the damping

coefficient and TL is the load torque.

( 2 )

3

Induction Motor Control

Before the thorough investigation on the observers and con-

trollers, several assumptions will be presented below to

make the problem more precise.

Assumptions

:

( A l ) All parameters of the motor are known, wcept the

(A2) The stator currents are measurable.

(A3) The load torque TL is an unknown constant.

(A4) The desired rotor speed should be a bounded smooth

function with known first and second order time deriva- tives.

rotor resistance R,.

Control Objective

:

Given the desired rotor speed trajectory w,d(t), our con-

trol goal is to design control laws such that the rotor speed

wr and flux Q can track the desired trajectory asymptoti- cally in finite time with all internal signals being bounded, subject to assumptions (Al)-(A4).

3.1

Observer Design

and

Analysis

The system mode is expressed as section 2. We set R,. =

%

+

81 and W r =

+

6%. where O1 stands for (unknown)

difference between the actual resistanve value and its nomi-

nal value, and 6 2 denotes the speed tracking error. For easy

reference, we define the notations for fihe-observed_ @ues

$a

-

$a,$b = $b

-

$al& = &

-

&,& = W r

-

8, where

the symbol A denotes that it is an observed value and the

symbol denotes an observation error. According to the

structure of the dynamics in (l), the observers are proposed

as in [4] :

and th: O@?NatiOn $ITOF as Ia = I % - I a , I b = I b - l a , $a =

where

k

= &,

+

&,ar

=

+

8,

and the constant

ko

>

0 is a control gain. In order to utilize this property

to cancel the unmeasurable terms, we design two auxiliary

observation errors [4] :

Another interesting characteristic @ studied_ here. Although

the auxiliary observation errors, Z a and-& ar: not m-a-

surable (because they are composed of I and $, and $ is

not measurable), their first-order time derivatives are mea-

surable. Two additional error signals qo and q b are then

defined as follows :

where ca and [ b are auxiliary control signals. Again, qo and

Q b are unmeasurable errors but with measurable first-order time derivatives. Motivated by how the coupling terms can

be canceled, we design the observer inputs VI

,

v2

,

v3 and

v4 to be

Now we are ready to perform the Lyapunov stability anal-

ysis to examine the stability condition of these observers.

Define a Lyapunov function mididate as :

where control gains k,,, kw and k R should satisfy

IC,, k w i k R

>

0

Here, to complete the final analysis an additional assump tion is requifed.

(A5) The speed tracking error and the change of the rotor

resistanceresistance from itd nominal value, i.e., 132 and

el,

varies slowly so that its first order time derivative

can be negligible.

Again, if

qa,

Ijb,

&

and

3,

are designed properly which

then leads to the design of those signal as

where rja and F e estimated valuer! of qo and 8 , respec-

tively, satisfying rj, = 7ja,7j,(O) = 0 ; r j b = ? j b , @ b ( o ) = 0 . By substituting the rotor resistance .estimator and the rotor

speed observer, we can reassess V as :

Clearly, the upper bound on V on the right hand side (RHS)

of equation (8) still does not have definite negative sign. To cope with that, we incorporate the variable structure design

(VSD) into our controller. We evaluate the four functions

Fi(t), a ' = 1,2,3,4whichsatisfy

Fl(t)

2

'54

I&/,

F3(t)

1

a b

lkl,

F2(t) _ > a b lGr1,

F4(t)

2

aa

l&l

Accordingly, we design the four control signals 215 and 216 as

where sgn(.) is the sign function defined as

As a result, Vo becomes

Vo

5

--(fZ

k0

+

fi)

L O

It follows that

fa,

!b1

A,,

4,

qa and q b are d l bounded. We now impose that &(t)

+

&,

2

>

0 , Vt 2 0. Denoting by &(t) the modification of & ( t ) given by the projection

algorithm, &,(t) is chosen such that

(el

-

& ( t ) l 2 2

(e1

-

& p ( t ) ) 2

which parantees that the value of Vo does not increase

when 81 is replaced by 81, given by the projection algo- rithm. Since % = R,,

+

01 is positive, the estimate & ( t )

is given as

+

€1 if 8 1 ( t )

I

-Rrn

-%n

&(t) = B,,(t) =

-

2

with E

>

0 a positive constant such that E

5

2(Rm

+

61) =

2%. Since ( I 4 , & ) are bounded in finite time ,i.e.(14, E

L,,), by assumption, ($J~, $ b , U,) are also bounded. Since ( $ 4 1 $ b , 1 4 , 1 b , W ~ ) axe bounded, if

&

= Rrn+& 2

>

0,

it follows thatA$, and d b are also bounded according i o

(3). So, ( $ 4 1 $ b ) are bounded, and therefore 2, and Zt,

are bounded. This implies that

Ca

and &, are bounded y d

that 211, 212, 213 and 214 are bounded

as

well. So, fa and f b

are bounded. On the other hand,

f,

and f b are $2 sig-

nals. By B-arbalat's lemma, it follows that l i t , , Ia(t) =

0 , limt-,, Ib(t) = 0. We can rewrite error system in matrix

form as

i

= A f + W T ( t ) X

+

B ( t ) , (9) X = _{- A W ( t ) j + A C f} (10) c . . where (12)

Lemma

1. such that

If, in addition, there exkt two positive constants T and E

with W ( t ) defined as in (Ii), then according to Appendiz A applied to

i

= A ( t ) f + W T ( t ) X ,

i

= - W ( t ) f (14)

0

We can get

f

asymptoically tend to zero and X - is

bounded by matrix

B

according t o Lemma 1. While I is

s m d , we can give s m d control signals ( 2 1 5 , ~ ~ ) . Finally, if I tend to zero, we give 215

,

216 zero. Then, the equilibrium point ( I = 0:

X

= 0) is asymptotically stable, namely

&,

&, q4,

Q,, I4 and I b tend aspptOtiCdjy to Zero, Which

implies that I a F d Ib

F e

bounded and @a and

$a

are also

bounded, that $4 and $b also tend asymptotically to zero.

in conclusion, equation ( 3 ) with ( V I

,

212, v3, 214) are given in

(6), and 8 1 , 6 2 , q4 and q b updated according t o (7) consti-

tute an adaptive observer provided that (13) is satisfied.

3.2

Controller Design and Analysis

In this section, we propose a state feedback control for in-

duction motor system which is adaptive with respect t o

the unknown constant load torque TA, assuming all states

(I4

,

I b

,

$a

,

$ b ) are measurable and the rotor resistance (&)

is a known constant. The controller objective is to guaran-

tee asymptotical zero convergence of rotor speed tracking error, rotor flux tracking error, and load torque estima-

of the induction motor is expressed as (1). And, we de- fine the speed tacking error, flux tacking error and load

torque estimation error as follows : e, = wr

-

WrdjeQ =

$f

+

$;

-

\Ei,eT = TL

-

T L , where w p d and Q d are the

reference signals and

f~

is a timevarying estimated of TL.

With the tracking errors and load torque estimation error

defined above, their dynamics can be derived from (1) as :

56, = kT($,Ib

-

$bI4)

-

DW,

-

TL

-

JWrd,

L,& = -2R,Q2-2L,&i

+2MRr($aIa +$bib) (15)

By designing the input signal I, and Ib, we first consider a

Lyapunov function candidate defined as :

1

2

VI = - ( P I J e i

+

rzL,e;

+

n e ? )

where gains P I , r2 and r3 are positive. The choice of the

desired currents l a d and Ibd are equivalent to the following

where K,,Kb 2 0, and U , and 'ub are to be designed later.

Given such design arrangement the error dynamics of I ,

and Ib become

Lo$b(Kw

-

D ) eT

Q2kTJ LOB, = -K4e, + U ,

+

The analysis is also based on the Lyapunov stability the-

ory. First, we define a Lyapunov function candidate for the

controller as :

1

2

V = -(r1 Je:

+

r,L,e$

+

r3e$

+

r4Lo(ef

+

e;)

+

EeTew)

for some positive constants T I , 7-2, r3 and r4 and for some

sufficiently small E (

>

0). After a careful choice, we following

payload estimation algorithn as follow:

I4d Ibd where

n,

=

n,

= (17)

Define the current errors, namely, the differences b e

tween the actual currents and their desired values, ase, =

I ,

-

l a d , eb = Ib

-

Ibd. Then, the error dynamics involving

(e,, eo, e,, ea) can be summerized as follows :

The derivatives of the reference currents are found as :

Lo$'b (Kw

- D)

eT \E2kTJ Iad = s / 3

-

There exist K,

>

0 then

Consequently, we know the signals (e,, eo, e,, ea, eT) in

the closed-loop system is bounded from Lyapunov stability

theory. since (I,, Ib, $,, $b, w,) are bounded and the

initial flux value 9 is not equal to zero, it follows that the

time derivative of the tracking errors

,

namely, (e,, 80,

e,,

signal

( 9 ~ )

are also bounded. On the other hand, e,, eo,

e,, eb and eT are L2 signals. So, they are asymptotically stable by Barbalat's lemma.

ea),

the control inputs

(v,,

vb, U,, U b ) and the estimated

where n3 and Cl4 are known function, Finally, we chose the

control input terms Va and v b as

4

Experimental

Result

1

v4 = &(M&14

+

P1I4

-

&$4 Experiments are done with a three horse power induction

motor which is manufactured by TECO Co. Ltd. Taiwan. In order to check the performance, we have done two ex- periments with exponential and sinusoidal speed command

as in figure 1 and figure 2. Obviously, experiments show

that the sensorless controller is indeed effective t o drive the

motor t o track a given smooth speed command.

-hwr$b

+

n3

-

K4e4

+

U,) 1 D3 + h & $ 4

+

0 4

-

Kbeb v b = -(M&Ib PlIb

-

Rr$b ub) (20) 2171

0 2 4 6 8 10 12 14 16 18 20 spwd Trasldq Ena 20, I 10 g o -10 -20 0 2 4 B 8 10 12 14 16 18 20

a-b Axis cunnu

20 10 g 0 -10 -20 E 0 2 4 0 8 10 12 14 16 18 20

Figure 1: Urd = 1200(1- e-t) RPM with no load

I I 0 2 4 0 8 10 12 14 10 18 a0 SpedTmddna ERor 50, I . . . . . . . -50

I

0 2 4 B 8 10 12 14 16 18 20 d A d S C U n n t . . .. .. . . . . . 2 0 _.! ,_{. .} . . . . . . . . . . . . . . . .

Figure 2: = 1200. sin(0.5t) RPM with no load

5

Conclusion

In this paper, we have presented a partial-state feedback adaptive sensorless speed and flux tracking controller for induction motors with Wth-order nonlinear dynamic model

which is actuacted by a voltage source. The main contri-

bution of the controller is that asymptotic tracking of rotor

speed and rotor fluxes are achieved without measurement of both the rotor fluxes and the rotor speed. Moreover, the

variations of the rotor resistance and load torque are also

taken into account.

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