Exponentially stable nonlinear field-oriented PI-controller for speed regulation of induction motor

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Expolientially

Stable Nonlinear Field-Qriented PI-Controller for Speed

Regulation of Induction Motor

Ja.ion-Shea

Chang'

and

Li-Chen

Ft1'9~

1. Department

of

Electrica.1 Engineering

2. Department

of

Computer Science and Information Engineering

National Taiwan University, Taipei,

Taiwan, R.O.C.

Abstract

A sin ple control law for speed regulation of induction motor is proposed in this paper. This controller is designed based on the concept of the field orientation and the property of linear PI-controller, and is shown to be rot.*.t with respect to variation of system parameters and load strnctrrre. In fact, the local exponential stability of the system around any desired speed can be proved by a simple iterative criterion derived in this paper. Numeri- cal simulations are also given to show the effectiveness of the presented control law.

Nomenclature

q-(d-)oxis stator current q-(d-)axis input stator voltage q-(d-)axis stator f 1112 rotor speed

electromagnetic torque mechanical load torque ststor ( r o f o r ) resistance stator (rotor) inductance

mtitual inductance nuniber of poles inertia of the rotor

(LSL?

-

L:n)

( R v / J ; r ) L"4

L m / D

( R , L t , ) / D

t

k

Z6,

1

ULa

_. 3PLm 4 L r a3 =

L , R , / D

1 Introduction

Pioneered by Blaschke [Z], field orientation is the most widely used method for servo contxol of induction motor, jiist the same as the

PID

coiitroller for linear systems.

In

most of the existing nonlinear controllers for induction motors [4][7J, the rotor flux is inade to converge asymp- totically to a fixed constant, which is t o prevent the set- nration of the magnetic circuit. But, the flux saturation remains due to the resultant flux in the stator and the rotor [ 5 ] , and hence the control schemes still can hardly achieve their origional goal satisfactorily.

The nonlinear adapt.ive control schemes are also applied when some imprecisely known parameters are con- cerned [6][7], siich as the rotor resistance, changing with temperature variation, and the coeffcients of the load characteristics. Bat since the control law will normally depend on the type of load structure, the form of the adaptive schemes will have to be changed when the load type is changed.

Given the above observat,ion, here we try to devise a simple controller for the speed regulation problem of in-

duction motor driving a more general class of load. The

flux tracking is not. consitlcred here. With the rohust.iwss of general PI-control, the iinkiiown parameters in the sys-

tem as mensioncd previously need not be e s t h i a t d . Be-

cause the transformation between the actiral input, volt- ages anti the d-q-axis volt~ges is stationary, which can be done easily by some appropriate analog circuit design, and because the computation time needed in the control process is relatively little, the proposed control scheme has a higher promise to be successfully imple- mented with more ease. So far to the best of the auther's knowledge, this is the first paper which analyzes the stability properf ly of the field-oriented linear PI-controller, tlioiigh it is already widly used in industry applicat.ions.

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2 Field-Oriented

P I-

Con

t r

o

11 er

(A) Dynamical model of induction motor:

As

being well-known, the dynamical model of an induction motor can he simplified

by

a d-q-axis coordinate transformation t o some rotation-reference frame [5]. But for iniplementation feasibility, the stationary-reference frame is more popiilarly used. ‘rhus, here wc adopt the following d-q-axis coordinat-transformed dynamical equations of an induction motor [4]:

iqn

=

- i i I i q *

+

a z ~ q r

-

i K J r ~ d r

+

c‘v~,

I d a = - 6 l i d a

+

BWrXqr

+

&Xdr

+

I?&

iqr

= a3 i q s

-

64 Aqr

+

W r Adr i d ,

=

n‘sids

-

WrXqc

-

ii4Adr

J,,,&

= T,

-

TL

2;

= l < t ( X d r i q g

-

X q r i d s ) , (1)

where the states and the parameters are defined as shown in tGhe nomenclature.

( R )

Analysis of mechanical load:

In many mechanical system, the load torque is a strirtly increasaing function of the speed w if the inertial

parb does not, exist or were to be neglected. For example,

t h c load torqiie of the fan-load systmn can be modeled as

” bo

+

b l w , + b 2 s i g n ( w r ) w ? ” which is a strictly increasing fiiuction of wr. For this reason, throiighocit the paper we will con. ider the class of mechanical loads which sat.isfies the following assumption:

( A l )

TL

=

J L & + ~ L ( w , )

with

>

6, Vur

E

It,

and f ~ ( 0 )

=

0, where

Jr.

is the mechanical inertia of the payload.

(C) Field-oriented control:

Th e basic concept of a field-oriented control of an induction motor at steady stare is to find the input voltages Vds,

vq,

such that

[A,,

Adr]*[Vds ~ q r ] is maximum

under the constraint that l$2s

+

\’is

is a fixed constant, whrre

Xqr

(or Xdr)

is

the q- (or d-)axis rotor lax[1][2].

ApNying such a control concept t o the input voltages design of an induction motor, an intuitive realization of

the field-oriented cont.ro1 law is the following:

which clearly satisfies the vokage constraint (V,”s

+

V$n)

=

(V/E)’ at any time instant. Of course, at present

V i$ no longer a constant. Instead, it. offers one D.O.F. (degree of freedom) control t o the system, but normally it

mill

converge t o a constant when the system approaches t,o the steady state.

On the other hand, we would like to show that, given a desired speed Wd, there exists a proper constant

v

such

that the steady state of the system exactly achieves the

pnrposc of sprcd rcgiilation, i.e., wr = w d . 7’0 this end,

wc fiirtlirr siiiiplify tlic dynariiics shown in (1) by intro- ducing a noiilinrar st.atc f ransformation as follows:

SI = h 2 ( i : s

+

it)

+

i 2 n 2 ( ~ i r

+

2 6 h 2 ( i q s ~ q r

+

idBAar) ~2

= A?,,

+

XZr

23

=

+

A i r >

+

( i q s x q r

+

i d s A d r )

~4

=

h(iq.Adr

-

idrAqr)

P5 = W l 1

where h

=

D/Lr,

under which the dynamical equat.ions shown in (1) can be transformed to the following ones:

k 2

=

- 2 a 4 2 2 + 2 a 3 2 3

53

t 4

=

-5523

-

(nl

+

a4)x4

+

&V

= n 3 2 1

+

n 2 5 2

-

(ni

+

a4)53

+

55x4 6 5 = n 5 ~ 4

-

f(rs), (2)

wliere the parameters a 1 , a2, a3, a 4 , a5 and the function

f(x5) are defined in the nomenclature. Thus, it can be

verified that the system (2) has [zls xzs 2 a s x4.. xss] as

its unique equilibrium point,, where

7n 2 2 2 1 s =

(R.l)

(a4

+

2)

x4* = t m 2 a3 . provided (3) (4)

In

ordcr to tiiake the conditions in (3) plansible, addi- f.iona1 assumption is needed as shown in the following:

(A2)

~ I I

<

3a4.

To justify this assumption, we can use IL linear current

feedback t o reduce t.he stator resistance

Re,

and hence

to reduce the parmeters (11 and (12. T he details abotit

t h e reason why the stator resistance can be reduced by a linear cnrrent feedback can be found in reference [SI.

(D)

PI-controller for speed regulation:

Controller design based on the system structure often in-

volves complicated algorithms which may likely lead t o high implementation cost. This fact thus motivates the need of designing a simple PI-controller that can be ap-

plicable t o a v w t class of system structure. Now, we con- sider the load which is modeled as in assumption (Al),

2

(3)

and let the amplitude of the input voltage V be designed as:

i 6 = k , ( w d - T 5 )

1’

=

2 6

+

k p ( w d

-

2 5 ) , ( 5 ) By combining systems (2) and (5), we can readily con- clnJP that x s = [ T I $ , x z S , zsr, 2ls, 2 5 s , X ~ S ] ~ , where ztr’s,

i = 1 ,

..,

5, are defined in (3) and

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m a3

2 6 s = (-)[a4Wd

+

(at

+

W ) z ] ,

is a ~ t u a l l y a unique equilibrium point of such an aug- mented system. In order t o analyze the stability of the property of this eqlilbrium point, we will require some

m a t h e m a t s a l preliminaries which will be shown in the

next section. In fact, later in Section 4, we will show

that under the following assumption:

(A3) 0 . 8 L s L ,

5 L k ,

the equilibrium point x s is exponentially stable under

proper design of the gains

k,

and

k,.

Remark: T h e air gap between the stator and the rO-

tor is generally made sufficiently small so as to increase the efficiency of energy conversion between the electrical part and the mechanical part, and to reduce the reluc- tance (mainly caused by the air gap) significantly, which, in turn, implies that. the ratio

&

will approach unity. As a consquence, the above assumption is quite reason- able.

emat

ical Preliminary

In this section, we will develop a simple criterion to check whether a matrix A, is Hurwitz or not, where the matrix

A, is extended by a given Hurwitz matrix, two vector,

and a scalar under a strictly positive real constraint. This result will be frequently used throughout the paper. P r o p q s i t i o n 1 Given a matriz A E

Rnxn,

two vectors

b E RnX’, c E

R’

Xn, and a scalar d

E

R, i j the matrix

A

i s Hurmitzand R e { - d - c ( j w l - A ) - ’ b )

>

0 j o r n l l w

E

R ,

i h c n the extended matrix A, =

[ ;

f;

]

is Hrrrroitz.

By some simple calcnlations, we can obtain tlie fol- lowing corollary whose proof is omitted here.

P r o p o s i t i o n 2 Given polynomials

P ( s ) ,

Ql(s), and

Qz(s).

If

both

Re{%}

and Re{%$$) are

larger than zero for all w

E

R ,

then we have

R r { e j )

>

0 for all w

E

R atid arty c

>

0 .

ility Analysis

Referring t o t he closed-loop ststem ( 2 ) and (S), which is rewritten as:

wi1.h the equilihriurii point x s as defined above, for any given W d

E

R , we are going t,o prove t hat the equilibrium point 2. is uniqne and is exponentially stable. T h e following theorem will give a complete statement of the conditions nnder which the aforement.ioned properties hold.

Theorem 1 For nrbitrary desired speed w d E

It,

suppose

that the assirntptions ( A l ) , (A2), and (A3) are safified. If the controller (5) satisfies

<

ki

<

*,

then x J is the rrtziqtre e~poncnfially stnble cqtidibrirt~n poinf of !Re system (7).

Proof:

Since 1 2 = A:,

+

A:, and 0.8LsLr

5 LE1,

we always have

d

>

0.2a1a4 and nz

2

0. Without lose of generality, we assume w , j

2

0.

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Proof

of iiniqiieness

For

any given desired speed w d , the equation n? =

4 -

and the assnmption on

f,

i.e., j ( 0 ) = 0 and

+$

>

0, imply that W d Z

2 o

and z

=

( - w d

+

& m ) / 2 . Then the equilibtium point x s is uniqiie if 2 6 6 is a one-to-one function of W d .

Define q l ( w d )

=

3 +

( 1 i-

%)&,

then we have the following relations:

kF

Since a1

<

3a4 from (A2), we have

Becanse g : ( w d )

>

o

for aIi W d

>

0, g z ( w d ) is a strictly

increasing function

of wd. Given the strictly increasing pr0pert.y of g l ( w s ) and f ( w d ) with respect to W d , we can infer that 2 6 s is aIso a strictly increasing function of wd.

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As a resnlt, rer is a one-to-one function of w d and hence

z b is uniquely determined by d.

(2lJ'roof of exponential stability:

Let

J

be t,he Jacobian matrix of the overall system a t the eqiiilibriiim point zs for an arbitrary desired speed W d , t.hen we define

J'

=

LJL-'

where

1"'

x 2 0

]

. Now, define A, t.o be the n'th principle minor of the matrix

J',

then we want t o prove that

J'

is Hurwitz through following steps based on the results of Proposition 1 and 2.

Step 1: A2 is Hurwitz for any given W d . This result can be checked easily.

Step 2: A3 is Hurwitz for any given wd.

liZ

1,

then we Rewrite the matrix A3 as

have the following inequality by assumption (AI) (A2) and (A3):

Real

+

a4 - c z ( j w 1 - A z ) - ' b z

>

0 V w

E

R.

By virtue of Proposition 1, we can conclude that, the matrix A3 is Hurwitz for all W d

E

R.

Step 3: By the same procedures, we can find that J'

and A, are Hurwitz for n=4, 5.

By Lyapunov's direct theorem, t.he eqiiilibrinm point T , is locally exponentially stable for any desired speed

W d

E

R. The proof is hence completed. Q.E.D.

I' "

-k,

1

L =

[

0 2 x 2

[

t:

-(a1 + o r )

5 Computer Simulation

In this section, t,he performance of the presented con- t.roller, its it is applied to an induction motor, will be demonstrated by a nomberof simulation examples. The characteristics of the motor are listed below:

R ,

=

3.745, R , = 3.583,

L ,

= 0.1633,

Lr

= 0.1633,

L,,

=

0.15467, P = 6 , J , = 0.0284, 6 poles, 220V AC,

1Hp.

The mechanical load torqne is chosen t.o be TL(w)

=

-71.h

+

bo

+

blw

+

b2aign(w)w2

+

T d whose paraineters

nrr list.ed as follows:

Jr. = 0.1, bo = 7, bl = 0.061, hz = 0.005, 7; = 12 at the time interal (0.4 0.61.

For practical implementation, t.he amplitude of the input voltage is limited. In our case, the npper bound of t h e inpiit voltage is 300

v.

The desired speed W d is 40 (rndlsec) (about 382 (tpm)) and the control gain

k,

i s selected as

k,

=

y.

I , if

k,

is

designed to be 0.04, it will induce significant stator currentSF and rotor fluxes which may cause the magnetic circuit to be saturated and hence, greater power rating

of the driver is required. Moreover, the disturbance re-

jection is mnch worse. Hence, we modify the control gain according to the criterion in the conventional PI-control

A s shown in Fig.

approach, in which case the gain ki is designed as 0.25.

It

is shown in Fig. 2 that under such gain selection the disturbance can be rejected perfectly, althongh the stator currents and rotor fluxes are still large. Hence, we pro-

posr a strategy to rcdnce the stator currents and rotor fluxes. This is similar to the so-called "soft-start" problem in industrial applications. Since the large transient currents are caused by the large initial speed error, we construct a modified speed command W d s by virture of Theorem 1 as follows:

Step 1: Select, a

Aw

>

0,

Step 2: Find the integer p that pAw

<

W d

5

(p

+

I)Aw,

Step 3: At any time, let m ( t ) be the integer such that Step 4: Choose the modified speed command wds as:

( m ( t )

-

I)AW

<

wr(t)

5

( m ( t ) A w - 0.3),

i f

m ( t )

<

p if m ( t ) = p

if

m ( t )

>

P Wdr =

{

t:'

( m ( t )

+

l ) A u ,

After replanning this new speed commend Wds with Aw =

2, the stator cnrrents can be redirced significant, but a t the expense of larger rising time, as shown in Fig. 3.

6 Conclusion

In

this paper, we have presented a ficld-oriented PI- controllcr for a n induction motor, wliich can drive wide class of loads. 'I'hongh it is dificult to prove the gloabl stability of the overall system through Lyapnnov analysis, we have proposed a simple iterative algorithm to

prove that. the system has a locally exponentially stable eqnilibrium point at. any desired speed. Moreover, the control law is independent of the rotor resistence (a parameter which normally varies with the temperature) and it. only uses the speed of rotor and the orientation of

rotor flux as feedback, which made it more feasible.

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[I]

E.

Bitmi,

F.

P. Denxi,

S.

Dolognani, and G.

S.

Buja,

A Field Oricntntion Scheme for Cnrrent-Fed 111-

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I E E E

'linn.9. lorfctst. Appl.,

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R.

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