A Rotor Flux Observer For The Vector Control Of The Induction Motor Drive

A R O T O R FLUX OBSERVER FOR THE V E C T O R

CONTROL O F THE INDUCTION MOTOR DR

D e p a r t m e n t o f Electrical E n g i n e e r i n Faculty o f Engineering O S A K A City U n i v e r s i t y *

2. Minimum-order

ABSTRACT An improved rotor flux observer was developed. This observer can be used effectively in vector control (Field oriented control) systems o f the current source inverter drive induction motor. The observer is composed of a 16 bit micro-computer which calculates the observer equation and A / D converters which measure currents and voltages o f the motor. As to the observer, the stability and correctness of the expected flux vector must be considered as most important things. Then, in our observer, the first-order hold and aexcellent pole-placement design are adopted. In this paper, simulated results o f estimated rotor flux and experimental results of the tested motor are presented. 1 . Introduction

Recently, some interesting work has been published concerning the application o f the rotor flux observer. For these application, the values of the rotor flux is expected to a accurate and a time lagging-less vector, then the observer is required to have characteristics of the quick and accurate calculation. In the case that the observer is composed in a micro-computer, the delay time of the estimated value is delayed cause o f both sampling observer signals and calculating the observer equation. Then,it is the most important problem for the rotor flux observer to make these delay times as short as possible. When a stable and accurate observer is composed on a micro-computer, the essential problem is how to arrange the poles o f the observer's equation. In this paper, a rot lux observer system for o f the induction motor the practical vector cont I S discussed. The observ s composed of the first -order hold to reduce th elay time which i s made by sampling the observer s I s , and a method o f the observer poles arrangement i s proposed. Finally, we show experimental results o f a vector controlled induction motor which i s driven by a controlled current type inverter.

(a,b axis)

is

ux observer.

(')

d

The state vector are fluxes of the primary and secondary clrcui The state vector supplied voltage Vs. V S = [ v a s , vbs] (2) I f the state vector o f the output si primary current o f I s= [ias, l b s l the output value

is shown as follows.

ri

where,

A = Ls

the induction m

A1 1= -{RsLr /A KEY WORDS: AC motor control: state observer; current source inverter; induction motor, *Address f o r correspondence: 3-138 SUGIMOTO 3 , SUMIYOSHI-KU OSAKA 558 JAPAN Phone 06-605-2586 IECON '88 I 4 7 2

A12=M/fl{Rr/Lr*

+ RrM'/LrA}*

I -w r-JI

A21=MRr/Lr+ I A22=-Rr/Lr. I +or.J Bi=Lr/A- I

I

The state equation ( 4 ) can be rewrite as follows. hr=A22*hr+A21* i s

(6)

Te=(3/2)*(P/Z>*(Lr/M) X(A areibs- A br*ias) (11) In the case of using the vector control in

From eq.(6) of the state equation we can understand that h r is the state vector, and that is i s the vector of the input signal and eq.(7) is the output equation in which (p. i s-All. i s-Bl.Vs) is the vector

the induction motor, the orthogonal currents i ds and i qs are controlled independently. In this case, these currents should be converted to the primary current vector of i as and i bs. In this conversion, we should have the angle L h r

of the output signal.

between an axis of the static coordinate and the rotor

ir-All.

is-Bl-Vs=A12*Ar

(7)

Then, we can show the minimum order observer as the equations (6) and ( 7 ) . in this observer, the expected value.of the rotor flux A r is obtained from eq.(6). and the error in the flux between the command value and the estimated one of eq.(7) is feedbacked to the system by the gain G shown in the following equation. h

h r=A22*x,r+A21* i s G { i s-AllLi s-Bl-Vs-A12*T r > =(A22-G.A12)Ar +(A21-G*All) i s - G * B l * V s + G * i s (8)

+

Ar=E+G.is *

(9)

In this manner, the lime derivative of i s can be ignored. The eq.(9) is the minimum order state equation of the rotor flux observer. The gain matrix G can be calculated as in the following eq.(lO) in order to make the pole of the flux observer - d j p .

*

g2=-

A

M

And the observer equation has to be converted from the differential equation of the continuous-time system to the difference equation of the discrete-time system. As the poles of

the observer equation can be

determined arbitrarily by a designer, eigenvalues of the observer equation can be given in advance. The

=

+ g2.J

Observer equation using the first-order hold. When we calculate the equation of the rotor flux

3-1.

observer by using the micro-computer, the detected in-

E=fir-G*is =(A22- G eA12)Tr +(A21-A11*G) i s - G*Bl*Vs

G=gl* I

3 . Design of the rotor flux observer.

put signals i s(t) and Vs(t) should be converted to the discrete-time input signals of i s(k) and Vsfk).

In the eq.(8), we must detect the value of the time-derivative of the primary current. Where, introduce a new variable F and express the equation of the A r as follows. A

flux vector A r axis. In the vector control system, the angle L A r is obtained by expecting the values of the rotor flux observer.

(10)

general solution of the observer equation can be obtained and the differential equation of the observer converted easily to the difference equation. However, as for the input signals, they have delay-time between continuous-time values and the discrete-time ones. We discussed the eliminating method of the delay-time of the observer signals, and we adopted the first-order hold instead of the ordinary zero-order hold. By adopting the first-order hold, the delay-time between the input signal of the motor f (t) and the input signal of the observer g (t) can be eliminated. At the sampling time KTs , the expected rotor flux by using the first-order hold is shown as follows.

w r *a -(Rr/Lr) p (Rr/LrI2 wr2

+

(12)

In the induction motor, the current component of the torque o r the rotor flux is shown in the primary current. And the torque equation can be shown in the same way as in the direct current motor. There, the reference flame is set on the rotor flux A r axis, and the primary current is divided into two components. One is the exciting current i ds, and the other is the current o f the torque component i qs. As well known, the generate torque of the induction motor can be calculated by eq.(ll).

In the above equations, conversion matrixes @ (Ts).

IECON '88 I 4 7 3

t7 (Ts) and

U

( T s ) are not

affected by

the rotating

speed of the motor, and they are constant values. When the input signal in the observer is hold by the first-order hold, the delay-time o f the expected output slgnal can be cancelled. However, the sampling time Ts of the observer increases because of the calculating time of the first-order hold in the observer. In practice, the percentage of the calculating time in the first-order hold is small In comparison with the sampling time of the observer. The construction of the above the observer system is shown in Fig.1.

real-axis. When the poles of the system are set on these straight line, the flux observer is set in the critical damping condition. And the maximum driving frequency less than 1/2 of the sampling frequency. When we adopt th ment, i t Is desira t the poles of the rotor flux observer should be placed on the in the complex-plane.

pT

Y

dt)

1s \

1 t( t)

S t a t e Observer

1%

l l ..._.... ...~..~.[ .k .- -. ... ...........................................

Bd=-C.Bl

Cd=AZI-AII- G -A22.C

\

___..

+ G * A OG

Fig.1. Farst-Oder Hold Flux Observer. 3-2,

Stabilization and pole-position o f the observer.

In the equation o f an observer, the real-part o f the pole a is related to the settling time of the observer output signals. When we make - a smaller, that is, the poles of the observer are placed on the far left side on the complex-plane, the output signal of the observer is settled i n the more short time. On the other hand, the imaginary part P o f the pole affects the angular ncy of the pulsation of the output signal. I f the of P is made a larger one, the observer is put in an unstable condition. However, the value of 0 is made too small, the step -response of the observer becomes vague. T o obtain the excellent output signals from t h e observer, i t is not perfect that the pole should be set in the left side on the complex-plane. In the closed loop system, the stable boundary of the pole position i s limited the slant line shown in Fig.2. In our observer system, the pole position in the open-loop i s the point of -Rr / L r + j o r , then, the real-part - a must be set smaller than -Rr / L r . And the pole position of the stable system must be the area In the inner one between 2 4 5 ' line to the

-

larger according to the di sition - R r / L r 2 j o the f l u x observer rotor flux observ

n the pole po-

damping conditi In that situation,

the observer s y disturbances. To keep the ga rotating speed,

nstant independent of the bserver must

should be placed i tlon of the open-1

the observer mu real axis

rotor flux correspond to the fundamental component.

IECON '88 I474

In Fig.J(a), the relation between components gl, g2 of

the gain

matrix

motor are shown.

and

the rotating speeds of the

And Fig.3(b)

shows the same relation

as in the case of the Fig.3(a), the case * o f Fig.3(a),

And the error of the expected flux vector

be divided into two parts such as

and

the phase

the amplitude

angle component.

Error of the amp1 i tude

6= I

Error of the Phase-angle

E

xr I / I Ar I

the variable Kaln de-

creases at lower rotating speeds of the motor. From

hr.

can

but these poles of the

observer are fixed on constant values of -400 k J 4 0 0 . In

vector

=Lxr

-LAr

these results, we concluded that by adopting

the pole placement method mentioned above, rotor flux observer even

sign a stable

at

we can de-

4. Characteristics of the rotor flux observer.

the lower In

rotating speeds of the motor.

rotor

this chapter,

the experimental results of the

flux observer are glven.

Fig.4

shows the construction of the vector control

system of the induction motor for the experiment. In this rotating speed control system, the switching devices of the current source inverter

are GTO's

and the inverter is supplied by the current controlled thyristor converter.

n

(b) Fixed Pole (-4002 1400)

U

(11r(

-

Ld

n

IOHz/d iv)

(a) Variable Pole

t

I

wr

16 b i t CPU

Fig.4. Fig.3.

Values of Gain Matrix G.

Error of the estimated flux. An induction motor is supplied by the primary current whose amplitude is Is and whose angular frequency is o e . The voltage vector Vs of the

3-3.

primary

and the flux vector A r of the h r can be calculated as follows.

voltage vs

secondary flux

A r =(A21 ' I s)/( j we- A22') V s = ( j o e * I s-All'* I s-A12'*Ar)/B1'

Induction M o t o r Drive System with Current Sorce Inverter.

A 100 V. 4 poles, 0.4 KW three-phase induction motor is directly connected to the torque transducer, the

loads of

the

eddy current brake

and the Prony

A rotary-encoder, whose output signal is 1024 PPR, was connected to the other end of the tested

-brake.

motor shaft. And these output signals are converted to the 4096 PPR signal by the multiplying circuit, then we obtained 1 . 8 RPM/bit accurately.

(13)

Where, each variable is a complex value. And ( I ) values of eq.(13) denote the practical values of the equivalent circuit or the values which are calculated

Control

-

from the equivalent circuit parameters. On

the other hand, the expected values of the sec/-

Ar

ondary flux vector equation.

These

"nr={(A21-

result

is obtained from the observer

1

are explained in eq.(14). Fig.5.

G - A l l + J o e * G ) I s- G*BI*Vs}

/ { j me-(A22-

G.Al2)I

(14)

The observer error can be obtained by the subtraction. that is, to subtract the practical value of the st-condary flux vector A r from the expected flux

r-

Construction of Vector Control System using Flux Obserber.

The thyristor converter is controlled by the phase control signals from the 8 bit CPU(Z-80A.4MH). The value of the DC reactor is about 250 mH. Gate slgnals

IECON '88 I475

of the GTO are supplied by a 8 bit CPUfZ-80A,4MH) which is controlled by the 16 bit personal-computer (8086CPU.8MH). And the wave forms o f the controlled current inverter are the 120 rectangular or the PWM wave, in which the switching frequency is unsynchronized to the fundamental frequency. The filtering capacitors of 1 fi F are connected to the output terminals of the inverter. in order to detect primary currents and voltages o f the inverter, 12 bit A / D converter system is adopted, which has the function of sampling in the six-channels A/D converter at the same time, and the conversion time is about 50 usec.. The 1 6 bit personal-computer (8086CPU,8MH) was used in calculating the data o f the rotor flux observer. The sampling time of a tested system is determined as 3 . 2 msec which includes the caiculating time o f the vector control for the induction motor. In this system, the upper limit of the inverter frequency is about 30 Hz, in which case the observer samples the data about 10 times t o one cycle o f these

the rotating speeds o

motor. In this and the DC link cu of the f s= & 1. results, at the values o f the measured ones, bu tion the expect decrease s 1i gh t 1 y

.

braking in comparison with the power running condition I t of the observer

.

signals The functional block diagram of the induction motor drive with the rotor flux observer i s given in Fig. 5. Expected wave forms of the rotor flux by the observer system are shown in Fig.6.

(a)

Measured Value

i

E

i

(b) First-Order Observer

(e) Zero-Order

-order hold,

s

a little

is maintained to lagging in the hold can improve

I 0

aos

om

01 1 I.e( 38c)

0

005

aro Tlr(JPI;)

Fig.6. Flux Wave Forms obtained from R o t o r Flux Observer. In figure 6,(a) and (b) show the expected flux wave forms on the fixed pole position (-125 +j125) of the observer. And the tested motor i s driven under steady-state conditlon in which the values of the DC link current and slip-f ncy of the motor are kept as constant values. Fig.6 (c) and (d) show the expected flux wave form under the condition in which the poles o f the observer change in accordance with

flux observe length of lag the observer the length of lagging angle o f

IECON '88 I476

the measured values, the d on the sampling-time o f us problem. In which case g time Is proportional to the the primary current. When the

sampling time based on

is set 3.2 msec.,

the lagging angle

this sampling time becomes

the 20 Hz of

at

about 23

the supply frequency and 35

at the 30

Hz.

This 1s a serious problem in the vector control system, therefore; we should make use of a state

estimator or the same one which acts as a compensator of the lagging angle. Where,we propose a compensating method of the lagging angle in the flux observer. In this method,

tating speed of the motor is about 600 rpm. Fig.9. shows the experimental results of deceleration and acceleration when

the tested motor is oper-

ated by the drive system shown In Fig.4 and the vector control system using the rotor flux observer shown in Fig.5.

the absolute value of the expected

flux I A r I is used to compensate the lagging angle with the expected rotor flux. That is, the error value between the absolute value of the expected flux I A r 1 and the command flux A r* is feedbacked to the phase command value of the primary current. When the command current of the torque is a positive value, and the phase angle of the expected flux is lagging from the current command, the value of the

z

0)

1.0

J:Homent of Inertia Rotat lng Assembly

-

0

I-..

8':

b.5°; 9 -1.0

20

-

30

40

50

Time(SEC1

exciting current becomes larger, and the value of the rotor flux comes to be a larger than the command value. Conversely, in the case that the phase angle of the expected flux is led to the command one. The values of the exciting current component becomes small than that o f the command one, and the values of the expected rotor flux are reduced in these of the flux command. In the case of the inverse polarity of the torque command signal, this relation is reversed.

In this experiment, the command values of the And in this case, the rotor flux maintained 0.15 Wb. zero-order hold and a state estimator ware adopted in order to save the calculating time. The command value of the rotating speed changes on the step from -600 to +600 rpm. The value of the DC link current i s set under 3 . 0 A.

In the experiment, the absolute value of the expected flux I A r I was used for the compensation of the phase angle in the primary current. Therefore,

6 . Conclusion

the value of I A r I can not be applied for compensating the amplitude of the flux. However, we actually use the above system, the command value of the exciting current i ds is kept constant. We suppose that this matter is not a serious problem in a practical application.

The rotor flux observer by the micro-computer was applied to the servosystem of the induction motor. In the rotor flux observer, the first order hold is adopted instead of the zero-order hold in the ordinary observer system. With this method, the lagging time which causes the sampling time of the And in our observer, the observer is eliminated pole positlon of the observer equation changes in

.

accordance with the motor speed and, quick and accurate response of the expected value of the rotor flux is obtained. When our proposed observer is applied to the vector control system of the induction motor, the phase angle of the primary current is compensated b y the expected flux. We suppose that our control method of the inducFig.8. Transient Response of Flux Observer. Fig.8. shows the step response of the estimated flux in the our experimental system. The flux command A r* is maintained as constant value of 0.15 Wb, and the current command of the torque component iqs* changes on a step from 1.5 A to 0.3 A. And the ro-

tion motor can be practically used in the current source inverter drive system. Reference (I)Y.Hori,V.Cotter and Y.Kaya.Control Theoretical Con siderations Relating to an lnduction Mashine Flux Obsever. Trans.J1EE.106-B,11,1001-1008,(1986.N0V.~

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