A New Ekf-based Algorithm For Flux Estimation In Induction Machines

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496

A New EKF-Based Algorithm for Flux Estimation in Induction Machines Luigi Salvatore, Silvio Stasi, a n d Lea Tarchioni Abstract-A new reduced-order algorithm to be used to estimate the rotor flux components of induction motors with schemes such as field-oriented control is described in this paper. The algorithm is based on the extended Kalman filter theory and estimates the desired quantities on line by using only measurements of the stator voltages and currents, and rotor speed. The on-line adaptation of the inverse rotor time constant makes it possible to obtain very accurate estimates of rotor flux components, in spite of temperature and magnetic saturation effects. The algorithm order reduction decreases the computational complexity and makes the proposed estimator superior to the other ones based on the EKF theory.

tion and the flux signal is distorted by slot harmonics. The direct schemes based on an indirect rotor flux sensing are sensitive to variations of motor parameters because the estimators need a machine model. Particularly, the rotor resistance variations due to temperature have a dominant negative effect on the estimation accuracy. Analogously, the indirect schemes are sensitive to variations of motor parameters, especially to those of the rotor resistance, because the vector controllers require a knowledge of the rotor time constant [2]. To develop high-performance induction motor drives the assumptions that the motor parameters are known I. INTRODUCTION and invariant are not valid. If the position of the rotor flux NDUCTION motor drives are now being used more vector cannot be known with high accuracy because of and more in process industries because of the applica- variations of motor parameters, the transformation of the tion of the vector control strategy. The induction motors stator current component references, dependent on the behave like dc machines when vector controllers are used desired values of rotor flux and torque, respectively, into to maintain a 90" electrical space phase angle between the the instantaneous phase current references is not correct rotor field and the torque-producing stator current com- and this fact leads to poor dynamic properties and saturaponent [l]. This control strategy permits one to obtain tion or under excitation of the machine in the stationary fast responses to load or speed changes without the dis- point of operation. Therefore, much attention has to be advantages of slip rings, brushes, and field supply of dc given to rotor flux estimators and rotor time constant machines. identification schemes. Generally, two types of field-oriented control schemes In recent years, many papers have been concerned with are available. In the direct scheme, the instantaneous the on-line identification of the rotor time constant [3]-[5], position of rotor flux has to be measured using either and many estimators of rotor flux have also been develsensors or estimators. By means of this information, the oped for monitoring and control of induction motors. A direct and quadrature axis stator current component ref- first lot of estimators in use for control purpose are based erences, dependent on the desired values of rotor flux and on the observer theory. There are full-order observers torque, respectively, can be transformed into the instanta- that estimate all the state variables in the model of the neous phase current references, which are compared to system and reduced observers that only estimate the rotor the actual ones to generate error signals for hysteresis or flux components. The observer structure and its gain allow PWM current controllers. In the indirect scheme, a model one to achieve fast convergence rate and reduced sensitivof induction motor is required to calculate the reference ity to parameter variations (see [6]-[lo]), but the error angular slip frequency that has to be added to the mea- dynamics contains, however, a driving term that is associsured rotor speed. The sum is integrated to calculate the ated with the uncertainty in the rotor resistance and instantaneous position of the rotor flux. causes a steady-state error. Moreover, the observers are The direct schemes, which use direct sensing of the air sensitive to any noise in the measurements of currents, gap flux by Hall probes or search coils, are insensitive to voltages, and speed. A second lot of estimators are based variations of motor parameters but they suffer from high on the extended Kalman filter theory (EKF) [lll-[14]. cost and unreliability of the measurements because the These estimators have been shown to be the best comHall elements are sensitive to heat and mechanical vibra- puter algorithms for processing noisy discrete measurements and obtaining high-accuracy estimates of dynamic Manuscript received September 28, 1991; revised November 16, system states. It is a fairly natural thing to include 1992 and March 14, 1993. This work was supported in part by the Italian unknown parameters, for example, the rotor resistance, in Ministry for University, Scientific, and Technological Research (MURST). the state vector and once this is done, the designed The authors are with the Department of Electrotechnics and Electronalgorithm also solves the parameter identification probics, Polytechnic of Bari, Bari 70125, Italy. lem. The EKF-based algorithms described in [11]-[14] are IEEE Log Number 9211425.

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very interesting, but they appear to be very complex because of the high order of the mathematical models. The applicability of the Kalman filter to real-time signal processing problems is generally limited by the complex mathematical operations, for example the matrix inversion, necessary in computing the designed algorithm. Reducing the order of the mathematical model simplifies the computational problems and makes it technologically feasible to implement Kalman filters in real time with DSP processors. Therefore reduced-order EKF-based algorithms are highly desirable. In this paper, a new reduced-order EKF-based estimator is presented. The estimator uses the mathematical model of the rotor circuit to perform the function of prediction of the rotor flux components. Moreover the estimator includes a third equation to predict the inverse rotor time constant. The predicted values are adjusted by adding the product of the Kalman gains and the differences between the measured quantities and their predictions. The adjustment gives the estimates of the rotor flux components and inverse rotor time constant. The main contributions of this paper may be divided into five parts. In the first part the mathematical model of an induction machine is recalled to evidence the stator and rotor equations and to illustrate the time-domain block of the motor in the d - q reference frame. In the second part the system configuration with speed and flux-regulating loops, Kalman filter, and direct field orientation is presented in conjunction with the transfer functions and the criteria of controller design. The third part contains the development of the new algorithm to estimate the state variables and parameter of interest. The estimator equations are written in complex form because the electrical engineers are more familiar with its use. The fourth part recalls some well established reducedorder observer equations (see [71 and [9]), which are also expressed in compact complex form, for sake of comparison with the proposed EKF-based algorithm. Finally simulation results of transient and steady-state performances of both the new estimator and the reduced-order observers are presented to highlight their effectiveness. MODELOF INDUCTION MOTOR 11. MATHEMATICAL

In this section we recall the mathematical model of a three-phase induction motor. The voltage equations may be written in the stationary reference frame a - p - 0 by using the following transformation:

and the zero variable is zero for balanced conditions. In (1), f can represent either voltage, current or flux ( U , i, or A). This change of variables and the introduction of the complex quantity

yield the following stator and rotor voltage equations 1151:

where p is the operator d/dt, L , is the stator inductance, L , is the rotor inductance, L,, is the magnetizing inductancc, R , is the stator resistance, R , is the rotor resistance, u, = K,/L,, (T = 1 - L',,/L,L,, w, is the electrical angular speed of the rotor, and

The motion of the mechanical system is described by p u r = itZ,,(Te- TI-)/J - R,w,/J

(7)

where J is the inertia of the rotor and connected mechanical load, R,,, is the damping coeffient, TL is the load torque, and n p is the number of pole pairs. The electromagnetic torque may be expressed as

r,

If a rotating reference frame d - g, oriented in such a way that the rotor flux vector h,,, points into d-axis direction, is defined, a simplification of the mathematical model is achieved. The transformation of the a - p variables to the d - q reference frame is (9)

wherc

and A,.

=

(AS,

+ AT,)

l/Z

.

In the d - q reference frame, (5) and (8) may be expressed as follows PA,

=

-?Ar

+ u,L,,i,,

(12)

(13)

where 1

K =

6

- _1

- _1

2

2

0

-

1

2 1

2 1

E

E

777

--

Equations (12) and (14) show that i,, and i,, are the rotor flux and the torque-producing components of the stator current. Field orientation is accomplished by controlling independently these two current components. Fig. 1 illustrates the time domain block diagram of a three-phase induction motor in the d - q reference frame.

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I

Fig. 1. Time-domain block diagram of an induction motor in the reference frame.

ti -

q

111. SYSTEM CONFIGURATION

Until today, the field-oriented control has been realized in many different technical forms. Fig. 2 shows a direct controlAaccompli;hed by estimating the rotor flux components A,, and Arp (where the “hat” denotes estimate) through Kalman filter-based algorithm using measured stator voltages uEb,U>:,., stator currents iy:,, i;;, and rotor speed w,“ (where denotes measurement). The current references ird,i:q (where * denotes reference quantity) are supplied by the speed and flux-regulating loops and transformed in the a /3 reference frame by means of the transformation M - I . Then, the current references iT,,iTp are transformed in the phase current references i ~ o , i ~ , , ,by i ~ cthe transformation K - ‘ . A current regulated pulse width modulator (CRPWM) is controlled in purpose that the stator currents i,,,i,,),i,, follow th: current references i:o, i r h ,iTc. If the estimated matrix M coincides with the actual matrix M and the performances of the CRPWM inverter and tachogenerator are ideal, that is their transfer functions are equal to one, the block diagram of the system results as shown in Fig. 3. In this diagram, s is the complex variable of Laplace transformation. The transfer function of the speed loop, with A w,(s) as its output and Am:($) as its input, is as follows:

Fig. 2. System configuration with speed and flux-regulating loops, Kalman filter, and direct field orientation using a CRPWM.

Fig. 3. System block diagram under idcal direct field orientation.

where k,,, and k , , are the proportional and integral gains of the flux controller. If one puts ( k 1 ! $ / k , , R , / k , k , , ) = 26, and J / k , k , , = 26* according to the absolute value optimum criterion [161, the time speed response to a speed reference step Aw; is

+

(15) where, k , = n;L,,,A,/L,, k , , , and k , , are the proportional and integral gains of the speed controller. The response of the system to a load disturbance AT,,(s) is ’I

Aw,(s)

=

and the time speed response to a load step AT, is

,,

(16) The transfer function of the flux loop with AA,(s) as its output and AAT(s) as its input is 1

If one puts k , , = ujk,,, the time flux response to a flux reference step AA: is

+ kpA -.F

IV. KALMAN FILTER-BASED ALGORITHM BUILDING (17)

To estimate the a-p components of the rotor flux, an algorithm based on the extended Kalman filter theory may be developed. This theory requires that the estima-

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SALVATORE et u l : A N E W EKF-BASE11 A L G O K I T H M

tion process is described by a state-space mathematical model and an observation equation, see [I71 and 1181. It is possible to assume that the estimation process is governed by the "current model" that is ( 5 ) [XI. Unfortunately the operational changes in the parameter q , which depends on temperature and saturation, have negative influence on the rotor flux estimation. As a consequence, the additional parameter a, has to be estimated in conjunction with the rotor flux components and a further equation must be appended to (5). The complete mathematical model of the estimation process has to be discretized when measurements at discrete points in time are available. It results in rcfP

(n

+

1)

=

h

I (1

p(

n ) e ~ ~ r , ~ ~ ~ ~ Tt + ~ ~ ~ ~ : " ' ~ ~ ) l

11 +

.[e-,r,(ll)?+Jm:"(fl)/

a,(n

-

EA(r/j(lz)

+ 1) = q ( n ) + E , , ( H )

Equation (25) is obtained from (4) and (6) by applying the classical Euler method to give a discrete form to (4), and the term Zap = vCx ju, is the measurement error. Equation ( 2 5 ) may be written in matrix form as follows:

+

z'"(n

0

+ 1) + J x ( n ) + v ( n + 1)

(28)

We assume at this point that we have an initial estimate of the process at the time ( n ) [181. With this assumption we now seek to use (21) and (22) to predict the rotor flux components and the parameter a, at the time ( n + 1). The prediction equations are

(21)

+ 2iY;,(tz)].

(23)

Equations (21) and (22) may be written in matrix form as follows:

+ 1) = f ( x ( n ) , n )+ E

(24)

where A,,

Hx(n

I

ap

( n + 1)

=

hr u p ( n ) e - e r ( n ) T + l u : ' ( n ) T

+

(22)

1 ;Tcf,(n)= f i i y i ( n ) +j--[iy:,(n)

X = [ A , ,

=

where

where T is the sampling time interval, during which the measured stator currents, : z :i and rotor speed w:' are considered to be constant, F A U P = 4 , + jcA0 and E,,, are the model errors, and

x(n

+ 1)

€=[EAn

EA@

.[e

-

f:,irl)r+ i w Y ( n ) T

-

11

+ 1) = < ( n )

(29) (30)

(3;(n

where (-) denotes prediction. Equations (29) and (30) may be written in a compact form as follows: f(n

+ 1) = f ( i ( n ) , n ) .

(31)

Correcting the predicted state variables h,,(n + l), hrp ( n + I), and Gr(n 11, by means of the measurements z:'(n + 11, and z$(n l), gives the estimates at the time instant ( n + 1) according to the following estimation equation:

+

+

E<,,]',

and f = [ f ,

f2

fill.

It is possible to assume that the estimation process is governed by the following complex observation equation:

where Zrp(n + 1) = zg'(n + 1) + jz;"(n + 1 ) is the complex measurement at time ( n + 1) given by: 2rp:;i(n+ 1)

=

[G&(n)

-

R,i3/1)]

where G is a 3 x 2 matrix called a gain matrix. Equation (32) may be written in a compact form as follows:

P(n

+ 1 ) = i ( n + 1) + C ( n + 1) .[z"(n

+ 1)

-

Hi(n

+ 1) -Ji(n)].

(33)

The problem now is to find the optimal gain matrix G . According to the Kalman filter theory we have to define the covariance matrices of the estimation and prediction errors

P ( n ) = E { ( i ( n ) x ( n ) ) ( i ( n >- x ( n > ) 7

(34)

P ( n ) = E { ( f ( n ) - x ( n ) ) ( i ( n )- x ( n ) I T }

(35)

-

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where E represents the expectation operator. It is possible to choose the gain matrix G so to minimize the estimation error variances for the elements of the state vector being estimated, ?hat is the individual terms along the major diagonal of P ( n ) . In the hypothesis that v and E are uncorrelated vectors of random noise, quantities having zero mean and covariances N and Q , the result is

where a,* is the reference value of the parameter 0; subjected to operational changes. Discretizing (41) yields

+ l)HT+ @(n)P(n)JT) + l ) H T + N + Jk(n)JT + H @ ( n ) r ; ( n ) J T + Jk(n)@T(n)HT]-'

G ( n + 1) = ( F ( n

.[HF(n

(36) where

+ 1) = @ ( n ) k ( n ) @ ' ( n +) Q

F(n and @ ( n )is a 3

X

(37) 3 matrix having the following elements

The error covariance matrix associated with the estimate may be computed as follows

P(n

+ 1)

=

F(n

+ 1)

-

G ( n + l ) L ( n + l ) G T ( n + 1) (39)

where

A straightforward calculation shows that the error in the rotor flux estimate is governed by the following state-space equation:

where Z is the identity matrix and L ( n + 1) is the bracketed term of (36) (not inverted). Equations (31), (33), (36), (37), and (39) are the recursive equations of "the delayed state filter algorithm" (see [19] 1. V. REDUCED-ORDER OBSERVERS The a-@ components of the rotor flux may be estimated by means of reduced-order observers. In this section we briefly examine some observers, by which we attempt to estimate the rotor flux components, to compare their effectiveness with that of the Kalman filter based algorithm. The mathematical model of the estimation process based on the observer theory can be obtained from the combination of (4) and (5) by introducing the expression of A,,, given by (6). The result is

(44) and that the error dynamics contains a driving term, which is the second one on the right-hand side of the above equation and depends on U, variations. The observer sensitivity to U, variations is 1/(1 - hL,/L,). If one considers the induction motor supplied by sinusoidal voltages, the steady-state error due to U, variations is Arm,

- -(U,

-

-

a,*

-

jw?

U,*

+ j w S ( 1 IL,/L,) -

+

Em

- Rs c a 8 ) +

z2)p:ra, -

1

=

j w r m )-h r a p u , L , ~ ' ~ ~ ,

L a

where h is a complex coefficient 161, and E.\, error. The estimation equation is (1 -

a, -

- 'rap

hULspEya,

(40) is the model

where w, is the frequency of the motor supply - angular voltages and A,,,, ZJap are complex phasors equal to A r u g e - l w ~ ' , is,: respectively. The choice h = 0 gives the discrete "current model" estimates [81. In this case the error time constant is l / q * and the "current model" sensitivity to a, variations is equal to 1. There is evidence that the choice h = Lr/2L,, suggested by Verghese and Sanders for ease of illustration [6], gives estimates that converge faster because the error time constant becomes 1/2a,*, and the observer sensitivity to a, variations equal to 2.

-

p'uy',

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SALVATORE et al: A NEW EKF-BASED ALGORITHM

The choice (46) suggested by Hori and Umeno 191 gives an error, in the rotor flux estimate, governed by the following state-space equation p('rap

- 'rap)

= -'('rep

- 'rap)

3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

rated voltage rated speed rated frequency pole number system inertia stator resistance rotor resistance stator inductance rotor inductance magnetizing inductance

380 V 1420 rpm 50 Hz 4 J = 0.062 kg m2 R, = 0.728 R R , = 0.706 Cl L , = 0.0996 H L , = 0.0996 H L m = 0.0969 H.

At first, the motor has been supposed to operate in steady-state condition with a load torque of 50.4 Nm, (47) rotor flux equal to 1.13 Wb, and electrical angular speed of 100 rad/s. The dc voltage has been chosen equal to The error time constant is l/a, and the observer sensitiv- 400 V, and the hysteresis band equal to 10% of the ity to a, variations is proportional to (a) and its modulus current reference. The speed regulator parameters have is constant and equal to q if ( a ) is chosen as follows been selected as follows: k , , = 122.8, k,, = 1.86. The flux regulator parameters have been selected as follows: k , , = (48) 188.2, k,, = 26.55. The EKF-based algorithm has been started at time where q has to satisfy the condition f = 0. The initial estimates of the rotor flux components a,* q < l + (49) a-/3 for the recursive estimation process have been [a,- a,* I, assumed to be zero (Arap@)= 0 j 0 ) because the actual to guarantee the stability at all the operational speeds. In value of the rotor flux ( A r a p ( 0 ) = 1.13) and its compothe above expression [a,- a,*Imax is the expected maxi- nents (A,, and &) have been considered to be unknown. The initial value of the inverse rotor time constant has mum increase of a,". been assumed to be equal to 50% of the actual value that The choice was 7.08 s-' (<(O) = 3.5). About the choice of the diago(50) a:1 elements of the estimation error covariance matrix P(O), it is to be noted that large values assigned to the initial variances give a fast convergence, but the initial suggested by Bellini, Figalli, and Ulivi [7] gives estimates may have a negative influence on the drive performance. Small values assigned to the initial variances, which means to have a some a priori knowledge of the expected magnitudes of the state variables to be estimated, give a slow convergence. By using common sense judgement, the estimation error variances of rotor flux components have been assumed to be 2 Wb2 and that . of the (51) of the inverse rotor time constant 100 s - ~ Choice variances of the model and measurement errors has been The above equation shows that the error time constant is based on the random noise, having a normal distribution aL,/bwT and the observer sensitivity to a, variations is with mean 0.0, added to the estimator inputs (voltage and current components, and electrical angular speed). equal to b / a L s , where b has to satisfy the condition Because the voltage component noise-variances have been (52) selected equal to 4 V2, the current component ones equal to 0.25 A', and the speed one equal to 1 rad2/s2, as a consequence we have put Q = lo-' diag(1 1 1) and N = to guarantee the stability at all the operational speeds. diag(1 11, where "diag" indicates a diagonal matrix and the argument is the main diagonal. It is to be noted VI. SIMULATION RESULTS that the estimates obtained from the EKF-based algorithm To verify the effectiveness of the proposed EKF-based have beenn supplied to the flux regulator and coordinate reduced-order algorithm, some simulations have been carchanger M - '. ried out by using the actual parameters of an induction At the same time, rotor flux has been estimated by the motor. We have the following: "current model," the reduced-order observer having h 1) rated power 7.5 kW equal to L,/2L,, the observer proposed by Hori and Umeno, and that suggested by Bellini, Figalli, and Ulivi. 2) rated current 16.5 A

+

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IEEE TRANSACTIONSON INDUSTRIAL ELECTRONICS, VOL. 40, NO. 5, OCTOBER 1993

In any case the initial values of the rotor flux components have been set equal to zero (AraP(O) = 01, and the reference of the inverse rotor time constant equal to 50% of the actual value (U,* = 3.5). A speed reference step has been applied at time t = 0.05 s. Fig. 4 shows the rotor flux estimate obtained from the “current model” with the reference of the inverse rotor time constant equal to the actual value and to 50% of it, respectively. Moreover, Fig. 4 shows the actual rotor flux, which is maintained constant by the flux-regulating loop and correct application of field-oriented control strategy. There is evidence that the incorrect reference a;* of the inverse rotor time constant causes a large estimation steady-state error. Fig. 5 shows the rotor flux estimated via reduced-order observer having a gain h = L r / 2 L , with the reference of the inverse rotor time constant equal to the actual value and to 50% of it, respectively. The comparison between the estimates of Figs. 4 and 5 evidences that, with this last choice of h, the rotor flux estimates converge significantly faster and the estimation steady-state error is largely reduced. It has to be noted that the choice of h equal to L r / 2L , causes a poorer transient performance because the observer sensitivity to U, variation is doubled. Figs. 6 and 7 show the rotor flux estimates obtained from the reduced-order observers proposed by Hori and Umeno [91, and Bellini, Figalli, and Ulivi [7],respectively, with the reference of the inverse rotor time constant equal to the actual value and to 50% of it. The transient and steady-state performances of these observers appear to be very good and similar in spite of the different choice of 2 (see (46) and (50)). We have chosen q = 1 for the observer proposed by Hori and Umeno (see 481, and b = a L , for the observer proposed by Bellini, Figalli, and Ulivi (see 50). These choices partially justify the performance similitude. When the inverse rotor time constant is assumed to be equal to 50% of the actual value, the actual rotor flux is not reached by the estimates because of the steady-state error due to a; - U,*. Several steps (over 20) are necessary to reach the steady-state condition. Fig. 8 shows the rotor flux estimated via EKF-based reduced-order algorithm with the initial value of the inverse rotor time constant equal to 50% of the actual value. It is to be noted that the actual rotor flux is reached by the estimated one after four steps, that is the convercence rate is very high. The estimates track the actual values very well. Fig. 9 shows the estimates of the inverse rotor time constant. There is evidence that the convergence rate is high and the estimates are very accurate. Then another simulation has been carried out by adding high-level random noise to the estimator inputs to show the optimal noise rejection of the EKF-based algorithm. The initial mqtor conditions, speed step, initial filter estimates, and P(0) have not been modified. Because the voltage component noise variances have been selected equal to 100 V2, the current component ones equal to

1,actual actual rotor flux

1

/ A0 * = 0.5 Ci

0.1

0

0.2

0.3

0.4

0.5

time, s Fig. 4. Rotor flux estimated via “current model” with the reference of the inverse rotor time constant equal to 50% and 100% of the actual value (50% increase of speed at t = 0.05 s).

2.5,

1

0.5

1

’

i

actual rotor flux

”

0

0.2

0.1

0.3

0.4

0.5

time, s Fig. 5. Rotor flux estimated via reduced-order observer having a gain h equal to L , / 2 L , with the reference of the inverse rotor time constant equal to 50% and 100% of the actual value (50% increase of speed at t = 0.05 s).

1.3 I

I

a 8

actual rotor flux

0

0.05

0.1

0.15

0.2

0.25

0.3

time, s Fig. 6. Rotor flux estimated via reduced-order observer proposed by Hori and Umeno, with the reference of the inverse rotor time constant equal to 50% and 100% of the actual value (50% increase of speed at t = 0.05 s).

6 A2, and the speed one equal to 27 rad2/s2, as a consequence we have put Q = l o p 6diag(1 1 1) and N = diag(1 1). Fig. 10 shows the actual rotor flux and estimates carried out by the EKF-based algorithm, and through the observer proposed by Hori and Umeno. The estimates obtained via

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SALVATORE et al: A NEW EKF-EASED ALGORITHM

1.3 1.25

I

1.2 1.15 l.l

I

$ 2

\

o

= 0.5

rotor flux estimated via Hori and Umeno

n/

actual rotor flux

6:

.-28

o

1j '

1.2

U

8

actual rotor flux

1.os

1.3

.*

1.1 rotor flux estimated via EKF algorithm

v1

e,

11

0

I

1 0.05

0.1

0.15

0.2

0.25

0

0.3

0.1

0.05

0.15

time, s Fig. 7. Rotor flux estimated via reduced-order observer proposed by Bellini, Figalli, and Ulivi, with the reference of the inverse rotor time constant equal to 50% and 100% of the actual value (50% increase of speed at t = 0.05 s).

1.3 I

g

1.25

I

1

0.25

0.3

Fig. 10. Rotor flux estimates carried out in presence of strong measurement noise: a) via reduced-order observer proposed by Hori and Umeno, with the reference of the inverse rotor time constant equal to 50% of the actual value, and b) via EKF algorithm, with the initial value of the inverse rotor time constant equal to 50% of the actual value.

. .

*

.

..

.

.

. .

I

injectea noise ana nave steaay-state errors wee ~

I"*\

(44)

..J

anu

(45)). The superior transient and steady-state performances of the new EKF-based reduced-order algorithm in comparison with the other ones (see Figs. 4-10) are due to the recursive estimation method of Kalman, and the on-line updating of the inverse rotor time constant that has been considered a further state variable (see (29)-(32)).

rotor flux estimated via EKF algorithm

actual rotor flux

I

0

0.2

time, s

0.05

0.1

0.15

0.2

0.25

VII. CONCLUSION

0.3

This paper has examined the problem of flux component and inverse rotor time constant estimation in induction machines. The new recursive algorithm developed in Fig. 8. Rotor flux estimated via EKF-based reduced-order algorithm this work is based on the EKF theory. It has shown with the initial value of the inverse rotor time constant equal to 50% of the actual value (50% increase of speed at t = 0.05 s). superior behavior to other well known methods. Its advantages may be resumed as follows: reduced order of the mathematical model, high convergence rate in the estimation of the rotor flux components, high convergence rate in the contemporaneous estimation of the inverse rotor time constant, estimates of high accuracy, good performance in transient conditions, very low steady-state error, optimal rejection of measurement noise. In spite of the inverse rotor time constant variation, the use of the flux estimates in the coordinate changer and flux-regulating loop has made it possible to keep constant rotor flux in the machine under direct field orientation. 4 The high accuracy of the estimate of the "actual" inverse rotor time constant also suggests the possibility of using 31 0 0.05 0.1 0.15 0.2 0.25 0.3 the new algorithm for correct implementation of indirect field orientation schemes. time, s time, s

.I

Fig. 9. Inverse rotor time constant estimated via EKF-based reducedorder algorithm (50% increase of speed at t = 0.05 SI.

the observer proposed by Bellini, Figalli, and Ulivi are very similar to those derived from the observer of Hori and Umeno and, therefore, have not been reported in Fig. 10. There is evidence that the EKF-based algorithm is able to reject the strong noise injected into the estimator inputs. On the contrary, the observers are sensitive to the

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers of this paper; the EKF-based algorithm has been improved because of their very useful comments. REFERENCES [l] W. Leonhard, Control of Electrical DriLm. Berlin: SpringerVerlag, 1985. [2] B. K. Bose, Power Electronics and AC Drices. Englewood Cliffs,

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Luigi Salvatore was born on March 4, 1945, in Lecce, Italy. He received the degree in electrical engineering from the University of Bari, Italy, in 1970. From 1976 to 1983, he worked in the Electrical and Electronic Department of the same University as a member of the research team on electrical machines. From 1983 to 1987, he was a Researcher of electrical machines in the same department. From 1987 to 1991, he was an Associate Professor of clectrical machines at the University of Bari. Since 1991, he has been an Associate Professor of electrical machines at the Polytechnic of Bari He has produced over 50 technical papers, 23 of which have been published in periodic journals. His current research interests include the control, monitoring, and dignostic of ac drives, and the areas of signal processing and digital measurements on power electronics systems. Prof. Salvatore is a member of the Italian Electrical and Electronic Association (AEI).

Silvio Stasi was born in Bari, Italy. He received the degree in electrical engineering from the University of Bari, in 1989. Currently he is pursuing the doctoral degree at the Polytechnic of Bari. His research interests are in electrical machinery, estimation techniques, and adaptive control.

Lea Tarchioni was born in Bari, Italy. She received the degree in electrical engineering from the University of Bari, in 1990. From 1990 to 1991 she worked in the Electrical and Electronic Department of the same University as a member of the research team on electrical machines. From 1991 to 1992, she worked in IBM in Raley R.T.P., N.C., as a systems designer. Since 1992, she has worked for ENEL the Italian Electrical Company. Her research interests are in the electrical machinery, especially induction motor drives and their nonlinear control.