Novel Rotor Flux Observer Using Observer Characteristic Function In Complex Vector Space For Field Oriented Induction Motor Drives

Novel Rotor Flux Observer Using Observer Characteristic Function in Complex Vector Space for Field Oriented Induction Motor Drives Jang-Hwan Kim, Jong-Woo Choi and Seung-Ki Sul School of Electrical Engineering, Seoul National University San Si-l,Shillim-Dong, Kwanak-Ku, Seoul, Korea(ZIP 151-742) htt~://eewl.muac.kr email :[email protected] ac.kr Abstrud

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current model and the desirable high speed attributes of the voltage model have been proposed[2-81. In this paper, generalized analysis method named “observer characteristic function method” is proposed to analyze all kinds of the linear flux observers in unified form. With the observer characteristic function, the estimated rotor flux error involved in the classical methods can be easily clarified. Moreover, the novel rotor flux observer based on this analysis is also presented and the effectiveness of the observer has been verified by simulation and experimental results.

This paper proposes a new strategy to estimate the rotor flux of an induction machine for the direct field oriented control. Electrical model of the induction machine presents the basic idea based on observer structure, which is composed of voltage model and current model. But the former has the defects in low speed range, the latter has the defects of sensitivity to machine parameters. In spite of these shortcomings, the closed loop flux observer based on two models has been a prevalent estimation method for the direct field oriented control. In this paper, generalized analysis method named ‘observer characteristic function method” is proposed to analyze the kinds of the linear flux observers in unified form. With the observer characteristic function, the estimated rotor flux error involved in the classical methods can be easily clarified. Moreover, the novel rotor flux observer based on this analysis is also presented and the effectiveness of the observer has been verified by the simulation and experimental results.

11. FLUX OBSERVERS The analysis of observers for an induction machines can be simplified considerably by the use of complex vector notation. The complex quantities used in the paper are written in the form, f ” = fi + jy; and the superscripts “ A ” denotes estimated quantities.

Index tenns: Induction Machines, Field Oriented Control, Flux Observer, Observer Characteristic Function

A. Voltage Model Flux Observer [2,4/ The voltage model utilizes the stator voltages and currents, but not the rotor speed. It is commonly used to implement direct field orientation without measured motor speed feedback for low cost drive application. From the stator voltage equation, the stator flux can be estimated by the integration of the following equation,

I. INTRODUCTION The methods based upon stator voltage and rotor current models are the traditional methods to estimate the rotor flux of an induction machine. And their benefits and drawbacks are well known. At high speed, the voltage model provides an accurate stator flux estimate because the machine back EMF dominates the measured terminal voltage. However at low speeds, voltage drop across stator resistance in stator voltage equation becomes significant, causing the accuracy of the flux estimate to be sensitive to the estimated stator resistance. Due to that sensitivity and to inherent signal integration problems at low excitation fiequency, direct field oriented control systems based solely upon the voltage model are generally not capable of achieving high dynamic performance at low and zero speed. Current model can make up the weak points of voltage model[ 11. But the current model would be expected to have the sensitivity to the rotor time constant and magnetizing inductance. The voltage model and current model are useful at nominal and low speed, respectively. Many closed loop flux observers utilizing the desirable low speed attributes of the 0-7803-6618-2/01/$10.00 0 2001 IEEE

ph: = v: -

fiom which the rotor flux is then obtained by

B. Current Model Flux Observer[1,2] From induction motor electrical dynamics, the governing equation of the open-loop rotor flux observer is (3).

(3)

615

Current model can also be implemented a linear system in the rotor reference h e .

particular reduced order flux observer is written by following form, (7). The state equation (7) can be expressed as (8) and (9), using the complex vectors.

C. Improved Gopinath model by Jamen[;! 31 (7)

The closed-loop flux observer is shown in Fig.1 and is seen to provide an automatic transition between the two most desirable open-loop flux observer models: the current model at low fiequency to the voltage model at high fiequency. The transition is now determined via the bandwidth of the flux loop.

D. Full OrdeT Flm Observer [5] From the induction motor electrical model, the state space equations of full order observer, which estimates both the stator current and the rotor flux has the form as (4)[5]. Equation (4) can be expressed as ( 5 ) and (6), using the complex vectors. rr

-I

rr

where the A and B are the matrices containing the information of plant, G is the gain matrix and gr, + jg,, is complex gain vector[7]. 111. OBSERVER CHARACTERISTIC FUNTION

-I

(4)

In this paper, a general form of all the linear flux observers is proposed by following equation, the combination form of voltage model and current model, where F ( s ) is referred as ''Observer Characteristic Function."

(6)

Atcm : Estimated rotor flux fiom current model

where the A and B are the matrices containing the information of piant., G is the gain matrix and g , + j g , ,g , + j g , are complex gain vectors[5].

I;.;vm : Estimated rotor flux fiom voltage model F(s)

: Observer Characteristic Function

E. Reduced orderflux observer [7] The observer characteristic functions of all linear flux observers can be summarized as Table 1 with some manipulation where c,are the complex numbers to variant with the rotor speed w, .

In full order observer, all the stator currents and the rotor flux are estimated. But, since the stator currents are already measured, only the rotor flux need to be estimated. Therefore many researchers have studied reduced order observers. The

Fig. 1. Closed-Loop "Improved Gopinath model'' flux observer 616

tl

I Observers

I

I I

Improved Gopinath Model

3

where F ( s ) =

Table 1. Observer CharacteristicFunctions

+

f Observer Characteristic Functions

I

+

sz K p s K,

*

At steady state, the fiequency response function (FRF) of the observer characteristic function is as follows and depicted in Fig. 3.

S’

s1 + Kps+K,

Full Order Observer Reduced Order Observer

c,s

axis

s+c,

U,

K~ = K , L , I L ~ , K=, K J ~ I ~ ~ C,

.

4

: complex numbers to variant with rotor angular speed 0, See Appendix

I

.

ale = w

b.

real axis

In the low speed range, all the linear observers using both current model and voltage model assume that the real rotor flux exists in the neighborhood of estimated rotor flux fiom the current model because F ( s ) is nearly zero in the low speed range. Similarly, in the high-speed range, all the linear observers estimate the rotor flux in the neighborhood of estimated rotor flux fiom voltage model because F ( s ) is nearly unity in the high speed range. To estimate the rotor flux in the mid-speed range, careful attention should be taken. The best choice is to assume that the rotor flux exists on the line connecting the estimated flux from current model to the estimated flux ii-om voltage model as shown in Fig. 2. imaginary axis

’.

=o

So, the estimated rotor flux trajectory according to the synchronous angular &equency is determined by (12) and shown in Fig. 4.

rotor flux

I

real axis

real axis

Fig. 4. The trajectory of estimated rotor flux in case of “Improved Gopinath model”

Fig. 2. Optimal estimation trajectory

IV. ANALYSIS OF FLUX OBSERVER USING OBSERVER CHARACTERISTIC FUNCTION

In Fig. 4, it is easily seen that the estimated rotor flux trajectory is far fiom the optimal estimation trajectory shown in Fig. 2 specially in the mid-speed range. So, the performance of the “Improved Gopinath model” flux observer is much deteriorated in case of parameter error and measurement error especially in the mid-fiequency region. Similarly, the other linear flux observers can be analyzed and also have the similar problems with “Improved Gopinath model”.

In case of “Improved Gopinath model” proposed by Jansen[2], estimated rotor flux can be expressed as (1 1).

617

V. PROPOSED FLUX OBSERVER The block diagram of proposed flux observer is shown in Fig.7, which adds angle compensation block to the “Improved Gopinath model”. Novel observer characteristic function is given as (13). F(s)=

“2 J

+

+

s2 K P s K ,

In equation (13), a is the angle of observer characteristic functions of “Improved Gopinath model”. The frequency response function of the observer characteristic function is as follows and depicted in Fig. 5. Note that theF(jme)lies only on the real axis.

I

Fig. 6. The trajectory of emhated rotor flux in case of proposed flux observer

current exists in the PWM inverter system, the effects can be easily analyzed by the observer characteristic function l i e (15). AA:-- and A A t , are the flux errors due to voltage and current offset in case of voltage model and current model, respectively.

axis me = o

real axis

0) =U3

i‘.

Y

r ’. r -

1

real axis

F(.ia, 1

In case of “ Improved Gopinath model” flux observer, AA: can be analyzed and solved out as (1 6). In (16), it can be seen that only the effect of oBet current influences in case of “ Improved Gopinath model”.

VI. EFFECTS OF OFFSET VOLTAGE AND CURRENT

Under the condition that the offset voltage and the o s e t

i: 0,

..

Current Model

I lAr

I

I

I

Fig. 7. Closed loop novel flux observer 618

I

estimated flux error of the proposed flux observer is nearly same as that of “Improved Gopinath model” flux observer in case of r, and oL, detuning. Considering that the tuning of r, and L, are more difficult than that of r, and OL, , the proposed rotor flux observer is much superior to the “Improved Gopinath model” in the real application.

Flux error of the proposed model due to the o&et voltage and offset current is same as “Improved Gopinath model”. In case of the full order and reduced order flux observers, the flux error can be analyzed and found as (1 7) and (1 8) respectively.

Flux error(Wb):bal flux - estimated flux) .-...-...-.

0.4

I

-

The steady state error due to the ofiet voltage is zero only in “Improved Gopinath model” and proposed model, and the flux error due to the o a e t current exists in all the linear flux observers. The flux error levels of full order and reduced order flux observer can be varied with the selection of the gain matxice. So in these flux observers (full order and reduced order flux observer), the necessity of accurately measured voltages at low and zero speeds is expected to be significant limitation for these observers at the implementation of real system. VII. SIMULATION

c,

-:I

,

I m p r o v e d aopinatfi case Proposed case

.

--.-e---*

. ..

Cument model case

,

0.3

0.2

0.1

0.0 0

20

80

60

40

synchronous speed(radls)

= 1.4r, & 50% load condition.

Fig. 9. Flux errors at

In simulation and experiments, 22kW induction machine is used whose parameters are summarized in Table 2. In the

Flux ermr(Wb):bal flux - estimated fluxl

simulation and experiments, transition fiequency between current model and voltage model is designed to be same, in both case “Improved Gopinath model” and proposed model.

Proposed case

...l..-._ ..l...-._ ..ll.ll...... 0.3

.

Current model case

.

......... .............. I

.

.

Table 2.22kW induction motor parameters

.

.

........Yi .........; .........:..................................................... ,

W,

I

: I

.

,

:

:

.

.

;

j

:the transition fiequency is set to 6Hz 40

20

0

In simulation and experiment, only the result of “Improved Gopinath model” flux observer compared with that of proposed flux observer. That is because similar result will be given with the other flux observers and the other flux observers have the defects of ofhet voltage like the voltage model as explained in section VI. The estimated flux errors (distance between the real flux and the estimated flux) due to the parameter( r, ,L, ,r, and oL, ) uncertainties are shown in Fig. 9-12 where the rated flux(1P.U) is 0.44 Wb. In Fig. 9 and 10, as the synchronous speed increases, estimated flux error is reduced because the estimated flux converged to the estimated flux fiom the voltage model. The estimated flux error of the proposed flux observer is much smaller than that of the “Improved Gopinath model” flux observer in case of r, and L, detuning in Fig. 9 and 10. Note that around the transition fiequency, the flux error of “Improved Gopinath model” is larger than that of current model in Fig. 9 and 10. In Fig. 11 and Fig. 12, the

60

80

Synchronqus speed(rad/s)

Fig.10. Flux errors at Lm= 0.7Lm & 50% load condition. Flux e r “ I ) : b l flux - estimated fluxJ

. ~ . :

-.

7

.

-:-

Improved Gopinath case : ,

ProposedCurrent model case

~-...

.

.

.

..

..

..

..

. . . . . . ....................................................................................... .. .. .. .. .. .. .

I

.

.-..-p..--..-.-...&-~

.

, I

. .

I

0

20

40

. .

I ,

.

60

Synchronous Speed(rad/s)

Fig. 11. Flux errors at t, = 1.3r, & 50% load condition. 619

80

-

Flux Enum):wflux estimated flux/

1

P.U

+

.

l”vedGopinathcase

.

-..-...

Proposed case Current model case

4- , - r -4 ..

........................

. -.- .

.........

..........

20

0

Fig.14 and 15 are obtained under the no load conditions 17 are obtained under the 50% load conditions(, ; = 1.4r, ). In Fig.14-17, The flux errors of the proposed flux observer are much smaller than those of “Improved Gopinath model” flux observer in steady state (200 and 1200 r/min) and even in the transient state (both the accelerating and decelerating ranges).

(im= 0.7Lm),and the Fig.16 and

40

0

80

60

synchrp”ous Speed(=W Fig.12. Flux errors at d, = 1.9015, & 50% load condition.

VIII. EXPERIMENTAL RESULTS

Various experiments have been carried out. The flux error has been measured by assuming that the real flux can be estimated from the well-tuned “Improved Gopinath model”. The Fig.13-17 are experimental results for the purpose of comparison with “Improved Gopinath model” and the proposed model. In Fig.13, all the points are obtained under the conditions that the measurement is performed with the no variation of speed and load torque. Note that the all flux error points of proposed model are smaller than those of “Improved Gopinath model” like the simulation results. In Fig. 14-17, the properties of dynamic and steady state can be confirmed, where the speed control is performed under the conditions that speed reference changes between 20Or/min and 120Or/min.

0

-15Pu (4.66 wb)

1 SCJdiv

Fig.14. Characteristicsof“1mprovedGopinath model”, ( im = 0.7Lm,No load) Detuned case of “Improved Gopinath model”. Speed reference: 200r/mino1200r/min step changes Upper: d q regulated currents. Lower: flux errors.

I

.

‘ I

............ b.

1

I

o

.

.

1.3 Pu

(0.66 wbl

w

-

9o.M

.

E

I

............... *

0

I

IW d b

I

Fig.15. Characteristicsof Proposed model, ( in= 0.7Lm,Noload) Detuned case of proposed model. Speed reference: 2OOr/minol2OOr/minstep changes. Upper: d q regulated currents. Lower: flux errors.

Fig.13. Flux error at steady state Upper: Lm = 0.7LI & no load case, Lower: ?, = 1.4r, & 50 % load. Triangular data: detuned case of “Improved Gopinath model”. Circular data: detuned case of proposed model. 620

IX.CONCLUSION

1 . 1 1 1 1 1 1 1

In this paper, generalized analysis method named “observer characteristic function method” has been proposed to analyze all kinds of linear flux observers in an unified form. With the observer characteristic function, the estimated rotor flux error involved in the classical methods can be easily clarified. Moreover, the novel rotor flux observer based on this analysis is also presented and the effectiveness of the proposed observer has been verified by simulation and experimental results. REFERENCES [l] Isao Takahashi and Toshihiko Noguchi, “A New Quick-Response and High-Eficiency Control Strategy of an Induction Motor ” IEEE Trans. Ind. Appl., vol. IA-22, NO.5, Sep/Oct. 1986.

(-0.m)

[2] Patrick L. Jansen and Robert D. Lorenz, “A Physically Insightti1 Approach to the Design and Accuracy Assessment of Flux Observers for Field Oriented Induction Machine Drives,” IEEE Trans. Znd. Appl., vol. 30, no. 1, pp. 101-110, Jan./Feb., 1994. [3] Patrick L. Jansen, Robert D. Lorenz and Donald W.Novotny, “ObserverBased Direct Field Orientation : Analysis and Compensation of Alternative Methods,” IEEE Trans. I d . AppL, vol. 30, no. 4, pp. 945-953, Jul./Aug., 1994. [4] Tsugtoshi Ohtani, Noriyuki Takada and Koji Tanaka, “Vector Control of Induction Motor without Shaft Encoder,” IEEE Trans. Ind. Appl., vol. 28, no. 1, pp. 157-164, Jan./Feb. 1992. [5] Hisao Kubota, Kouki Matsuse and Takayoshi Nakano, “DSP-Based Speed Adaptive Flux Observer of Induction Motor,” IEEE Trans. Ind. Appl., vol. 29, no. 2, pp. 344-348, MarJApr., 1993. [6] Hisao Kubota, Kouki Matsuse and Takayoshi Nakano, ’Wew Adaptive Flux Observer of Induction Motor for Wide Speed Range Motor Drives,” IEEE, IECON90, pp. 921-926. [7l Hirokazu Tajima and Yoichi Hori, “Speed Sensorless Field-Orientation Control of the Induction Machines,” IEEE Trans. Ind. AppZ., vol. 29, no. 1, pp. 175-180, Jan./Feb. 1993. [8] George C. Verghese and Seth R. Sanders, “Observers for F l u Estimation in Induction Machines,” IEEE Trans. Ind. elect, vol. 35, no. 1, pp. 85-94, Feb., 1988.

lsaydiv

Fig.16. Characteristics of “Improved Gopinath model”, ( ?, = 1.45 50% load) Detuned case of‘hproved Gopinath model”. Speed reference: 200r/mina1200r/min step changes Upper: d q regulated currents. Lower: flux errors.

APPENDIX 1S d &

Fig.17. Characteristics of Proposed model, ( F, = 1.4r, 50% load) Detuned case of proposed model. Speed reference 200r/mino1200r/min step change. Upper: d q regulated currents. Lower: flux errors

Gain matrices are selected with reference to [5] and [7], respectively. F(s) of ‘‘Full Order Observer[5J”

In case of the “Improved Gopinath model”, the speed control is nearly failed in case of Fig.16. Contrary to the “Improved Gopinath model” in Fig.16, the flux error of the proposed model in Fig. 17 is nearly zero, except with the decelerating ranges. From the experiments, the validity of proposed flux observer is verified.

62 1