\

Dynamic performance limitations for MRAS based sensorless induction motor drives. Part 1 : Stability analysis for the closed loop drive

R. Blasco-Gimenez G.M. Asher M.Sumner K.J.Bradley

Indexing terms: Sensorless induction motor drives, Dynamic performance limitation

Abstract: The paper deals with a speed sensorless direct rotor flux orientated vector controlled induction motor drive in which the speed and rotor flux angle are derived from a closed loop MRAS estimator. A small signal stability analysis is carried out on the combined estimator- controller, and allows for the stability effects of incorrect estimator parameters to be studied. The small signal stability analysis is sufficient to show that incorrect parameters give rise to oscillatory drive behaviour that worsens, even to the point of instability, as the closed loop speed bandwidth of the drive is increased. Experimental results are given to validate the analysis. It is concluded that speed and load torque rejection bandwidths comparable to sensored vector controllers cannot be obtained without online parameter tuning. A companion paper addresses this problem and experimentally verifies the transient oscillatory phenomena.

List of symbols

w,>we = rotor speed (electrical), excitation fre-

w,, wad = closed loop natural frequencies of speed,

L,, L,, Lo = rotor, stator, mutual inductance refered to

R,, R, = rotor, stator resistance referred to stator

T, = L,/R,

J , B

A = denotes estimated value

0 = motor leakage coefficient

quency (rad/s or Hz)

adaptive MRAS loop (radls or Hz)

stator (H)

(a>

= drive inertia (kgm2) and viscous friction (Nm s)

~

0 IEE, 1996 ZEE Proceedings online no 19960104 Paper first received 23rd May 1995 and EI revlsed form 11th October 1995 The authors are with the University of Nottingham, Department of Elec- tncal& Electromc Engmeenng, University Park, Nottmgham NG7 2RD, UK

b

iS

,. _h, , - R h

= rotor flux vector in synchronous co- ordinates (Wb)

= stator current vector in synchronous co-ordinates (A)

= estimated rotor flux vector from cur- rent model, voltage model in synchro- nous co-ordinates

= suffices denoting quantities in stator fixed co-ordinates

(a, P)

do, q0, dq0, 70 = suffices denoting quiescent value P = number of poles

1 Introduction

In recent years a number of manufacturers have intro- duced to the market the sensorless vector induction motor drive which does not require a speed sensor affixed to the shaft. The performance of these sensor- less drives is generally compared with the open loop V-f PWM drive over which the sensorless vector exhib- its improved flux holding both at low speeds and under disturbance torque rejection. In the research commu- nity however, recent years have also seen the develop- ment of effective speed and flux estimators [I-71, which allow good rotor-flux orientated (RFO) performance at all speeds except those close to zero. The independent control of torque and flux implied by RFO itself implies that a good speed and load torque rejection bandwidth should be attainable and that, in these respects, sensored vector drives rather than open loop PWM drives should provide the benchmark for sensor- less vector performance. However such a comparison (or results which allow a comparison to be deduced) has not hitherto been reported. This paper, together with a companion paper, provides such a comparison. The first paper analyses the small signal stability of a speed estimator embedded in a closed loop vector con- trolled drive and illustrates that the speed bandwidth of a sensorless drive is limited by stability considerations arising from incorrect estimator parameters. These lim- itations are verified experimentally. The results also show how the dynamic performance of sensored and sensorless drives may be compared. The companion paper will describe an experimental sensorless drive having on-line parameter tuning and thus an enhanced dynamic performance. The extent to which a practical sensorless drive can achieve the performance of a sen- sored one will be discussed.

IEE Proc.-Electr Power Appl., Vol. 143, No. 2, March 1996 113

Broadly, speed estimators fall into two categories. The first are observer based and are normally designed using the extended Kalman filters (EKF) technique [8, 91. Observers are based on a linearised state-space motor model whose states include rotor flux and rotor speed. The states are estimated through forcing an error vector (nominally the difference between the model stator currents and the measured stator currents) to zero with prescribed dynamics. Like all model based approaches, the EKF speed estimator accuracy is sensi- tive to incorrect parameters [8, 101. However their effect on stability and closed loop sensorless perform- ance has not been assessed. The main shortcoming of the EKF, however, is that design goals do not relate readily to the design (weighting) parameters so that the design procedure tends to be iterative.

The second type of estimator is based on the MRAS principle, in which an error vector is formed from the outputs of two models both dependent on different motor parameters. The error vector is driven to zero through adjustment of a parameter that influences one model and not another. A comparative study between the observer and MRAS approaches for speed and flux estimation has yet to be made. However, the MRAS approach has the immediate advantage in that the models are simple, very easy to implement and have direct physical interpretation. There is a choice of error vectors which may, or may not, give a wider flexibility in achieving design goals. The most common choice of error vector is that of rotor flux [l, 2, 3, 51 which also has the advantage of producing rotor flux angle esti- mate that could be used for the vector controller. At very low speeds, however, all flux calculators are very sensitive to the effects of stator resistance and integra- tor drift and, to overcome these problems, back EMF and reactive power have both been suggested as error vectors [I 11. However these quantities themselves disap- pear at low speed and also give rise to highly nonlinear gains in the adaptive MRAS controllers. The authors have achieved poor results with these alternative MRAS schemes [la, 131 and have thus restricted them- selves to rotor-flux based MRAS structures.

.* EJ

‘‘.ZL!EqJ observer IM

Fig. 1 General DRFO Structure

The general structure of a DRFO vector controller with speed and flux estimation is shown in Fig. 1. In common with sensored drives, a speed filter is included in the estimated speed feedback loop to take into account the possibility of noise or ripple.

114

2 Rotor-flux based MRAS structure

The use of the rotor flux MRAS for estimating the rotor speed was principally developed by Schauder [2]. The principle is based on the fact that the rotor flux (or whatever quantity is being used for the error vector) can be obtained from the dynamic equations of either the stator circuit or the rotor circuit. The former, known as the ‘voltage model’ solves the rotor flux from

111, . = - “7 U, - R,i, - aL,i,) (1) LO whilst the latter, known as the adjustable ‘current model’, solves for the same quantity,

e’@ rh r? mi!:

I w +is

Fig. 2 MRAC-CLFO speed and flux observer including mechanical model

Both the equations are in the 2-axis stationary stator frame. Only eqn. 2 is dependent on the rotor speed which can thus be adjusted to force the instantaneous phase between the two flux estimates (i.e. the vector cross product) to zero. The controller (loop) which adjusts this estimated speed is termed the adaptive MRAS controller (loop). In eqn. 1, the rotor flux is obtained by integration so that, in practice, DC offsets must be removed by employing a DC-blocking filter so that the error vector becomes an AC-coupled rotor flux signal. This itself causes the dynamics of the adaptive MRAS loop to become speed dependent, a problem which was solved in [3]. At low speeds, however, misi- dentification of Rs and the distorting effects of the DC blocking filters cause the rotor flux estimate to become inaccurate and both the speed estimate and the field orientation breaks down below 1 or 2Hz. An improve- ment in both the rotor flux and speed estimate at very low speed may be obtained by using a closed loop con- troller between the flux estimates of the two models 131. This structure, depicted in Fig. 2, is termed a closed loop flux observer MRAS (MRAS-CLFO) and con- tains a coupling control loop between the two flux esti- mates. This is termed the flux-coupling loop and its natural frequency ocpl is designed to be between 0.5Hz and 1Hz. The output flux estimate thus follows the voltage model at frequencies above ocpl and the current model at frequencies below it. Since the current model output has no DC level, the flux-coupling PI controller ensures zero DC level at the output. However, as the frequency approaches zero, the cross product error E also approaches zero and speed estimate forcing is lost. A mechanical model can compensate for this effect in that flux and speed estimates are produced even when E is zero. The structure of the mechanical model is shown in Fig. 2 in which an estimated torque signal is used to

IEE Proc -Electr Power Appl , Val 143, No 2, March 1996

drive a first order drive train model. This model is also driven by the E signal through a PI controller which will help compensate model errors. The feedforward term K3 weights the effect of the pure MRAS-CLFO and the mechanical model upon Q,. If the mechanical parameters are not accurately known, the compensa- tion will merely be less effective but still an improve- ment over the case when no mechanical model is used at all. In practice, it is found that field orientation and the speed estimate start to deteriorate for excitation fre- quencies below qLm, a frequency slightly higher than ocpl (e.g. if mcpl is designed at 0.8Hz, olLm is about 1.5Hz). The degree of deterioration depends on the accuracy of the estimated voltage model and mechani- cal parameters. Through simulation and experiment, the authors have substantiated the good performance of the MRAS-CLFO estimator [12, 131 and as such this structure, considered the best MRAS flux and speed estimator available, will form the basis for which an enhanced parameter-tuned system can be developed. In particular, the small signal stability analysis will be done with the MRAS-CLFO structure. The justifica- tion for not doing a detailed analysis on other MRAS structures is discussed later. The global stability of the adaptive MRAS loop based on the structure due to Schauder has been reported [l , 21. A similar proof for the MRAS-CLFO structure can proceed on similar lines, but has not been included here since the authors have not been able to take the effect of inaccurate model parameters into account. A small signal stability analysis, however, yields numerical stability margins for specific model parameter errors and is thus more useful from an engineering view point. The small signal stability analysis in Section 4 will be subject to case study using motor and estimator parameters listed in Appendix 10.1. These parameters are those of a 4kW experimental rig.

10 '"i C I I

L. O t ........................................................................................................................................ 0

-I5 t 1 -201 I

0 500 1000 1500 actual speed, revlrnin

Fig. 3 Estimated speed error for inaccurate T,

The effect of incorrect estimator parameters on the accuracy of the speed estimator is shown in Figs. 3-6 and agrees with previous results [3]. The conclusion there, is that, after errors in T,., errors in oL, have most effect on steady state accuracy. The effect of R, and Lo is only significant at low speeds. However, the effect of the adaptive MRAS loop bandwidth on this sensitivity has not been considered in any depth. Although the effect of incorrect parameters on speed accuracy is important, the effect on drive stability is even more so and will be the focus of the paper. Since the stability

IEE Proc.-Electr. Power Appl., Vol. 143, No. 2, March 1996

and speed bandwidth of the closed loop drive is, to some degree, affected by the design of the adaptive speed loop within the estimator, the control design of this adaptive loop will be considered below.

l 5 t

- 1 5 1

-20 1 0 500 1000 1500

actual speed, rev/rnin Fig. 4 Estimated speed error for inaccurate oL,

10 l5 t

-1 51

-201 I 0 500 1000 1500

actual speed, revlrnin Fig. 5 Estimated speed error for inaccurate Lo

t

- 1 5 1 -20 0 500 1000 1500

actual speed, revlmin Fig. 6 Estimated speed error for inaccurate R,

3

The guidelines for designing the parameters of the MRAS-CLFO are included here since the controller equations are used later in the stability analysis. Also an explicit treatment has not hitherto appeared in liter- ature.

Design of adaptive control parameters

115

For excitation frequencies above ocpl the voltage model gives an accurate estimate of the rotor flux vec- tor and hence can be considered as the reference model for the MRAS. The control structure is shown in Fig. 7. It is shown in Appendix 10.2 that the adaptive controller in Fig. 7 is equivalent to a PID controller, resulting in the equivalent control loop of Fig. 8. To derive the PID parameters, a linearised transfer func- tion between T(s) and E($) is obtained. Since all volt- ages, currents and fluxes of Fig. 7 are 2-axis sinusoidal quantities, the linearisation is facilitated by transform- ing the defining equations of the estimator into a syn- chronously rotating reference frame. Appendix 10.2 derives the plant transfer function as

and the dynamics varies only with the motor slip. The PID controllcr dcmands thc placcmcnt of two zeros. One possibility is to cancel the slip dependent poles which will make the control independent of the operat- ing point albeit with a slip dependent controller. Alter- natively, one of the zeros can be used to cancel the mechanical pole. This approach is the one used hence- forth and is depicted in the root loci of Fig. 9a and Fig. 9b. At very low gains a slow real pole will domi- nate. As the gain is increased the dominant real pole approaches the zero at UT, and since this zero is also in the closed loop transfer functions for col (see eqn. 36 in Appendix 10.2), the effect of this slow real pole will become negligible. Therefore, the dominant poles can be freely placed in order to obtain a given bandwidth. One criteria for the placement of these poles is to obtain two real poles in such a way that one of them lies, at the high gain, very close to the second PID zero. The closed loop natural frequency is then determined by the second fast pole. For the parameters of Appen- dix 10.1, a typical ‘fast adaptive loop’ can be designed with the PID parameters of Appendix 10.2 giving a natural frequency of 125radis (or 20Hz) for the case of full load. Fig. 6b shows the no-load case of w,/ = 0 and in which the slip-dependent poles lie on the 1iC zero. The position of the ‘fast’ closed loop poles are almost identical to the full load case and the design is practi- cally load independent.

w\ Fig. 7 Adaptive controller and mechanical compensation

current mqdel,

Fig. 8 Equivalent adaptive controller

To allow for the fact that the ‘fast’ design may be noisy or destabilising, a slower design can of course be made. In this case both the residual from the slow real pole and the slip dependence of the closed loop poles

116

are larger for the slow design; both effects are small [12],

ilz practice, however,

a Fig. 9 Root locifor adaptive loop

b

4 State equations and linearised model

To study the effects that incorrect estimator parameters have on the stability of the MRAS-CLFO, a state space analysis is carried out using a small signal line- arised model. The system pole-zero positions will thus vary with speed and load. However, it will be shown that for excitation frequencies greater than wllm, the pole-zero positions move in a predictable and experi- mentally verifiable manner. For frequencies below all,, the movement is large (even for very small changes in the quiescent frequency). In practice, the usefulness of the linearisation is restricted to excitations frequencies greater than all,.

For the following analysis it has been assumed that the machine is current fed (due to the large bandwidth of the current controllers), and that field orientation is always kept. This is valid since with DRFO, the flux angle calculation from the flux estimates is largely inde- pendent of poor speed estimates.

4. I Machine dynamics The rotor dynamics of the machine can be expressed as

we - w,) A, + -is However, if the flux in the machine is constant under field orientation (and therefore isd is constant as well), this equation has no dynamics and can be used to obtain a relationship between the quiescent values hyo and isdo.

(4) 1 LO

& = - [ - + l ( Tr 1 Tr

The mechanical dynamics for the machine is

(5) B J

KT(X, x i s ) -- Trn J J

w, = --wr + where K, is the torque constant and T, the load torque.

Fig. 10 Voltage model equivalent circuit

4.2 Estimator dynamics The equation for the current model is

IEE Proc -Electr Power Appl , Vol 143, No 2, March 1996

To derive the equations for the voltage model, the block diagram in Fig. 10 is used. To reflect errors in voltage model parameters, the following variables are defined:

AR, = R, - R,

(7 ) AaL, = & - c7Ls

A’ z ~ -r ioLr Ar

X s ( a , p ) = K2xe(a,p) + Klxe(a,p) - A R s i s ( a , p )

Xe(a,p) = l $ a , p j --r(a,p) A”

G L o

Hence the equations for the voltage model in stator fixed co-ordinates are:

(8)

(9)

Therefore, the corresponding equations in synchronous co-ordinates are;

ir Lr ke = -jwexe + ,iz - - x, + yAoLsis (11)

The equations for the adaptive loop and mechanical model may be obtained from Fig. 7. Two states x2 and x3 are defined and have state equations:

Lo Lo

2.2 = ( ip x 2:) (12)

23 = Kt (Ay x is) + K5x2 + K4i2 - B& (13)

4.3 Combined equations The resulting state equations in synchronous co-ordi- nates are linearised to give the standard form:

61i: = A6x + B6u Sy = C6x + D6u (14)

where the state, input and output vectors are defined as:

A 6x = (6wr

6u = sasq 6y = 6th.

s i $ bX,d 6 X s q 6 X , d bXeq 6x2 bx3)T

(15)

(16)

(17)

a

n

In general, the values of the different matrices depend on the operating point and on the different parameter errors (Appendix 10.3).

It is seen that Sy(s)/Su(s) can form the transfer func- tion of the estimator dynamics and thus the open loop transfer function for designing the speed controller. However, to use this approach for speed controller design is only possible when the estimator and motor parameters are equal. When they are not, 6y(s)/6u(s) yields root loci or Bode gain-phase plots which are too complex to be useful. The approach taken therefore, is to design the speed controller on the basis of perfect estimator parameters (see Section 5) and to study the resulting closed loop system. The above matrix equa-

IEE Proc -Electr Power Appl, Vol 143, No 2, March 1996

tions are thus augmented by a speed controller which for simplicity is assumed to be a conventional PI Con- troller:

Hence the resulting (still open loop) system can be writ- ten: Sz’ = ( 6 x 6x1 ) T ; Sy’ = 6y; 6th’ = bUl (19)

C ’ = ( C 0 ) ; D’ = 0 Finally, the estimated speed feedback signal can include a first-order filter expressed as

(21) 2 4 = -ax4 + au4

Y4 = 2 4

where x4 is the filtered value of 6 j r and u4 the unfil- tered. Incorporating eqn. 23 and closing the loop gives: 62” = (Sx’ 6x4 ) T ; 6y” = by’; 6th’’ = 6u’ t by4 (22)

C’’=(C 0 ) ; D“ 0 From eqn. 23 the pole-zero positions of the closed loop transfer function 6y” (s)/6u‘(s) = 6 6 j r / 6 0 , * can be derived. In all studies that follow, the pole-zero posi- tions are plotted with oro as a parameter. The principle pole-zero movement derives from the o,-dependent pole-zero pairs of the estimator deriving, in turn, from the implicit transformation into synchronous co-ordi- nates. These estimator pole-zero pairs can be shown to be cancelling if the estimation parameters are correct. If not, the pole and zero of each pair will diverge caus- ing the loci of the closed loop roots (for variable loop gain and at a given value of ~ 0 , ~ ) to branch between them. It is not possible to infer from such plots what the shape of these branches will be. This is why all studies will show the loci of the closed loop poles and zeros for varying or0 and fixed controller gain. It is also noted that since the pole-zero pairs derive from the estimator, they will be present in any closed loop transfer function that includes it. Thus there is no loss of generality in using the reference speed demand as the closed loop input.

-

100 :::! -

ol.;T---, \! -200 -100 -10 0 10 100 200

Fig. 1 1 Pole-zero loci for perfect estimator parameters

The pole-zero loci for the case when the estimator parameters are perfect are shown in Fig. 11. Only the closed loop poles and zeros deriving from the speed controller design remain. The loci are plotted for ora varying from 0 to 50Hz (electrical). However, oro is not

117

shown on the loci, since the value is very close to the frequency of the oscillatory poles and thus can be approximated from the imaginary axis of the plot (in rad/s). For perfect estimator parameters, the transfer function of the MRAS-CLFO reduces to

(24) Gr - K t X r d O -~ - is, s J f B

and the speed controller can be designed assuming a first-order mechanical pole.

It may be noted that the moment of inertia of the experimental drive is 0.3kgm2 which is large for the power rating. This arises from the motor being loaded by a DC machine dynamometer whose inertia is nearly 90% of the total. Since experimental results are used to verify the analysis, the large inertia is used in the simu- lations also. Fortunately, the effect of inertia on the results is simple to predict (and is discussed in Section 6), so that the use of a high inertia is of no particular import.

5

In Sections 5.1 to 5.4, the closed loop pole-zero posi- tions are plotted for variations in wTO. The results are for the adaptive MRAS loop design with mad = 20Hz as described in Section 3, and for a speed loop natural frequency of about 4radls. No filter is included in the speed estimate feedback path.

Effect of incorrect estimator parameters

-7

0 1.; ---,-$t--~ 1 !

! -200 -100 -10 0 IO 100 200

Fig. 12 Pole-zero locifor varying speed and inaccurate io Lo = 0.9 Lo

300

200

100

-200 -100 -10 0 10 100 200 Fig. 13 Pole-zero loci for varying speed and inaccurate 2, L , = 1.1 Lo

-

-

-

5.7 Figs. 12 and 13 show the closed loop pole-zero posi- tions when the estimated value of Lo varies by -10% and +lo% from the true value, respectively. The non- cancellation of the (nearly) we-dependent poles is immediately evident. For the case when Lo is underesti- mated (Fig. 12), the separation is not severe and the residuals of the closed loop poles are small. Neverthe- less, since the poles cause very lightly damped oscilla- tions at near CO,, corresponding transient oscillations may appear in the response. The overestimated case is more serious in that one of the zeros has significantly

Variations in the magnetising inductance, Lo

118

diverged. The transient oscillations caused by its pole- pair will therefore be increased. The close proximity of the closed loop poles to the imaginary axis is also cause for concern.

2 00

100

"LAd 100 200 -200 -100 -10 0 10

Fig. 14 Pole-zero loci for varying speed and inaccurate R , R , = 0.9 R,

, 100 200 -200 -100 -10 0 10

Fig. 15 Pole-zero locifor varying speed and inaccurate R, R , = 1.1 R,

5.2 Variations in the rotor resistance R, Figs. 14 and 15 show the loci when the estimate of R, is -10% and +lo% from the true value. Of interest is the right-hand half zero in Fig. 15 (which also occurred in Fig. 13) which implies that the estimated speed will ini- tially go in an opposite direction to that demanded by a step input in speed demand. This is a very common observation in experimental closed loop sensorless drives and will be seen in the results of the companion paper.

100 200 -200 -100 -10 0 10 F[g. 16 Pole-zero locifor varying speed and inaccurate oi, O L s = 0.9 CL$

100 200 LA,"' -200 -100 -10 0 10

Ftg. 17 Pole-zero loci for varying speed and inaccurate 02, O L , = 1.1 CLs

IEE Proc.-Electr. Power Appl., Vol. 143, No. 2, March 1996

5.3 Variations in the motor leakage, oL, A variation of -10% and +lo% in this parameter causes the loci of Figs. 16 and 17, respectively. For both under and overestimates there is a large residual from the one of the closed loop poles for speeds above approximately 15Hz, so that oscillations in the system behaviour are easily induced. Further, the residual and hence the size of transient oscillation will increase for an increase in speed.

' - ' ' x A ' ' ' 7 300 -

200 .

A -200 -100 -10 0 10 100 200

Fig. 18 Pole-zero loci for varying speed and inaccurate R , R, = 0.9 R,

300r-- - ' ' '

2oo t 3 0 L _

I

, ,..,- , , , 200 100 -10 0 10 100 200

Fig. 19 Pole-zero loci for varying speed and inaccurate R, R, = 1.1 R,

300 I 1

.F 200 E

2

:: 100

. >

U a,

0 1 2 3 r , 5 6 1 8 9 time, s

Fig. 20 Instability in real and estimated speeds when R , = 1.1 R,

5.4 Variations in the stator resistance, R, The 10% variations are shown in Figs. 18 and 19. Of all the parameters, R, is found to have the most influ- ence on system stability. For the underestimated case, the poles are always seen to shift to the left; however the pole-residuals are large causing substantial, albeit stable oscillations. For the overestimated case, the poles travel in the opposite direction and in this case instability occurs for all but very low frequencies. This result is experimentally verified in Fig. 20 shows the estimated and real speed when the value of R, is changed from 6.1 to 6.72Q (i.e. a 10% increase). As a consequence, the real and estimated speeds begin to

IEE Proc.-Electr. Power Appl., Vol. 143, No. 2, March 1996

oscillate, reaching a limit cycle when the speed control- ler saturates. The presence of this limit cycle will cause the average speed of the motor to drop. When the esti- mated R, is restored to its original value, the oscilla- tions decrease and eventually disappear. At the same time, the machine returns to its original speed.

2001 !\I

-200 -100 -10 0 10 100 200 Fig. 21 Pole-zero loci for dgerent mad with R, = 1.1 R, wad = lOHz

' -- I I

300 -

200 -

Fig. 22 Pole-zero loci for dgerent wad with R , = 1.1 R, mad = 20Hz

2001

I , , , , * , ;RI , , , d -200 -100 -10 0 10 100 200

Fig. 23 Pole-zero loci for dqerent mad with R , = 1.1 R, mod = 40Hz

6 Effect of loop bandwidths

The movement of the pole-zero pairs with incorrect parameters is itself dependent on the natural frequen- cies of the adaptive loop, wad, and the main speed loop, a,. Since the movements due to overestimation of R, are always the most serious, results are shown for the condition that R, = 1.1 R,. Figs. 21-23 show the effect with wad as 10Hz, 20Hz and 40Hz, respectively. It can be surmised that, with o, = 4rad/s, the maximum wad that can be attained before instability is about 15Hz. With varying wad a similar qualitative pole-zero move- ment with variations in the other parameters are also obtained. Since the movements are of less consequence than that due to R,, they are not given here.

For investigating the variation with w,, the damping factor was set at 0.8 and mad at 20Hz. The results for w, of 2, 4 and 8radIs are shown in Figs. 24-26. Increas- ing instability is seen which is not surprising since the speed controller can be viewed as a feedback gain

119

between the output and input of the estimator. For a PI controller controlling a purely inertial plant, the proportional and integral gains can be easily shown to be:

K p = 25w, J K , = KpCw, (25)

3001

2001 i

1 , , , L - - , - - 7 - - , - / d l , , , ' -200 -100 -10 0 10 100 200

Fig. 24 Pole-zero loci for dEfferent on with R , = I 1 R, and ood = 20H: w, = 2radis

zoo1

-200 -100 -10 0 10 100 200 Fig. 25 Pole-zero locifor different on with R , = 1.1 R, ando,, = 20Hz w, = 4radis

- ' , ' I ' " / " ' I

-200 -100 -10 0 -10 100 200 Fig. 26 Pole-zero loci for different a,, with R, = 1.1 R, undo,, = 20Hz a, = Xrad/s

Thus, if purely proportional control is used then, for a given damping factor, the degree of oscillation or insta- bility (for a given error in the estimator parameters) increases with the quantity o,J (and not on). Other statements follow: (i) For a given set of parameter errors, if onJ is held constant the degree of oscillations increases with mad.

(ii) For a given set of parameter errors, w,J and mud have an inverse relationship for a given stability (i.e. if one is increased the other must be decreased). (iii) The less the error in the estimator parameters, the greater o,J can be before a specified degree of system oscillation occurs. For a given parameter error, 5 and mud, the adjustment of o, and J keeping o,J constant does not result in exactly the same pole-zero movement due to the inte- grating term whose constant is not dependent on J . This can be seen in Fig. 27 in which J is reduced by a factor of 10 and on is increased by 10, all other condi-

120

tions being identical to the case of Fig. 22. Compari- son with Fig. 22 shows the difference to be relatively small. However, both sets of oscillatory poles have become marginally less oscillatory. This is a general trend; the integrating term has a secondary but stabilis- ing influence.

1 ' ' ' ' , r ' ' I ' ' " ' ' ' I

100 200 -200 -100 -10 0 10 Fig. 27 Pole-zero loci for J reduced by factor of 10

1 , , , , *,.-;--,--fl , , , -200 -100 -10 0 10 100 200

Fig. 28 Effect ofl5Hzfilter in feedback path

Finally, an increased value of o,J can be attained at higher operating frequencies through filtering the esti- mated speed signal. With the conditions pertaining to that of Fig. 22, Fig. 28 shows the effect of a 1st order 15Hz filter. The poles are bent left at the higher fre- quencies. Care must be taken of course to ensure acceptable operation at frequencies that are within the filter bandwidth.

7 Discussion

The results of Sections 5 and 6 show that, from a prac- tical viewpoint, a MRAS speed estimator with incorrect parameters can be considered as an encoder with inher- ent ripple. The improving effects of the speed estimate filter and the integrating component of the speed con- troller can be viewed in this light. The ripple is, how- ever, closely related to the motor speed and at low speeds will be inside the speed controller bandwidth. It also constrains the magnitude of the design parameter onJ which if too high results in the ripple magnitude growing into instability. In an effort to reduce this rip- ple, the results provide a number of ground rules: (i) The sensorless system is most sensitive to errors in R,. Unless this parameter is identified online, it is important that it is underestimated rather than overes- timated. Fortunately, a 'cold' R, measure obtained dur- ing self-commissioning fortuitously provides this since the motor R, can only increase during operation. Although underestimation may avoid instability, the ripple induced by even small errors in this parameter is likely to be the most significant factor in limiting o,J and thus obtaining comparable performance with sen- sored vector drives.

IEE Proc -Electr Power Appl , Val 143, No 2, March 1996

(ii) The estimated values of Lo, R,. and oL, are of less importance but still cannot be ignored. Underestima- tion is preferred with respect to R,. and a ‘cold’ value is appropriate. With respect to oL,, overestimation is pre- ferred, therefore the value of oL, obtained through self- commissioning [14] should be obtained under no load current since the reduced stator tooth saturation pro- vides for a higher leakage value. A slightly ‘overfluxed’ value of Lo is also appropriate during parameter identi- fication. Flux orientation is relied upon to keep Lo and oL, reasonably constant during motor operation. How- ever, it is noted that the ripple dependence on Lo and oL, may cause problems during field weakening opera- tion. Errors in the R, estimate are well known to be the biggest single factor effecting the accuracy of the speed estimate. Online tuning for Rr [I51 can thus provide both for increased accuracy and the removal of the rip- ple dependence on this parameter. (iii) The adaptive loop natural frequency w,d should be kept as low as possible, whilst still obeying the natural observer condition wfld > o, for good tracking. It is emphasised that o,&induced ripple is dependent on o,J and not on on. The nominal value of ofld used in this paper (which is high with respect to the nominal o, used) envisages on values of up to lOHz which is still feasible for smaller drives. (iv) The use of extra controller zeros, either with PID or with lead-type controllers, should be avoided as these will only amplify the ripple in the system. A very significant result derives from the statement (iii) of Section 6. For a specified ripple, the less the error in the estimated parameters, the greater o,J can be. Thus onJ can be used as a measure of goodness for the sen- sorless drive. The quantity o,J can be termed the closed loop natural angular momentum of the drive. Since it can also be derived for sensored vector drives, the measure provides for a direct quantitative compari- son of sensorless and sensored drives. This will be con- sidered in [15].

Although the stability analysis has been carried out on the MRAS-CLFO estimator, the results have a wider significance. For the rotor flux based MRAS estimators developed by Schauder [2], an almost identi- cal behaviour is observed [12]. It is felt that the oscilla- tory poles arise from an implicit transform of the motor equations into synchronous co-ordinates. If this is true, the system oscillation is likely to affect speed estimators based on observer or EKF techniques. In observer theory, it is known that the poles and zeros of the plant are cancelled by those of the observer model (so that the closed loop system poles are the union of the closed loop observer and controller poles consid- ered separately). However, the cancellation is incom- plete if the observer model parameters are in error. The effect of this for observer based sensorless drives hav- ing large o,J values certainly merits further study.

8 Conclusions

In an effort to obtain closed loop speed bandwidths from sensorless vector controlled induction motor drives that are comparable to sensored drives, it has been found that MRAS based speed estimators yield speed estimates corrupted by transient oscillations at very near the excitation frequency. This paper has ana- lysed the small signal stability of a MRAS-CLF esti- mator embedded in a closed loop drive. It has been

IEE Proc -Elect,. Power Appl, Vol 143, No 2, Murch 1996

shown that the transient oscillations derive from incor- rect estimator parameters. The oscillations, which can lead to unstable operation, are also dependent on the natural frequency of the adaptive MRAS loop and, more importantly, upon the closed loop natural angu- lar momentum of the drive, onJ. This parameter repre- sents a goodness measure of the closed loop sensorless drive and one which can be used for comparisons between sensorless and sensored drives. To increase this measure towards that of a sensored drive, online parameter tuning is necessary. This will be the subject of a companion paper.

9

1

2

3

4

5

6

7

8

9

References

TAMAI, S., SUGIMOTO, H., and YANO, M.: ‘Speed sensor- less vector control of induction motor with model reference adap- tive system’. Proceedings of the IEEE-IAS annual meeting, 1987, pp. 189-195 SCHAUDER, C.: ‘Adaptive speed identification for vector con- trol of induction motors without rotational transducers’. Proceed- ings of the IEEE-IAS annual meeting, 1989, pp. 493499 JANSEN, P.L., and LORENZ, R.D.: ‘Accuracy limitations of velocity and flux estimation in direct field orientated induction machines’. Proceedings of the EPE Conference, 1993, (Brighton), pp. 312-318 OHTANI, T., TAKADA, N., and TANAKA, K.: ‘Vector con- trol of induction motor without shaft encoder’, ZEEE Trans., 1992, IA-28, (l), pp. 157-164 TAJIMA, H., and HORI, Y.: ‘Speed sensorless field-orientation control of the induction machine’, ZEEE Trans., 1993, IA-29, (l), pp. 175-180 KUBOTA, H., MATSUSE, K., and NAKANO, T.: ‘DSP-based speed adaptive flux observer of induction motor’, ZEEE Trans.,

YANG, G., and CHIN, T.H.: ‘Adaptive speed identification scheme for a vector controlled speed sensorless inverter-induction motor drive’, ZEEE Trans., 1993, IA-29, (4), pp. 820-825 PTKINSON, D.J., ACARNLEY, P.P., and FINCH, J.W.: Observers for induction motor state and parameter estimation’,

ZEEE Trans., 1991, IA-27, (6), pp. 1119-1127 HENNEBERGER, G., BRUNSBACH, B.J., and KLEPSCH,. T.: Field oriented control of synchronous and asynchronous drives

without mechanical sensors using a Kalman filter’. Proceedings of the EPE’91 conference, 1991, (Firenze), Vol. 3, PP. 664-671

1993, IA-29, (2), pp. 344348

10 PENA, R.S., and ASHER, G.M.: ‘Parameter -sensitivity studies for induction motor parameter identification using extended Kalman filters’. Proceedings of the EPE’93 conference, 1993, (Brighton), pp. 306-31 1

11 PENG, F.Z., and FUKAO, T.: ‘Robust speed identification for speed sensorless vector control of induction motors’. Proceedings IEEE-IAS annual meeting, 1993, Vol. 1, pp. 419426

2 CILIA, J., AND ASHER, G.M.: ‘Modelling of MRAS estima- tors using various error vectors’. Internal report, Department of Electrical Engineering, University of Nottingham, UK

3 STUBENER, D.: ‘Simulation of MRAS speed estimators’. Stu- dent project, 1995, Department of Electrical Engineering, Univer- sity of Nottingham, UK

4 SUMNER, M., and ASHER, G.M.: ‘Autocommissioning for voltage-referenced, voltage-fed, vector controlled induction motor drives’, ZEE Proc. B, 1993, 140, (3), pp. 187-200

5 BLASCO-GIMENEZ, R., .ASHER, G.M.,, SUMNER, M., and BRADLEY, K.J.: ‘Dynamic performance limitations for MRAS based sensorless induction motor drives. Part 2: Online parameter tuning and dynamic performance studies’, ZEE Proc. Electr. Power Appl., 1996, 143, (2), pp. 123-134

10 Appendix

IO. 1 Machine parameters

Rated power: 4kW Rated frequency: 50Hz R, = 5.32Q Lo = 0.6H oL, = 0.071H B = 0.02kgm2s-‘

R, = 3.17Q Lr = 0.633H

J = 0.3kgm2 p = 4

121

10.2 With reference to Fig. 8, we have

h P ( T ' + Q 2 ( J s + B )

w, = -

In the dq synchronously rotating frame, we have

= L d X r q - i r q X r d (27) 0 = Rrirdq + SAT& + jwslAr,dq (28)

and writing eqn. 28 in terms of stator current yields:

b& = 6 & d X r q + 6 X T q i T d - s i , q x r d - i T q b X r d = -birqXr,dO

(31) assuming field orientation and that the voltage model flux is ideal and constant. Linearising eqn. 26,

P 6T' b;, = -

2 ( J s + B) Assuming field orientation and that isq is constant, we can linearise eqn. 30 to yield:

1

and eliminating 6h,d gives

6X,, = - (35) ( ( s + k ) 2 + w i l o )

Substituting eqn. 32 into eqn. 35 and eqn. 35 into eqn. 31, noting that hvdO = and 6ws = -6a,., gives

If the PID controller is written as k(s+x)(s+y)/s where y = BIJ, the controller constants for Fig. 7 are given by:

K3 = k / J K4 = k ( ~ + B / J ) (37) 115 = k B / J

10.3 The deduction of the linearised equations has been car- ried out as follows. Let the MRAS-CLFO be defined as follows:

/ f l ( X , . ) \

B KT(Ar x i s ) -- T, s1 (x, U) = --Ur J + J J

I L, er -&-Fxs+-rAoLsi,

Lo Lo (46)

(47) f S ( Z , u ) = (A: x A y ) fg ( 2 , U) = Kt (Ay x is) + Ksx2 + K4X2 - B G , (48)

1 J

g(x , U ) = 7 2 3 + K3&2 (49)

From these equations, the coefficients of the linearised system matrices eqn. 14 are obtained conventionally:

The matrices A, B and C are evaluated at the operating point (xo, U& obtained by solving

The actual state space matrices depend on the errors on the observer parameters and on the operating point, that can be determined by setting the values of i,,!, arO and hrd. An explicit expression for A, B and C is, in general, too cumbersome, and therefore will not be included here. For this work, the explicit expressions have been obtained by solving the above equations using MAPLE and by subsequent numerical evaluation with MATLAB.

0 = f Z ( X 0 , UO) (51)

122 IEE P~oc.-Electr. Power AppL, Vol. 143, No. 2, March 1996