Estimation of speed and armature temperature in a brushed DC drive using the extende Kalman filter

P. P. Aca rn I ey J. K. AI-Tayie

Indexing terms: DC drives, Extended Kalman filter

Abstract: Closed-loop speed control of a DC drive requires a rotor-speed feedback signal which can be obtained using a speed measuring device. However this device adds to the cost of the drive and the number of electrical connections to the motor. The paper describes a method of speed estimation using armature voltage and current measurements that eliminates the speed measuring device. By including thermal effects in the estimation process the effect of temperature variations on the speed estimte are minimised and an estimate of average armature temperature is produced. An extended Kalman filter observer for real-time estimation of speed and armature temperature is formulated using electrical, mechanical and thermal models of the motor. The observer is implemented using a TMS 320C30 digital signal processor system. A range of experimental results demonstrating the observer’s steady-state and transient performance is presented for the particular case of a 3kW machine with constant field current supplied from a phase-controlled rectifier, although the estimation method is applicable to DC drives of any power rating and configuration.

List of symbols

b = viscous friction constant i, = armature current k = sample index k, = torque constant k, = iron loss constant k, = thermal power transfer coefficient k, = variation of thermal power transfer coefficient

la = armature inductance ts = sampling time v, = armature voltage H = armature thermal capacity

with speed

0 IEE, 1997 IEE Proceedings online no. 19970927 Paper first received 25th April and in revised form 20th September 1996 The authors arc with the Electric Drives & Machines Group, Department of Electrical & Electronic Engineering, University of Newcastle upon Tyne, NE1 7RU, UK

J = total inertia K = Kalman filter gain R, = armature resistance RaO = armature resistance at ambient T = motor torque Tl = load torque U = input vector x = state vector y = output vector v = measurement noise vector w = system noise vector A = continuous-time system matrix B = continuous-time input matrix C = output matrix F = discrete-time system matrix G = discrete-time input matrix 0 = observability matrix P = error covariance matrix Q R * E{ .} = expectation operator a 8 = temperature above ambient w = armature speed

1 Introduction

DC motor speed controllers frequently use feedback from a speed measuring device, such as a tachogenera- tor, but this device adds to the cost of the drive and the number of electrical connections to the motor. A fur- ther limitation of the tachogenerator is that the output signal can be distorted by effects such as erratic contact between the brushes and commutator. A common solu- tion to the noise problem is to introduce some lowpass filtering in the feedback path, but this filtering also restricts the transient performance of the speed control loop. Speed measurement by other means, such as an incremental optical encoder, has its own problems at low speed where the number of encoder pulses in the sampling period becomes very small, so the resolution of speed measurements is poor. Increasing the sampling period affects the dynamic speed response of the drive because of the long delay time inserted in the feedback loop. Attempts to counteract the problem by using an encoder with higher resolution adds to the expense of the drive system.

= process noise covariance matrix = measurement noise covariance matrix = estimated value of a state vector

= temperature coefficient of resistance

13 IEE ProcElectr. Power Appl., Vol. 144, No. 1, January I997

Some effort has been made to calculate the speed from position signal obtained by an inexpensive low- precision shaft encoder [l, 21, but this solution still uses a speed measurement device and extra electrical con- nections to the motor. An alternative approach, which avoids the need for extra connections, is to estimate the motor speed from measurements of voltage and current at the converter output. The simplest estimation method is based on the steady-state voltage equation

V, = I,R, + k , ~ which can be rearranged to give the rotor speed in terms of armature voltage and current

(1) v, - I,&

ke However, the speed estimate w is based on the steady- state values of armature current I, and armature volt- age V,, so in a switching converter application some prefiltering of these signals is required. The cut-off fre- quency of the lowpass filter must exclude switching fre- quency components and therefore impacts on the frequency response of the speed estimation, especially in the transient state of the drive. A further difficulty for speed estimation is the variability of the armature resistance R, with temperature introduced by the arma- ture heating. If the value of R, in the estimator is incorrect, this may cause error in the resultant speed estimate, particularly when the drive is operating in low-speed, high-current mode.

This paper describes a technique for combined speed and temperature estimation in a DC motor drive. By including an armature temperature estimation proce- dure temperature-related variations in armature resist- ance are included in the calculations so the speed estimates remain accurate over the full range of operat- ing conditions. The armature temperature estimate itself may be used for condition monitoring or for setting current limits in a drive with erratic duty cycles.

In the combined speed and armature temperature estimator shown in Fig. 1, armature current and volt- age measurements from the motor are used in a closed- loop observer incorporating a motor model, which pro- duces estimates of armature current, speed and arma- ture temperature. The estimated armature current is compared with the measured armature current to form an estimation error, which is used to correct all three estimated quantities using the extended Kalman filter (EKF) algorithm [3, 41.

w -

+ -

armature current

estimated speed and temperature

Fig. 1 Observer fo r estimation of speed and armature temperature

2

The extended Kalman filter requires a dynamic model of the DC motor, that takes account of electrical, mechanical and heating effects. For a motor with con- stant field excitation, and neglecting voltage drops

State space model of DC motor

14

across the brushicommutator contact, the armature electrical equation can he written

(2) di d t

W, = i,R,o(l + ~ 4 ) +I," + kew

The mechanical equation is

( 3 )

The thermal equation is d e

i:R,O(l + 04) + kirw2 = k o ( l + ~ T W ) O + H - d t (4)

in which the left-hand side represents electrical power dissipated in the motor by copper and iron-loss effects, while the right-hand side represents the effect of this power loss on the motor temperature. Examining each term in eqn. 4 in more detail, (i) i2Ra0(l + a0) represents the power dissipated by the armature current flowing through the armature resist- ance which varies in proportion to the temperature. (ii) k i r d represents the variation with speed of iron loss in the armature body. For constant excitation, this loss is proportional to speed squared. (iii) ko(l + k T w ) 8 represents the thermal power flow from the armature surface which is proportional to the temperature difference between the armature and the ambient air temperature. The effect of the cooling fan is approximated by introducing a speed dependence of the thermal transfer coefficient. (iv) HpQ represents the rate of temperature rise in the armature which depends on the thermal capacity of the armature body. Thermal modelling of an electrical machine is a com- plex issue and the model represented by eqn. 4 is a sim- plified representation of the processes involved. For example, no account is taken of thermal paths on the armature such as those involved in the transfer of heat energy from the armature conductors, through the winding insulation and onto the armature body. How- ever, the model is being used here for estimation of the bulk armature temperature so detailed thermal model- ling of the armature is considered unnecessary.

Eqns. 2-4 represent the continuous state space model of the DC motor, this model must be discretised for using in the digital computer. Also eqns. 2 and 4 are nonlinear, because of the multiplication of the states, so this state model must be linearised around a steady- state operating point or nominal state. The processes of discretisation and linearisation are described in Section 10.1. The discrete linearised state equations of the aug- mented state model can be written as follows in terms of the three states (armature current, speed and arma- ture temperature) i a ( k + 1) = aooi,(k) + ao1w(k) + ao2Q(k) + uosv,(k) ( 5 )

td(k + 1) = U l O i a ( k ) + a l l w ( k ) + a l z Q ( k ) + al& (6)

Q ( k + 1) 1 azoi,(k) + aa1w(k) + a228(k) (7) the coefficients amn are given in Section 10.2. Eqns. 5-7 are formulated in the state-space representation required for application of the extended Kalman filter algorithm which takes account of process and measure- ment noise in a general nonlinear system

x ( k + 1) = f (x (k ) , 4% + w(k)

Y ( k ) = C(k)X(k) + v(k)

( 8 )

(9)

IEE Pvoc -Ebctr. Powev A&, Vol. 144, No. I , Januirry 1997

In the formulation of eqn. 9, the output equation, it is assumed that the system outputs y(k) are a linear com- bination of the system states x(k), which is the case for speed and armature temperature estimation in the DC motor. The process noise w(k) is a 3 x 1 vector and the measurement noise v(k) is a 1 x 1 vector. The vectors w(k) and v(k) are random white-Gaussian variables and can be described by statistical measures (mean and variance). The mean of the vectors are zero and the variance can be described by covariance matrices. The process noise covariance matrix is defined as

E { W ( ~ ) W ( ~ ) ~ > = Qb (10)

E { v ( k ) ~ ( k ) ~ } = RGik (11)

and the measurement noise covariance matrix is defined as

where Q is 3 x 3 constant matrix and R is 1 x 1 con- stant matrix. The initial value of the state vector x(0) is described in terms of its mean value and covariance matrix.

X(0) = E{x(O)} (12)

P(0) = {[x(o) - W ) l [ X ( 0 ) - WIT) (13)

3 Observability

Before attempting to implement a closed-loop observer for the estimation of speed and armature temperature in a DC motor it is appropriate to confirm the observ- ability of these two quantities. In the proposed state estimator (Fig. 1) the error between measurements and estimates of armature current is used to correct esti- mates of both rotor speed and armature temperature. A test for observability establishes whether the current error contains sufficient information to correct any errors which may occur in speed and temperature esti- mates: the separate effects of speed and temperature must be observable in the current estimate. The condi- tions for observability are investigated in Section 10.3, where it is established that the system is observable for all steady-state operating conditions in which the arma- ture current is nonzero.

4 Application of EKF estimator

The linearised model of the DC motor, developed in Section 2, is used by the EKF algorithm to estimate the armature current, speed and armature temperature from measurements of the armature voltage and cur- rent. The measured armature current is used as the out- put vector to be compared with the estimated armature current to produce an estimation error which is used to correct the three estimated quantities, as shown in Fig. 1.

The EKF algorithm implemented to perform the esti- mation of speed and temperature is described below. The prediction stage is

X(k + 1) = . f (X (k ) , u(k) )

P ( k + 1) = F ( k ) P ( k ) F T ( k ) + Q

(14)

(15) The correction stage is

K ( k + 1) = P ( k + l)CTICP(k + l )CT + RI-' (16)

P(k + l / k ) = P ( k + I) - K ( k + l )CP(k + 1) (17)

x ( k + 1/k) = x ( k + I) + K ( k + 1) [i,(k + 1) - ; a ( k + l)] (18)

IEE Proc.-Electr. Power Appl., Vol. 144, No. 1, January 1997

where F(k) is the partial derivative or the Jacobian matrix

and after expansion

the coefficients are given in Section 10.2. The structure of the extended Kalman filter employed for speed and armature temperature estimation is shown in Fig. 2.

-w correct ion

Fig. 2 perature estimation

Structure of extended Kalman$lter employed for speed and tem-

Load bank resistor U

I r

unit

data

system buffer circuits acquisition

DSP card

act ih-h, emu lato r

Fig.3 motor

Experimental set up for speed and temperature estimation in DC

5 Implementation

The motor model described above has been implemented on a TMS32C30 digital signal processor system, shown schematically in Fig. 3 . The motor is a standard 3 kW DC machine in which the field current is kept constant throughout the series of experiments.

15

The armature winding is supplied by a commercial phase-controlled thyristor converter. The DC motor is coupled to a DC generator driving a switchable resistance loading bank.

The required signals for the estimation algorithm are digitised and processed by the data acquisition system based on a Texas Instrument TMS320C30 DSP operat- ing at 33MHz and providing up to 33.3 MFLOPS (mil- lion floating-point operations per second).

6 Experimental results

6. I Evaluation of thermal parameters To calculate the thermal parameters of the motor, three tests were performed. The first test was carried out with the armature locked and no field voltage applied. The motor was operated at rated armature current until a steady-state armature resistance was reached. This test satisfied the relation (from eqn. 4)

i : f l on ( l + QB) = koB (21) using RaO = 3.43Q and a = 0.004i"C (the temperature coefficient of resistance for copper materials), 8 = 69°C the final temperature above ambient (obtained from a thermocouple embedded in the armature winding), and armature current of 8.2A, the constant I to was calculated:

The second test was carried out with no-load current ial (0.4A) and the motor operated at speed wl (96.3 radis) until a steady temperature rise (7°C) was attained. From eqn. 4 the thermal equation becomes

ko = 4.33\.1:/"C

z;lRa()(l +ole,) + karLJ? = k o ( l + kTLJ1)Bl (22) The final test was conducted at full-load current iOz (7.75A) and speed wz (86 radis) until a constant arma- ture temperature 0, (56.5"C) was reached. These condi- tions satisfied the relation

&Ra"(l + 00.2) + L W ; = ko(1 + k^TCV2)& (23) from eqns. 22 and 23 k,, and kr were calculated as

k . ~ = 0.0028s/rad

k,, = 0.0041 W/(racl/s)' The last parameter needed in the model is the thermal capacity (m. During the full-load test the thermal time constant was measured from the temperature/time characteristics and was found to be 56 min. From eqn. 4 the thermal time constant is Hiko(l + k,u). Substituting the values already calculated, H = 18KJI"C.

process noise covariance matrix

[Qon 0 0 1 = 11.3 0 0

'= 1 0 0 0.21 1 0 0 1 e - 6 1

measurement noise covariance matrix

0 0.001 0 1 (26) 0 Qii 0

R = [0.65] (27) The values for the matrices P, Q, R are set to optimum values, determined by trial-and-error.

The experimental results for speed and temperature estimation are given in Figs. 4-8. As mentioned in Sec- tion 4, the estimation process is based on comparison of measured and estimated armature currents. Typical full-load current waveforms are shown in Fig. 4. These waveforms have a fundamental frequency of l00Hz and demonstrate that the speed and temperature esti- mation processes do not require prefiltering of input voltage and current waveforms.

''Or

'. a 200 1

l a

15 Q.

0 -

b

15 Q 0 -

0 60 120 t t m e , m s

C

Fig.4 (a) Measured armature voltage (6) measured armature current ( e ) estimated armature current

Meusured and estimated waveforms at rated speed and loud

I estimated

6.2 Temperature and speed estimation Real time online estimation for armature speed and temperature has been carried out using the EKF algo- rithm developed in Section 4. At the start of the esti- mation process, initial values for the three states should be given together with values for the state and noise covariance matrices Q, R, P. The EKF settings are

state vectoi %(o) = [i] = [ (24)

initial error covariance matrix

= ri :b R1 (25) [PO" 0

P(0) = 0 PI1 0 1 o 0 l;J 10 0 41

16

2 o v I I , I 0 50 100 150 200

time,min Estirnuted and meusured temperature Fig. 5

The estimated speed and temperature are shown in Figs. 5 and 6 for operation at rated speed and current as the armature temperature increases. In Fig. 5 , the estimated and measured temperature is shown. The temperature measurements were obtained from a ther- mocouple embedded in the armature winding. At 10-15 minute intervals it was necessary to bring the motor to rest for a short period while the thermocouple was con- nected to the instrumentation. Observations of the rate of armature temperature change during cooling (shown in Fig. 7) were used to correct the measurements to

IEE Pro.-Electr. Power Appl., Vol. 144, No. 1. January 1557

take account of the temperature reduction caused by the delay (approximately 30s) in taking the measure- ment. The measured temperature settled at 80°C, while the estimated temperature settled at 83"C, and both temperature variations exhibit similar time constants. The agreement between the estimated and the meas- ured temperature in Fig. 5, shows the accuracy of the model used to represent the DC motor and the ability of the EKF to separate the effects of speed and temper- ature variations in the current and voltage waveforms. Small differences between the estimated and measured temperature are to be expected since the estimated tem- perature is an average for the entire winding, whereas the measurement is taken at a particular winding location.

v) \

60- U : 1 0 - a v)

20-

\" E

: 1 0 -

60-

U

a v)

20

01 I I I

CI

loo; 80

-

01 4 I I 1

0 50 100 150 2 00 time,min

6 Fig. 6 a Estimated h Measured

Estimated and measured .speed

Fig. 6a shows the estimated speed and Fig. 66 shows the measured speed obtained from a tachogenerator. These results were obtained under identical operating conditions as the results shown in Fig. 5, except that the motor was run continuously: stops for armature temperature measurement were not required. After the initial transient when the estimation process is switched on, the estimated speed maintains its correct constant value of 86 radis, despite the large changes in the arma- ture temperature shown in Fig. 5. There is a difference of approximately 2% between the instantaneous values of measured and estimated speed. Therefore the speed estimation technique will not be sufficiently accurate to replace the tachogenerator in high-performance servo drives. However, the technique will be useful in gen- eral-purpose applications, including electric vehicles, where speed signals are used for monitoring and limit- ing functions, rather than as primary control variables.

Fig. 7 shows the estimated and measured tempera- ture during cooling when there is no armature current, so according to the observability criteria described in Section 3 the estimator is operating open-loop. There is good agreement between the estimated and measured temperature, again demonstrating the accuracy with which thermal effects have been modelled.

time,min Estimated and meusured armature temperature under cooling Fig. 7

conditions

20: I I 8

0 60 120 180 2LO 300 360 time,min

Eslimuled temperature during heating and cooling Fig. 8

Fig. 8 shows the estimated armature temperature during temperature cycling. Here the machine was operated at rated speed and load for about 160 min, turned off for about 60 min and then turned on for a further 120 min. As shown, the armature temperature variation is estimated during both heating and cooling intervals, over a timescale during which unacceptable drift would be evident in an open-loop estimator. This result demonstrates that temperature estimation is pos- sible for all operating conditions including erratic duty cycles.

loor U 0 60 U

20

2ot I I I

0 50 100 150 200 time,"

6

0:

Fig.9 Ef ic t o j Q-matrix elements on estimation of speed (a) Estimated speed at Q,, =. le-2 (b) Estimated speed at Q,, = le-4

6.3 Effects of covariance settings The effects of changes in the covariance settings are demonstrated in Figs. 9-12. In Figs. 9 and 10 the effect of the Q-matrix element values is illustrated. The proc- ess noise covariance defines the uncertainty in the esti- mation process for each of the corresponding states.

17 IEE Proc -Eleclr Power Appl , Vol 144, No I , January 1997

The @matrix controls the convergence speed of the estimated quantities and also affects their uncertainty. In Fig. 9a and b ell, which defines the uncertainty in the estimation of the speed, has values of le-2 and le- 4. Higher values of covariance in the speed estimation process indicate more uncertainty in the speed estima- tion process so changes in speed estimates are allowed to occur more rapidly: transient changes are tracked more easily, but some filtering effect is lost. Thus in Fig. 9a, the steady-state value for the speed is noisy and the amount of uncertainty in the speed estimation is greater than is the case for a lower value of ell. In Fig. 10 the uncertainty in the temperature estimation which is represented by Q22 has alternative values of le-7 and le-3 compared with the optimum value of le-6. For the smaller value of Q22 the convergence time is longer and there is appreciable steady-state error, whereas the larger value causes the estimated tempera- ture to overshoot the measured temperature, as seen by comparing Fig. 106 with Fig. 6a. The value chosen for general use in the EKF algorithm = le-6) is there- fore a compromise between these extremes of perform- ance.

a

2 0 v I I I I

0 50 100 150 200 time,min

b Fig. 10 (a ) Estimated temperature at QZZ = le-3 (b) Estimated temperature at Q22 = 1 e-7

Effect of Q-matrix elements on estimation of temperature

the real-time constant. So the bigger the values for the elements in the P-matrix the faster convergence occurs, except that an excessively high value may lead to a loss of accuracy.

01 I I

a l o o r

60

73 jni 20

01 I I I

0 2 1 6 ti me, min

b Fig. 11 speed (a) Estimated speed P,, = 1 (b) Estimated speed P,, = 20

Effect of initial values of P-matrix elements on estimation of

The values of the initial condition covariance matrix P control the initial convergence of the estimation process. Figs. 11 and 12 show the effect of changes in the elements of the P-matrix on the estimation of speed and temperature. A higher value indicates to the Kalman filter that there is a potentially large error in the initial estimate. Therefore when an initial estima- tion error occurs it is corrected rapidly, as shown in Fig. I lb , for the case of P I 1 = 20. Conversely, a small covariance value indicates that the initial estimate has a higher probability of being correct, so the estimation error is corrected more slowly as shown in Fig. l l a for the case of P I , = 1.

A similar effect can be observed in Fig. 12a and b, which explore the effect of adjusting the covariance of the initial temperature estimate. The larger value for P22 (= 100) in Fig. 126 allows the temperature to reach the steady-state value faster, but with a lot of noise and the apparent thermal time constant is different from

18

201 I I I I

0 50 100 150 200 time, min

b Fig. 12 estimation (U ) Estimated temperature P2z = 2 (b) Estimated temperature P22 = 100

Effect of initial values of P-matrix elements on temperature

7 Conclusions

A general method of estimating DC machine speed and armature temperature, from measurements of armature voltage and current, has been presented. Speed and temperature estimation is possible for all operating conditions.

An important feature of the estimator is that, in con- trast to steady-state estimators and tachogenerators, it does not require any prefiltering of the input signals. Therefore there is no speed signal delay caused by signal filtering.

IEE Proc -Elecn Power Appl , Vol 144, No 1, January 1997

Although the digital signal processing hardware required to obtain the experimental results in this paper is expensive, the future availability of low-cost digital signal processing [5] will ensure that the EKF approach is extremely cost effective. In addition, the processing hardware costs must be weighed against the costs of the tachogenerator and its installation since the need for speed signal connections to the motor is avoided with the technique described.

8 Acknowledgments

The authors acknowledge financial assistance provided by the UK Engineering & Physical Sciences Research Council via the LINK PEDDS initiative, and the sup- port of the industrial partners in the Concerted Action on Power Electronic Control.

References

HIRO, Y., UNENO, T., UCHIDA, T., and KONNA, Y.: 'An instantaneous speed observer for high performance control of dc servomotor using DSP and low precision shaft encoder', EPE Firenze, 1991, 3, pp. 647-652 BRANSHACH, B.J., HENNEBERGER, G., and KLEP- SCH, TH.: 'Speed estimation with digital position sensor'. Conf. Record of ICEM 92 Manchester, 1992, pp. 577-581 ATKINSON, D.J., ACARNLEY, P.P., and FINCH, J.W.: 'Observers for induction motor state and parameter estimation', IEEE Trans. Znd. AppL, 1991, 276, pp. 1119-1127 ANDERSON, B.O., and MOORE, J.B.: 'Optimal filtering' (Pren- tice-Hall, 1979) 'Texas Instruments: Industry first-floating point DSP barrier', New Products Showcase, 1995, Issue 1 RICHARDS, R.J.: 'An Introduction to Dynamics and Control' (Longman, 1979)

10 Appendix

IO. I Linearisation and discretisation of motor model Eqns. 2-4 are nonlinear and must be linearised around a nominal steady-state operating point. For a general state variable x

x = X + A x (28) where X is the steady-state value of x and Ax is the perturbation from the steady-state value. So by apply- ing this formula to the variables IJ,, i,, ra, CO and 8,

U, = v, + AV, i, = I , + ai, w = W + A w 6'=O+AO

Substituting into eqns. 2-4, then separating steady-state terms, vu = I,R,o(l+ a@) + keW (29)

keIa = bW (30)

and first-order terms (after rearranging): dAi, Ai,R,o(l+ a@) A w k ,

dt 1, I , -__ -

(32 )

(33)

AOR,od, AV, +- -

1, 1,

d a w Ai,k, Awb TL dt J J J

dA6' Ai,21,Rao(1 + a@) A w ( k 0 k ~ O - 2ki,W) dt H H

- - -

~ - - -

(34) - A B k o ( l + ~ T W ) + AOIZR,oa

H H IEE Psoc.-Electr. Powes Appl., Vol. 144, No. I , January 1997

These linearised equations are in the standard state- space format

= [A][z] + [B][u] dt (35 )

where the state vector

ai, (36)

the input vector

[ U ] =

and the output vector

[Yl = [@it] (37 )

(38 )

Therefore the output matrix

[ C ] = [ l 0 01 (39) and the system matrix

A00 A01 A02

where the matrix elements are as follows:

A00 = -R,o(l + a@)/Z,

Ai1 = -b/ J A20 = 21u,Rao(l+&)/H A 2 2 = -ko(l + k T W ) / H + I ~ R , ~ c x / H

A01 1 -ke/Za A02 = -R,offL,/l, A10 = k , / J

A12 = 0 A21 = - ( k o k ~ O - 2k,,W)/H

(41) Since the system uses sampled data, the system equa- tions have to be formulated as difference equations. For example, if Ai,(k) and Ai,(k + 1) are successive val- ues of the armature current variation separated by the sampling interval t,, the rate of change of current can be written

~ (42)

(43)

dAi, - Ai,(k + 1) - Ai,(k) - d t t ,

or Ai,(k + 1) = Ai,(k) + { d A i , / d t } t ,

Substituting from eqn. 43 into eqn. 32 yields the rela- tions shown in eqn. 5. A similar process of discretisa- tion can be applied to eqns. 33 and 34 to produce eqns. 6 and 7.

where I,, W and 0 represent the nominal values of the states.

19

10.3 Formal test for observability A formal test for observability [6] can be expressed in terms of the system matrix [A] and output matrix [Cl, both of which are derived in Section 10.1 for the DC motor drive. The condition for observability of an nth- order system is that the observability matrix

[O] = [C' ATCT . . . (AT)"plCT 1 (44) should have rank n. The system under consideration has three states (n = 3), so the observability matrix

[0] = [ C' AT@ ( A T ) 2 C T ] (45) should have rank 3.

The output matrix [C] = [ 1 0 01 (46)

[Aoo Aoi &a]

LAO AX A221

and the system matrix [A] = AI0 All Ala (47)

where the matrix elements are as defined in eqn. 41 of Section 10.1 Substituting into the observability matrix [U] from the expressions for [Cl and [ A ] :

[OI = 0 Aoi AoiAoo + AllAoi + A21A02~ (48) + A12Aol + Az2A02 1

The rank of this matrix is equal to three if the rows or columns are not linear combinations of each other. The first row of the matrix cannot be a linear combination of the other two rows because the first element { l} cannot be formed from a linear combination of the first element in the second row {0} and the third row CO}. The other possibility is that the second and third rows are proportional, i.e. the corresponding elements in these rows share a common constant of proportion- ality. Hence the rank of the observability matrix is less than three if

11 AOO A& + AloAoi + A2oA02

1 0 A02

(49) Aoi - Aoi Aoo + AiiAoi + A21A02 A02 Ao2Aoo + A12A01 + AaaAoa

-

which can be rearranged and simplified to give the con- dition

AilA12 + AoiAoa(A22 - Ail} + A & A z ~ = 0 (50) and substituting from eqn. 41 for the system matrix elements in terms of the motor's parameters and oper- ating point

{ -+} { -R;:aIa} { - k o ( l + k ~ W ) +- I:Baoa + - H H

In eqn. 51, the nominal armature current I, is an over- all multiplier. Therefore the most significant condition for which the rank of the observability matrix reduces to two, implying that armature temperature and rotor speed are not observable, is when the armature current is zero. In this situation it is only possible to form an open-loop estimate of the three motor states (armature current, speed and temperature). Clearly from eqn. 51 there may be other combinations of motor parameters and working point for which the rank of the observa- bility matrix also reduces to two. However, in practice, such combinations are encountered only during tran- sient operation so there is no sustained loss of the esti- mator's functionality.

10.4 Details of DC motor Rated voltage = 240V Rated current = 10.4A Power 3kW Maximum speed = 4000 revlmin Rated torque = 11" Armature resistance R, = 3.5Q Armature inductance, L, = 34mH EMF constant, k, Mechanical time constant = 4.6s.

= 0.9348 Vlradls

20 IEE Proc.-Electr. Power Appl., Vol. 144, No. I , January 1997