Kalman Filter Estimators Applied to Robust Control of Induction Motor Drives A. Dell’Aquila
F. Cupertino
L. Salvatore
S. Stasi
Politecnico di Bari Via E. Orabona 4 70125 Bari, Italy
[email protected]
Politecnico di Bari Via E. Orabona 4 70125 Bari, Italy
Politecnico di Bari Via E. Orabona 4 70125 Bari, Italy
[email protected]
Politecnico di Bari Via E. Orabona 4 70125 Bari, Italy
Abstract - This paper proposes the contemporaneous application of two Kalman filter-based algorithms to realise a robust control of induction motor drives. The first one (EKF) is used to obtain a correct implementation o f direct vector control, since it estimates the rotor flux components, rotor time constant, rotor resistance, and leakage reactance. The on-line adaptation of electrical parameters makes it possible to obtain very accurate estimates of rotor flux components, in spite of temperature and saturation effects. The second one (LKF) estimates the equivalent disturbance to realise the rejection of external disturbances and robustness to mechanical-parameter variations.
I. INTRODUCTION Vector-controlled induction motor (IM) drives are widely replacing the dc ones in industrial applications where high performance, like fast torque and speed responses, are demanded. However, the full advantage of vector control, which consists in separately controlling the rotor flux- and torque-producing stator current components, is available only if the instantaneous position of the rotor flux space vector in a stationary reference frame can be determined anyhow. In direct vector control, this position has to be measured using either sensors or estimators. Direct sensing of the air-gap flux by Hall probes or search coils, placed inside the motors, makes vector control implementation almost insensitive to motor parameter variations. However, this solution suffers from high cost and unreliability of the measurements, because the Hall elements are sensitive to heat and mechanical vibration, and the flux signal is distorted by slot harm on ics. A possible solution to this problem consists in estimating the rotor flux components from other quantities, like rotor speed, stator voltages and/or currents that can be easily measured. In this case, however, high estimation accuracy has to be reached to guarantee, over the whole range of machine operation, a correct field orientation. But rotor flux estimators need a machine model, so that changes in model parameters can severely affect the estimation accuracy of rotor flux components. An on-line identification and adaptation of IM model parameters is then required to realise insensitiveness of vector control schemes to parameter variations. The machine-converter assembly in electric drives causes a great amount of electromagnetic noise, leading to interference phenomena with measurement instruments. Therefore, more realistic stochastic models of IM have to be used to account for these random phenomena. Many state and parameter identification algorithms, based on least-squares method, observer theory, Kalman filter, and so on, have been proposed till now [l]. Among 0-7803-4503-7/98/$10.00
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them, the Kalman filter based algorithm has demonstrated to be the best one for processing noisy discrete measurements and obtaining high accuracy estimates [2]. In this paper the authors propose a new algorithm based on the extended Kalman filter (EKF) theory for the contemporaneous estimation of rotor flux components, rotor time constant, rotor resistance, and leakage reactance. When fast-trajectory tracking control is an important task to perform, insensitiveness of the control system to changes in mechanical parameters and good rejection capability of external disturbances have to be guaranteed. In this paper the robustness features of the IM drive are obtained by estimating the “equivalent disturbance”, which takes into account for external load torque, changes of mechanical parameters, process non linearities, and model uncertainties. A feedforward compensating action, proportional to the estimated equivalent disturbance, is injected to preserve the desired dynamic performance of the drive. The estimation algorithm used for this purpose is a linear time invariant Kalman filter (LKF) that is able to estimate also the rotor speed, by processing speed measurements only. In this way, computational efforts are minimised, since the estimation algorithm has minimum order.
It. DELAYED-STATE EKF ESTIMATOR As previously stated, the EKF algorithm has the function of estimating the a-P rotor flux components and the electrical parameters o,.,i,., and oL,. To derive the discrete-time mathematical model of the EKF it is necessary to resume the induction motor model in the stationary reference frame a-p-0: -
-
-
-
p (hrap + 0 L\ i\ ap) = V\ ab - R\ i, ap -!
- !
phrap=-
(Or-jmi)
I
(1)
-
hi ab + R , i\ap
(2)
where
(3) (4) -
-
and ~ , is~ thep stator voltage vector, i , a p is the stator current vector, hrap is the rotor flux vector, L , , L,, and LIT, are stator, rotor, and magnetising inductance, respectively, R S and R, are stator and rotor resistance, or= R 2 is the inverse rotor time constant, L,
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where
is the leakage coefficient, and oris the L , L,. electrical rotor speed. The electromagnetic torque is:
o=1-
- Ill
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iap(n+l)=[v,ap(n) - ~ , i , ~ p ( nT) ] (17)
3 To = - n,, (Ara i,sp - h,.p i,s a ) 2
-
Independent control of rotor flux and torque can be achieved by imposing the stator current components i,d and i,, in the reference frame d-q synchronously rotating with the rotor flux space vector. In this case (2) and (5) become
and qap (n+l) is the measurement error, which depends on sensor operation, analogue/digital conversions, quantisation and discretisation procedures. The measured - 111 stator voltages v,,p(n) are the mean values in the sampling interval from time (n-1) to time (n). The observation equation can be written in matrix form as follows: z(n+l) = H ( n + l ) x(n+l)
+ J(n)
x(n)
+ q(n+l) (18)
where
(7)
z where 0 is the rotor flux vector position with respect to the a-axis. The discrete time state-space model used for EKF is then derived from (2), using the Euler method, and adding three further equations to estimate the model parameters:
o L , (n + 1)
= o L , (n) + E ~ L(n) ,
(12)
where T is the sampling time interval. The measured - 171 stator currents i\,p,(n) and rotor speed co':'(n) are considered to be constant and equal to the mean values in the sampling interval from time (n-1) to time (n), and c(n) is the model equation error, which depends on the supply 111 signals (i, ae(n), 0':(n)), ' and discretisation procedure. The EKF model in compact matrix form is:
J(n) =
zplT
-I o O O -i.:h(n)
o
-1
o o
I
-i.;"r4(n)
The state noise vector E and measurement noise vector 9 are admitted to be white, Gaussian, and uncorrelated with zero mean values. The observation equation has the same structure of the delayed-state Kalman filter [3], since it is function of the state at time (n+l) and at time (n). The state variables at time (n+l) are predicted by means of the mathematical model using the updated estimates at time (n). The state prediction equation is: F ( n + l ) =f(x^(n), n)
(19)
where (-) denotes prediction and (") denotes estimation. The predicted state at time (n+l) is corrected by adding the weighted difference between the measurement vector z (n) and the predicted measurement vector H(n+l)y(n+l) + J(n)x(n) The result is the updated state estimation at time (n+l):
~
x (n+I) =f (x (n), n) + E (n)
=[za
A
(13)
A
-
-
x(n+l)=x (n+l)+G(n+l) [z(n+l)-H(n+l)x(n+l)
where
- J(n);(n)]
(20)
where the gain matrix C is chosen to minimise the estimation error variances of the state variables being estimated. The optimal Kalman gain matrix G is expressed as: A
G(n+l) =[Y(n+l)HT(n+ 1) + @(n)P(n)JT(n)] L-'(n+I) (21) where
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With this aim the authors propose the application of a linear Kalman filter-based algorithm to estimate the equivalent disturbance. The discrete-time state-space model of the LKF is derived from (27) by using the Euler's method and considering that the q-axis current reference is4*is the sum of the speed P-controller output U, and the compensating signal
P and P are the covariance matrices of the prediction and estimation errors, respectively, and Q, (n) is the Jacobian matrix of the function f (n), computed in the estimated state-space point x (n). The prediction and estimation error covariance matrices are computed as follows: A
-
A
P ( n + l ) = cb (n) P ( n ) Q,'(n) A
+Q
Wr(n+l) = o,(n)-T' n Tey(n)
-
P ( n + l ) = P(n+l) - G(n+l)L(n+l)GT(n+l)
Ji,
(24)
+T
111. EQUIVALENT DISTURBANCE ESTIMATOR
11
kon
[u,(n)
+ iYL(n)]+ co(n)
Jn
When current-vector-control method is applied to induction motor drives, the torque control is obtained by simply regulating the q-axis stator current component isq, provided by a current-controlled PWM inverter. In the hypothesis of ideal field orientation, the mechanical equation of the induction motor is as follows:
where T,,is the external load torque, J is the inertia moment of system, Bk is the k-th order viscous-friction coefficient, k, is the torque constant (proportional to the rotor flux), and parameter 6 takes into account for unknown or unmodelled dynamics. We define the equivalent disturbance as:
Te4(n+l) = Teq(n) + cT(n)
(28) (29)
This model contains only two variables, i.e. the rotor speed and the equivalent disturbance, so that computational efforts are minimised. Moreover, its properties of linearity and time-invariance guarantee the asymptotic stability and convergence of the estimation process when the conditions for complete system observability and controllability are satisfied and state transition matrix, system model error and measurement noise covariance matrices are bounded. The LKF model in matrix form is: x(n+l = Fx(n)
+ Lu(n) +E(n)
where
(26) where Jn is the nominal value of inertia of the system, k , is the nominal torque constant, isq' is the q-axis current reference, and A is the unknown variation referred to the nominal or reference value. Then the non-linear time varying mechanical process governed by (25) can be substituted by the following linear time-invariant equation: The observed variable is rotor speed and the measurement equation is:
The terms on the right-hand side of (26) represent the external load torque, the viscous-friction torque, the equivalent disturbance terms due to the parameter variations, and that due to the current ripple, respectively. In the following, the mechanical process described by (27) without the equivalent disturbance will be referred to as the "reference model". The speed control of the reference model can be simply realised by any linear constant-gain control method. In presence of the equivalent disturbance Te4 the system dynamics will not follow the reference one, and transient- and steady-state errors will occur in the rotor A
speed. Estimating the equivalent disturbance Teq on-line is then an effective mean to compensate or, at least, minimise these errors with the injection of the feedforwad signal iqc= Tes/ k,,, . Furthermore, this compensation allows designing the speed control loop in a very simple way using a proportional controller only.
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where H=[l
01
The equations constituting the recursive LKF algorithm are well known, so it is unnecessary to recall here [4]-[5]. The only differences between LKF and EKF consist in what follows: the LKF state prediction equation is linear and time invariant; the LKF is not a delayed-state one, so that the state estimate equation at time (n+l) does not depend on the state estimated at time (n); the LKF gain matrix at time (n+l) depends only on quantities at the same time step. Another advantage of the LKF is that the prediction and estimation error dynamics and Kalman gain dynamics don't need any measurement data acquisition to be
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determined; therefore they may be pre-computed off-line and stored in the computer memory. The recursive on-line linear Kalinan filter algorithm is then reduced to stateprediction and state-estimation equations. Another advantage of the proposed LKF consists in its capability of filtering the noisy measurements of rotor speed. This fBct leads one to prefer the use of this estimated variable as feedback signal for control rather than the measured one. Originality of the designed linear Kalman filter lies in the independence of the prediction model fi-om the equivalent disturbance, that is the prediction model coincides with the discrete-time reference model, as the following equation evidences:
valuing the effectiveness of the proposed control system. The induction motor used in the simulations has the following nominal parameters: rated power 7.5 kW, rated current 16.5 A, rated voltage 380 V, rated speed 1420 rpm, rated torque 50.4 Nm, rated frequency 50 Hz, pole pairs 2, system inertia 0.062 kg.m 2 , stator resistance 0.728 R, rotor resistance 0.706 R, stator inductance 0.0996 H, rotor inductance 0.0996 H, magnetising inductance 0.0969 H. Each algorithm based on Kalman filter theory needs the assignment of an initial state prediction, the initial
-
covariance matrix of the prediction errors P(O), and the covariance matrices Q and R of model and measurement errors, respectively. As regards the diagonal elements of N
Since the Kalman gain minimises the estimation error variances and permits one to obtain the optimal prediction, the actual system dynamics follows the reference model one as better as possible. When tracking of speed reference trajectories is required, as it happens for robot manipulators, feedfonvard actions compensating the reference model dynamics have to % provided. The tracking of the speed reference trajectory o, is then obtained by injecting, just aRer the P-controller, the following feedforward signal if:
(33) where a,.* is the accelerati2n reference trajectory corresponding to the speed one a,..
V. SIMULATION RESULTS The algorithms proposed above, applied to the vector control of an IM drive, have been developed in MatlabSiinulink environment. Computer simulations allow
Estimated equivalent disturbance
the matrix P(O), they are chosen approximately equal to the maximum possible square deviations of the state variables from the relevant initial prediction values. This allows avoiding divergence problems obtaining, at the same time, a sufficiently fast convergence of the estimates. Estimate accuracy could be compromised due to the different sizes of the state variables. High-percentage estimation errors, for the variables of small size, occur in consequence of trace minimisation of estimate covariance matrices. To overcome this problem, the following weighted state vector has been selected:
=[A:, 1,J.p or 10R: 1OOOoL,] T The initial values of the inverse rotor time constant, rotor resistance, and leakage reactance have been assumed to be equal to 50% of the actual values that are 7.08 s-', 0.668 R,and 0.0059 H, respectively. The EKF has been started with the following initial conditions:
x
X(0) = [ 0 0 3.54 3.34 2.9517 F(0)= diag[ 0.01 0.01 10 10 IO]
R = diag [
]
FIux ommand
Estimated rotor
Rotorsoeed
IU
Y
Fig. 1 - Siinulink block diagram of the IM drive.
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As regards the LKF algorithm it has been started with the following initial conditions:
.L (0) = [ 0
N
PL (0) = diag [ 0 251
01'
QL = diag [
1]
R L =[9]
Two different simulations are now described. In the first test the rated load toque has been applied at time 0 s. The results are shown in figures 2a, 2b, and 2c. The performances of the described EKF-based algorithm are compared to those of a similar one that estimates only the 400
inverse rotor time constant with the other parameters held constant and equal to 80% of the relevant actual values. Figures 3a, 3b, 3c show the results. There is evidence that the contemporaneous estimation of all the parameters (compare figures 2 and 3) guarantees an effective field orientation. In the second test the performances of the drive with the LKF-based estimator of disturbances are compared to those obtained with a traditional speed PI controller and feedforward action compensating the reference model dynamics. A load torque (TL=12.4.sin(100t)+38.u(t-0.2)) has been applied.
' 400
-400b
0'1
0'2
0'3
0'4
time, s
0'5
0'6
0'7
i
0.1
0]8
0.2
0.3
04 time, s
os
0.6
0.7
0'5
0'6
0'7
(34
(24 12
1
ne a
206
04
ti
o
01
02
03
04 time, s
os
06
07
Qc) Fig. 2 - Speed tracking with estimates of all the parameters and equivalent disturbance: (2a) speed tracking response, (2b) comparison between actual and estimated rotor flux, referred to the reference one, (2c) model parameter estimates.
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0'1
0'2
0'3
0'4
tme, s
(3c) Fig. 3 - Speed tracking with estimates of orparameter and equivalent disturbance: (3a) speed tracking response, (3b) comparison between actual and estimated rotor flux, referred to the reference one, (3c) orestimates.
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4U0
01
0.2
time, s
03
(44
400,
I
04
U
g -1 00
P -200
-300
I
-4001
05
01
0.2
time, s
03
04
5
(54 25
20 -
15-
d
10-
-1 5 -20
I
(4b) Fig. 4 - Speed tracking with estimates of all the parameters and equivalent disturbance: (4a) speed tracking response, (4b) speed tracking errors.
(33) Fig. 5 - Speed tracking with estimates of all the parameters and PI speed controller: (5a) speed tracking response, (5b) speed tracking errors.
Figures 4 and 5 show that the speed tracking errors obtained by compensating the equivalent disturbance are considerably less than those using the PI controller. The on-line estimation of the equivalent disturbance allows an optimal rejection of load torque, variations of mechanical parameters, and process non-linearities.
compensated by injecting a feedfonvard signal provided by the LKF; field-oriented control and speed trajectory tracking are better than those obtained by other known schemes; speed, voltage, and current measurement noises are rejected. It is expected that the scheme based on two Kalman filters will be applied to induction motor drives successfully.
VI. CONCLUSION
In this paper a new scheme for vector controlled induction motor drives has been proposed. This scheme is based on two Kalman filters. The first one estimates all the model parameters (excepted stator resistance), and the rotor flux components. The execution time of this algorithm is about 200 ps when it is implemented on a 32 bit, 33 MHz, floating-point TMS 320C30 DSP chip. The second one estimates the equivalent disturbance and rotor speed. This last is more simple and can be executed within 50 ps. The main advantages of the proposed control scheme are: the estimation errors of rotor flux components are strongly reduced because of the contemporaneous estimation of all the model parameters carried out by the EKF; 0 the mechanical-parameter variations, load torque, unmodelled dynamics, and viscous-friction torque are
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VII. REFERENCES
K. Rajashekara, A. Kawamura, K. Matsuse, Sensorless Control of AC Motor Drives, IEEE PRESS, 1996. L. Salvatore, S . Stasi, and L. Tarchioni, "A New EKF-Based Algorithm for Flux Estimation in Induction Machines," IEEE Trans. on Industrial Electronics, vol. 40, N. 5, pp. 496-504, 1993. R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering. New York: Wiley & Sons, 1992. A. Gelb, Applied Optimal Estimation. Cambridge, MA: M.I.T. Press, 1988. L. Salvatore and S. Stasi, "LKF-Based Robust Control of Electrical Servo Drives" - IEE Proceedings -Electric Power Applications, vol. 142, N. 3, pp. 161-168, May 1995.
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