A Comparison of Extended and Unscented Kalman Filters for the State Estimation of Induction Motor Drives

A Comparison of Extended and Unscented Kalman Filters for the State Estimation of Induction Motor Drives

D.Sleeva Reddy1 K.Ankalamma2 M.Vijaya Kumar3

1Loyola Institute of Technology and Management Dhulipalle-522412.Guntur.A.P.India

2&3 J.N.T.U.A College of Engineering Ananatapur. A.P.India .

Abstract-This paper investigates the application of Unscented Kalman Filter (UKF) for induction motor (IM) sensorless drives and compares the general UKF with Extended Kalman Filter (EKF) in detuned conditions. The speed and rotor resistance estimation results are compared. Simulation results for Unscented Kalman Filter are presented and compared with those of Extended Kalman Filter. It evaluates the very low speed performance of general UKF.Only General UKF is presented as it provides the best performance compared to other UKFs (basic,simplex,spherical) and compared with respect to EKF.It is concluded that the UKF provides more robust performance than the conventional EKF.

Keywords-Extended Kalman Filter, general Unscented Transformation, state estimation.

1.INTRODUCTION

Estimation in nonlinear systems is extremely important because almost all practical systems involve nonlinearities of one kind or another. In estimation theory, the Extended Kalman Filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In case of well defined transition models, the EKF has been considered the de facto standard in the theory of nonlinear state estimation and navigation systems.Unlike its linear counterpart, the Extended Kalman Filter in general is not an optimal estimator (of course it is optimal if the measurement and the state transition model are both linear, as in that case the extended Kalman Filter is identical to the regular one). In addition,

if the initial estimate of the state is wrong, or if the process is not modeled correctly, the filter may quickly diverge, owing to its linearization. Another problem with the Extended Kalman Filter is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilising noise.

Although the EKF maintains the elegant and computationally efficient update form of KF,it suffers from number of serious limitations[1].EKF is difficult to tune, the jacobian can be hard to derive,and it can only handle limited amount of nonlineariy. A nonlinear Kalman Filter which shows promise as an improvement over the EKF is the Unscented Kalman Filter (UKF). In the UKF, the probability density is approximated by a deterministic sampling of points which represent the underlying distribution as a Gaussian. The nonlinear transformations of these points are intended to be an estimation of the posterior distribution, the moments of which can then be derived from the transformed samples. The transformation is known as the Unscented Transform(UT).The UKF tends to be more robust and more accurate than the EKF in its error estimation .

The paper is organized as follows. In section II, the general UT concept is discussed. In section III, the UKF and the discrete UKFs are discussed. The IM model and observer are described in section IV. Section V provides the simulation

results.Conclusions and scope for future work are given in section VI.

II.GENERAL U.T

The Unscented Transformation is a method for calculating the statistics of a random variable,

(0) = (3)

(0) = + () = 1, ,2 (4)

()

while undergoes a nonlinear transformation[2].The UT was developed to

address the deficiencies of linearization by providing a more direct and explicit mechanism

=

(+)

+

()

= 1, , (5)

for transforming mean and covariance

=

+

()

= 1, . . , (6)

information.

Throughout the paper, the superscript (i) is

Unlike the basic UT, the sigma points are assigned unequal weights in the calculation of the mean and covariance as follows:

used to denote the sigma point. Suppose the mean mx and covariance Px of a n × 1 stochastic vector x are known and moreover if one is

0

= (7)

+

interested in the mean and covariance of the output of a known nonlinear function y= h (x), A

set of sigma points (), i = 0, 1, 2,N, with

() = 1

2(+)

= 1, ,2 (8)

the same mean and covariance as the vector can be selected. Then, the sigma points are transformed through the known nonlinear function h (x) to obtain a set of projected sigma

points (), i = 0, 1, 2,N. The weighted sample mean and sample covariance of the projected sigma points give a good approximation of the true mean and covariance of the output y. If the weight associated with the ith sigma point is denoted by W(i) then the approximate mean and covariance of the output are calculated as

The design parameter k determines the

degree of emphasis on point (0), and reduces the higher-order approximation errors. For k = 0, the general UT reduces to basic UT.

1. UNSCENTED KALMAN FILTER

   The Kalman filter (KF) was originally developed for linear systems [7] but later applied to nonlinear systems using the linearized or extended KF (EKF) [8]. Although the performance of the EKF is poor in some situations, its performance is acceptable if the system nonlinearity is not severe. Its simplicity, together with the popularity of the KF, makes it

   =0

   =

   .

   ()()

   (1)

   the most widely applied nonlinear state estimator. Nonlinearly mapping an input random variable typically results in a complex distribution with a

   =

   =0

   ()

   ()

   (2)

   large number of associated parameters. Hence, optimal nonlinear state estimation requires

   The general UT [5] uses a set of 2n+1 sigma

   knowledge of higher order statistics and the exact estimation of the states of a nonlinear system is

   points which lie on the

   covariance

   often impossible in practice [9].

   contou[6]. This is taken as the general UT, since it includes the basic UT as a special case. The general UT uses the 2n+1 sigma points

   The Unscented Kalman Filter belongs to a bigger class of filters called Sigma-Point Kalman Filters or Linear Regression Kalman Filters,which are using the statistical linearization technique[3,4].

   (a)

   (b)

   (c)

   (d)

   Fig.1-Simulation results for high speed operation of IM with general UKF with speed of 200rad(a) estimated speed, (b) measured speed,(c)estimated torque,(d)estimated stator and rotor flux magnitudes

   The discrete KF uses the first two statistical moments and updates them with time. This is the key idea when combining the UT and KF to obtain UKF. The UKF is basically the discrete KF in which an UT is used to obtain the mean and covariance updates. The UKF as presented here is a simplified UKF which is suitable for estimation of IM states. In general, the observation model can also be nonlinear and all parameters and functions can be time-varying. Moreover the UKF can be extended to the case of non-additive noise [8].

   Given the discrete-time nonlinear system

   (a)

   (b)

   (c)

   (d)

   Fig.2-Simulation results for low speed operation of IM with square speed referenceof 12 rad/s and general UKF(a)estimated speed (b)measured speed (c)estimated torque(d)estimated stator and rotor flux magnitudes

   and a linear measurement model

   = . + (10)

   Where is an n×1 state vector, is an m×1 measurement vector, H is the measurement matrix (m×m) and f ( , ) is a known nonlinear state transition vector. It is assumed that the process noise is white and zero mean with covariance matrix Q and measurement noise

   is also white and zero mean with covariance matrix R. Estimates of the initial state +and the initial error covariance matrix+ are available. The iterations in the classic KF consist of a prediction

   (a)

   (b)

   (c)

   (a)

   (c)

   (b)

   (d)

   Fig-Simulation results for high speed operation of IM with speed of 200 rad/s(a)estimated speed with EKF ,(b)measured speed with EKF,(c)estimated speed with general UKF,(d)measured speed with general UKF

   step followed by a correction step. For the correction step the discrete KF equations are used.

   = . . (. . + )1 (11)

   (d)

   Fig.4: Simulation results for low speed operation of IM with speed of 60 rad/s(a)estimated speed with EKF(b)measured speed with EKF(c)estimated speed with general UKF(d)measured speed with general UKF

   state equation (recall that the KF estimates are unbiased). For the nonlinear system of (9), the state equation is a nonlinear transformation of a

   stochastic input . Hence, the UT can be used

   to obtain the mean + and covariance of

   + = + . ( . ) (12)

   its output.

   +

   +

   The mean + and the covariance + of

   + = . .

   (13)

   the stochastic input are used to obtain a set of

   +

   sigma points(), (i= 0, 1, 2,., N) and the

   Where is the Kalman gain.

   The prediction step in the KF is the projection of the mean + and covariance + in time using the

   corresponding weights (W(i), i=0,1, .., N). Then the sigma points are projected in time using the non linear transformation(9).

   (a)

   (

   (b)

   (c)

   (d)

   Fig:5- Simulation results for low speed operation of IM with speed of 120 rad/s (a)estimated speed with EKF(b)measured speed with EKF(c)estimated speed with general UKF(d)measured speed with general UKF

   Given the projected sigma points() , the

   +

   +1

   predicted mean +1using (1) and the predicted error covariance using the following modified version of (2) are calculated.

   obtaining the covariance of the projected sigma points. While the latter is an inevitable costly operation, Cholesky factorization can be simplified then the error covariance matrix is

   (a)

   (b)

   (c)

   (d)

   Fig:6-Simulation results for low speed operation with speed of 300rad/s(a)estimated speed with EKF (b)measured speed with EKF(c) estimated speed with general UKF (d)measured speed with general with general UKF

   sparse. Since the load torque TL appears only in the swing equation, its cross-correlation with both currents and both fluxes is negligible and it

   =

   () () ()

   can be considered independent of both the

   +1

   =0

   +

   +

   +

   currents and the fluxes. In addition, the

   +

   + (15)

   orthogonality of the axes implies that s and s are independent, as are is and is. Consequently, the covariance matrix is sparse. For application to

   Although the UKF is computationally costly, its

   computational load is acceptable for modern microprocessors. The most costly operations are the Cholesky factorization and outer products in

   the IM, symbolic manipulation can be used to simplify the expressions off-line and thereby significantly reduce the computational load.

   = 0 (19)

   Where s is stator flux space vector, is is stator current vector, p is the pole pairs, J is the inertia, Rs and Rr are the stator and rotor resistances, Ls and Lr are the stator and rotor inductances, and

   L is magnetizing inductance. = , =

   m

   , = .

   is the rotor speed, and u =

   r s

   (a)

   (b)

   Fig.7-Simulation results for rotor resistance

   us+ j us is the stator voltage vector, which is the system input. The load torque is TL, it is usually unknown, and in this model is assumed constant. The choice of stationary reference frame results in a simpler mathematical model and a simpler UKF design. However the UT is applicable to any state-space representation of the IM, in any reference frame, and is not limited to the one chosen here.

   To use the discrete KF, the IM model is discretized by solving the systems state equation to determine the states at the sampling instants. To avoid cross-coupling problems, the forward Euler method is used which provides an acceptable approximation of the system dynamics for a short sampling period ts. The resulting system is

   + 1 = . . + . ()

   (20)

   (a) rotor resistance with EKF (b)rotor resistance with UKF

   + =

   +

   + +

2. INDUCTION MOTOR MODEL

The IM state space model in the stator reference frame is

() (21)

+ 1 =

=

+

(16)

+ 3 2 .

2

=

+

+

. () (22)

+ 1 = () (23)

+ (17)

The state vector is chosen as

= [

] (24)

(18)

The stator current is the measured output and the measurement is

= 0 0 1

0 0 0

0 0 0 (25)

1 0 0

deliver superior performance under various operating conditions compared to EKF.

Thus, the measurement model is linear.

1. SIMULATION RESULTS

   are:

   APPENDIX

   The IM name plate data and parameters

   General UKF is a good choice in terms of performance and computational load. Only the speed estimation and rotor resistance results are compared. For an unbiased comparison, the same parameter values are used for the general UKF and EKF in each simulation. The simulation results for general UKF are shown in Fig.1 where thirteen sigma points are used and k=1.The speed estimation errors are low during both transients and steady-state operation, and its overall performance in terms of torque and noise is excellent. The observers low speed performance is tested with square wave speed reference.The low speed operation of general UKF is shown in fig.2.

   PN=0.75hp,USN=240V,fSN=60HZ,nN=1725rpm

   ,p=2,TeN=3.1Nm,Rs=2.3,Rr=2.5,Ls=Lr=0.25H

   and Lm =0.24H.

   The noise covariance matrices used are Q=diag{1.3x 10-6, 1.3x 10-6,1.4×10-5, 1.4×10-5,5.17×10-7} and R=diag{120,120} and the initial setting is selected as a zero-mean vector with an identity error covariance matrix.

   Fig.3 shows the estimation results for general UKF in slow reversal operation. Figs- 4,5,6 shows the comparison of EKF and UKF at different speeds. Fig .7 shows the rotor resistance of IM with EKF and UKF.

2. Conclusions

This paper proposes the general unscented Kalman Filters and Extended Kalman filter for IM drives state estimation.The comparison includes results for low speed operation of the drive. The results of extended kalman filter and unscented kalman filter are compared for speed an rotor resistance estimation.

The variation of rotor resistance is negligible as the thermal model is not used. If the thermal model is used the variation is upto 50% due to temperature variations.

The general UKF which provides the best performance was compared with the conventional EKF under detuned conditions. It is concluded that the UKF provides more robust performance than the conventional EKF. The simulation prove that UKF has the capability to

REFERENCES

[1] C. Lascu, I. Boldea, F. Blaabjerg, A class of speed-sensor less sliding-mode observers for high-performance induction motor drives, IEEE Trans. Industrial Electronics, vol. 56, no. 9, Sep. 2009, pp. 3394-3403.

[2]E.Wan and R. vander Merwe.The Unscented Kalman Filter.Wiley publishing,2001.

1. Arthur Gelb.Applied Optimal Estimation.M.I.T.Press,1974.

2. Tine Lefebvre and Herman Bruyninckx.Kalman Filter for Nonlinear systems:A comparison of performance.

[5]S.Julier,J.Uhlmann and H.F.Durrant-Whyte,simple derivative free non linear state observerfpr sensorless ac drives,Vol.11,no:5,pp.634-643,oct 2006.

[6]J.K.Uhlmann,simultaneous map building and localization for real time applications,transfer thesis,Univ.Oxford,U.K,1994.

[7]H.J.Kushner,Dynamical equations for optimum nonlinear filtering,,J.Different Equations,Vol.3,pp.179-190,1967

1. D. Simon, Optimal state estimation: Kalman, H_, and nonlinear approaches, John Wiley & Sons, 2006.

2. E. S. Santana, E. Bim and W. C. Amaral, A predictive algorithm for controlling speed and rotor flux of induction motor, IEEE Trans. Industrial Electronics, vol. 55, no. 2, pp..4398-4407, Dec. 2008.