kalman filters for nonlinear systems

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Kalman filters for non-linear systems: a comparison of

performanceTine Lefebvre

a, Herman Bruyninckx

a& Joris De Schutter

a

aKatholieke Universiteit Leuven, Dept. of Mechanical Eng., Division P.M.A., Celestijnenlaa

300B, B-3001 Heverlee, Belgiumb

Katholieke Universiteit Leuven, Dept. of Mechanical Eng., Division P.M.A., Celestijnenlaa

300B, B-3001 Heverlee, Belgium E-mail: [email protected]

Available online: 19 Feb 2007

To cite th is article: Tine Lefebvre , Herman Bruyninckx & Joris De Schutter (2004): Kalman filters for non-linear systems: a

comparison of performance, International Journal of Control, 77:7, 639-653

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Kalman filters for non-linear systems: a comparison of performance

TINE LEFEBVREy*, HERMAN BRUYNINCKXy and JORIS DE SCHUTTERy

The Kalman filter is a well-known recursive state estimator for linear systems. In practice, the algorithm is often usedfor non-linear systems by linearizing the systems process and measurement models. Different ways of linearizing themodels lead to different filters. In some applications, these Kalman filter variants seem to perform well, while for otherapplications they are useless. When choosing a filter for a new application, the literature gives us little to rely on. Thispaper tries to bridge the gap between the theoretical derivation of a Kalman filter variant and its performance in practicewhen applied to a non-linear system, by providing an application-independent analysis of the performances of thecommon Kalman filter variants.

This paper separates performance evaluation of Kalman filters into (i) consistency, and (ii) information content of theestimates; and it separates the filter structure into (i) the process update step, and (ii) the measurement update step. Thisdecomposition provides the insights supporting an objective and systematic evaluation of the appropriateness ofa particular Kalman filter variant in a particular application.

1. Introduction

During recent decades, many research areas

looked into the matter of on-line state estimation. Theuncertainty on the state value varies over time due to

the changes in the system state (the process updates) and

due to the information in the measurements (the meas-

urement updates). The uncertainty can be represented in

different ways, e.g. by intervals or fuzzy sets.

In Bayesian estimation (Bayes 1763, Laplace

1812), a state estimate is represented by a probability

density function (pdf ). Fast analytical update algo-

rithms require the pdf to be an analytical function

of a limited number of time-varying parameters,

which is only true for some systems. A well-known

example is systems with linear process and measure-ment models and with additive Gaussian uncertain-

ties. The pdf is then a Gaussian distribution, which is

fully determined by its mean vector and covariance

matrix. This mean and covariance are updated ana-

lytically with the Kalman filter (KF) algorithm

(Kalman 1960, Sorenson 1985).

For most non-linear systems, the pdf cannot be

written as an analytical function with time-varying

parameters. In order to have a computationally

interesting update algorithm, the KF is used as an

approximation. This is achieved by linearization of

the process and measurement models of the system.

It also means that the true pdf is approximated

by a Gaussian distribution. Different ways of lineari-

zation (different KF variants) lead to different

results.This paper describes (i) how the common KF

variants differ in their linearization of the process and

measurement models; (ii) how they take the linearization

errors into account; and (iii) how the quality of their

state estimates depends on the previous two choices.

The studied algorithms are:

1. The extended Kalman filter (EKF) (Gelb et al.

1974, Maybeck 1982, Bar-Shalom and Li 1993,

Tanizaki 1996);

2. The iterated extended Kalman filter (IEKF) (Gelb

et al. 1974, Maybeck 1982, Bar-Shalom and Li

1993, Tanizaki 1996);

3. The linear regression Kalman filter (LRKF)

(Lefebvre et al. 2002). This filter comprises the

central difference filter (CDF) (Schei 1997),

the first-order divided difference filter (DD1)

(Nrgaard et al. 2000a, b) and the unscented

Kalman filter (UKF) (Uhlmann 1995, Julier and

Uhlmann 1996, 2001, Julier et al. 2000).

The paper gives the following new insights:

1. The quality of the estimates of the KF variants

can be expressed by two criteria, i.e. the consis-

tency and the information content of the estimates(defined in } 3). This paper relates the consistency

and information content of the estimates to

(i) how the linearization is performed and

(ii) how the linearization errors are taken into

account.

2. Although the filters use similar linearization

techniques for the linearization of the process

and measurement models, there can be

International Journal of Control ISSN 00207179 print/ISSN 13665820 online # 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00207170410001704998

INT. J. CONTROL, 10 MAY 2004, VOL. 77, NO. 7, 639653

Received 1 September 2003. Revised and accepted 1 April2004.

* Author for correspondence. e-mail: [email protected]

y Katholieke Universiteit Leuven, Dept. of MechanicalEng., Division P.M.A., Celestijnenlaan 300B, B-3001Heverlee, Belgium.


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a substantial difference in their performance for

both updates:

(a) for the linearization of the process update

(} 4), which describes the evolution of the

state, the state estimate and its uncertainty

are the only available information;

(b) the measurement update (} 5), on the other

hand, describes the fusion of the information

in the state estimate with the information in

the new measurement. Hence, in this update

also the new measurement is available and

can be used to linearize the measurement

model.

Therefore, it can be interesting to use different

filters for both updates.

3. Two new insights on the performance of specific

KF variants are: (i) the IEKF measurement

update outperforms the EKF and LRKF updates

if the stateor at least the part of it that causes

the non-linearity in the measurement modelisinstantaneously fully observable (} 5.2); and

(ii) for large uncertainties on the state estimate,

the LRKF measurement update yields consistent

but non-informative state estimates (} 5.3).

These insights are obtained because:

1. This paper describes all filter algorithms as the

application of the basic KF algorithm to linear-

ized process and measurement models. The differ-

ence between the KF variants is situated in the

choice of linearization and the compensation for

the linearization errors. In previous work this

linearization was not always recognized, e.g. the

UKF is originally derived as a filter which does

not linearize the models.

2. The analysis clarifies how some filters automati-

cally adapt their compensation for linearization

errors, while other filters have constant (devel-

oper-determined) error compensation.

3. Additionally, the paper compares the filter per-

formances separately for process updates and

measurement updates instead of their overall

performance when they are both combined. In

the existing literature, the performances of the

KF variants are often compared by interpretingthe estimation results for a specific application

after executing a large number of process and

measurement update steps.

The analysis starts from a general formulation of

non-linear process and measurement models, making the

results application independent. The obtained insights

are important for all researchers and developers who

want to apply a KF variant to a specific application.

They lead to a systematic choice of a filter, where pre-

viously the choice was mainly made based on success in

similar applications or based on trial and error.

Examples of 2D systems are provided. The models

are chosen such that they provide a clear graphical

demonstration of the discussed effects. For the measure-

ment update, the filters performances are system

dependent, hence, in that case several models are usedfor illustration.

2. The Kalman filter algorithm

2.1. The (linear) Kalman filter

The Kalman filter (KF) (Kalman 1960, Sorenson

1985) is a special case of Bayesian filtering theory. It

applies to the estimation of a state x if the state space

description of the estimation problem has linear process

and measurement equations subject to additive

Gaussian uncertainty

xk Fk1xk1 bk1 Ck1qp, k1 1

zk Hkxk dk Ekqm, k: 2

z is the measurement vector. The subscripts k and k 1

indicate the time step. F, b, C, H, d and E are (possibly

non-linear) functions of the system input. qp denotes the

process uncertainty, being a random vector sequence

with zero mean and known covariance matrix Q. qm is

the measurement uncertainty and is a random vector

sequence with zero mean and known covariance matrix

R; qp and qm are mutually uncorrelated and uncorre-

lated between sampling timesy. Furthermore, assume a

Gaussian prior pdf px0 with mean xx0j0 and covariance

matrix P0j0.

For this system, the pdfsz pxkjZZk1 and pxkjZZk

are also Gaussian distributions. The filtering formulas

can be expressed as analytical functions calculating the

mean vector xx and covariance matrix P of these pdfs

xxkjk1 EpxkjZZk1xk

Fk1xxk1jk1 bk1 3

Pkjk1 Epxk

jZZk1

xk xxkjk1 xk xxkjk1 Th i

Fk1Pk1jk1FTk1 Ck1Qk1C

Tk1 4

y Correlated uncertainties can be dealt with by augmentingthe state vector, this is the original formulation of the KF(Kalman 1960). Expressed in this new state vector, theprocess and measurement models are of the form (1) and (2)with uncorrelated uncertainties.

zpxkjZZj denotes the pdf of the state x at time k, given the

measurements ZZj fzz1, . . . , zzjg up to time j.

640 T. Lefebvre et al.


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xxkjk EpxkjZZkxk

xxkjk1 Kkgk 5

Pkjk EpxkjZZkxk xxkjk

xk xxkjk Th i

Inn KkHk Pkjk1 6

where

gk zzk Hkxxkjk1 dk 7

Sk EkRkETk HkPkjk1H

Tk 8

Kk Pkjk1HTk S

1k : 9

g is called the innovation, its covariance is S. K is the

Kalman gain. Equations (3)(4) are referred to as the

process update, equations (5)(9) as the measurement

update. xxkjk1 is called the predicted state estimate and

xxkjk the updated state estimate. If no measurement zzk is

available at a certain time step k, then equations (5)(9)reduce to xxkjk xxkjk1 and Pkjk Pkjk1.

2.2. Kalman filters for non-linear systems

The KF algorithm is often applied to systems with

non-linear process and measurement modelsy

xk fk1xk1 Ck1qp, k1 10

zk hkxk Ekqm, k 11

by linearization

xk Fk1xk1 bk1 q

p, k1 Ck1qp, k1 12

zk Hkxk dk qm, k Ekqm, k: 13

The difference between these models and models (1) and

(2) is the presence of the terms qp and qm representing

the linearization errors. The additional uncertainty on

the linearized models due to these linearization errors

is modelled by the covariance matrices Q and R.

Unfortunately, applying the KFz (3)(9) to systems

with non-linear process and/or measurement models

leads to non-optimal estimates and covariance matrices.

Different ways of linearizing the process and measure-

ment models, i.e. different choices for F, b, Q, H, d

and R, yield other results. This paper aims at making

an objective comparison of the performances of the

commonly used linearizations (KF variants).

3. Consistency and information content of the

estimates

The KF variants for non-linear systems calculate

an estimate xxkji and covariance matrix Pkji for a pdf

which is non-Gaussian. The performance of these KFs

depends on how representative the Gaussian pdf with

mean xxkji and covariance Pkji is for the (unknown) pdf

pxkjZZi. Figure 1 shows a non-Gaussian pdf pxkjZZi

and three possible Gaussian approximations p1xkjZZi,

p2xkjZZi and p3xkjZZi. Intuitively we feel that p1xkjZZi

is a good approximation because the same values of xare probable. Similarly p3xkjZZi is not a good approxi-

mation because a lot of probable values for x of the

original distribution have a probability density of

approximately zero in p3xkjZZi. Finally pdf p2xkjZZi

is more uncertain than pxkjZZi because a larger

domain of x values is uncertain.

These intuitive reflexions are formulated in two

criteria: the consistency and the information content of

the state estimate. The consistency of the state estimate

is a necessary condition for a filter to be acceptable. The

information content of the state estimates defines an

ordering between all consistent filters.

3.1. The consistency of the state estimate

A state estimate xxkji with covariance matrix Pkji is

called consistent if

EpxkjZZi

xk xxkji

xk xxkji Th i

Pkji: 14

For consistent results, the matrix Pkji is equal to or

larger than the expected squared deviation with respect

to the estimate xxkji under the (unknown) distribution

pxkjZZi. The mean and covariance of pdfs p1xkjZZi

and p2xkjZZi in figure 1 obey equation (14). Pdf

p3

xk

jZZi, on the other hand, is inconsistent.

Inconsistency of the calculated state estimate xxkjiand covariance matrix Pkji (divergence of the filter) is

the most encountered problem with the KF variants.

In this case, Pkji is too small and no longer represents a

reliable measure for the uncertainty on xxkji. Even more,

once an inconsistent state estimate is met, the sub-

sequent state estimates are also inconsistent. This is

because the filter believes the inconsistent state estimate

to be more accurate than it is in reality and hence it

y Models which are non-linear functions of the

uncertainties qp and qm, can be dealt with by augmenting thestate vector with the uncertainities. Expressed in this new statevector, the process and measurement models are of the form(10) and (11).

z Ck1qp, k1 of equation (1) corresponds to

qp, k1 Ck1qp, k1 of equation (12); hence instead of

Ck1Qk1CTk1 in equation (4), Q

k1 Ck1Qk1C

Tk1 is used.

Ekqm, k of equation (2) corresponds to qm, k Ekqm, k of

equation (13); hence instead of EkRkETk in equation (9),

Rk EkRkETk is used.

Kalman filters for non-linear systems 641


5/16

attaches too much weight to this state estimate when

processing new measurements.

Testing for inconsistency is done by consistency

tests such as tests on the sum of a number of normalized

innovation squared values (SNIS) (Willsky 1976, Bar-

Shalom and Li 1993).

3.2. The information content of the state estimate

The calculated covariance matrix Pkji indicates how

uncertain the state estimate xxkji is: a large covariance

matrix indicates an inaccurate (and little useful) state

estimate; the smaller the covariance matrix, the larger

the information content of the state estimate. For

example both pdfs p1xkjZZi and p2xkjZZi of figure 1

are consistent with pxkjZZi, however, p1xkjZZi has a

smaller variance, hence it is more informative than

p2xkjZZi (a smaller domain of x values is probable).

The most informative, consistent approximation is the

Gaussian with the same mean and covariance as the

original distribution, i.e. p1xkjZZi for the example.

There is a trade-offbetween consistent and informa-

tive state estimates: inconsistency can be avoided by

making Pkji artificially larger (see equation (14)).

However, making Pkji too large, i.e. larger than neces-

sary for consistency, corresponds to losing information

about the actual accuracy of the state estimate.

The different KF variants linearize the process and

the measurement models in the uncertainty region

around the state estimate. Consistent estimates are

obtained by adding process and measurement uncer-

tainty on the linearized models to compensate for the

linearization errors. In order for the estimates to be

informative, (i) the linearization errors need to be as

small as possible; and (ii) the extra uncertainty on the

linearized models should not be larger than necessary

to compensate for these errors. The following sections

describe how the extended Kalman filter, the iterated

extended Kalman filter and the linear regression

Kalman filter differ in their linearization of the process

and measurement models; how they take the lineariza-

tion errors into account; and how the qualityy of the

state estimates, expressed in terms of consistency and

information content, depends on these two choices.

4. Non-linear process models

This section contains a comparison between the

process updates of the different KF variants when deal-

ing with a non-linear process model (10) with lineariza-tion (12). The KF variants differ by their choice of Fk1,

y The more non-linear the behaviour of the process ormeasurement model in the uncertainty region around thestate estimate, the more pronounced the difference in qualityperformance (consistency and information content of the stateestimates) between the KF variants.

Figure 1. Non-Gaussian pdfpxkjZZi with three Gaussian approximations p1xkjZZi, p2xkjZZi and p3xkjZZi. p1xkjZZi and

p2xkjZZi are consistent, p3xkjZZi is inconsistent. p1xkjZZi is more informative than p2xkjZZi.



6/16

bk1 and Qk1. After linearization, they all use process

update equations (3) and (4) to update the state estimate

and its uncertainty.

Section 4.1 describes the linearization of the process

model by the EKF and IEKF, } 4.2 by the LRKF. The

formulas are summarized in table 1. Section 4.4 presents

some examples.

4.1. The (iterated) extended Kalman filterThe EKF and the IEKF linearizey the process model

by a first-order Taylor series around the updated state

estimate xxk1jk1

Fk1 @fk1@x

xxk1jk1

15

bk1 fk1xxk1jk1 Fk1xxk1jk1: 16

The basic (I)EKF algorithms do not take the lineariza-

tion errors into account (n is the dimension of the state

vector x)

Q

k1 0nn: 17

This leads to inconsistent state estimates when these

errors cannot be neglected.

4.2. The linear regression Kalman filter

The linear regression Kalman filter (LRKF) uses

the function values of r regression points Xjk1jk1 in

state space to model the behaviour of the process

function in the uncertainty region around the updated

state estimate xxk1jk1. The regression points are chosen

such that their mean and covariance matrix equal the

state estimate^xxk1jk1 and its covariance matrixPk1jk1. The CDF, DD1 and UKF filters correspond

to specific choices. The function values of the regression

points are

Xjkjk1 fk1X

jk1jk1: 18

The LRKF algorithm uses a linearized process

function (12) where Fk1, bk1 and Qk1 are obtained

by statistical linear regression through the

Xjk1jk1, X

jkjk1 points, j 1, . . . , r; i.e. the deviations

ej between the function values in Xjk1jk1 for the

non-linear and the linearized function are minimized in

least-squares sense

ej Xjkjk1 FX

jk1jk1 b 19

Fk1, bk1 arg minF, b

Xrj1

eTj ej: 20

The sample covariance of the deviations ej

Qk1 1

r

Xrj1

ejeT

j 21

gives an idea of the magnitude of the linearization errors

in the uncertainty region around xxk1jk1.

Intuitively we feel that when enoughz regression

points are taken, the state estimates of the LRKF proc-

ess update are consistent and informative. They are

consistent because Qk1 gives a well-founded approxi-mation of the linearization errors (equation (21)). They

are informative because the linearized model is a good

approximation of the process model in the uncertainty

region around xxk1jk1 (equations (19) and (20)).

4.3. Extra process uncertainty

In all of the presented filters, the user can decide to

add extra process uncertainty Qk1 (or to multiply the

calculated covariance matrix Pkjk1 by a fading factor

larger than 1 (Bar-Shalom and Li 1993)). This is useful if

the basic filter algorithm is not consistent. For example,

this is the case for the (I)EKF or for an LRKF with anumber of regression points too limited to capture the

y The EKF and IEKF only differ in their measurementupdate (} } 5.1 and 5.2).

z This depends on the non-linearity of the model in theuncertainty region around the state estimate. A possibleapproach is to increase the number of regression points untilthe resulting linearization (with error covariance) does notchange any more. Of course, because the true pdf isunknown, it is not possible to guarantee that the iterationhas converged to a set of regression points representative forthe model behaviour in the uncertainty region.

Fk1 bk1 Qk1

EKF@fk1@x

xxk1jk1

fk1xxk1jk1 Fk1xxk1jk1 0nn

IEKF@fk1@x

xxk1jk1

fk1xxk1jk1 Fk1xxk1jk1 0nn

LRKF arg minF, bPr

j1 eT

j ej argminF, bPr

j1 eT

j ej1r

Prj1 eje

Tj

Table 1. Summary of the linearization of the process model by the extended Kalman filter (EKF),the iterated extended Kalman filter (IEKF) and the linear regression Kalman filter (LRKF).



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non-linear behaviour of the process model in the uncer-

tainty region around xxk1jk1.

For a particular problem, values for Qk1 that result

in consistent and informative state estimates are

obtained by off-line tuning or on-line parameter learning

(adaptive filtering, Mehra 1972). In many practical cases

consistency is assured by taking the added uncertainty

too large, e.g. by taking a constant Q over time which

compensates for decreasing linearization errors. This,

however, results in less informative estimates.

4.4. Examples

The different process updates are illustrated by a

simple 2D non-linear process model (xi denotes the

ith element of x)

xk1 xk11 2

xk2 xk11 3xk12)

22

with no process uncertainty: qp, k1 02 1. xk1

depends non-linearly on xk1. The process update of

xk2 is linear. The updated state estimate and its uncer-

tainty at time step k 1 are

xxk1jk1 10

15

" #; Pk1jk1

36 0

0 3600

" #: 23

4.4.1. Monte Carlo simulation. The mean value and

the covariance of the true pdf pxkjZZk1 are calculated

with a (computationally expensive) Monte Carlo

simulation based on a Gaussian pdf pxk1jZZk1 with

mean^xxk1jk1 and covariance matrix Pk1jk1. Theresults of this computation are used to illustrate the

(in) consistency and information content of the state

estimates of the different KF variants. The mean and

covariance matrix of the pxkjZZk1 calculated by the

Monte Carlo algorithm are

xxkjk1 136

55

" #; Pkjk1

16 9 94 721

721 32 4 36

" #: 24

4.4.2. (I)EKF. Figure 2 shows the updated and (I)EKF

predicted state estimates and their uncertainty ellipsesy.

The dotted line is the uncertainty ellipse of the distri-bution obtained by Monte Carlo simulation. The IEKF

y The uncertainty ellipsoid

xk xxkjiTP

1kjixk xxkji 1 25

is a graphical representation of the uncertainity on the stateestimate xxkji. Starting from the point xxkji, the distance to theellipse in a direction is a measure for the uncertainty on xxkji inthat direction.

Figure 2. Non-linear process model. Uncertainty ellipses for the updated state estimate at k 1 (dashed line), for the (I)EKFpredicted state estimate (full line) and the Monte Carlo uncertainty ellipse (dotted line). The predicted state estimateis inconsistent due to the neglected linearization errors: the uncertainty ellipse of the IEKF predicted estimate is shiftedwith respect to the Monte Carlo uncertainty ellipse and is somewhat smaller.



8/16

state prediction and its covariance matrix are

xxkjk1 100

55

!; Pkjk1

14 400 720

720 32 4 36

!: 26

Due to the neglected linearization errors, the state esti-

mate is inconsistent: the covariance Pkjk1 is smaller

than the covariance calculated by the Monte Carlo

simulation for the first state component x1 which

had a non-linear process update. For consistent results

this covariance should even be larger because the IEKF

estimate xxkjk1 differs from the mean of the pdf

pxkjZZk1, calculated by the Monte Carlo simulation.

4.4.3. LRKF. Figure 3 shows the Xjk1jk1 points (top

figure) and the Xjkjk1 points (bottom figure), the

updated state estimate (top) and the predicted state

Figure 3. Non-linear process model. Uncertainty ellipses for the updated state estimate at k 1 (dashed line, top figure), for theLRKF predicted state estimate (full line, bottom figure), and Monte Carlo uncertainty ellipse (dotted line which coincideswith the full line, bottom figure). The LRKF predicted state estimate is consistent and informative: its uncertainty ellipse

coincides with the Monte Carlo uncertainty ellipse.



9/16

estimate (bottom) and their uncertainty ellipses for

the LRKF. The Xjk1jk1 points are chosen with the

UKF algorithm of Julier and Uhlmann (1996)

where 3 n 1. This corresponds to choosing six

regression points, including two times the point xxk1jk1.

The uncertainty ellipse obtained by Monte Carlo simu-

lation coincides with the final uncertainty ellipse of

the LRKF predicted state estimate (bottom figure).

This indicates consistent and informative results. The

LRKF predicted state estimate and its covariancematrix are

xxkjk1 136

55

!; Pkjk1

16 992 720

720 32 4 36

!: 27

4.5. Conclusion: the process update

The LRKF performs better than the (I)EKF when

dealing with non-linear process functions:

1. The LRKF linearizes the function based on its

behaviour in the uncertainty region around the

updated state estimate. The (I)EKF on the other

hand only uses the function evaluation and itsJacobian in this state estimate.

2. The LRKF deals with linearization errors in a

theoretically founded way (provided that enough

regression points are chosen). The (I)EKF on the

other hand needs trial and error for each particu-

lar example to obtain good values for the covari-

ance matrix representing the linearization errors.

3. Unlike the (I)EKF, the LRKF does not need the

function Jacobian. This is an advantage where

this Jacobian is difficult to obtain or non-existing

(e.g. for discontinuous process functions).

5. Non-linear measurement models

The previous section contains a comparison between

the (I)EKF and LRKF process updates; this section

focuses on their measurement updates for a non-linear

measurement model (11) with linearization (13). The

EKF, IEKF and LRKF choose Hk, dk and Rk in a

different way. After linearization they use the KF

update equations (5)(9).

The linearization of the measurement model by the

IEKF (} 5.2) takes the measurement into account; the

EKF (} 5.1) and LRKF (} 5.3) linearize the measure-

ment model based only on the predicted state estimate

and its uncertainty. For the latter filters, the lineariza-

tion errors (Rk) are larger, especially when the measure-

ment function is quite non-linear in the uncertainty

region around the predicted state estimate. A large

uncertainty on the linearized measurement model Rk

EkRkETk (due to a large uncertainty on the state esti-mate) results in throwing away the greater part of the

information of the possibly very accurate measurement.

The different linearization formulas are summarized in

table 2. Section 5.5 presents some examples.

5.1. The extended Kalman filter

The EKF linearizes the measurement model around

the predicted state estimate xxkjk1

Hk @hk@x xxkjk1 28

dk hkxxkjk1 Hkxxkjk1: 29

The basic EKF algorithm does not take the linearization

errors into account

Rk 0mm 30

where m is the dimension of the measurement vector zk.

If the measurement model is non-linear in the uncer-

tainty region around the predicted state estimate, the

linearization errors are not negligible. This means that

the linearized measurement model does not reflect the

relation between the true state value and the measure-ment. For instance, the true state value is far fromy the

linearized measurement model. After processing the

measured value, given the linear measurement model

and the measurement uncertainty, the state is believed

y Far from (and close to) must be understood as: thedeviation of the true state with respect to the linearizedmeasurement model is not justified (is justified) by the

measurement uncertainty ~RRk ~EEk ~RRkETk .

Hk dk Rk

EKF @hk

@x

xxkjk1hkxxkjk1 Hkxxkjk1 0mm

IEKF @hk@x

xxkjkhkxxkjk Hkxxkjk 0mm

LRKF minH, dPr

j1 eT

j ej minH, dPr

j1 eT

j ej1r

Prj1 eje

Tj

Table 2. Summary of the linearization of the measurement model by the extended Kalman filter(EKF), the iterated extended Kalman filter (IEKF) and the linear regression Kalman filter (LRKF).



10/16

to be in a region which does not include the true state

estimate, i.e. the updated state estimate is inconsistent.

5.2. The iterated extended Kalman filter

The EKF of the previous section linearizes the

measurement model around the predicted state estimate.

The IEKF tries to do better by linearizing the measure-

ment model around the updated state estimate

Hk @hk

@x

xxkjk

31

dk hkxxkjk Hkxxkjk: 32

This is achieved by iteration: the filter first linearizes

the model around a value xx0kjk (often taken equal to

the predicted state estimate xxkjk1) and calculates the

updated state estimate. Then, the filter linearizes the

model around the newly obtained estimate xx1kjk and cal-

culates a new updated state estimate (based on xxkjk1,

Pkjk1 and the new linearized model). This process

is iterated until a state estimate xxikjk is found for which

xxikjk is close to xx

i1kjk . The state estimate xxkjk and uncer-

tainty Pkjk are calculated starting from the state estimate

xxkjk1 with its uncertainty Pkjk1 and the measurement

model linearized around xxikjk.

Like the EKF algorithm, the basic IEKF algorithm

does not take the linearization errors into account

Rk 0mm: 33

If the measurement model is non-linear in the uncer-

tainty region around the updated state estimate xxkjk,

state estimates will be inconsistent. In case of a meas-urement model that instantaneously fully observes the

state (or at least the part of the state that causes the

non-linearities in the measurement model), the lineariza-

tion errors will be smally in the uncertainty region

around xxkjk. The true state estimate is then close to

the linearized measurement function and the updated

state estimate is consistent. The result is also informa-

tive because no uncertainty due to linearization errors

needs to be added.

5.3. The linear regression Kalman filter

The LRKF evaluates the measurement function in r

regression points Xjkjk1 in the uncertainty region

around the predicted state estimate xxkjk1. The Xjkjk1

are chosen such that their mean and covariance matrix

are equal to the predicted state estimate xxkjk1 and its

covariance Pkjk1. The CDF, DD1 and UKF filters

correspond to specific choices. The function values

of the regression points through the non-linear func-

tion are

Zjk hkX

jkjk1: 34

The LRKF algorithm uses a linearized measurement

function (13) where Hk, dk and Rk are obtained by

statistical linear regression through the points

Xjkjk1, Z

jk, j 1, . . . , r. The statistical linear regres-

sion is such that the deviations ej between the non-linear

and the linearized function in the regression points are

minimized in least-squares sense

ej Zjk HX

jkjk1 d

35

Hk, dk arg minH, d

Xrj1

eTj ej: 36

The sample covariance matrix of the deviations ej givesan idea of the magnitude of the linearization errors

Rk 1

r

Xrj1

ejeT

j : 37

Intuitively we feel that when enough regression

points (Xjkjk1, Z

jk) are taken the state estimates of

the LRKF measurement update are consistent because

Rk gives a well-founded approximation of the lineariza-

tion errors (equation (37)). However, if the measurement

model is highly non-linear in the uncertainty region

around xxkjk1, the (Xjkjk1, Z

jk) points deviate substan-

tially from a hyperplane. This results in a large R

k andnon-informative updated state estimates (see the

example in } 5.5).

5.4. Extra measurement uncertainty

In order to make the state estimates consistent, the

user can tune an inconsistent filter by adding extra

measurement uncertainty Rk.

Only off-line tuning or on-line parameter learning

can lead to a good value for Rk for a particular prob-

lem. In many practical cases consistency is assured by

choosing the added uncertainty too large, e.g. by taking

a constantR

over time which compensates for decreas-

ing linearization errors. This reduces the information

content of the results.

5.5. Examples

5.5.1. First example. The comparison between the dif-

ferent measurement updates is illustrated with the meas-

urement function zk h1xk m, k

h1xk xk1 2 xk2

2 38

y This assumes that the iterations lead to an accurate xxikjk.

The linearizations are started around a freely choosen xx0kjk. In

order to assure quick and correct iteration, (part of) this valuecan be chosen based on the measurement information if thisinformation is more accurate than the predicted state estimate.



11/16

where

xk 15

20

" #

is the true value and

^xxkjk1

10

15" #

is the predicted state estimate with covariance matrix

Pkjk1 36 0

0 3600

" #:

The processed measurement is zzk 630 and the meas-

urement covariance is Rk 400.

5.5.2. Second example. To illustrate the consistency of

the state estimate of an IEKF when the measurement

observes the state completely, a second example is used.

The measurement function is

zk hxk qm, k h1xk qm, k1

h2xk qm, k2

" #39

with

h1xk xk1 2 xk2

2

h2xk 3 xk2 2=xk1

)40

where

xk 15

20

" #

is the true value and

xxkjk1 10

15

" #

is the predicted state estimate with covariance matrix

Pkjk1 36 0

0 3600

" #:

The processed measurement and the measurement

covariance matrix are

zzk 630

85" #; Rk 400 0

0 400" #: 41

In all figures, the true state value xk is plotted; if

this value is far outside the uncertainty ellipse of a

state estimate, the corresponding estimate is inconsis-

tent. Because the measurement is accurate and the

initial estimate is not, the uncertainty on the state esti-

mate should drop considerably when the measurement

is processed. The updated state estimate is not informa-

tive if this is not the case.

5.5.3. EKF. Figure 4 shows the state estimates, uncer-

tainty ellipses and measurement functions for the EKF

applied to the first example (equation (38)). The true

measurement function is non-linear. xk is the true

value of the state, and is close to this function. The

linearization around the uncertain predicted state esti-

mate is not a good approximation of the function

around the true state value: the true state value is farfrom the linearized measurement function. The result-

ing updated state estimate

xxkjk 10

25

" #; Pkjk

36 24

24 16

" #42

is inconsistent.

5.5.4. IEKF. Figure 5 shows the measurement function,

the linearized measurement function around the point

xxikjk, the true state value xk and the state estimates for

the IEKF applied to the first example (equation (38)).

The measurement model does not fully observe the state.

This results in an uncertain updated state estimate xxikjkaround which the filter linearizes the measurement func-

tion. As was the case for the EKF, the linearization

errors are not negligible and the true value is far

from the linearized measurement function. The updated

state estimate

xxkjk 10

23

" #; Pkjk

36 16

16 7:0

" #43

is inconsistent.

If however the measurement model fully observes the

state, the IEKF updated state estimate is accurately

known; hence, the linearization errors are small andthe true state value is close to the linearized measure-

ment function. In this case, the updated state estimate is

consistent. Figure 6 shows the measurement function,

the linearized measurement function, the true state

value xk, the state estimates and the uncertainty ellipses

for the IEKF applied to the second example (equations

(39) and (40)). The updated state estimate and covari-

ance matrix

xxkjk 14

21

" #; Pkjk

2:6 1:7

1:7 1:3

" #44

are consistent and informative due to the small, ignored,linearization errors.

5.5.5. LRKF. An LRKF is run on the first example

(equation (38)). The Xjkjk1 points are chosen with the

UKF algorithm with 3 n 1 (Julier and Uhlmann

1996). This corresponds to choosing six regression

points, including two times the point xxkjk1. Figures 7

and 8 show the non-linear measurement function, the

Xjkjk1-points and the linearization. The predicted



12/16

Figure 5. Non-linear measurement model zz h1xk that does not observe the full state, and its IEKF linearization around xxkjk(dotted lines). The true state xk is far from this linearization, leading to an inconsistent state estimate xxkjk (uncertaintyellipse in full line).

Figure 4. Non-linear measurement model zz h1xk and EKF linearization around xxkjk1 (dotted lines). The true state xk is farfrom this linearization and the obtained state estimate xxkjk (uncertainty ellipse in full line) is inconsistent.



13/16

Figure 6. Non-linear measurement model zz hxk that fully observes the state, and its IEKF linearization around xxkjk (dottedlines). The true state xk is close to this linearization, leading to a consistent state estimate (uncertainty ellipse in full line).

Figure 7. Non-linear measurement model zz h1x and LRKF linearization. The linearization errors are large.



14/16

state estimate is uncertain, hence the Xjkjk1-points are

widespread. Rk is large (Rk 2:6 10

7) due to the large

deviations between the (Xjkjk1, Z

jk) points and the

linearized measurement function (see figure 8). The

updated state estimate and its covariance matrix are

xxkjk 10

2:6

" #; Pkjk

36 0

0 3600

" #: 45

Figure 9 shows the Xjkjk1 points, the measurement

function, the LRKF linearized measurement function,

the true state value xk, the state estimates and the

uncertainty ellipses. The updated state estimate is

consistent. However, it can hardly be called an improve-

ment over the previous state estimate (Pkjk % Pkjk1).

The information in the measurement is neglected due

to the high measurement uncertainty Rk EkRkETk

on the linearized function.

Figure 9. Non-linear measurement model zz h1x and LRKF linearization (dotted lines). The large linearization errors result

in a large measurement uncertainty Rk EkRkETk . The updated state estimate (uncertainty ellipse in full line) is consistent

but non-informative.

Figure 8. Figure 7 seen from another angle.



15/16

Note that some kind of iterated LRKF (similar to

the iterated EKF) would not solve this problem: the

updated state estimate xxkjk and its covariance matrix

Pkjk are more or less the same as the predicted state

estimate xxkjk1 and its covariance matrix Pkjk1.

Hence, the regression points and the linearization

would be approximately the same after iteration.

5.6. Conclusion: the measurement update

Measurements which fully observe the part of the state

that makes the model non-linear, are best processed by the

IEKF. In this case (and assuming that the algorithm

iterates to a good linearization pointy), the IEKF lin-

earization errors are negligible.

In the other cases, none of the presented filters out-

performs the others. A filter should be chosen for each

specific application: the LRKF makes an estimate of its

linearization errors (Rk), the EKF and IEKF on the

other hand require off-line tuning or on-line parameter

learning of R

kto yield consistent state estimates.

Because the IEKF additionally takes the measurement

into account when linearizing the measurement model,

its linearization errors are smaller than those of the

EKF and LRKF. This means that once a well-tuned

IEKF is available, the state estimates it returns can

be far more informative than those of the LRKF or a

well-tuned EKF.

Finally, note that the LRKF does not use the

Jacobian of the measurement function, which makes

it possible to process discontinuous measurement

functions.

6. Conclusions

This paper gives insight into the advantages and draw-

backs of the extended Kalman filter (EKF), the iterated

extended Kalman filter (IEKF) and the linear regression

Kalman filter (LRKF). These insights are a result of the

distinct analysis approach taken in this paper:

1. The paper describes all filter algorithms as the

application of the basic KF algorithm to linear-

ized process and measurement models. The differ-

ence between the KF variants is situated in the

choice of linearization and the compensation for

the linearization errors. In previous work thislinearization was not always recognized, e.g. the

UKF is originally derived as a filter which does

not linearize the models.

2. The analysis clarifies how some filters automati-

cally adapt their compensation for linearization

errors, while other filters have constant (devel-

oper-determined) error compensation.

3. The quality of the state estimates is expressed by

two criteria: the consistency and the information

content of the estimates. This paper relates the

consistency and information content of the esti-mates to (i) how the linearization is performed

and (ii) how the linearization errors are taken

into account. The understanding of the lineariza-

tion processes allows us to make a well-founded

choice of filter for a specific application.

4. The performance of the different filters is com-

pared for the process and measurement updates

separately, because a good performance for one

of these updates does not necessarily mean a good

performance for the other update. This makes it

interesting in some cases to use different filters

for both updates.For process updates the LRKF performs better

than the other mentioned KF variants because

(i) the LRKF linearizes the process model based

on its behaviour in the uncertainty region around

the updated state estimate. The (I)EKF on the

other hand only uses the function evaluation

and its Jacobian in this state estimate; and

(ii) the LRKF deals with linearization errors in a

theoretically founded way, provided that enough

regression points are chosen. The (I)EKF on the

other hand needs trial and error for each particu-

lar application to obtain a good covariancematrix representing the linearization errors.

The IEKF is the best way to handle non-linear

measurement models that fully observe the part

of the state that makes the measurement model

non-linear. In the other cases, none of the pres-

ented filters outperforms the others: the LRKF

makes an estimation of the linearization errors;

the EKF and IEKF on the other hand require

extensive off-line tuning or on-line parameter

learning in order to yield consistent state esti-

mates. However, unlike the EKF and LRKF,

the IEKF additionally uses the measurement

value in order to linearize the measurementmodel. Hence, its linearization errors are smaller

and once a well-tuned IEKF is available, the state

estimates it returns can be far more informative

than those of the LRKF or a well-tuned EKF.

The insights described in this paper are important

for all researchers and developers who want to apply a

KF variant to a specific application. They lead to a

systematic choice of a filter, where previously the choice

y This assumes that the iterations lead to an accurate xxikjk.

The linearizations are started around a freely choosen xx0kjk. In

order to assure quick and correct iteration, (part of) this valuecan be chosen based on the measurement information if thisinformation is more accurate than the predicted state estimate.



16/16

was mainly made based on success in similar applica-

tions or based on trial and error. Further work should

report on practical applications using these insights

and an effort should be made to analyse and include

future KF algorithms in this framework.

Acknowledgements

T. Lefebvre is a Postdoctoral Fellow of the Fund

for Scientific ResearchFlanders (FWO) in Belgium.

Financial support by the Belgian Programme on

Inter-University Attraction Poles initiated by the

Belgian StatePrime Ministers OfficeScience

Policy Programme (IUAP), and by K. U. Leuvens

Concerted Research Action GOA/99/04 are gratefully

acknowledged.

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