Aalborg Universitet
Industrial applications of the Kalman filter
a review
Auger, François ; Hilairet, Mickael; Guerrero, Josep M.; Monmasson, Eric; Orlowska-
Kowalska, Teresa; Katsura, Seiichiro
Published in:
I E E E Transactions on Industrial Electronics
DOI (link to publication from Publisher):
10.1109/TIE.2012.2236994
Publication date:
2013
Document Version
Early version, also known as pre-print
Link to publication from Aalborg University
Citation for published version (APA):
Auger, F., Hilairet, M., Guerrero, J. M., Monmasson, E., Orlowska-Kowalska, T., & Katsura, S. (2013). Industrial
applications of the Kalman filter: a review. I E E E Transactions on Industrial Electronics, 60(12).
https://doi.org/10.1109/TIE.2012.2236994
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1
Industrial Applications of the Kalman Filter:
A Review
François Auger, Senior member, IEEE, Mickael Hilairet, Member, IEEE, Josep Guerrero, Senior member, IEEE,
Éric Monmasson, Senior member, IEEE, Teresa Orlowska-Kowalska , Senior member, IEEE
and Seiichiro Katsura Member, IEEE
Abstract—The Kalman filter has received a huge interest from
the industrial electronics community and has played a key role
in many engineering fields since the 70s, ranging, without being
exhaustive, trajectory estimation, state and parameter estimation
for control or diagnosis, data merging, signal processing and
so on. This paper provides a brief overview of the industrial
applications and implementation issues of the Kalman filter in six
topics of the industrial electronics community, highlighting some
relevant reference papers and giving future research trends.
Index Terms—Kalman filter, state estimation, implementation
issues, industrial applications.
I. INTRODUCTION
M
ANY industrial applications require to measure a large
number of physical variables so as to own a sufficient
quantity and quality of information on the system state and
to ensure the required level of performance. However, the
measurement of some physical quantities may not be possible
or desired, mainly because of the cost reduction and/or the
increase in system reliability. In this context, the Kalman filter
(KF), whose 50th anniversary occured in 2010, has played a
key role in many industrial applications of the engineering
professions since the 70s, including without being exhaustive,
Manuscript received Apr. 2, 2012; Accepted for publication Dec. 3, 2012.
Copyright (c) 2009 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
F. Auger is with the “Institut de Recherche en Énergie Électrique de
Nantes-Atlantique” (IREENA) of the University of Nantes Angers Le Mans
(LUNAM), Saint-Nazaire, France (email: francois.auger@univ-nantes.fr). M.
Hilairet is now with the “Franche-Comté Electronique Mécanique Ther-
mique et Optique - Sciences et Technologies” (FEMTO-ST), University
of Franche-Comté, Belfort, France (email: mickael.hilairet@univ-fcomte.fr).
Most of this research was done while he was with the Laboratoire de Génie
Electrique de Paris, University of Paris-Sud, Gif-sur-Yvette, France. J. M.
Guerrero is with the Institute of Energy Technology, Aalborg University,
Aalborg East DK-9220, Denmark (e-mail: joz@et.aau.dk). E. Monmasson
is with the “Systèmes et Applications des Technologies de l’Information
et de l’Energie” (SATIE), University of Cergy-Pontoise, Cergy-Pontoise,
France (email: eric.monmasson@u-cergy.fr). T. Orlowska-Kowalska is with
the Institute of Electrical Machines, Drives and Measurement Systems,
Wroclaw University of Technology, Wroclaw, Poland (email: teresa.orlowska-
kowalska@pwr.wroc.pl). S. Katsura is with the Department of System
Design Engineering, Keio University, Yokohama 223-8522, Japan (email:
katsura@sd.keio.ac.jp)
trajectory estimation, state prediction for control or diagnosis,
data merging, and so on.
Many researches have been dedicated to the implementation
and performance improvement of the KF, namely the numer-
ical stability improvement, the computation time reduction or
the study of effective implementations. The main objective
of this paper, designed as a concluding paper to the Special
Section of these Transactions on the industrial applications
and implementation issues of the Kalman filter [1], is to
highlight the latest theoretical and experimental advances and
to emphasize practical implementation issues of this state
estimator.
The scope of this paper is dedicated to the KF applica-
tions in five topics covered by the industrial electronics soci-
ety, namely i) sensorless control, diagnosis and fault-tolerant
control of AC drives, ii) distributed generation and storage
systems, iii) robotics, vision and sensor fusion techniques,
iv) applications in signal processing and instrumentation and
v) real-time implementation of a KF for industrial control
systems. Therefore, this paper is organized in seven sections:
section II gives a brief overview of Kalman filtering theory,
and sections III to VI are dedicated to the items cited above.
Finally, conclusions and future trends are discussed in the last
section.
II. A BRIEF OVERVIEW OF KALMAN FILTERING THEORY
In his famous and now 50-year-old publication [2], Rudolf
Emil Kalman proposed an optimal recursive estimator of the
state of an uncertain dynamic system. Although it is based
on advanced results of probability theory, its final formulation
is remarkably simple and effective to implement on a digital
target. The first derivation was made for a discrete-time finite-
dimensional linear stochastic process
X[k+1] = A X[k] + B U[k] + G V [k] (1)
Y [k+1] = C X[k+1] + W [k+1] (2)
where X ∈ IR
n
is the state vector, U ∈ IR
l
is a deter-
ministic process input and y ∈ IR
m
is the measurement.
The two random variables V and W respectively represent
the process and the measurement noises: V bears the model
2
uncertainty, whereas W bears the sensor uncertainty and dig-
ital quantization effects. These noises are assumed to be zero
mean, white and independent of each other, with respective
covariance matrices Q anr R. All the matrices A, B, G and
C are deterministic and may also depend on time. Since the
measurement y does not exhaustively inform us on the current
situation of the process, the Kalman filter aims at providing
an estimate of the process state X. This filter is made of two
groups of equations:
• the time update equations, which try to predict the state
value at time k+1 based on the transition equation (Eq. 1)
and on the set of all the measurements until time k,
Y[k] = {Y [0], Y [1], . . . Y [k] }. This prediction is deduced
from a previously derived estimation of the state at time
k, X
e
[k]:
X
p
[k+1] = IE [X[k+1] | Y[k]] (3)
= A X
e
[k] + B U [k] (4)
P
p
[k+1] = IE
h
e
X
p
[k+1]
e
X
p
[k+1]
t
| Y[k]
i
(5)
= A P
e
[k] A
t
+ Q (6)
with
e
X
p
[k+1] = X[k+1] − X
p
[k+1] (7)
In Eq. 5 and 6, P
e
[k] and P
p
[k+1] are respectively the
estimation error covariance matrix at time k and the
prediction error covariance matrix at time k + 1. Both
provide a quantitative evaluation of the quality of this
estimation and of this prediction.
• the measurement update equations, which try to improve
the prediction X
p
[k+1] thanks to the measurement avail-
able at time k + 1:
X
e
[k+1] = X
p
[k+1] + K[k+1]
e
Y
p
[k+1] (8)
with
e
Y
p
[k+1] = Y [k+1] − C X
p
[k+1] (9)
This correction of the prediction will be optimal if the
estimation error is statistically orthogonal [3] to the mea-
surement prediction error
e
Y
p
[k+1], which is sometimes
called the measurement innovation. This way, all the
information that the current measurement Y [k+1] has
about the current value of the state and that is not
conveyed by the set of the previous measurements Y[k]
will be used to derive an estimate of X[k+1]:
IE
h
e
X
e
[k+1]
e
Y
p
[k+1]
t
i
= 0 =⇒
K[k+1] = P
p
[k+1] C
t
C P
p
[k+1] C
t
+ R
−1
(10)
The covariance matrix of the estimation error can then
be computed as
P
e
[k+1] = P
p
[k+1] − K[k+1] C P
p
[k+1] (11)
These equations are repeated at each time sample, the previous
state estimate being first used to compute a state prediction
(Eq. 4 and 6), then a new state estimation (Eq. 8, 10 and 11).
In some publications, X
p
[k+1] and X
e
[k+1] are written as
X[k+1|k] and X[k+1|k+1], but this notation increases the length
of the equations and may frighten some students. As they
can be considered as the a priori and the a posteriori state
estimates, since they can respectively be computed before and
after the availability of the measurement Y [k+1], they may
also be written as X
−
[k+1] and X
+
[k+1]. Unfortunately, the
notation of vectors and matrices is a major concern for the
understanding of discrete-time Kalman filtering.
This first derivation of the Kalman filter has been extended
to linear continuous-time finite-dimensional stochastic pro-
cesses: if the state equations can be written as
˙
X(t) = A X(t) + B U (t) + G V (t) (12)
Y (t) = C X(t) + W (t) (13)
then an optimal state estimate
ˆ
X can be obtained by a Kalman-
Bucy filter [4] defined as
˙
ˆ
X(t) = A
ˆ
X(t) + B U(t) + K(t)
y(t) − C
ˆ
X(t)
(14)
K(t) = P (t) C
t
R
−1
(15)
˙
P (t) = A P (t) + P (t) A
t
+ Q − P (t) C
t
R
−1
C P (t) (16)
Finally, the original Kalman filter has also been extended to
a discrete-time non-linear stochastic process. In such a frame-
work, the optimal Kalman filter [5] can often not be computed,
and approximations, such as the well known extended Kalman
filter, must be used. The set of all of these filters allow
engineers and researchers to solve many problems in a wide
range of applications.
To illustrate this overview with a simple example, we may
consider the case of a target moving in a one-dimensional
space whose position x(t) is observed with both an accel-
eration sensor and a position sensor. This motion observer
is called a disturbance observer in robotics [6] and an angle
tracking observer in electrical engineering [7]. From Taylor
approximations, one may modelize the target motion as a
linear stochastic process:
X[k+1] = A X[k] + G v[k] (17)
Y [k+1] = C X[k+1] + W [k+1] (18)
with A =
1 1 1
0 1 2
0 0 1
, G
t
=
1
3
3
, C
t
=
1 0
0 0
0 1
, (19)
where the three components of the state are defined as x(kT
s
),
dx
dt
(kT
s
) T
s
,
d
2
x
dt
2
(kT
s
) T
2
s
/2, T
s
being the sampling period
and where v[k] =
d
3
x
dt
3
(kT
s
) T
3
s
/6 is derived from the third-
order derivative of the position, sometimes called the jerk, and
considered as a scalar zero-mean random variable of variance
Q. Since this process is observable, a Kalman filter can be
designed from this model using Eqs. 4, 6, 8, 9, 10 and 11.
For constant values of Q and R, the Kalman correction gain
3
K goes to a constant which does not depend on the initial
value of the covariance matrices and can be computed offline,
reducing the computational cost of this Kalman filter to a
few elementary arithmetic operations. Since the measurement
noises can reasonably be regarded as uncorrelated, the R
matrix is diagonal and can be written as R = diag
σ
2
p
, σ
2
a
.
Finally, since the Kalman correction gain is left unchanged
when Q, R and P
e
(0) are all multiplied by the same scalar
[8], this means that the final value of K only depends on σ
2
p
/Q
and σ
2
a
/Q.
Other academical examples, more detailed explanations and
implementation issues may be found in [9]–[11]. Historical is-
sues may also be found in [12], [13], whereas actual industrial
applications of the KF in six fields of the industrial electronics
community are reviewed in the next sections.
III. SENSORLESS CONTROL, DIAGNOSIS, PROGNOSTIC
HEALTH MONITORING AND FAULT-TOLERANT CONTROL OF
AC DRIVES
A. Motivation and background
As in many application fields, KF have been used for over
twenty years in intelligent electrical drives for state variable
estimation. Nowadays, standard requirements for industrial
drives of induction motors (IM) or permanent magnet syn-
chronous motors (PMSM) include sensorless speed control,
which means that the system can be used without a po-
sition sensor [14], [15]. The advantages of speed/position
sensorless control are reduced hardware complexity, lower
cost, reduced size of the machine drive, elimination of the
sensor cable, better noise immunity, increased reliability and
lower maintenance requirements. Moreover, a motor without
a position sensor is more suitable in case of harsh operating
environments.
For this, the rotor speed or position has to be estimated, and
many methods are now available. Numerous estimation meth-
ods have been developed so far, based on various techniques,
like signal injection, based on rotor saliency, or algorithmic
methods, based on a motor mathematical model or on a black
box model [14]–[22]. Model-based methods are sometimes
called Fundamental Wave Models [17]. Among the algorithmic
methods, some are using state estimators, including MRAS
(Model Reference Adaptive System) [18], or state observers,
based on a deterministic approach [14]–[16], [19]–[21], while
others are using a stochastic approach based on extended
Kalman filtering (EKF), which will be discussed in details
in the next subsections.
B. Overview of sensorless control for IM
Some of the first applications of EKF for the rotor flux and
speed estimation as well as for the rotor flux and rotor time
constant estimation for IM drives can be found in [23], [24]
and [25], [26], respectively. In these works, only simulation
results were presented.
In the first research works concerned with rotor flux and
speed estimation [23], [24], [27]–[29], the motion equation
of the drive system was omitted in the model used to build
the KF, and the motor speed was considered as a randomly
varying parameter. This led to a significant speed estimation
error during transients, particularly during instantaneous load
variations, although the performance was improved in steady
state. A similar approach was used in [30], [31], where a
reduced-order EKF was applied for the rotor flux and speed
estimation.
All these methods provide an estimation of the rotor flux
and speed based on the assumption that there is no change in
the resistances of the motor windings. Similarly, none of these
studies estimated the load torque, thus the proposed solutions
showed some sensitivity to the variation of those parameters.
The state vector of the IM was extended to the rotor time
constant for the first time in [25], proposing a simultaneous
state and parameter estimation.
The rotor resistance was also estimated in [32]. However,
the rotor resistance estimation was performed by the injection
of low-amplitude high-frequencysignals into the flux reference
in the direct vector control of the IM, causing fluctuations in
the motor flux, torque, and speed.
Another approach was developed in [33], where the authors
estimate the motor speed by taking the motion equation into
consideration for the design of the EKF. The authors also
propose to estimate the rotor resistance and mechanical load
torque, thus demonstrating improved results over a wide speed
range. However, these results are sensitive to stator resistance
variations, indicating the necessity of an approach to estimate
both winding resistances of the motor as well as the load
torque.
Studies achieving the simultaneous estimation of stator
and rotor resistances in the sensorless control of IMs are
very few and show well-known difficulties in steady state,
due to a lack of identifiability of the IM model parameters.
Several approaches combining extended state observers, neural
networks, high-frequency signal injection methods or MRAS
techniques with switching models depending on the actual
operation state of the drive, were proposed, e.g. [34]. The main
drawback of these techniques is that the algorithm identifying
the resistances can only be used when the sensorless speed
control system is in steady state and not when the load torque
is largely varying or when the speed reference is changed.
So the proposed solutions can compete with a speed-sensor-
equipped drive only if accuracy in steady state is not essential
and operation under high loads and low speeds is not a
requirement.
Studies achieving the simultaneous estimation of stator and
rotor resistances for the sensorless control of IMs was reported
4
in [35], where two EKF algorithms were consecutively used
at every time step, without the need for signal injection or for
algorithm changes, as in most previous studies. This technique
was called a “braided” technique. The two EKF algorithms
have exactly the same configuration and are derived from
the same extended model, except for one state, namely the
stator resistance in one replaced by the rotor resistance in the
other. The braided EKF technique exploits the persistency of
excitation required for parameter convergence in steady state,
fulfilled by the system noise (or modeling error), as well as the
fast convergence of EKFs. An improvement of this technique
was reported in [36], where a so called bi-input EKF was pro-
posed. This algorithm consists of a single EKF algorithm using
consecutively two inputs based on two extended IM models
developed for simultaneous stator and rotor resistances. Such
a solution, requiring less memory and computation time, is
more suitable for real-time implementation
C. Overview of sensorless control for PMSM
Thanks to their ability to perform state estimation of non-
linear systems, EKFs have also found wide application for the
estimation of rotor position and speed in synchronous motor
drives. Initial attempts to combine flux linkage and position
estimation for brushless PMSM machines were frustrated by
the real-time processing power available at that time [37]–
[39]. Subsequent advances in DSP technology have allowed
these estimation principles to be effectively implemented in
[40], [41], including stator-resistance estimation joined to an
algorithm to counter the effects of flux-linkage estimation
errors caused by an incorrect value of resistance as the motor
temperature rises during continuous operation.
Although last generation floating-point DSPs can easily
overcome the EKF real-time calculations, they are not suitable
for low-cost PMSM applications. Moreover, long computation
requirements disturb other program service routines such as
fault diagnosis or custom programs implemented in industrial
products. Therefore, some efforts have been made to reduce
the computation time of EKF algorithms for PMSM by using a
reduced-order EKF [42]–[44]. A third-order EKF using back-
EMF detection algorithm is also proposed in [42] and [43], but
the output state equation used is complex. In [43], a second-
order EKF is proposed to estimate stator resistance and flux
linkage, but not for a sensorless control purpose.
Some recent achievements on the use of EKF for online
estimation of state variables in sensorless IPMSM control
applications are reported in [45], [46]. In [45], the EKF
is used for the permanent magnet flux identification of an
IPMSM, combined with a rotor speed and stator resistance
estimation performed with an MRAS technique. The authors
showed that the convergence and stability problems generally
encountered when simultaneously estimating the flux, speed
and stator parameters are avoided this way. In [46], the ease
of implementation and the robustness to parameter uncertainty
of an EKF and of an adaptive sliding-mode observer are
compared. The authors claim that for IPMSM drives, this latter
solution is much simpler.
EKF has also been proposed for the joint estimation of
mechanical variables and parameters of systems with complex
mechanical parts, including elastic couplings [47]–[52]. In
these works, the estimation of the load side speed, torsional
and load torque as well as the load side inertia have been
estimated effectively, using linear and nonlinear EKFs. In
[49], an original method was proposed for the simultaneous
estimation of mechanical state variables and of the load side
inertia. The elements of the covariance matrix Q are adapted
according to the estimated value. In [50], [51], an evolution-
nary algorithm associated with a µ-analysis for the stability
analysis of the closed-loop system was used to tune the
observer and controller. The µ-analysis theory helps to cancel
known unstable set of parameters before running iterations in
the optimization algorithm.
D. Diagnosis, prognostic health monitoring and fault-tolerant
control overview
In order to guarantee a safe and efficient operation of control
systems against various failures, computer-based failure de-
tection algorithms have been developed. Various approaches
have been applied, i.e. observer-based techniques, artificial
intelligence techniques, etc. In this context, the KF has played
a major role. See for example [25], [25], [53]–[55].
The KF relies on a system model with uncertainties that are
assumed to be Gaussian white centered random variables with
known covariance properties. Nevertheless, this assumption
is not generally satisfied and the tuning of the covariance
matrices is not obvious. This point is especially sensible
for diagnostic. In fact, the covariance matrices can provide
information about the quality of the estimates. However, if
the covariance matrices of the noise and state are not well
defined, the estimation error covariance matrix is therefore
meaningless. In practice, two methods exist for the tuning
of the KF: the first one relies on the evaluation of the state
and measurement noises, allowing to assess the quality of the
estimates with the covariance matrices. Yet, this approach is
often difficult, if not impossible [53]. The second one relies
on the tuning of the dynamic convergence with or without
autotuning methods [35], [44], [56]–[58]. In practice this latter
method is often used. In [49], [59] some guidelines for a
more systematic way of covariance matrix selection have
been proposed, including genetic algorithms. Therefore, the
evaluation of the covariance matrices Q and R which take into
account the physical approach, i.e. the model approximation
(discretization, parameters’uncertainties) and the measurement
noises (quantification error) is still an open issue.
![Fig. 1. Experimental results of a PMSM fault-tolerant contrller in the event of a power failure at no load from paper [65] (A. Akrad, M. Hilairet and D. Diallo).](/figures/fig-1-experimental-results-of-a-pmsm-fault-tolerant-2xe97lns.webp)
![Fig. 2. Extrapolation and estimation with a KF from paper [66] (O. Ondel, E. Boutleux, E. Blanco, G. Clerc).Ω1 healthy mode,Ω2 unbalance stator supply 5%, Ω3 unbalance stator supply 10%,Ω4 unbalance stator supply 20%,Ω5 unbalance stator supply 40%.](/figures/fig-2-extrapolation-and-estimation-with-a-kf-from-paper-66-o-10h1iih7.webp)
![Fig. 4. Simulation results from paper [69] (V. Smidl and Z. Peroutka). Electrical rotor speed estimation error for all investigated EKF algorithms.](/figures/fig-4-simulation-results-from-paper-69-v-smidl-and-z-gfhruwnr.webp)