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Kalman fiter

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New Results in Linear Filtering and
Prediction Theory1
R. E. K A L M A N
Research Institute for Advanced
Study,2 Baltimore, Maryland

R. S. BUCY
The Johns Hopkins Applied Physics
Laboratory, Silver Spring, Maryland

1

A nonlinear differential equation of the Riccati type is derived for the covariance
matrix of the optimal filtering error. The solution of this "variance equation" completely specifies the optimal filter for either finite or infinite smoothing intervals and
stationary or nonstationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential
equations of the calculus of variations. Analytic solutions are available in some cases.
The significance of the variance equation is illustrated by examples which duplicate,
simplify, or extend earlier results in this field.
The Duality Principle relating stochastic estimation and deterministic control
problems plays an important role in the proof of theoretical results. In several examples,
the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive
systems. Some aspects of this are considered briefly.

Introduction

A T PRESENT, a nonspecialist might well regard the
Wiener-Kolmogorov theory of filtering and prediction [1, 2]3 as
"classical' —in short, a field where the techniques are well
established and only minor improvements and generalizations
can be expected.
That this is not really so can be seen convincingly from recent
results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7].
Using a variety of time-domain methods, these investigators have
solved some long-stauding problems in nonstationaryfilteringand
prediction theory. We present here a unified account of our own
independent researches during the past two years (which overlap
with much of the work [3-71 just mentioned), as well as numerous
new results. We, too, use time-domain methods, and obtain
major improvements and generalizations of the conventional
Wiener theory. In particular, our methods apply without
modification to multivariate problems.

The following is the historical background of this paper.
In an extension of the standard Wiener filtering problem, Follin
[8] obtained relationships between time-varying gains and error
variances for a given circuit configuration. Later, Hanson [9]
proved that Follin's circuit configuration was actually optimal
for the assumed statistics; moreover, he showed th...
R.
E.
KALMAN
Research Institute for Advanced
Study,
2
Baltimore, Maryland
R. S. BUCY
The Johns Hopkins Applied Physics
Laboratory, Silver Spring, Maryland
New Results
in
Linear Filtering and
Prediction Theory
1
A nonlinear differential equation of the Riccati type is derived for the covariance
matrix of the optimal filtering error. The solution of this "variance equation" com-
pletely specifies the optimal filter for either finite or infinite smoothing intervals and
stationary or nonstationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential
equations of the calculus of variations. Analytic solutions are available in some cases.
The significance of the variance equation is illustrated by examples which duplicate,
simplify, or extend earlier results in this field.
The Duality Principle relating stochastic estimation and deterministic control
problems plays an important role in the proof of
theoretical
results. In several examples,
the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive
systems. Some aspects of this are considered briefly.
1 Introduction
AT PRESENT, a nonspecialist might well regard the
Wiener-Kolmogorov theory of filtering and prediction [1, 2]
3
as
"classical' —in short, a field where the techniques are well
established and only minor improvements and generalizations
can be expected.
That this is not really so can be seen convincingly from recent
results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7].
Using a variety of time-domain methods, these investigators have
solved some long-stauding problems in
nonstationary
filtering
and
prediction theory. We present here a unified account of our own
independent researches during the past two years (which overlap
with much of the work [3-71 just mentioned), as well as numerous
new results. We, too, use time-domain methods, and obtain
major improvements and generalizations of the conventional
Wiener theory. In particular, our methods apply without
modification to multivariate problems.
The following is the historical background of this paper.
In an extension of the standard Wiener filtering problem, Follin
[8] obtained relationships between time-varying gains and error
variances for a given circuit configuration. Later, Hanson [9]
proved that Follin's circuit configuration was actually optimal
for the assumed statistics; moreover, he showed that the differen-
tial equations for the error variance (first obtained by Follin)
follow rigorously from the Wiener-Hopf equation. These results
were then generalized by Bucy [10], who found explicit rela-
tionships between the optimal weighting functions and the error
variances; he also gave a rigorous derivation of the variance
equations and those of the optimal filter for a wide class of non-
stationary signal and noise statistics.
Independently of the work just mentioned, Kalman [11] gave
1
This research was partially supported by the United States Air
Force under Contracts AF 49(638)-382 and AF 33(616)-6952 and by
the Bureau of Naval Weapons under Contract NOrd-73861.
2
7212 Bellona Avenue.
3
Numbers in brackets designate References at the end of paper.
Contributed by the Instruments and Regulators Division of THE
AMERICAN SOCIETY OP MECHANICAL ENGINEERS and presented at
the Joint Automatic Controls Conference, Cambridge, Mass.,
September 7-9, I960. Manuscript received at ASME Headquarters,
May 31, 1960 Paper No. GO—JAC-12.
a new approach to the standard filtering and prediction problem.
The novelty consisted in combining two well-known ideas:
(i) the "state-transition" method of describing dynamical sys-
tems [12-14], and
(ii) linear filtering regarded as orthogonal projection in Hilbert
space [15, pp. 150-155].
As an important by-product, this approach yielded the Duality
Principle [11, 16] which provides a link between (stochastic)
filtering theory and (deterministic) control theory. Because of
the duality, results on the optimal design of linear control systems
[13, 16, 17] are directly applicable to the Wiener problem. Dual-
ity plays an important role in this paper also.
When the authors became aware of each other's work, it was
soon realized that the principal conclusion of both investigations
was identical, in spite of the difference in methods:
Rather than to attack the Wiener-Hopf
integral equation
directly,
it is
better
to
convert
it into a nonlinear
differential equation,
whose
solution
yields the
covariance
matrix of
the
minimum
filtering
error,
which
in turn
contains
all
necessary information
for
the
design
of
the
optimal filter.
2 Summary of Results: Description
The problem considered in this paper is stated precisely in
Section 4. There are two main assumptions:
(Ai) A sufficiently accurate model of the message process is
given by a linear (possibly time-varying) dynamical system
excited by white noise.
(A2) Every observed signal contains an additive white noise
component.
Assumption (Aj) is unnecessary when the random processes in
question are sampled (discrete-time parameter); see [11]. Even
in the continuous-time case, (A2) is no real restriction since it can
be removed in various ways as will be shpwn in a future paper.
Assumption (Ai), however, is quite basic; it is analogous to but
somewhat less restrictive than the assumption of rational spectra
in the conventional theory.
Within these assumptions, we seek the best linear estimate of
the message based on past data lying in either a finite or infinite
time-interval.
The fundamental relations of our new approach consist of five
equations:
Journal
of
Basic Engineering
march 1 96 1 / 95
Copyright © 1961 by ASME
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