688
~EEE
TRANSACTIONS ON ATJTON.~IC CONTROL, VOL.
AC-16,
NO.
6,
DECEMBER
1971
A
Tutorial
Introduction
to
Estimation and Filtering
Absiracf-In
this
tutorial paper the basic principles of least
squares
estimation are introduced and applied to
the
solution of some
filtering, prediction, and smoothing problems involving stochastic
linear dynamic systems. In particular, the paper includes derivations
of the discrete-time and Continuous-time Kalman aters and their
prediction and smoothing
counterparts,
with remarks on the modiii-
cations that
are necessary if the noise
processes are colored and cor-
related. The examination of these
state estimation
problems is pre-
ceded by a
derivation
of
both the
unconstrained
and
linear
least
estimator
one
random vector in
terms of another,
an
examination
the properties
each, with
particular attention
to
the
case
jointly Gaussian vectors. The paper
concludes
with
a
discus-
sion
the duality
between least
estimation problems and
squares optimal control problems.
T
I. INTRODUCTION
HIS paper
contains
a
tutorial
introduction
to
t,he
basic principles of least
squares estimat.ion and
their
application to
the solution of some stat.e-est.imation prob-
lems associated wit.h finite-dimensional linear
dynamic
systems
operating
in
a
stochast.ic environment.
. The ex-
position begins n-it11 t,he
problem of cstimat.ing on(: random
variable or vector
in
t,erms
of anot>her, and proceeds
through
a
derivation
of the discrete-time Kalman filter
a.nd predict.or to a
derivat.ion of the continuous-t.ime Ihl-
man-Bucy filter and predictor, and
an examination of t.he
continuous-t,ime smoothing problem. The
paper concludes
vit.11 a discussion of t,he duality
bet.wen
least squares
esti-
mation
squares opt.ima1 control problems.
The development, in each section dram nontrivially on
that in preceding semions and on the following a.ssumed
prerequisites.
1) Familiarity v,-it,h the elements of probability t.heory
through
t,he concept of jointly
distributed
random
vari-
ables and
random vectors described by
their joint. prob-
.
ability
density
function,
and t,he associated means, co-
.
variances, and
condhional expectations. For a. discussion
of these
topics, see, for example,
[l].
2)
Beginning
xvith Section VII, a.n exposure t.o t.he
state-space
description of linear
dynamic
systems,
t.he dual
concepts of
controllability and observability, and t,he
least. squares
(“linear-quadrat.ic”)
regulator
problem
in
both
finite
infinite t.ime. For an exposition of these
t.opics the reader is referred t.o the appropriate
papers
in
this issue or,
for
example, t.o
[2].
In Section
11, we esanune t.he least. squares
estimation of
one random
vector
in
t.erms of anot.her. Some important,
3Iannscript received July 19, 19il.
Paper recommended by D. G.
Luenberger, Associate Guest Editor.