688
~ E E E
TRANSACTIONS
ON
ATJTON.~IC CONTROL,
VOL.
AC16,
NO.
6,
DECEMBER
1971
A
Tutorial
Introduction
to
Estimation and Filtering
AbsiracfIn
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 discretetime and Continuoustime 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
the
linear
least
squares
estimator
of
one
random vector in
terms of another,
and
an
examination
of
the properties
of
each, with
particular attention
to
the
case
of
jointly Gaussian vectors. The paper
concludes
with
a
discus
sion
of
the duality
between least
squares
estimation problems and
least
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.eest.imation prob
lems associated wit.h finitedimensional linear
dynamic
systemsoperatingina
stochast.ic environment.. The ex
position begins nit11 t,he
problem of cstimat.ing
on(:
random
variable or vector
in
t,erms
of
anot>her, and proceeds
throughaderivation
of the discretetime Kalman filter
a.nd predict.or
to
a
derivat.ion of the continuoust.ime Ihl
manBucy filter and predictor, and
an examination of t.he
continuoust,ime smoothing problem. The
paper concludes
vit.11
a discussion of t,he duality
bet.wen
least squares
esti
mation
and
least
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
statespace
description of linear
dynamic
systems,
t.he dual
concepts of
controllability and observability, and t,he
least. squares
(“linearquadrat.ic”)
regulator
problem
in
both finite and
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.
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 Fall '08
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 Least Squares, Normal Distribution, Variance, Probability theory

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