Kalman
Filter
5
vector with covariance
E[u(~)u~(T)]
=
Q6(t
-
T)
where
iS(t
-
z)
is
a
Dirac delta function. The output of the measurement
system is given by
z
=
Hx
+
v
(1.17)
z
is an m-dimensional vector and
v
is an m-dimensional white noise vector
with covariance given by
E[v(r)vT(,)]
=
Rd(t
-
z)
The problem
is
to find the best estimate in the mean square sense
of
x(t),
i(tlt),
given
Z(T)
for
0
5
z
5
t.
The optimal filter is a linear dynamical system
of the form
-i(tlt)
=
Fi(t(t)
+
K(?)[Z
-
H;(tlt)]
(1.18)
whose initial state is
SO,
and
where
K(t)
=
P(t)H*R-'
(1.19)
P(t)
is
the covariance matrix
of
the optimal error
P(t)
=
E{[s(t)
-
"?(tIt)][X(t)
-
(1.20)
This covariance matrix is given by a solution to the matrix Riccati equation
P
=
FP
+
PF~
-
PH~R-~HP
+
GQG~
(1.21)
1.4
CONTINUOUS-DISCRETE-TIME KALMAN-BUCY FILTER
The linear dynamical system is described by the differential equation
[2,3]
as
dx,
=
F(t)x,
dt
+
G(t)
dp,
t
2
to
(1.22)
where
x,
is
an
12-dimensional state vector,
F
and
G
are
11
x
n
and
11
x
r con-
tinuous-matrix-time functions, and
f3,
is
an r-dimensional process noise
vector with covariance
E{d/l,
df3;')
=
Q(t)
dt
The discrete linear observations are taken at time instants
tk
and the
measurement equation is described by
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