Graphics Programs Reference
In-Depth Information
z
x
()
Ax
()
Bw
()
z
x
()
=
+
+
(9.51)
y
()
Cx
()
Dw
()
=
+
(9.52)
Manipulating Eqs. (9.51) and (9.52) yields
1
1
x
()
z
I
=
[
A
]
Bw
()
z
I
+
[
A
]
z
x
()
(9.53)
1
1
y
()
C
z
I A
=
{
[
]
BD
+
}
w
()
C
z
I A
+
[
]
z
x
()
(9.54)
It follows that the state transition matrix is
1
1
I
z
1
Φ
()
zz
I A
=
[
]
=
[
A
]
(9.55)
and the system impulse response in the z-domain is
1
h
()
C
Φ
()
z
=
BD
+
(9.56)
9.7. The LTI System of Interest
For the purpose of establishing the framework necessary for the Kalman fil-
ter development, consider the LTI system shown in
Fig. 9.18.
This system
(which is a special case of the system described in the previous section) can be
described by the following first order differential vector equations
ß
()
Ax
()
u
()
=
+
(9.57)
y
()
Gx
()
v
()
=
+
(9.58)
where
y
is the observable part of the system (i.e., output),
u
is a driving force,
and
v
is the measurement noise. The matrices
A
and
G
vary depending on the
system. The noise observation
v
is assumed to be uncorrelated. If the initial
condition vector is
x
t
()
, then from Eq. (9.36) we get
t
∫
x
()
Φ
t
=
(
)
x
t
()
Φ
t
+
(
τ
)
u
()
d
(9.59)
0
t
0
The object (abstract) is observed only at discrete times determined by the
system. These observation times are declared by discrete time where is
the sampling interval. Using the same notation adopted in the previous section,
the discrete time representations of Eqs. (9.57) and (9.58) are
nT
T
x
()
Ax
n
1
=
(
)
+
u
()
(9.60)
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