of the dynamic system.
Observability
and
controllability
are further properties of dynamic systems.
Example 1
: A video camera captures an object moving along a
straight line. Its centroid (location) is described by coordinate
x
(on this line), and its move by speed
v
and a constant
acceleration
a
. We do not consider start or end of this motion.
The process state is characterized by vector
x
= (
x, v, a
)
T
, and
we have that
˙
x
= (
v, a,
0)
T
because of
˙
x
=
v,
˙
v
=
a,
˙
a
= 0
It follows that
˙
x
=
v
a
0
=
0
1
0
0
0
1
0
0
0
·
x
v
a
This defines the
3
×
3
system matrix
A
. It follows that
det(
A

λ
I
) =

λ
3
,
i.e.
λ
1
,
2
,
3
= 0
(”very stable”)
Page 18
September 2006
.
.
Subject MI37: Kalman Filter  Intro
(D) Goal of the TimeDiscrete Filter
Given is a sequence of noisy observations
y
0
,
y
1
, . . . ,
y
t

1
for a
linear dynamic system. The goal is to estimate the (internal)
state
x
t
= (
x
1
,t
, x
2
,t
, . . . , x
n,t
)
of the system such that the
estimation error is minimized (i.e., this is a recursive estimator).
Standard Discrete Filtering Model
We assume
•
a
state transition matrix
F
t
which is applied to the (known)
previous state
x
t

1
,
•
a
control matrix
B
t
which is applied to a
control vector
u
t
, and
•
a
process noise vector
w
t
whose joint distribution is a
multivariate Gaussian distribution with variance matrix
Q
t
and
μ
i,t
=
E
[
w
i,t
] = 0
, for
i
= 1
,
2
, . . . , n
.
We also assume an
•
observation vector
y
t
of state
x
t
,
•
an
observation matrix
H
t
, and
•
an
observation noise vector
v
t
, whose joint distribution is also
a multivariate Gaussian distribution with variance matrix
R
t
and
μ
i,t
=
E
[
v
i,t
] = 0
, for
i
= 1
,
2
, . . . , n
.
Page 19
September 2006
.
.
Subject MI37: Kalman Filter  Intro
Kalman Filter Equations
Vectors
x
0
,
w
1
, . . . ,
w
t
,
v
1
, . . . ,
v
t
are all assumed to be mutually
independent.
The defining equations of a Kalman filter are as follows:
x
t
=
F
t
x
t

1
+
B
t
u
t
+
w
t
with
F
t
=
e
Δ
t
A
=
I
+
∞
X
i
=1
Δ
t
i
A
i
i
!
y
t
=
H
t
x
t
+
v
t
Note that there is often an
i
0
>
0
such that
A
i
equals a matrix
having zero in all of its components, for all
i
≥
i
0
, thus defining
a finite sum only for
F
t
.
This model is used for deriving the
standard Kalman filter
 see
below. This model represents the linear system
˙
x
=
A
·
x
with respect to time.
There exist modifications of this model, and related
modifications of the Kalman filter (not discussed in these lecture
notes).
Note that
e
x
= 1 +
∞
X
i
=1
x
i
i
!
Page 20
September 2006
.
.
Subject MI37: Kalman Filter  Intro
Continuation of Example 1
: We continue with considering
linear motion with constant acceleration. We have a system
vector
x
t
= [
x
t
, v
t
, a
t
]
T
(note:
a
t
=
a
) and a state transition
matrix
F
t
defined by the following equation:
x
t
+1
=
1
Δ
t
1
2
Δ
t
2
0
1
Δ
t
0
0
1
·
x
t
=
x
t
+ Δ
t
·
v
t
+
1
2
Δ
t
2
a
v
t
+ Δ
t
·
a
a
Note that “time
t
” is short for time
t
0
+
t
·
Δ
t
, that means,
Δ
t
is
the actual time difference between time slots
t
and
t
+ 1
.
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 Spring '17
 Derya ALTUNAY
 Normal Distribution, Probability theory, probability density function, Kalman filter