In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.
The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)
where
x
is the "state vector",
y
is the "output vector",
u
is the "input (or control) vector",
A
is the "state matrix",
B
is the "input matrix",
C
is the "output matrix",
D
is the "feedthrough (or feedforward) matrix".
Similarly, a discrete-time linear control system can be described as
x
(
k
+
1
)
=
A
x
(
k
)
+
B
u
(
k
)
y
(
k
)
=
C
x
(
k
)
+
D
u
(
k
)
with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices
(
A
,
B
,
C
,
D
)
. Let the order of the system be
n
.
Then, the Kalman decomposition is defined as a transformation of the tuple
(
A
,
B
,
C
,
D
)
to
(
A
^
,
B
^
,
C
^
,
D
^
)
as follows:
A
^
=
T
−
1
A
T
B
^
=
T
−
1
B
C
^
=
C
T
D
^
=
D
T
is an
n
×
n
invertible matrix defined as
T
=
[
T
r
o
¯
T
r
o
T
r
o
¯
T
r
¯
o
]
where
T
r
o
¯
is a matrix whose columns span the subspace of states which are both reachable and unobservable.
T
r
o
is chosen so that the columns of
[
T
r
o
¯
T
r
o
]
are a basis for the reachable subspace.
T
r
o
¯
is chosen so that the columns of
[
T
r
o
¯
T
r
o
¯
]
are a basis for the unobservable subspace.
T
r
¯
o
is chosen so that
[
T
r
o
¯
T
r
o
T
r
o
¯
T
r
¯
o
]
is invertible.
By construction, the matrix
T
is invertible. It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then
T
=
T
r
o
, making the other matrices zero dimension.
By using results from controllability and observability, it can be shown that the transformed system
(
A
^
,
B
^
,
C
^
,
D
^
)
has matrices in the following form:
A
^
=
[
A
r
o
¯
A
12
A
13
A
14
0
A
r
o
0
A
24
0
0
A
r
o
¯
A
34
0
0
0
A
r
¯
o
]
B
^
=
[
B
r
o
¯
B
r
o
0
0
]
C
^
=
[
0
C
r
o
0
C
r
¯
o
]
D
^
=
D
This leads to the conclusion that
The subsystem
(
A
r
o
,
B
r
o
,
C
r
o
,
D
)
is both reachable and observable.
The subsystem
(
[
A
r
o
¯
A
12
0
A
r
o
]
,
[
B
r
o
¯
B
r
o
]
,
[
0
C
r
o
]
,
D
)
is reachable.
The subsystem
(
[
A
r
o
A
24
0
A
r
¯
o
]
,
[
B
r
o
0
]
,
[
C
r
o
C
r
¯
o
]
,
D
)
is observable.
