In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying) matrices
[
A
(
t
)
,
B
(
t
)
,
C
(
t
)
,
D
(
t
)
]
{\displaystyle [A(t),B(t),C(t),D(t)]}
such that
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t)}
with
(
u
(
t
)
,
y
(
t
)
)
{\displaystyle (u(t),y(t))}
describing the input and output of the system at time
t
{\displaystyle t}
.
Contents
LTI System
For a linear time-invariant system specified by a transfer matrix,
H
(
s
)
{\displaystyle H(s)}
, a realization is any quadruple of matrices
(
A
,
B
,
C
,
D
)
{\displaystyle (A,B,C,D)}
such that
H
(
s
)
=
C
(
s
I
−
A
)
−
1
B
+
D
Canonical realizations
Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
H
(
s
)
=
n
3
s
3
+
n
2
s
2
+
n
1
s
+
n
0
s
4
+
d
3
s
3
+
d
2
General System
D = 0
If we have an input
u
(
t
)
{\displaystyle u(t)}
, an output
y
(
t
)
{\displaystyle y(t)}
, and a weighting pattern
T
(
t
,
σ
)
{\displaystyle T(t,\sigma )}
then a realization is any triple of matrices
[
A
(
t
)
,
B
(
t
)
,
C
(
t
)
System identification
System identification techniques take the experimental data from a system and output a realization. Such techniques can utilize both input and output data (e.g. eigensystem realization algorithm) or can only include the output data (e.g. frequency domain decomposition). Typically an input-output technique would be more accurate, but the input data is not always available.