Civil Engineering Reference
In-Depth Information
y
k
¼
Cx
k
þ
Du
k
þ
v
k
(8.11)
where
x
k
¼
t
is the time-step. In the discrete-time state-space model,
A
is the discrete-time system matrix, and
B
is the discrete-time input matrix
.u
k
is the input of the system, which could be assumed to be zero in this application. As the
system is stochastic,
w
k
and
v
k
are Gaussian white noise processes. Therefore the system is restated as:
x
ð
k
Δ
t
Þ
Δ
and
¼
x
k
1
x
kþ
1
y
k
Aw
k
C
k
(8.12)
The Numerical Algorithms for Subspace State Space System Identification (N4SID) [
9
,
12
] toolbox is applied to
determine the optimal system matrices A and C in this study. Block Hankel matrices, oblique projections and singular
value decomposition are all important computational component to the N4SID algorithm for SSI. The output block Hankel
matrix is constructed and defined as follows:
0
@
1
A
y
0
y
1
...
y
j
1
...
y
i
2
...
y
i
1
...
...
...
y
iþj
3
def
Y
0
ji
1
Y
ij
2
i
1
y
i
1
y
i
...
y
iþj
2
Y
p
Y
f
Y
0
j
2
i
1
def
def
(8.13)
y
i
y
iþ
1
...
y
j
1
y
iþ
1
...
y
iþ
2
...
...
...
y
iþj
...
y
2
i
1
y
2
i
...
y
2
iþj
2
while an alternative form is:
0
@
1
A
y
0
y
1
...
y
j
1
...
y
i
2
...
y
i
1
...
...
...
y
iþj
3
y
i
1
y
i
y
i
y
iþ
1
...
...
y
iþj
2
y
j
1
def
Y
p
þ
Y
f
Y
0
ji
Y
iþ
1
j
2
i
1
Y
0
j
2
i
1
def
def
(8.14)
y
iþ
1
y
iþ
2
...
y
iþj
...
...
...
...
y
2
i
1
y
2
i
...
y
2
iþj
2
Denotation of
p
means “past” and
f
means “future” variables. The joint space of the past and future can be defined as
W
p
¼½
T
. The oblique projection, P
i
,of
Y
f
along the row space of input
U
f
onto the row space of
W
p
is
U
p
T
Y
p
T
P
i
¼
Y
f
=
U
f
W
p
. As the input
U
f
is zero in the stochastic system, P
i
, is stated as:
P
i
¼
Y
f
=
Y
p
(8.15)
The oblique projection is also equal to the product of the extended observability matrix,
Γ
i
, and the state sequence
estimate,
X
i
;
:
T
T
X
i
¼½
T
C
T
CA
i
1
P
i
¼ Γ
i
ðÞ
CA
... ð
Þ
½b
x
i
b
x
iþ
1
...b
x
iþj
1
(8.16)
The singular value decomposition of
P
i
is:
V
1
T
V
2
T
S
1
0
00
USV
T
U
1
S
1
V
1
T
P
i
¼
¼
ð
U
1
U
2
Þ
¼
(8.17)
