Geology Reference
In-Depth Information
changing the parameters according to the design
conditions. To design the active control system the
following input parameters have been selected:
PGA = 0.3g, BW = 30 Hz,
t
f
= 50 sec.
After the artificial earthquake is generated,
the structure is calibrated based on the modi-
fied LQG method (Agranovich, 2010) described
below. Control energy distribution between the
floors where active control devices are attached
is obtained from the calibration of the building
with simulated ground motion. At the next stage,
based on this energy distribution, the quasi-optimal
dampers' location at the stories of the structure
is found. The basic assumption for the dampers
location stage is that it is more efficient to place
dampers at stories with maximum control energy
contribution.
The algorithm includes a set of successive
improving steps (Agranovich, 2010). At each
of these steps a new floor becomes “active” ac-
cording to the above described assumption, and
the dampers' distribution at the “active” floors is
updated. After updating the number of “active”
floors structural response to the artificial ground
motion is simulated. Then, based on the peak con-
trol forces at every story, the number of dampers
per each of the “active” floors is obtained.
According to the LQG approach, a system,
described by Equation (1) and the optimal control
forces
u
( )
should minimize the performance
index (2). An additional assumption is that the
optimal feedback is a function of the measurement
vector, containing the noised floor accelerations
y t
=
H d t
+
D u t
+
F x t
+
v t
( )
( )
( )
( )
( )
m
m
m
m g
(3)
Matrices
H
m
, D
m
and
F
m
describe the param-
eters of the measurement subsystem. Detailed
description of these parameters is given in (Spen-
cer, 1999). The ground acceleration
x t
g
( )
and the
measurement noise
v(t)
are assumed to be station-
ary white noises with known intensities. Hence
the optimal control force
u(t)
is a function of the
measurement vector, containing the noised floors
accelerations. The optimal control force is calcu-
lated as
ˆ
( )
u t
( )
= − ⋅
K d t
(4)
where
K
is the optimal feedback gains matrix,
depending on dampers distribution and perfor-
mance index
J
given in Equation (2),
ˆ
( )
d t
is the
optimal estimation of structure's state vector
d(t)
,
generated by Kalman filter algorithm using the
floors' accelerations
y
m
(t)
.
The performance index, defined in Equation
(2), contains weighting parameters matrices
Q
nxn
and
R
mxm
. In this investigation matrices
Q
and
R
are assumed to be diagonal with positive diagonal
elements. The dimension
m
equals to the number
of floors with LQG active devices. It is obvious,
that the performance index (2) is a function of
the weighting parameters
Q
and
R
. The optimal
choice of their values, according to modified
LQG optimization method (Agranovich et al.
2004), is performed by minimizing the global
performance index
MODIFIED LQG METHOD
The modified LQG method is based on the state
model of a structure
d t
=
Ad t
+
Bu t
+
Ex t
g
(1)
( )
( )
( )
( )
and the performance index
T
∫
J
=
lim
E
d t Qd t
T
( )
( )
+
u t Ru t dt
T
( )
( )
1
T
T
→∞
0
(2)
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