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n
CRITERIA OF DAMPERS
CONFIGURATION EFFICIENCY
∑
max
J
=
d t
( )
(5)
G
i
0
≤ ≤
t
t
i
=
1
f
For evaluating the efficiency of control systems
with limited set of dampers, connected to Chev-
ron braces, the following criteria were proposed
(Ribakov, in press):
considering the following control forces restric-
tions
max
u t
( )
≤
U
(6)
i
i
,max
0
≤ ≤
t
t
f
NDOF
NDOF
∑
∑
d
−
d
i
,
0
i m
,
For solution of the modified LQG optimiza-
tion problem (2) and (5) the following algorithm
is proposed:
i
=
1
i
=
1
J
=
⋅
1
00%
(7)
1
NDOF
NDOF
∑
∑
d
−
d
i
,
0
i NDOF
,
i
=
1
i
=
1
•
Step 1:
k = 1
. Specifying initial values of
Q
k
=Q
0
, R
k
=R
0
.
where
d
i,0
is the peak inter-story drift at floor
i
in
a structure without dampers,
d
i,m
is the peak inter-
story drift at floor
i
in a structure with optimally
distributed dampers located at
m
active floors
and
d
i,NDOF
is the peak inter-story drift at floor
i
in a structure with optimally distributed dampers
located at all floors.
•
Step 2:
Calculation of
u(t)
,
d(t)
, which
minimize the performance index
J = J(Q
k
,
R
k
)
. For the obtained optimal solution find
the value of the global performance index
J
Gk
(5) and verify the constraints (6)
•
Step 3:
k = k +1
. Updating the values of
Q
k
,
R
k
, that decrease the global performance
index
J
Gk
under the control constraints (6).
NDOF
NDOF
∑
∑
a
−
a
•
Step 4:
Repeat steps 1 - 3 until condition
J
i
,
0
i m
,
i
=
1
i
=
1
J
=
⋅
1
00%
(8)
+
−
J
<
ε
is satisfied.
2
NDOF
NDOF
G k
,
1
G k
,
∑
∑
a
−
a
i
,
0
i NDOF
,
i
=
1
i
=
1
Following the above described algorithm, peak
optimal control forces and inter-story drifts should
be calculated at Step 2. For this reason response
of the optimal controlled structure to the artificial
white noise ground motion is simulated.
For carrying out Step 3, any parametrical
optimization numerical method can be used. The
optimization procedure (Agranovich, 2004) is
realized using MATLAB functions “dlqry”, “Kal-
man” (Control System Toolbox) and “fmincon”
(Optimization Toolbox).
where
a
i,0
is the peak acceleration at floor
i
in a
structure without dampers,
a
i,m
is the peak ac-
celeration at floor
i
in a structure with optimally
distributed dampers located at
m
active floors
and
a
i,NDOF
is the peak acceleration at floor
i
in
a structure with optimally distributed dampers
located at all floors.
BS
−
−
BS
0
m
J
100
%
=
⋅
(9)
3
BS
BS
0
NDOF
where
BS
i
is the peak base shear force in a structure
without dampers, BS
m
is the peak base shear force
in a structure with optimally distributed dampers
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