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defined as 114 - [ J* [•] 2 dt /t ] l ' 2 , where T is sufficiently large to allow the response of the structure
to attenuate, and [•] is the quantity whose norm is to be calculated. Note that Ji to Ju are bridge
response quantities normalized by the corresponding uncontrolled responses, and J^ to J^are the
required actuator capacities normalized by different quantities. Hence, the control performance is better
if the value of indices J t (i =1,2,...,18) is smaller.
Evaluation criteria Ji to Jis for each control strategies are shown in Table 1, in which the result
denote the maximum value due to 3 earthquakes, i.e., El Centra, Gebze, and Mexico. Columns (2) and (6)
of Table 1 show the results for the LQG sample controller considered in [Dyke et al (2002)], except that
all the diagonal elements of the weighting matrix R are reduced from 1.0 to 0.62, so that the maximum
actuator control force is increased to 996kN (see Table 1) to improve the control performance.
For both H 2 B-EB and H 2 B-PB controllers, we use the same state-reduced order system and the
Kalman-Bucy filter as that used hi the LQG sample controller. Our control output vector z^t) is
selected as, zTz^z^z, where z T Qz was used in the objective function of the LQG sample
controller. The constraint output vector z 2 (t) in Eq. (3) is equal to u(t), i.e., z 2 (t) = u(t) = [MI, u 2 ,...,
ug] T where u l is the control force from one actuator at the ith location and it has a peak force
limitation of 1000 kN. Hence, only the constraints on the actuator peak forces are considered in the
constraint output vector z 2 (t), i.e., B l =1000 kN for i = 1, 2, ..., 8. Finally, measured output vector y(t)
in Eq.(3) is identical to that used in the LQG controller, i.e., four accelerometers are installed on the top
of the tower legs and one accelerometer on the deck at the mid-span, whereas two displacement sensors
are located between the deck and pier 2 and other two displacement sensors are located between the deck
and pier 3. Based on three earthquake records, the maximum total "energy" is computed as p 2 = 3.37
m/sec 3/2 (El Centro earthquake), whereas the maximum peak ground acceleration (PGA) is p00= 3.42
m/sec 2 (El Centro earthquake). With the input parameter 8j, P 2 and p^ above, the design parameters
for the H 2 B-EB controller are y j = y = 4.76 for i = 1,2,..., 8, whereas that for the H 2 B-PB controller are
y> =y = 68.95 for i = 1,2,..., 8 and cc=0.01.
The resulting performance indices for these two controllers are presented in Table 1. A comparison
of the results in Table 1 indicates that the performances of the proposed new controllers are superior to
that of the LQG sample controller. Peak values of control forces for all controllers in Uj (i= 1,2,..., 8)
Table 1: Evaluation Criteria and Peak Control Forces For LQG, H 2 B-EB and H 2 B-PB Controllers
Criteria
(1)
Ji
:->
J3
J 4
h
J6
J
J 8
J 9
Force
ui
U2
U 3
U4
LQG
(2)
0456
1.340
0.547
1.073
0.174
3.043
0.396
1.348
0.436
LQG
471
471
914
914
H 2 B-EB
(3)
0.454
1.225
0.494
0.964
0.162
2.406
0.394
1.192
0.398
H 2 B-PB
(4)
0.453
1.218
0.493
0.956
0.161
2.374
0.394
1.183
0.397
Criteria
(5)
JIG
Jn(xlO- z )
Ji2(xlO' J )
Jl3
Ji 4 (xi