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semi-active and hybrid control systems. An extensive review of various control systems, including their
advantages and limitations, can be found in [ He et al (2002) ]. Based on the latter approach, a
benchmark problem for the seismic response control of a cable-stayed bridge has been established
recently [ Dyke et al (2000) ].
For the benchmark cable-stayed bridge, control systems, such as actuators, semi-active dampers,
passive viscous dampers, etc , can be installed between the deck and the towers or piers to reduce the
seismic response. A sample LQG control strategy has been used when actuators are installed in the
bridge [ Dyke et al (2000) ]. In this paper, we present two H2 based control strategies for applications to
the benchmark cable-stayed bridge and their performances in reducing the seismic response of the
bridge are evaluated and compared with that of the LQG controller.
PROBLEM STATEMENTS
The equation of motion of an n-DOF structure equipped with control systems and subject to earthquake
or wind excitations can be written as
Mi(t) + Ci(t) + Kx(t) = Hu(t) + t|w(t) (1)
in which x(t) is the displacement vector with the ith element xj being the displacement of the ith DOF
relative to the ground; M, C and K are mass, damping and stiffness matrices of the structure; H is the
location matrix of controllers; r\ is the influence coefficient matrix of the excitations; and u(t) and w(t)
are the vectors of control forces and disturbances, respectively. For a structure subject to earthquake
excitations, r\ is a (nxl) matrix and w(t)=x g (t) is a scalar denoting the earthquake ground
acceleration.
In the state space, Eq.(l) can be expressed as
Z(t) = AZ(t) + Bu(t) + Ew(t) (2)
in which Z(t) = {x T (t) , i (t)} T is a 2n state vector, A is a system matrix, B and E are appropriate
matrices, and the superscript T denotes the transpose of a vector or matrix. In general, a ni -dimensional
controlled output vector z\, a ^-dimensional constraint output vector zi, and a m-dimensional measured
output vector y can be expressed, respectively, by
Zl(t) = C 2l Z(t)-hD zl u(t)-hE zl w(t)
(t) (3)
where v is a measurement noise vector. Let z^t) be the ith element of the vector Z2(t), where all
elements of za(t) denote the constraint variables (i.e., physical constraints). Since we are interested in
limiting or penalizing the peak values of these variables individually and since each of Z2,i(t) is a scalar,
the peak values of these physical quantities are the LQO norms, i.e.,
1 2 2,i W I oo = sup i Z2 >» (t) I '
for [ = ls 2) •"' n 2 ( 4 )
For a given structure in Eq.(2), our goal is to find a controller u(t) such that: (i) the closed-loop system
is asymptotically stable, (ii) the Hi performance index J*2 is minimized, i.e.,
•7*2 = Kw(»|| 2 = [ ~ C trace [T^OJo) T ZlW (jco)] do> ] 1/2 (5)
and (iii) the constraints for the peak values of all components of 22(1) are satisfied, i.e.,
|Ki(t)L^YiSi> i = l,2,...,n 2 , Vw(t)eW (6)
In Eqs.(5>(6), T ZlW (s) is the transfer matrix from w(t) to zi(t), the superscript H is the conjugate and