Earthquake Engineering Research - HKU Libraries - The University ...

455
ratio of the zth mode; /, -{ T cME} t is the participation factor of the z'th mode; v, = {
denotes the control force corresponding to the ith mode.
The seismic response control of the structure can be achieved through the corresponding energy
control of structural modal vibration. By applying the stochastic averaging technique for quasiintegrable-Hamiltonian
system (Zhu & Lin 1991; Zhu et al 1997), the averaged Ito stochastic
differential equations with respect to modal energies can be obtained in the form of
_
dH t ^[m l (H)--^v ] }dt + a i (E}dW l (t) (f,;=l,2,...,/) (5)
dQj
in which the model energy vector H , the reduced drift vector m(H) and the diffusion matrix a(H)
are represented as
H,=U2+a>Q)l2
m,(H) = -2^ca,H, +Lfis t «o t )
(6a)
(6b)
and W, (r) is unit Wienner process.
a;(H) = ft-S l (a),)H, (6c)
The objective of the study is to find out an optimal feedback control law to minimize a finite horizon
performance index. Assuming the ground excitation be stationary and ergodic, the performance index
of the stochastic optimal control in the finite time horizon ( r 0 , T ) is expressed in the form of
J = \imlL(Q c (T\Q c (r),U(r))dr (7)
7"_»oo J
0
where L denotes a continuous differentiable convex function.
In order to minimize the average energy diffusion of the structure, the following stochastic Hamilton-
Jacobi-Bellman (HJB) equation for optimal ergodic control is established according to the stochastic
optimal dynamical programming principle (Stengel 1986)
_i t 2 o H t
(8)
where V represents a value function with respect to H corresponding to optimal control force.
The optimal control force U* is then obtained by minimizing the right hand side of Equation (8).
When the convex function has the form
the expression of U* can be obtained as
U (9)