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with the parameters A,fi,y and n govern the shape of the hysteresis loop. *andi are the
interstory drift and velocity, respectively. The state-space representation of Eqn. 2.1 can be written in
the form
(2.4)
where A-
-M~ I F(X,X]
0 X
-M' I CX\'
matrix, I denote the identity matrix, and -1 is a matrix with -1 as the elements.
The instantaneous optimal control strategy for nonlinear structures can be obtained as follows
(2.5)
where A/ is the sampling time, and superscript -1 denote the inverse matrix. The control force in Eqn.
2.5 will miriirnize the following objective
J(t) = Z T (i)QZ(t)+ U(t} T RU(t) (2.6)
where Q and R are positive-semi-definite and positive-definite weighing matrices, respectively. 0
is selected as follows in general
M
(2.7)
The weighing matrix Q in Eqn.2.7 will make the kinetic energy and strain energy of controlled
structure minimum However, another type of weighing matrix Q has showed more effectively to
reduce responses of linear structures subjected to earthquakes and described here
(2.8)
in which Q n can be arbitrary positive-semi-definite matrix. The above weighing matrix Q
generated following control force (MC for short)
(2.9)
The weighing matrix R is frequently selected as diagonal matrix, so the control force in Eqn.2.9
would supply the optimal control force with classical proportional damping matrix to the controlled
building when the diagonal elements of R are equal to each other. As a result, the value of the
elements in R implies the additional damping ratio to the controlled building by the control devices
and can be conveniently determined according to the response reduction demand. The additional