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(LQR/LQG), H M and the instantaneous optimal control strategies are frequently employed as active
control and semi-active control strategies. However, the weighing matrices in the performance index
are more difficult to choose. The stirrhess and mass matrices were ever proposed and widely used as
the weighing matrix with the strain energy and kinetic energy of the controlled building being
minimum in nature.
However, damping matrix or stifihess matrix can be chosen as the linear quadratic objective weighing
matrix for the instantaneous optimal control algorithm due to the special expression of the control
force. The control force will provide a classical proportional damping to the controlled building when
the mass multiplying by damping matrix (MC) is selected as the weighing of the instantaneous optimal
control algorithm that results in a reasonable damping distribution for the controlled structures.
Simulation results indicated that the control forces obtained from the instantaneous optimal control
algorithm with the suggested weighing matrix can further reduce the drifts and absolute responses
significantly than that obtained from the instantaneous optimal control algorithm with the conventional
weighing matrix (MM) and LQR under the same control force. In addition, the control strategy is more
suitable for semi-active variable damping control (such as variable damping damper and MR damper
and ER damper) because it just acts like passive damper with adjustable parameters in nature. The
simulation results showed that the semi-active control based on the instantaneous optimal control
algorithm with proposed weighing matrix is more trackable than that based conventional instantaneous
optimal control and LQR strategies.
2, INSTANTANEOUS OPTIMAL CONTROL WITH FULL STATE OBSERVER
For a structure with n degrees of freedom (DOFs), the equation of motion is given by
)= -MX S (f)+ DU(t) (2.1)
whereMand Care the nxn mass and damping matrices, respectively, X(t),X(t),X(t) are the
relative displacement, velocity and acceleration vectors, respectively, U(t) is the pxl active
control force vector, D is the n x p distribution matrix that relates the effect of each control force
with each DOF, X g (t) is the earthquake ground acceleration vector at each DOF, and F(x(t),X(t))
is the restoring force of the structure and can be described by many hysteretic models, herein,
Bouc-Wen model is employed to investigate the control strategy in this study as follows
F(x(t\ X(t))= ofcc + (1 - a}kz (2.2)
where k is the pre-yielding stiffiiess, a is the ratio of post-yielding stirrhess to pre-yielding
stiffness, and (l-a)fc is the hysteretic part of the restoring force in which z is described by the
following nonlinear differential equation
~ (2.3)