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damping ratio provided by the control devices is described
where f a and C are the additional and the original structural damping ratios, respectively, and s is
the diagonal element of the weighing matrix R . The instantaneous optimal control force in Eqn. 2.9 is
(2.11)
The control force described in Eqn. 2.11 provides more reasonable damping distribution and is more
suitable for semi-active fluid control devices considering semi-active fluid control devices acting
essentially as passive dampers with adjustable parameters.
The control force in Eqn. 2.1 1 can further reduce the linear responses of structures to compare with the
linear optimal control force and other type of control force, however, it will be further investigated in
this study that the control force in Eqn. 2.11 can establish further reduction for the nonlinear structures
with hysteretic restoring force in comparison with the linear optimal control strategy and other type of
control strategies. The optimal linear control force (LQR or LQG) can be obtained in Eqn. 2.12 with
the positive definite symmetric matrix P determined by the matrix Riccati equation.
= -R' l B T PZ(t) (2.12)
For simplicity, A will be replaced with A^, and
0
where K is the
pre-yielding stiffness matrix. And the weighing matrices Oand R in Eqn. 2.12 and Riccati equation
will be determined by Eqn. 2.7 and the demanding control force, respectively. The matrix in Eqn. 2.7
will also be used to determine the instantaneous optimal control force in the Eqn. 2.5 (MM for short) in
order to compare the performance with that established in Eqn.2.11.
3. INSTANTANEOUS OPTIMAL CONTROL WITH OBSERVER
It is impossible to install sensors at all DOFs to measure all responses from the practical application
view. However, it is a realistic way that few sensors are only installed at limit locations to measure
responses and other unobserved responses could be estimated by some filter methods, such as the
Kalman filter and the extended Kalman filter (EKF). The Kalman filter is a stochastic filter that allows
the estimation of the states of a system based on a linear state space model and the EKF uses a local
linearisation to extend the scope of the Kalman filter to systems described by nonlinear ordinary
differential equations (Maybeck 1982).
Since the Kalrnan filter is based on linear model that inherently implies the structure to be always in
elastic phase, it is not suitable for the nonlinear structure control problem. Comparing with the Kalman
filter, the EKF is based on nonlinear model, its time varying model parameter can track the stiffness