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Gaussian) methods are advocated because of their successful application in previous studies (Dyke et
al. 1996). For the controller design, x g is taken to be a stationary white noise, and an infinite horizon
performance index is chosen that weights the modal states by controlled modes such as
J slimier f \w T cQw c + u T Ru)dt] (7)
*-»- r L J ° J
where R is a 2x2 identity matrix because the numerical example has two MR dampers, and Q is a
2Zx2/ diagonal matrix. It should be noted that the size of Q is reduced from 2nx2n to 21x21 because the
limited lower modes are controlled. Therefore, it can be said that it is more convenient to design the
smaller weighting matrix of modal control. For example, when the lowest one mode is controlled for
calculating the modal control action, Q is a 2x2 diagonal matrix.
Modal State Estimation
An observer for modal state estimation should be provided, since real sensors may not estimate the fu]l
modal states directly or the system may be expensive to prepare the sensors for the full states. To
estimate the modal state vector H>cW from the measured output y (£), we consider a Kalman-Bucy filter
as an observer(Meirovitch, 1990). Not only, in this paper, the state feedback including velocities or
displacements is considered, but also the acceleration feedback is implemented for the modal state
estimation using a Kalman-Buch filter. In any case, we can write a modal observer in the form
$ c (t) = A c w c (t) + B c f(t) + E c x g +L[y(t)~C c w c (t}-D c f(t)] (8)
where w c (t) is the estimated controlled modal state and L is the optimally chosen observer gain matrix
by solving a matrix Riccati equation, which assumes that the noise intensities associated with
earthquake and sensors are known. C c is changeable according to the signals which are used for the
feedback and D c is generally zero except the acceleration feedback. For modal state estimation from
the displacements, C c = [4> C 0 ]. For control with the velocity feedback, C c = [ 0