## 3 Optimum Seismic Control of an MDOF System with CBISs  
### Evaluation of the Seismic Response  
The state-space approach [40] is employed here to evaluate the seismic responses on their high efficiencies. The seismic input action is simulated as a Kanai-Tajimi filtered white noise [41]:  
\[S_{\rm g}(\omega)=\frac{\omega_{\rm g}^{4}+4\zeta_{\rm g}^{2}\omega^{2}\omega_ {\rm g}^{2}}{\left(\omega_{\rm g}^{2}-\omega^{2}\right)^{2}+4\zeta_{\rm g}^{2 }\omega^{2}\omega_{\rm g}^{2}}S_{0} \tag{28}\]  
where \(\omega_{\rm g}\), \(\zeta_{\rm g}\) are the frequency and damping properties of the supporting ground, respectively; \(S_{0}\) is the intensity of the white noise excitation at the bedrock, associated with the ground motion acceleration \(\ddot{u}_{0}\):  
\[S_{0}=\frac{0.141\zeta_{\rm g}\ddot{u}_{0}{}^{2}}{\omega_{\rm g}\sqrt{1+4 \zeta_{\rm g}{}^{2}}} \tag{29}\]  
Rewriting the Kanai-Tajimi power spectrum (Equation (28)) into the form of state-space equations gives:  
\[\left\{\begin{array}{c}\dot{x}_{\rm q}={\bf A_{\rm q}}x_{\rm q}+{\bf E_{\rm q }}w\\ \ddot{x}_{\rm g}={\bf C_{\rm q}}x_{\rm q}\end{array}\right. \tag{30}\]  
where \({\bf A_{\rm q}}=\left[\begin{array}{cc}0&1\\ -\omega_{\rm g}^{2}&-2\zeta_{\rm g}\omega_{\rm g}\end{array}\right]\), \({\bf C_{\rm q}}=\sqrt{2\pi S_{0}}\left[-\omega_{\rm g}^{2}-2\zeta_{\rm g} \omega_{\rm g}\right]\), \({\bf E_{\rm q}}=\left[\begin{array}{c}0\\ 1\end{array}\right]\).  
Let the \(x_{\rm R}=\left\{{\bf X}^{T},\dot{{\bf X}}^{T}\right\}^{T}\). Rewriting Equation (16) with a state-space form gives:  
\[\dot{x}_{\rm R}={\bf A_{\rm R}}x_{\rm R}+{\bf E_{\rm R}}w \tag{31}\]  
where \(x_{\rm R}=\left\{{\bf X}^{T},\dot{{\bf X}}^{T}\right\}^{T}\), \({\bf A_{\rm R}}=\left[\begin{array}{cc}0_{({\rm n}+{\rm_{d}})\times({\rm n}+{ \rm_{d}})}&{\bf I_{({\rm n}+{\rm_{d}})\times({\rm n}+{\rm_{d}})}}\\ -{\bf M^{-1}K}&-{\bf M^{-1}C}\end{array}\right]\), \({\bf E_{\rm R}}=\left[\begin{array}{cc}0_{1\times({\rm n}+{\rm_{d}})}&-{\bf r }^{T}\end{array}\right]^{T}\), and \(w\) denotes the white noise input.  
Combining Equations (31) and (30), extended state-space equations considering the Kanai-Tajimi power spectrum are obtained:  
\[\left\{\begin{array}{l}\dot{x}_{\rm s}=\mathbf{A}_{\rm s}x_{\rm s}+\mathbf{E}_{ \rm s}w\\ z=\mathbf{C}_{\rm s}x_{\rm s}\end{array}\right. \tag{32}\]  
where \(x_{\rm s}=\left\{\begin{array}{l}x_{\rm R}\\ x_{\rm q}\end{array}\right\}\), \(\mathbf{A}_{\rm s}=\left[\begin{array}{cc}\mathbf{A}_{\rm R}&\mathbf{E}_{ \rm R}\mathbf{C}_{\rm q}\\ 0_{2\times(2n+2n_{\rm d})}&\mathbf{A}_{\rm q}\end{array}\right]\), \(\mathbf{E}_{\rm s}=\left[\begin{array}{c}0_{(2n+2n_{\rm d})\times 1}\\ \mathbf{E}_{\rm q}\end{array}\right]\); the output matrix \(\mathbf{C}_{\rm s}\), used to calculate the output vector \(z\) which includes the inter-story drifts of the structure and forces of the CBISs, is defined as:  
\[\mathbf{C}_{\rm s}=\left[\begin{array}{cc}\mathbf{T}&0_{n\times(2n+2n_{\rm d }+2)}\\ -\mathbf{K}_{\rm e}{}^{T}\mathbf{K}_{\rm c}{}^{T}&\mathbf{K}_{\rm e}&0_{n_{ \rm d}\times(n+n_{\rm d}+2)}\end{array}\right] \tag{33}\]  
The direct stochastic analysis method [40] ensures that the covariance matrix of the output vector \(z\) is:  
\[\mathbf{K}_{zz}=\mathbf{C}_{\rm s}\mathbf{P}\mathbf{C}_{\rm s}^{T} \tag{34}\]  
where the state covariance matrix \(\mathbf{P}\) is the solution of the following Lyapunov equation:  
\[\mathbf{A}_{\rm s}\mathbf{P}+\mathbf{P}\mathbf{A}_{\rm s}^{T}+\mathbf{E}_{\rm s }\mathbf{E}_{\rm s}^{T}=0 \tag{35}\]  
Note that the diagonal terms of the covariance matrix \(\mathbf{K}_{zz}\) contain all the response information on the inter-story drifts of the structure and forces of the CBISs. The root of first \(n\) diagonal terms are the root mean square (RMS) structural responses \(\sigma_{\rm u}\) of each layer, and the root of the remaining items are the RMS control forces \(\sigma_{\rm F}\) of each CBIS.