# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 2 Mathematical Model
### State-Space Model and Output Variables  
By considering the state vector \[\mathbf{x}(t)=\left[ \begin{array}{c}\mathbf{q}(t)\\ \dot{\mathbf{q}}(t)\end{array} \right], \tag{29}\]  
the overall dynamical response of the multibuilding system can be described by a first-order state-space model  
\[\dot{\mathbf{x}}(t)=\big{(}\mathbf{A}_{b}+\mathbf{A}_{d}\big{)}\mathbf{x}(t)+ \mathbf{B}\,\hat{w}(t), \tag{30}\]  
where the matrices  
\[\mathbf{A}_{b}=\left[ \begin{array}{cc}[\mathbf{0}]_{\mathbb{H}\times \mathbb{H}}&\mathbf{I}_{\mathbb{H}}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right],\quad \mathbf{B}=\left[ \begin{array}{c}[\mathbf{0}]_{\mathbb{H}\times 1}\\ -[\mathbf{1}]_{\mathbb{H}\times 1}\end{array} \right] \tag{31}\]  
are constant matrices that model the dynamical response of the nonactuated MBS and  
\[\mathbf{A}_{d}=\left[ \begin{array}{cc}[\mathbf{0}]_{\mathbb{H}\times \mathbb{H}}&[\mathbf{0}]_{\mathbb{H}\times\mathbb{H}}\\ [\mathbf{0}]_{\mathbb{H}\times\mathbb{H}}&-\mathbf{M}^{-1}\big{(}\widehat{ \mathbf{C}}+\widetilde{\mathbf{C}}\big{)}\end{array} \right] \tag{32}\]  
is a matrix that reflects the action of the overall system of added dampers. The vector of interstory drifts \(\mathbf{\hat{r}}^{(j)}(t)=[\mathbf{\hat{r}}^{j}_{1}(t),\ldots,\mathbf{\hat{r}} ^{j}_{n}(t)]^{T}\) corresponding to building \(\mathcal{B}^{(j)}\) has the following components:  
\[\left\{ \begin{aligned} \mathbf{\hat{r}}^{j}_{1}(t)& =q^{j}_{1}(t),\\ \mathbf{\hat{r}}^{j}_{i}(t)&=q^{j}_{i}(t)-q^{j}_{i-1}(t), \quad i=2,\ldots,n.\end{aligned} \right. \tag{33}\]  
Using the matrix \(\mathbf{P}\) in Equation (5), the vector \(\mathbf{\hat{r}}^{(j)}(t)\) can be computed as  
\[\mathbf{\hat{r}}^{(j)}(t)=\mathbf{P}^{T}\mathbf{q}^{(j)}(t), \tag{34}\]  
and the overall vector of interstory drifts corresponding to the multibuilding system  
\[\mathbf{\hat{r}}(t)=\left[ \begin{array}{c}\mathbf{\hat{r}}^{(1)}(t)\\ \vdots\\ \mathbf{\hat{r}}^{(m)}(t)\end{array} \right] \tag{35}\]  
can be obtained in the form \(\mathbf{\hat{r}}(t)=\mathbf{C}_{p}\,\mathbf{x}(t)\) with the output-matrix  
\[\mathbf{C}_{p}=\left[\text{diag}[\mathbf{P}_{p},\overset{(m)}{\mathbf{m}}, \mathbf{P}]^{T}\;\;[\mathbf{0}]_{\mathbb{H}\times\mathbb{H}}\right]. \tag{36}\]  
The vector of total accelerations of the \(j\)th building  
\[\mathbf{a}^{(j)}(t)=[a^{j}_{1}(t),\ldots,a^{j}_{n}(t)]^{T}=\mathbf{\hat{q}}^{ (j)}(t)+[\mathbf{1}]_{n\times 1}\hat{w}(t) \tag{37}\]  
contains the story accelerations with respect to an inertial reference frame. Using the state vector \(\mathbf{x}(t)\), the overall vector of total accelerations  
\[\mathbf{a}(t)=\left[ \begin{array}{c}\mathbf{a}^{(1)}(t)\\ \vdots\\ \mathbf{a}^{(m)}(t)\end{array} \right] \tag{38}\]  
can be computed in the form \[\mathbf{a}(t)=-\left[ \begin{array}{cc}\mathbf{M}^{-1}\mathbf{K}&\mathbf{M}^{-1} \big{(}\mathbf{C}+\widehat{\mathbf{C}}+\widetilde{\mathbf{C}}\big{)}\end{array}  \right]\mathbf{x}(t). \tag{39}\]  
The vector of interbuilding approaches between buildings \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\)  
\[\bar{\mathbf{r}}^{(j)}(t)=[\bar{r}_{1}^{j}(t),\ldots,\bar{r}_{n}^{j}(t)]^{T}=- \big{\{}\mathbf{q}^{(j+1)}(t)-\mathbf{q}^{(j)}(t)\big{\}} \tag{40}\]  
describes the variation of the interbuilding separation at the different story levels. As schematically illustrated in Figure 3, for a given interbuilding gap \(\Delta_{j}\), the value \(\Delta_{j}-\bar{r}_{i}^{j}(t)\) indicates the separation between stories \(s_{i}^{j}\) and \(s_{i}^{j+1}\). Hence, positive values of the interbuilding approaching \(\bar{r}_{i}^{j}(t)\) will produce a reduction of the interbuilding separation, and a value \(\bar{r}_{i}^{j}(t)>\Delta_{j}\) will indicate an interbuilding impact at the \(i\)th story level. To avoid the computational complexity associated to building impacts, we will assume that the interbuilding gaps \(\Delta_{j},j=1,\ldots,m-1\) are large enough to avoid bounding.