# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 3 Optimization Procedure
### Objective Function and Optimization Constraints  
For a linear system \(\mathcal{S}\) with the state-space model  
\[\mathcal{S}:\left\{ \begin{aligned} \dot{\mathbf{x}}(t)& =\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{w}(t),\\ \mathbf{z}(t)&=\mathbf{C}_{z}\mathbf{x}(t),\end{aligned} \right. \tag{60}\]  
the \(H_{\infty}\) system-norm  
\[\gamma(\mathcal{S})=\gamma(\mathbf{A},\mathbf{B},\mathbf{C}_{z})=\sup_{\| \mathbf{w}\|_{2}\
eq 0}\frac{\|\mathbf{z}\|_{2}}{\|\mathbf{w}\|_{2}} \tag{61}\]  
indicates the maximum energy-gain from the external disturbance \(\mathbf{w}(t)\) to the controlled output \(\mathbf{z}(t)\)[26; 33]. Broadly speaking, the \(H_{\infty}\) approach aims at minimizing the effect of the worst-case scenario by designing a system \(\mathcal{S}\) with a minimum \(\gamma\)-value. In our case, we consider the seismic ground acceleration \(\hat{w}(t)\) as external disturbance and take the overall vector of interstory drifts \(\hat{\mathbf{r}}(t)\) given in Equation (35) as controlled output, which can be computed from the state vector using the controlled-output matrix \(\mathbf{C}_{\hat{r}}\) in Equation (36). As described in the previous section, a particular configuration of the DDS is specified by a list of optimization variables \(v=[\overline{\alpha},d]\), where \(\overline{\alpha}=[\overline{\alpha}_{1},\ldots,\overline{\alpha}_{\overline {\eta}}]\) is the list of dampers allocation and \(d=[d_{1},\ldots,d_{n_{d}}]\) is the list of damping coefficients. According to Equation (30), the corresponding damped system \(\mathcal{S}(v)\) admits the state-space representation  
\[\mathcal{S}(v):\left\{ \begin{aligned} \dot{\mathbf{x}}(t)& =\{\mathbf{A}_{b}+\mathbf{A}_{d}(v)\}\mathbf{x}(t)+\mathbf{B} \hat{w}(t),\\ \hat{\mathbf{r}}(t)&=\mathbf{C}_{\hat{r}}\mathbf{x}(t), \end{aligned} \right. \tag{62}\]  
where \(\mathbf{A}_{b}\), \(\mathbf{B}\) and \(\mathbf{C}_{\hat{r}}\) are constant matrices and \(\mathbf{A}_{d}(v)\) has the form indicated in Equation (32). The associated \(H_{\infty}\)-cost of that DDS configuration is  
\[\gamma(v)=\gamma(\mathbf{A}_{b}+\mathbf{A}_{d}(v),\mathbf{B},\mathbf{C}_{\hat {r}}). \tag{63}\]The GA solver included in the Matlab Global Optimization Toolbox allows working simultaneously with discrete and continuous optimization variables. Taking advantage of that feature, the allocation variables \(\overline{\pi}_{1},\ldots,\overline{\pi}_{\overline{\eta}}\) are declared as discrete variables and the constraints  
\[0\leq\overline{\pi}_{i}\leq 1,\ i=1,\ldots,\overline{\eta} \tag{64}\]  
are used to define them as binary variables. To specify the total number of dampers, we have to set the condition \(\sum_{i=1}^{\overline{\eta}}\overline{\pi}_{i}=n_{d}\). As the selected optimization solver does not admit equality constraints on the discrete variables, we use the following pair of linear constraints to impose that condition  
\[n_{d}-\epsilon\leq\sum_{i=1}^{\overline{\eta}}\overline{\pi}_{i},\quad\sum_{i= 1}^{\overline{\eta}}\overline{\pi}_{i}\leq n_{d}+\epsilon, \tag{65}\]  
where \(\epsilon\) is a small positive number. We also consider the sublists of interbuilding dampers allocation \(\bar{\pi}^{(j)}=[\bar{\pi}_{1}^{(j)},\ldots,\bar{\pi}_{\bar{\eta}_{j}}^{(j)}]\), \(j=1,\ldots,m-1\), defined in Equation (51) and enforce a full-linked configuration by setting the linear constraints  
\[\sum_{i=1}^{\bar{\eta}_{j}}\tilde{\pi}_{i}^{(j)}\geq 1,\ j=1,\ldots,m-1. \tag{66}\]  
As to the continuous optimization variables \(d_{1},\ldots,d_{n_{d}}\), in this work we assume that all the dampers are FVDs with a maximum damping capacity \(c_{\text{max}}\) and the DDS has a maximum overall damping \(c_{\text{tot}}\). Accordingly, we set the bound constraints  
\[0\leq d_{i}\leq c_{\text{max}},\ i=1,\ldots,n_{d} \tag{67}\]  
and the linear constraint  
\[\sum_{i=1}^{n_{d}}d_{i}\leq c_{\text{tot}}. \tag{68}\]  
In summary, the DDS optimal configuration can be obtained by solving the constrained optimization problem:  
\[\mathcal{P}_{\Gamma}: