# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 1 Introduction
Moreover, as rows of adjacent identical buildings is a quite common arrangement in residential areas, the selected MBS configuration can be considered as a case of potential practical interest [23]. Also for simplicity, the buildings are modeled as linear planar frames and the damping devices are assumed to be fluid viscous dampers (FVDs), which have proved to be effective energy-dissipation elements in structural vibration control and bring the modeling advantage of admitting a reasonably linear representation [7]. Attending to the structural placement, the DDS can contain two different kinds of dampers--(i) _interstory dampers_, which are implemented between consecutive stories of the same building and produce a resistant force proportional to the corresponding interstory velocity and (ii) _interbuilding dampers_, which are implemented as linking elements between stories located at the same level in adjacent buildings and produce a resistant force proportional to the relative velocity of the linked stories. The DDS can also include two types of building damping configurations of particular interest--(i) _nonactuated buildings_, which do not contain any interstory damper and (ii) _linked buildings_, which are linked to all their adjacent buildings by means of interbuilding dampers. For instance, in the MBS presented in Figure 1, buildings \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(3)}\) are nonactuated; buildings \(\mathcal{B}^{(1)}\), \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(3)}\) are linked; \(\mathcal{B}^{(4)}\) is only partially linked and \(\mathcal{B}^{(5)}\) is unlinked. It should be observed that the DDS implementation can be carried out without internal modifications of nonactuated buildings, which can be an important factor in retrofitting. The relevance of linked configurations lies in the fact that they can help to mitigate the vibrational response of nonactuated buildings and can also provide an effective protection against interbuilding impacts (rounding) [22; 23]. To incorporate those aspects in the DDS design and reducing the number of optimization variables, we introduce the _schemes of allowed damper positions_, which specify the interstory and interbuilding locations where the dampers can be implemented. Finally, we complete the technical setup determination by setting the maximum damping capacity of the dampers, the overall maximum damping capacity of the DDS and the total number of allowed damping elements. In order to define the ATOP, the generic goal of improving the overall seismic response of the MBS is formulated as a triplet of particular design objectives: (i) reducing the magnitude of the buildings interstory drifts, (ii) reducing the magnitude of the buildings floor accelerations and (iii) reducing the interbuilding approaches. Objectives (i) and (ii) are applicable to single-building designs, and are respectively associated to the protection of the buildings structural and nonstructural elements [16; 24]. Objective (iii) is specific to MBS designs and is associated to avoiding interbuilding impacts, which can produce severe damage to both structural and nonstructural elements. In order to obtain a computationally effective procedure, we select the overall vector of interstory drifts as controlled output and follow a single-objective \(H_{\infty}\) approach [25; 26]. That choice sets the reduction of interstory drifts as the primary objective and, at the same time, is able to produce positive results in mitigating the story accelerations [22]. As for design objective (iii), reduction of interbuilding approaches can be attained by enforcing a full-linked configuration in the optimization constraints [27; 28]. Computationally, the selected \(H_{\infty}\) objective-function avoids conducting numerical simulations of seismic time-responses and admits a fast evaluation using the hinfnorm function of the Matlab Robust Control Toolbox [29; 30].