## 2 Connections between Interstory FVDs and Decentralized Velocity-Feedback Controllers  
Let us consider the \(n\)-story building model schematically depicted in Figure 1(a) where \(m_{i}\), \(k_{i}\) and \(c_{i}\) respectively denote the mass, stiffness and damping coefficients of the \(i\)th story, and the red dashpots represent a system of supplemental interstory FVDs with damping constants \(\hat{c}_{i}\), \(i=1,\ldots,n\). The interstory FVD implemented in the \(i\)th building-level produces a resistant force  
\[u_{i}(t)=-\hat{c}_{i}\left\{\dot{q}_{i}(t)-\dot{q}_{i-1}(t)\right\}, \tag{1}\]  
Figure 3: Multistory building equipped with a complete set of ideal interstory force-actuation devices and collocated interstory-velocity sensors. where \(q_{0}(t)\equiv 0\), \(q_{i}(t)\), \(i=1,\ldots,n\) is the top-level displacement of the \(i\)th story \(s_{i}\) with respect to the ground level \(s_{0}\) (see Figure 2b), and \(\dot{q}_{i}(t)\) is the time derivative of \(q_{i}(t)\). To illustrate the connections between interstory FVDs and decentralized SVF controllers, let us assume that the building is equipped with a system of interstory force-actuation devices \(d_{i}\), \(i=1\ldots,n\). As schematically displayed in Figure 3a, the actuation device \(d_{i}\) produces opposed control forces of magnitude \(|u_{i}(t)|\) on the adjacent stories \(s_{i}\) and \(s_{i-1}\). The dynamical response of the controlled building can be described by the second order model  
\[\mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{C}_{d}\dot{\mathbf{q}}(t)+\mathbf{K}_{s}\mathbf{q}(t)=\mathbf{P}_{n }\mathbf{u}(t)-\mathbf{M}[\mathbf{1}]_{n\times 1}\ddot{w}_{g}(t), \tag{2}\]  
where \(\mathbf{q}(t)=[q_{1}(t),\ldots,q_{n}(t)]^{\mathrm{T}}\) is the vector of story displacements with respect to the ground; \(\mathbf{u}(t)=[u_{1}(t),\ldots,u_{n}(t)]^{\mathrm{T}}\) is the vector of control actions; \(\ddot{w}_{g}(t)\) is the seismic ground-acceleration disturbance; \(\mathbf{M}\in\mathbb{R}^{n\times n}\), \(\mathbf{C}_{d}\in\mathbb{R}^{n\times n}\) and \(\mathbf{K}_{s}\in\mathbb{R}^{n\times n}\) are the building mass, damping and stiffness matrices, respectively; \([\mathbf{1}]_{n\times 1}\) is a column vector of size \(n\) with all its entries equal to one; and \(\mathbf{P}_{n}\in\mathbb{R}^{n\times n}\) is the control-input matrix, which has an upper band-diagonal structure with elements  
\[ \begin{cases}[P_{n}]_{i,i}=1,&i=1,\ldots,n,\\ [P_{n}]_{i,i+1}=-1,&i=1,\ldots,n-1,\\ [P_{n}]_{i,j}=0,&\text{otherwise}.\end{cases}  \tag{3}\]  
The mass matrix has the diagonal form  
\[\mathbf{M}=\operatorname{diag}[m_{1},\ldots,m_{n}]=\left[ \begin{array}{ccc}m_{1 }&&\\ &\ddots&\\ &&m_{n}\end{array} \right], \tag{4}\]  
and the stiffness matrix can be computed as  
\[\mathbf{K}_{s}=\mathbf{P}_{n}\operatorname{diag}[k_{1},\ldots,k_{n}]\,\mathbf{P}_{n}^{ \mathrm{T}}. \tag{5}\]  
When the story damping coefficients \(c_{i}\), \(i=1,\ldots,n\) are known, the building damping matrix can also be computed in the form  
\[\mathbf{C}_{d}=\mathbf{P}_{n}\operatorname{diag}[c_{1},\ldots,c_{n}]\,\mathbf{P}_{n}^{ \mathrm{T}}, \tag{6}\]  
otherwise, approximate damping matrices can be computed following different approaches as, for example, Rayleigh or Caughey damping [39]. By considering the state vector  
\[\mathbf{x}(t)=\left[ \begin{array}{c}\mathbf{q}(t)\\ \dot{\mathbf{q}}(t)\end{array} \right] \tag{7}\]  
of dimension \(n_{x}=2n\), the dynamical response of the controlled building can be described by the first-order state-space model  
\[\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{E}\,\ddot{w}_{g}(t), \tag{8}\]  
with  
\[\mathbf{A}=\left[ \begin{array}{ccc}[\mathbf{0}]_{n\times n}&&\mathbf{I}_{n}\\ -\mathbf{M}^{-1}\mathbf{K}_{s}&&-\mathbf{M}^{-1}\mathbf{C}_{d}\end{array} \right],\quad\mathbf{B}= \left[ \begin{array}{c}[\mathbf{0}]_{n\times n}\\ \mathbf{M}^{-1}\mathbf{P}_{n}\end{array} \right],\quad\mathbf{E}=\left[ \begin{array}{c}[ \mathbf{0}]_{n\times 1}\\ -[\mathbf{1}]_{n\times 1}\end{array}