## 2 Connections between Interstory FVDs and Decentralized Velocity-Feedback Controllers
\right], \tag{9}\]  
where \([\mathbf{0}]_{p\times q}\) denotes a zero matrix of dimensions \(p\times q\) and \(\mathbf{I}_{n}\) represents the identity matrix of dimension \(n\). In the static state-feedback (SSF) control approach, we assume that the state information can be fully accessed and the control actions are computed as  
\[\mathbf{u}(t)=\mathbf{G}\mathbf{x}(t) \tag{10}\]by means of a constant control-gain matrix \(\mathbf{G}\in\mathbb{R}^{n\times n_{x}}\). Static output-feedback (SOF) control provides a more realistic and general approach by restricting the feedback information to a vector of measured outputs \(\mathbf{y}(t)=[y_{1}(t),\ldots,y_{n_{y}}(t)]^{T}\) (typically with \(n_{y}<n_{x}\)), which can be expressed as linear combination of the states in the form  
\[\mathbf{y}(t)=\mathbf{C}_{y}\mathbf{x}(t) \tag{11}\]  
by using the output matrix \(\mathbf{C}_{y}\in\mathbb{R}^{n_{y}\times n_{x}}\). In that case, the desired control actions are computed as  
\[\mathbf{u}(t)=\mathbf{K}\mathbf{y}(t), \tag{12}\]  
where \(\mathbf{K}\in\mathbb{R}^{n\times n_{y}}\) is a constant control gain matrix. Structured SOF controllers can be designed by setting a specific zero-nonzero pattern on the gain matrix \(\mathbf{K}\). In particular, a fully decentralized SOF controller is obtained when the gain matrix \(\mathbf{K}\) is diagonal. Obviously, SSF controllers can be considered as a case of SOF controllers by setting \(\mathbf{C}_{y}=\mathbf{I}_{n_{x}}\). The interstory drifts  
\[ \begin{cases}r_{1}(t)=q_{1}(t),\\ r_{i}(t)=q_{i}(t)-q_{i-1}(t),\quad i=2,\ldots,n\end{cases}  \tag{13}\]  
are the relative displacements between consecutive stories (see Figure 3b). The vector of interstory drifts \(\mathbf{r}(t)=[r_{1}(t),\ldots,r_{n}(t)]^{T}\) can be computed in the form  
\[\mathbf{r}(t)=\mathbf{C}_{r}\mathbf{x}(t) \tag{14}\]  
with the output matrix  
\[\mathbf{C}_{r}= \begin{bmatrix}\mathbf{P}_{n}^{T}&[\mathbf{0}]_{n\times n}\end{bmatrix} , \tag{15}\]  
where \(\mathbf{P}_{n}\) is the upper band-diagonal matrix defined in Equation (3). Analogously, the vector of interstory velocities \(\mathbf{v}(t)=[v_{1}(t),\ldots,v_{n}(t)]^{T}=\dot{\mathbf{r}}(t)\) can be computed as  
\[\mathbf{v}(t)=\mathbf{C}_{\mathbf{v}}\mathbf{x}(t) \tag{16}\]  
by means of the output matrix  
\[\mathbf{C}_{\mathbf{v}}= \begin{bmatrix}[\mathbf{0}]_{n\times n}&\mathbf{P}_{n}^{T}\end{bmatrix} . \tag{17}\]  
By taking \(\mathbf{v}(t)\) as measured-output vector, we can obtain a static velocity-feedback (SVF) controller \(\mathbf{u}(t)=\mathbf{K}\mathbf{v}(t)\), which will produce control actions of the form  
\[u_{i}(t)=k_{ii}v_{1}(t)+\cdots+k_{in}v_{n}(t),\quad i=1,\ldots,n, \tag{18}\]  
where \(k_{ij}\) denote the elements of the control gain matrix \(\mathbf{K}\). When, additionally, the SVF controller is fully decentralized, \(\mathbf{K}\) is a diagonal matrix and the control actions take the simplified form  
\[u_{i}(t)=k_{ii}v_{i}(t),\quad i=1,\ldots,n. \tag{19}\]  
Considering Equation (1), if the diagonal gains \(k_{ii}\) are all negative, then the decentralized SVF controller can be passively implemented by a set of FVDs with damping coefficients  
\[\mathbf{c}_{i}=-k_{ii},\quad i=1,\ldots,n. \tag{20}\]  
Typically, the design objectives in optimization-based procedures are defined by means of a vector of controlled outputs \(\mathbf{z}(t)=[z_{1}(t),\ldots,z_{n_{z}}(t)]^{T}\) that describes some relevant properties of the system behavior. The controlled-output vector \[z(t)=C_{z}x(t)+D_{z}u(t) \tag{21}\]  
with  
\[C_{z}=\left[ \begin{array}{cc}\mathbf{P}_{n}^{T}&[\mathbf{0}]_{n\times n}\\ \left[\mathbf{0}\right]_{n\times n}&\left[\mathbf{0}\right]_{n\times n}\end{array}  \right],\quad D_{z}=\alpha\left[ \begin{array}{cc}[\mathbf{0}]_{n\times n}\\ \mathbf{I}_{n}\end{array} \right], \tag{22}\]  
satisfies  
\[z^{T}(t)\,z(t)=r^{T}(t)\,r(t)+\alpha^{2}u^{T}(t)\,u(t) \tag{23}\]  
and allows setting the mitigation of interstory-drift responses and the reduction of control actions as design objectives.