## 3 Design of Interstory FVD Systems Using a Decentralized SOF \(H_{\infty}\) Approach  
### Decentralized SOF \(H_{\infty}\) Controllers  
Let us consider a linear system  
\[\mathcal{S}: \begin{cases}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{ E}\mathbf{w}(t)\\ \mathbf{z}(t)=\mathbf{C}_{z}\mathbf{x}(t)+D_{z}\mathbf{u}(t)\\ \mathbf{y}(t)=\mathbf{C}_{y}\mathbf{x}(t)\end{cases}  \tag{27}\]  
where \(\mathbf{x}(t)\in\mathbb{R}^{n_{x}}\), \(\mathbf{u}(t)\in\mathbb{R}^{n_{u}}\), \(\mathbf{w}(t)\in\mathbb{R}^{n_{w}}\), \(\mathbf{z}(t)\in\mathbb{R}^{n_{z}}\), \(\mathbf{y}(t)\in\mathbb{R}^{n_{y}}\) and \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{E}\), \(\mathbf{C}_{z}\), \(\mathbf{D}_{z}\) and \(\mathbf{C}_{y}\) are constant matrices of appropriate dimensions. A SOF controller \(\mathbf{u}(t)=\mathbf{K}\mathbf{y}(t)\) defines the closed-loop system  
\[\mathcal{S}_{K}: \begin{cases}\dot{\mathbf{x}}(t)=\mathbf{A}_{K}\,\mathbf{x}(t)+\mathbf{E}\mathbf{ w}(t)\\ \mathbf{z}(t)=\mathbf{C}_{K}\,\mathbf{x}(t)\end{cases}  \tag{28}\]  
with  
\[\mathbf{A}_{K}=\mathbf{A}+\mathbf{B}\mathbf{K}\mathbf{C}_{y},\quad\mathbf{C}_{K}=\mathbf{C}_{z}+\mathbf{D}_{z} \mathbf{K}\mathbf{C}_{y}, \tag{29}\]  
and the closed-loop frequency response function (FRF)  
\[\mathbf{T}_{K}(\omega)=\mathbf{C}_{K}\left(2\pi\omega j\mathbf{I}_{n_{x}}-\mathbf{A}_{K} \right)^{-1}\mathbf{E}, \tag{30}\]  
where \(\omega\) is the frequency in Hertz and \(j=\sqrt{-1}\). In the \(H_{\infty}\) approach, the controller performance is assessed by means of the \(H_{\infty}\) norm  
\[\gamma_{K}=\sup_{\|\mathbf{w}\|_{2}\
eq 0}\frac{\|\mathbf{z}\|_{2}}{\|\mathbf{w}\|_{2}}, \tag{31}\]where \(\|f\|_{2}=\left\{\int_{0}^{\infty}f^{T}(t)f(t)dt\right\}^{1/2}\) denotes the continuous 2-norm. Using the maximum singular value \(\sigma_{\max}\) of the closed-loop FRF, the \(H_{\infty}\)-norm \(\gamma_{K}\) can be computed in the frequency domain as  
\[\gamma_{K}=\sup_{\omega}\ \sigma_{\max}\left[T_{K}(\omega)\right]. \tag{32}\]  
When the closed-loop system \(\mathcal{S}_{K}\) is asymptotically stable, \(\gamma_{K}\) has a finite value that can be interpreted as the worst-case energy gain from the disturbance input to the controlled output. In that sense, the \(H_{\infty}\) controller design approach aims to obtain a control gain matrix \(K\) with an asymptotically stable closed-loop system and a minimum \(H_{\infty}\)-norm \(\gamma_{\text{opt}}\). According to the Bounded Real Lemma (BRL) [40], the closed-loop system \(\mathcal{S}_{K}\) is asymptotically stable and \(\gamma_{K}^{2}<\eta\) if and only if there exists a matrix \(\mathbf{X}\in\mathbb{S}^{n_{x}}\) that satisfies  
\[\mathbf{X}>0,\quad \begin{bmatrix}\text{sym}(\mathbf{A}_{K}\mathbf{X})&*&*\\ \mathbf{E}^{T}&-\eta\mathbf{I}_{n_{w}}&*\\ \mathbf{C}_{K}\mathbf{X}&[\mathbf{0}]_{n_{z}\times n_{w}}&-\mathbf{I}_{n_{z}}\end{bmatrix} <0, \tag{33}\]  
where \(\mathbb{S}^{n_{x}}\) is the set of symmetric real matrices of dimension \(n_{x}\), the matrix inequality \(\mathbf{M}>0\) (\(\mathbf{M}<0\)) indicates that \(\mathbf{M}\) is a positive (negative) definite matrix, \(\text{sym}(\mathbf{M})\) is a shorthand for \(\mathbf{M}+\mathbf{M}^{T}\), and the asterisks represent the transpose of the matrices located in the symmetric position. Considering the BRL, the design of an optimal SOF \(H_{\infty}\) controller can be formulated as the following bilinear matrix inequality (BMI) optimization problem:  
\[\mathcal{P}_{\text{soft}}:\left\{ \begin{array}{ll}\text{Minimize}&\eta\\ \text{Subject to}&\mathbf{X}>0\text{ and the BMI in Equation \eqref{eq:SSF}}.\end{array} \right. \tag{34}\]  
\[ \begin{bmatrix}\text{sym}(\mathbf{A}\mathbf{X}+\mathbf{B}\mathbf{K}\mathbf{C}_{y}\mathbf{X})&*&*\\ \mathbf{E}^{T}&-\eta\mathbf{I}_{n_{w}}&*\\ \mathbf{C}_{z}\mathbf{X}+\mathbf{D}_{z}\mathbf{K}\mathbf{C}_{y}\mathbf{X}&[\mathbf{0}]_{n_{z}\times n_{w}} &-\mathbf{I}_{n_{z}}\end{bmatrix} <0, \tag{35}\]  
where \(\mathbf{X}\), \(\mathbf{K}\) and \(\eta\) are the optimization variables.