## 3 Design of Interstory FVD Systems Using a Decentralized SOF \(H_{\infty}\) Approach
### Decentralized SOF \(H_{\infty}\) Controllers
If the BMI optimization problem \(\mathcal{P}_{\text{soft}}\) attains an optimal solution for the triplet \((\widetilde{\mathbf{X}},\widetilde{\mathbf{K}},\tilde{\eta})\), then the SOF controller \(\mathbf{u}(t)=\widetilde{\mathbf{K}}\mathbf{y}(t)\) has an optimal \(H_{\infty}\)-norm \(\gamma_{\widetilde{K}}=\tilde{\eta}^{1/2}\). In the particular case of a SSF controller \(\mathbf{u}(t)=\mathbf{G}\mathbf{x}(t)\), we have \(\mathbf{C}_{y}=\mathbf{I}_{n_{x}}\) and the BMI problem \(\mathcal{P}_{\text{soft}}\) takes the form  
\[\mathcal{P}_{\text{ssf}}:\left\{ \begin{array}{ll}\text{Minimize}&\eta\\ \text{Subject to}&\mathbf{X}>0\text{ and the BMI in Equation \eqref{eq:SSF}}.\end{array} \right. \tag{36}\]  
\[ \begin{bmatrix}\text{sym}(\mathbf{A}\mathbf{X}+\mathbf{B}\mathbf{G}\mathbf{X})&*&*\\ \mathbf{E}^{T}&-\eta\mathbf{I}_{n_{w}}&*\\ \mathbf{C}_{z}\mathbf{X}+\mathbf{D}_{z}\mathbf{G}\mathbf{X}&[\mathbf{0}]_{n_{z}\times n_{w}}&-\mathbf{I}_{n _{z}}\end{bmatrix} <0, \tag{37}\]  
which, by setting \(\mathbf{G}\mathbf{X}=\mathbf{Y}\), can be transformed into the following linear matrix inequality (LMI) optimization problem:  
\[\widehat{\mathcal{P}}_{\text{ssf}}:\left\{ \begin{array}{ll}\text{ Minimize}&\eta\\ \text{Subject to}&\mathbf{X}>0\text{ and the LMI in Equation \eqref{eq:SSF}}.\end{array} \right. \tag{38}\]  
\[ \begin{bmatrix}\text{sym}(\mathbf{A}\mathbf{X}+\mathbf{B}\mathbf{Y})&*&*\\ \mathbf{E}^{T}&-\eta\mathbf{I}_{n_{w}}&*\\ \mathbf{C}_{z}\mathbf{X}+\mathbf{D}_{z}\mathbf{Y}&[\mathbf{0}]_{n_{z}\times n_{w}}&-\mathbf{I}_{n_{z}} \end{bmatrix} <0, \tag{39}\]where \(\mathbf{X}\), \(\mathbf{Y}\) and \(\eta\) are the optimization variables. In this case, if the LMI optimization problem \(\widehat{\mathcal{P}}_{\text{ssf}}\) attains an optimal solution for the triplet \((\widetilde{\mathbf{X}},\widetilde{\mathbf{Y}},\widetilde{\eta})\), then the control gain matrix can be computed as  
\[\widetilde{\mathbf{G}}=\widetilde{\mathbf{Y}}\widetilde{\mathbf{X}}^{-1} \tag{40}\]  
and the SSF controller \(\mathbf{u}(t)=\widetilde{\mathbf{G}}\mathbf{x}(t)\) has an \(H_{\infty}\)-norm \(\gamma_{\widetilde{G}}=\gamma_{\text{ssf}}\), where  
\[\gamma_{\text{ssf}}=\tilde{\eta}^{1/2} \tag{41}\]  
is the optimal \(H_{\infty}\) norm corresponding to the SSF controllers. Moreover, structured control gain matrices can be obtained by setting a proper zero-nonzero pattern on the LMI variables \(\mathbf{X}\) and \(\mathbf{Y}\). Unfortunately, this line of solution cannot be applied to the BMI optimization problem \(\mathcal{P}_{\text{sof}}\), as the substitution \(\mathbf{K}\mathbf{C}_{\mathbf{y}}\mathbf{X}=\mathbf{Y}\) does not allow obtaining a general explicit expression for \(\mathbf{K}\). According to the results in [36], when the measured-output matrix \(\mathbf{C}_{y}\) has full row-rank and \(n_{y}<n_{x}\), we can obtain an LMI formulation for SOF controllers by considering the matrix transformation  
\[\mathbf{X}=\mathbf{Q}\mathbf{X}_{Q}\mathbf{Q}^{T}+\mathbf{R}_{L}\mathbf{X}_{R}\mathbf{R}_{L}^{T}, \tag{42}\]  
where \(\mathbf{X}_{Q}\in\mathbb{S}^{n_{x}-n_{y}},\mathbf{X}_{R}\in\mathbb{S}^{n_{y}}\), \(\mathbf{Q}\in\mathbb{R}^{n_{x}\times(n_{x}-n_{y})}\) is a matrix whose columns contain a basis of \(\ker\bigl{(}\mathbf{C}_{y}\bigr{)}\), and \(\mathbf{R}_{L}\in\mathbb{R}^{n_{x}\times n_{y}}\) can be written in the form  
\[\mathbf{R}_{L}=\mathbf{C}_{y}^{\dagger}+\mathbf{Q}\mathbf{L}, \tag{43}\]  
where \(\mathbf{C}_{y}^{\dagger}=\mathbf{C}_{y}^{T}(\mathbf{C}_{y}\mathbf{C}_{y}^{T})^{-1}\) is the Moore-Penrose pseudoinverse of \(\mathbf{C}_{y}\) and \(\mathbf{L}\in\mathbb{R}^{(n_{x}-n_{y})\times n_{y}}\) is a matrix whose values can be arbitrarily selected. By applying that transformation in \(\mathcal{P}_{\text{sof}}\) and performing the substitution \(\mathbf{K}\mathbf{X}_{R}=\mathbf{Y}_{R}\), for a selected \(L\)-matrix, we obtain the LMI optimization problem  
\[\mathcal{P}_{\text{sof}}\left(\mathbf{L}\right):\left\{