## 3 Design of Interstory FVD Systems Using a Decentralized SOF \(H_{\infty}\) Approach
### Decentralized SOF \(H_{\infty}\) Controllers
\begin{array}{l}\text{ Minimize}\ \ \eta\\ \text{Subject to the LMI in Equations (\ref{eq:LMI}) and (\ref{eq:LMI})}.\end{array} \right. \tag{44}\]  
\[\mathbf{Q}\mathbf{X}_{Q}\mathbf{Q}^{T}+\mathbf{R}_{L}\mathbf{X}_{R}\mathbf{R}_{L}^{T}>0, \tag{45}\] \[ \begin{bmatrix}\text{sym}\bigl{(}\mathbf{A}\mathbf{Q}\mathbf{X}_{Q}\mathbf{Q}^{T} +\mathbf{A}\mathbf{R}_{L}\mathbf{X}_{R}\mathbf{R}_{L}^{T}+\mathbf{B}\mathbf{Y}_{R}\mathbf{R}_{L}^{T}\bigr{)} &*&*\\ \mathbf{E}^{T}&-\eta\mathbf{I}_{n_{w}}&*\\ \mathbf{C}_{z}\mathbf{Q}\mathbf{X}_{Q}\mathbf{Q}^{T}+\mathbf{C}_{z}\mathbf{R}_{L}\mathbf{X}_{R}\mathbf{R}_{L}^{ T}+\mathbf{D}_{z}\mathbf{Y}_{R}\mathbf{R}_{L}^{T}&[\mathbf{0}]_{n_{z}\times n_{w}}&-\mathbf{I}_{n_{z}} \end{bmatrix} <0, \tag{46}\]  
where the optimization variables are \(\mathbf{X}_{Q}\), \(\mathbf{X}_{R}\), \(\mathbf{Y}_{R}\) and \(\eta\). According to the results in [36], if the LMI optimization problem \(\mathcal{P}_{\text{sof}}\left(\mathbf{L}\right)\) attains an optimal solution for the quartet \(\bigl{(}\widehat{\mathbf{X}}_{Q},\widehat{\mathbf{X}}_{R},\widehat{\mathbf{Y}}_{R}, \hat{\eta}\bigr{)}\), then the triplet \(\bigl{(}\widehat{\mathbf{X}},\widehat{\mathbf{K}},\hat{\eta}\bigr{)}\) with  
\[\widehat{\mathbf{X}}=\mathbf{Q}\widehat{\mathbf{X}}_{Q}\mathbf{Q}^{T}+\mathbf{R}_{L}\widehat{\mathbf{ X}}_{R}\mathbf{R}_{L}^{T},\quad\widehat{\mathbf{K}}=\widehat{\mathbf{Y}}_{R}\widehat{\mathbf{X}}_{R}^{ -1} \tag{47}\]  
is a feasible solution of the BMI problem \(\mathcal{P}_{\text{sof}}\), and the \(\gamma\)-value corresponding to the SOF controller \(\mathbf{u}(t)=\widehat{\mathbf{K}}\mathbf{y}(t)\) satisfies the inequality  
\[\gamma_{\widehat{K}}\leq\tilde{\eta}^{1/2}. \tag{48}\]  
In the case \(n_{u}=n_{y}\), if the LMI optimization problem \(\mathcal{P}_{\text{sof}}\left(\mathbf{L}\right)\) can be successfully solved with the additional constraints  
\[\mathbf{X}_{R}\in\mathbb{D}_{n_{u}},\quad\mathbf{Y}_{R}\in\mathbb{D}_{n_{u}}, \tag{49}\]  
where \(\mathbb{D}_{n}\) denotes the set of diagonal real matrices of dimension \(n\), then the obtained gain matrix \(\widehat{\mathbf{K}}\) is also diagonal and defines a fully decentralized SOF controller. **Remark 1**. : _It can be easily proved that the LMI in Equation (45) implies that the matrix \(\mathbf{X}_{R}\) is positive definite (see Proposition 1 in [36]). Hence, the matrix \(\widehat{\mathbf{X}}_{R}\) in Equation (47) is non-singular and the SOF gain matrix \(\widehat{\mathbf{K}}\) is well-defined._  
**Remark 2**. : _It should be observed that the value \(\hat{\eta}^{1/2}\) provided by the LMI optimization problem \(\mathcal{P}_{\mathrm{soft}}(\mathbf{L})\) is an upper bound of \(\gamma_{\widehat{K}}\). The actual value of \(\gamma_{\widehat{K}}\) can be computed by setting \(\mathbf{G}=\hat{\mathbf{K}}\mathbf{C}_{y}\) in the BMI problem \(\mathcal{P}_{\mathrm{ssf}}\) and solving the resulting LMI optimization problem \(\mathcal{P}_{\mathrm{ssf}}\left(\mathbf{G}\right)\) with the optimization variables \(\mathbf{X}\) and \(\eta\). If an optimal solution of \(\mathcal{P}_{\mathrm{ssf}}\left(\mathbf{G}\right)\) is achieved for the pair \(\left(\overline{\mathbf{X}},\tilde{\eta}\right)\), then \(\gamma_{\widehat{K}}=\tilde{\eta}^{1/2}\). Moreover, as the \(\gamma\)-value in Equation (41) is the optimal \(H_{\infty}\) norm for SSF controllers, we will have \(\gamma_{\mathrm{ssf}}\leq\gamma_{\widehat{K}}\leq\tilde{\eta}^{1/2}\)._