## 3 Design of Interstory FVD Systems Using a Decentralized SOF \(H_{\infty}\) Approach
### ILMI Design Procedure
#### 3.2.2 Iterations
Therefore, if the quartet \(\big{(}\mathbf{X}_{\mathbb{Q}},\mathbf{X}_{R},\mathbf{Y}_{R},\eta\big{)}\) is a feasible solution of the LMI problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}\right)\), then the triplet \(\big{(}\mathbf{L},\mathbf{Z}_{\mathbb{Q}},\eta\big{)}\) with \(\mathbf{Z}_{\mathbb{Q}}=\mathbf{X}_{\mathbb{Q}}+\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T}\) is a feasible solution of the LMI problem \(\mathcal{P}_{\text{\rm{aux}}}(\mathbf{X}_{R},\mathbf{Y}_{\mathbb{R}})\). Analogously, if the triplet \(\big{(}\mathbf{L},\mathbf{Z}_{\mathbb{Q}},\eta\big{)}\) is a feasible solution for the auxiliary LMI problem \(\mathcal{P}_{\text{\rm{aux}}}(\mathbf{X}_{R},\mathbf{Y}_{\mathbb{R}})\), then the quartet \(\big{(}\mathbf{X}_{\mathbb{Q}},\mathbf{X}_{R},\mathbf{Y}_{\mathbb{R}},\eta\big{)}\) with \(\mathbf{X}_{\mathbb{Q}}=\mathbf{Z}_{\mathbb{Q}}-\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T}\) is a feasible solution of the LMI problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}\right)\). Let us now assume that the problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(i-1)},\mathbb{D}\right)\) is feasible and it has attained an optimal solution for the quartet \(\big{(}\mathbf{X}_{\mathbb{Q}}^{(i-1)},\mathbf{X}_{\mathbb{R}}^{(i-1)},\mathbf{Y}_{ \mathbb{R}}^{(i-1)},\eta_{i-1}\big{)}\) with diagonal matrices \(\mathbf{X}_{R}^{(i-1)}\) and \(\mathbf{Y}_{\mathbb{R}}^{(i-1)}\). Then, the triplet \(\big{(}\mathbf{L}^{(i-1)},\widetilde{\mathbf{Z}}_{\mathbb{Q}}^{(i-1)},\eta_{i-1}\big{)}\) with  
\[\widetilde{\mathbf{Z}}_{\mathbb{Q}}^{(i-1)}=\mathbf{X}_{\mathbb{Q}}^{(i-1)}+\mathbf{L}^{(i-1 )}\mathbf{X}_{R}^{(i-1)}\big{(}\mathbf{L}^{(i-1)}\big{)}^{T} \tag{59}\]  
is a feasible solution of the LMI problem \(\mathcal{P}_{\text{\rm{aux}}}\big{(}\mathbf{X}_{R}^{(i-1)},\mathbf{Y}_{R}^{(i-1)} \big{)}\) in Step \(i\).a, which can be solved to obtain an optimal triplet \(\big{(}\mathbf{L}^{(i)},\mathbf{Z}_{\mathbb{Q}}^{(i)},\widetilde{\eta}_{i}\big{)}\) with a minimum \(\eta\)-value \(\widetilde{\eta}_{i}\) that satisfies  
\[\eta_{i-1}\geq\widetilde{\eta}_{i}. \tag{60}\]  
Next, we observe that the quartet \(\big{(}\widetilde{\mathbf{X}}_{\mathbb{Q}}^{(i)},\mathbf{X}_{R}^{(i-1)},\mathbf{Y}_{R}^{(i-1 )},\widetilde{\eta}_{i}\big{)}\) with  
\[\widetilde{\mathbf{X}}_{\mathbb{Q}}^{(i)}=\mathbf{Z}_{\mathbb{Q}}^{(i)}-\mathbf{L}^{(i)}\mathbf{X} _{R}^{(i-1)}\big{(}\mathbf{L}^{(i)}\big{)}^{T} \tag{61}\]  
is a feasible solution of the structured LMI problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(i)},\mathbb{D}\right)\) in Step \(i\).b, which can be solved to obtain an optimal quartet \(\big{(}\mathbf{X}_{\mathbb{Q}}^{(i)},\mathbf{X}_{R}^{(i)},\mathbf{Y}_{\mathbb{R}}^{(i)},\eta_ {i}\big{)}\) with diagonal matrices \(\mathbf{X}_{R}^{(i)}\) and \(\mathbf{Y}_{\mathbb{R}}^{(i)}\) and a minimum \(\eta\)-value \(\eta_{i}\) that satisfies \(\widetilde{\eta}_{i}\geq\eta_{i}\) and, considering Equation (60), we obtain \(\eta_{i-1}\geq\eta_{i}\). Finally, if we assume that the initial structured LMI problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(0)},\mathbb{D}\right)\) is feasible, then the LMI optimization problems \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(i)},\mathbb{D}\right)\), \(i\geq 1\), are all feasible by induction. According to the theorem and the discussion in Remark 2, after a proper initialization, the proposed ILMI procedure produces a sequence of diagonal gain matrices \(\mathbf{K}^{(i)}\) with associated \(\gamma\)-values that satisfy  
\[\gamma_{\text{ssf}}\leq\gamma_{\mathbf{K}^{(i)}}\leq\big{\{}\eta_{i}\big{\}}^{1/2}, \tag{62}\]  
where \(\gamma_{\text{ssf}}\) is the optimal \(H_{\infty}\)-norm of the SSF controllers. The decreasing character of the values \(\eta_{i}\) indicates that the proposed procedure can be effective in producing improved controllers, whose performance level can be assessed by taking the sharp lower bound \(\gamma_{\text{ssf}}\) as a reference.