## 3 Design of Interstory FVD Systems Using a Decentralized SOF \(H_{\infty}\) Approach
### ILMI Design Procedure
#### 3.2.2 Iterations  
After the initialization phase, the computations in step \(i\geq 1\) include two parts:  
**Step \(i.\)a**: Solve the auxiliary LMI optimization problem \(\mathcal{P}_{\mathrm{aux}}(\mathbf{X}_{R},\mathbf{Y}_{R})\) with fixed matrices \(\mathbf{X}_{R}=\mathbf{X}_{R}^{(i-1)}\) and \(\mathbf{Y}_{R}=\mathbf{Y}_{R}^{(i-1)}\) to obtain an optimal triplet \(\left(\mathbf{L}^{(i)},\mathbf{Z}_{Q}^{(i)},\widetilde{\eta}_{i}\right)\). **Step \(i.\)b**: Solve the structured LMI optimization problem \(\mathcal{P}_{\mathrm{soft}}\left(\mathbf{L},\mathbb{D}\right)\) with fixed matrix \(\mathbf{L}=\mathbf{L}^{(i)}\) to obtain an optimal quartet \(\left(\mathbf{X}_{Q}^{(i)},\mathbf{X}_{R}^{(i)},\mathbf{Y}_{R}^{(i)},\eta_{i}\right)\) and a diagonal gain matrix \(\mathbf{K}^{(i)}=\mathbf{Y}_{R}^{(i)}\{\mathbf{X}_{R}^{(i)}\}^{-1}\) with associated \(\gamma\)-value \(\gamma_{\mathbf{K}^{(i)}}\leq\{\eta_{i}\}^{1/2}\). Assuming that the initialization step has been successfully completed, the following theorem guarantees that the iterations are well defined and can produce a sequence of improved \(\eta\)-values. **Theorem 1**. : _If the initial LMI optimization problem \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(0)},\mathbb{D}\right)\) is feasible, then the sequence of LMI optimization problems \(\mathcal{P}_{\text{\rm{soft}}}\left(\mathbf{L}^{(i)},\mathbb{D}\right)\), \(i\geq 1\), are all feasible, and the sequence of optimal \(\eta\)-values satisfies \(\eta_{i-1}\geq\eta_{i}\)._  
Proof. : Let us consider the LMI in Equation (45). By using the expression of \(\mathbf{R}_{L}\) in Equation (43), we have  
\[ \begin{split}\mathbf{Q}\mathbf{X}_{\mathbb{Q}}\mathbf{Q}^{T}+\mathbf{R}_{L}\mathbf{X }_{R}\mathbf{R}_{L}^{T}&=\mathbf{Q}\mathbf{X}_{\mathbb{Q}}\mathbf{Q}^{T}+(\mathbf{C} _{y}^{\dagger}+\mathbf{Q}\mathbf{L})\mathbf{X}_{R}(\mathbf{C}_{y}^{\dagger}+\mathbf{Q}\mathbf{L})^{T} \\ &=\mathbf{Q}\mathbf{X}_{\mathbb{Q}}\mathbf{Q}^{T}+\mathbf{C}_{y}^{\dagger}\mathbf{X}_{ R}(\mathbf{C}_{y}^{\dagger})^{T}+\text{\rm{sym}}\left\{\mathbf{Q}\mathbf{L}\mathbf{X}_{R}(\mathbf{C}_{ y}^{\dagger})^{T}\right\}+\mathbf{Q}\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T}\mathbf{Q}^{T}\\ &=\mathbf{Q}\big{(}\mathbf{X}_{\mathbb{Q}}+\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T} \big{)}\mathbf{Q}^{T}+\mathbf{C}_{y}^{\dagger}\mathbf{X}_{R}(\mathbf{C}_{y}^{\dagger})^{T}+ \text{\rm{sym}}\left\{\mathbf{Q}\mathbf{L}\mathbf{X}_{R}(\mathbf{C}_{y}^{\dagger})^{T}\right\} \end{split}  \tag{55}\]  
and, by introducing the new variable  
\[\mathbf{Z}_{\mathbb{Q}}=\mathbf{X}_{\mathbb{Q}}+\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T}, \tag{56}\]  
we obtain  
\[\mathbf{Q}\mathbf{X}_{\mathbb{Q}}\mathbf{Q}^{T}+\mathbf{R}_{L}\mathbf{X}_{R}\mathbf{R}_{L}^{T}=\mathbf{Q} \mathbf{Z}_{\mathbb{Q}}\mathbf{Q}^{T}+\mathbf{C}_{y}^{\dagger}\mathbf{X}_{R}(\mathbf{C}_{y}^{ \dagger})^{T}+\text{\rm{sym}}\left\{\mathbf{Q}\mathbf{L}\mathbf{X}_{R}(\mathbf{C}_{y}^{\dagger })^{T}\right\}. \tag{57}\]  
Hence, if the matrices \(\mathbf{X}_{\mathbb{Q}}\) and \(\mathbf{X}_{R}\) satisfy the LMI in Equation (45) for a given matrix \(\mathbf{L}\), then the LMI in Equation (51) holds for the matrices \(\mathbf{Z}_{\mathbb{Q}}\), \(\mathbf{X}_{R}\) and \(\mathbf{L}\). Conversely, if the LMI in Equation (51) is satisfied by the matrices \(\mathbf{Z}_{\mathbb{Q}}\), \(\mathbf{X}_{R}\) and \(\mathbf{L}\), then the LMI in Equation (45) holds for the matrices \(\mathbf{X}_{R}\), \(\mathbf{R}_{L}=\mathbf{C}_{y}^{\dagger}+\mathbf{Q}\mathbf{L}\) and  
\[\mathbf{X}_{\mathbb{Q}}=\mathbf{Z}_{\mathbb{Q}}-\mathbf{L}\mathbf{X}_{R}\mathbf{L}^{T}. \tag{58}\]  
Similar relationships can be easily derived for the LMIs in Equations (46) and (52).