## 4 Extensions of the Loewner Framework
#### Petrov-Galerkin Projections Applied to the Bilinear Model Computed via Carleman's Linearization  
In what follows, we will connect the structure of the projection matrices introduced in (52) to the particular compressed quantities of the original QBTI system. For this, consider that the projection matrices are of required dimension, i.e., \(\mathbf{V},\mathbf{W}\in\mathbb{C}^{N\times r}\) with \(\mathcal{N}=n^{2}+n\), corresponding to the dimension of the bilinear model computed via [PERSON]'s linearization (for \(N=2\)). Additionally, the projection matrices are split as:  
\[\mathbf{V}= \begin{bmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix} ,\ \ \mathbf{W}= \begin{bmatrix}\mathbf{W}_{1}\\ \mathbf{W}_{2}\end{bmatrix} , \tag{54}\]  
with \(\mathbf{V}_{1},\mathbf{W}_{1}\in\mathbb{C}^{n\times r}\) and \(\mathbf{V}_{2},\mathbf{W}_{2}\in\mathbb{C}^{n^{2}\times r}\). These two matrices are used to construct the realization of a bilinear system of reduced dimension computed by means of Petrov-Galerkin projections applied to the bilinear model in (31):  
\[\mathbf{\Sigma}_{\text{B}}^{\otimes}:\ \begin{cases}\hat{\mathbf{x}}^{\otimes}(t)= \hat{\mathbf{A}}^{\otimes}\hat{\mathbf{x}}^{\otimes}(t)+\hat{\mathbf{N}}^{ \otimes}\hat{\mathbf{x}}^{\otimes}(t)\mathbf{u}(t)+\hat{\mathbf{B}}^{\otimes} \mathbf{u}(t),\\ \hat{\mathbf{y}}^{\otimes}=\hat{\mathbf{C}}^{\otimes}\hat{\mathbf{x}}^{\otimes }(t),\end{cases}  \tag{55}\]  
where the matrices are computed as follows:  
\[\hat{\mathbf{A}}^{\otimes} = \begin{bmatrix}\mathbf{W}_{1}^{\star}&\mathbf{W}_{2}^{\star} \end{bmatrix}\begin{bmatrix}\mathbf{A}&\mathbf{Q}\\ \mathbf{0}&\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A} \end{bmatrix}\begin{bmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix}  \tag{56}\] \[=\mathbf{W}_{1}^{\star}\mathbf{A}\mathbf{V}_{1}+\mathbf{W}_{1}^{ \star}\mathbf{Q}\mathbf{V}_{2}+\mathbf{W}_{2}^{\star}(\mathbf{A}\otimes \mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}_{2},\] \[\hat{\mathbf{N}}^{\otimes} = \begin{bmatrix}\mathbf{W}_{1}^{\star}&\mathbf{W}_{2}^{\star} \end{bmatrix}\begin{bmatrix}\mathbf{N}&\mathbf{0}\\ \mathbf{B}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{B}&\mathbf{N} \otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{N}\end{bmatrix}\begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix} \] \[=\mathbf{W}_{1}^{\star}\mathbf{N}\mathbf{V}_{1}+\mathbf{W}_{2}^{ \star}(\mathbf{B}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{B})\mathbf{ V}_{1}+\mathbf{W}_{2}^{\star}(\mathbf{N}\otimes\mathbf{I}_{n}+\mathbf{I}_{n} \otimes\mathbf{N})\mathbf{V}_{2},\]while the other matrices are given by:  
\[ \begin{split}\hat{\mathbf{B}}^{\otimes}&=\begin{bmatrix} \mathbf{W}_{1}^{*}&\mathbf{W}_{2}^{*}\end{bmatrix}\begin{bmatrix}\begin{array} []{c}\mathbf{B}\\ \mathbf{0}\end{array}\end{bmatrix}=\mathbf{W}_{1}^{*}\mathbf{B},\;\;\hat{ \mathbf{C}}^{\otimes}=\begin{bmatrix}\begin{array}{cc}\mathbf{C}&\mathbf{0} \end{array}\end{bmatrix}\begin{bmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix}=\mathbf{C}\mathbf{V}_{1},\\ \hat{\mathbf{E}}^{\otimes}&=\begin{bmatrix}\mathbf{W}_{1}^{*}&\mathbf{W}_{2}^{* }\end{bmatrix}\mathbf{I}_{n^{2}+n}\begin{bmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix}=\mathbf{W}_{1}^{*}\mathbf{V}_{1}+\mathbf{W}_{2}^{* }\mathbf{V}_{2}.\end{split}  \tag{57}\]  
It is to be noted that matrices \(\mathbf{V}\) and \(\mathbf{W}\) in (54) can indeed be chosen so that the relation \(\mathbf{W}^{*}\mathbf{V}=\mathbf{I}_{r}\) holds. This implies that \(\mathbf{W}_{1}^{*}\mathbf{V}_{1}+\mathbf{W}_{2}^{*}\mathbf{V}_{2}=\mathbf{I}_ {r}\) and hence, it follows that \(\hat{\mathbf{E}}^{\otimes}=\mathbf{I}_{r}\). The formulas in (56) and (57) show that the system matrices of the reduced-order bilinear model \(\mathbf{\Sigma}_{\mathbb{B}}^{\otimes}\) in (55) can be written as sums of appropriately compressed quantities, as enumerated below:  
1.