## 4 Extensions of the Loewner Framework  
The Loewner framework [31] constructs ROMs that approximate the transfer function of the underlying model, by using data, i.e., samples of the transfer function at particular evaluation points. We refer the reader to the recent handbooks [43; 44; 45] for a more comprehensive description. Extensions of the Loewner framework have been proposed in recent years (some of them collected in [39]); these were mainly based on matching values of the generalized transfer functions introduced in Section 3, at certain evaluation points. Here, we present a simplified presentation of the main approaches and refer to the original works in [20; 23] for a more detailed analysis, the general interpolation scheme, and many other derivations and complementary results. As in the classical linear case [31], the interpolation points are partitioned into two disjoint sets of left and right interpolation points. Since the transfer functions corresponding to the original bilinear or to the quadratic-bilinear system depend on multiple frequencies, the interpolation points need to be arranged in a suitable way. To simplify the presentation, we show the procedure for BITs only, and we refer the reader to [23] for the algorithm in the case of QBTIs. Additionally, assume that \(k=2\bar{k}\) left and right interpolation points are available, which are grouped as follows:  
\[ \begin{split}\text{left points}:&\mu_{1}^{(1)},\mu_{2 }^{(1)},\ldots,\mu_{1}^{(\bar{k})},\mu_{2}^{(\bar{k})},\\ \text{right points}:&\lambda_{1}^{(1)},\lambda_{2}^{(1 )},\ldots,\lambda_{1}^{(\bar{k})},\lambda_{2}^{(\bar{k})}.\end{split}  \tag{41}\]  
Next, the left and right interpolation points are grouped in multi-tuples for \(j=1,\ldots,k\):  
\[\mu^{(j)}=\{(\mu_{1}^{(j)}),\ (\mu_{1}^{(j)},\ \mu_{2}^{(j)})\},\ \ \ \lambda^{(j)}=\{(\lambda_{1}^{(j)}),\ ( \lambda_{2}^{(j)},\ \lambda_{1}^{(j)})\}. \tag{42}\]The generalized controllability matrix \(\mathcal{R}\in\mathbb{C}^{n\times k}\) associated with the right multi-tuples \(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(\bar{k})}\) is  
\[\mathcal{R}=\Big{[}\mathcal{R}^{(1)},\;\mathcal{R}^{(2)},\;\cdots,\;\mathcal{R} ^{(\bar{k})}\Big{]}, \tag{43}\]  
where the matrices \(\mathcal{R}^{(j)}\in\mathbb{C}^{n\times 2},j=1,\ldots,\bar{k}\), are associated with the \(j\)-th multi-tuple \(\boldsymbol{\lambda}^{(j)}\) in (42) are given by  
\[\mathcal{R}^{(j)}=\Big{[}(\lambda^{(j)}_{1}\mathbf{E}-\mathbf{A})^{-1}\, \mathbf{B},\;\;\;(\lambda^{(j)}_{2}\mathbf{E}-\mathbf{A})^{-1}\,\mathbf{N}\,( \lambda^{(j)}_{1}\mathbf{E}-\mathbf{A})^{-1}\,\mathbf{B},\Big{]}. \tag{44}\]  
Similarly, the generalized observability matrix \(\mathcal{O}\in\mathbb{C}^{k\times n}\) associated with the left multi-tuples \(\boldsymbol{\mu}^{(1)},\boldsymbol{\mu}^{(2)},\ldots,\boldsymbol{\mu}^{( \bar{k})}\) is given by  
\[\mathcal{O}=\Big{[}\big{(}\mathcal{O}^{(1)}\big{)}^{T},\;\big{(}\mathcal{O}^{ (2)}\big{)}^{T},\;\ldots\big{(}\mathcal{O}^{(\bar{k})}\big{)}^{T}\Big{]}^{T} \in\mathbb{C}^{k\times n}, \tag{45}\]  
where \(\mathcal{O}^{(i)}\in\mathbb{C}^{2\times n}\), \(i=1,\ldots,\bar{k}\), correspond to the \(i\)-th multi-tuple \(\boldsymbol{\mu}^{(i)}\) in (42) and  
\[\mathcal{O}^{(i)}=\Bigg{[} \begin{array}{c}\mathbf{C}^{T}\,(\mu^{(i)}_{1} \mathbf{E}-\mathbf{A})^{-1}\\ \mathbf{C}^{T}\,(\mu^{(i)}_{1}\mathbf{E}-\mathbf{A})^{-1}\,\mathbf{N}\,(\mu^{( i)}_{2}\mathbf{E}-\mathbf{A})^{-1}\end{array} \Bigg{]}. \tag{46}\]  
Next, the Loewner matrix \(\mathbb{L}\) and the shifted Loewner matrix \(\mathbb{L}_{s}\) are defined using the generalized controllability (43) and observability (45) matrices as  
\[\mathbb{L}=-\mathcal{O}\,\mathbf{E}\,\mathcal{R},\;\;\mathbb{L}_{s}=-\mathcal{ O}\,\mathbf{A}\,\mathcal{R}\,. \tag{47}\]  
The fact that the Loewner matrices are factorized in terms of the pairs of matrices \((\mathbf{E},\mathbf{A})\) and \((\mathcal{O},\mathcal{R})\) is an inherent property of the Loewner framework, which holds true for both the bilinear and quadratic-bilinear extensions of the method.