## 3 Deriving Input-Output Mappings for the Two Classes of Systems
### Frequency Domain Mappings
 \tag{26}\]  
_More precisely, in (26), it follows that \(\mathbf{\Gamma}(\mathbf{N},s_{1})=(s_{1}\mathbf{E}-\mathbf{A})^{-1}\) and by \(\mathbf{\Gamma}(\mathbf{Q},(s_{1},s_{2}))=\ (s_{1}\mathbf{E}-\mathbf{A})^{-1}\mathbf{B}\otimes(s_{2} \mathbf{E}-\mathbf{A})^{-1}\)._Denote with \(\mathrm{Y}^{\ell}\) the set of all ordered tuples of length \(\ell\) with entries from \(\mathrm{Y}\), i.e., for all \(\ell\geqslant 1\), write \(\mathrm{Y}^{\ell}=\{(w_{1},w_{2},\ldots,w_{\ell})|w_{k}\in\mathrm{Y},1\leqslant k \leqslant\ell\}\). Moreover, \(\mathrm{Y}^{0}=\{\epsilon\}\) contains only the null symbol. Definition 6: Let \(\ell,h\geqslant 0\) be two integers so that \(\ell\leqslant h\leqslant 2\ell\) and consider the ordered tuple \(\mathcal{S}=(s_{0},s_{1},\ldots,s_{h})\), where \(s_{0},s_{1},\ldots,s_{h}\in\mathbb{C}\). Additionally, let \(\mathcal{S}_{k}\), \(1\leqslant k\leqslant\ell\) be a tuple of 1 or 2 complex numbers from the set \(\{s_{0},s_{1},\ldots,s_{h}\}\) so that \(\mathcal{S}_{1}\odot\mathcal{S}_{2}\odot\cdots\odot\mathcal{S}_{\ell}= \mathcal{S}\). Finally, let \(\mathbf{w}=(w_{1},w_{2},\ldots,w_{\ell})\in\mathrm{Y}^{\ell}\) be a tuple of length \(\ell\geqslant 0\). Associate the following level \(\ell\) rational transfer function to the tuple \(\mathbf{w}\in\mathrm{Y}^{\ell}\):  
\[\mathbf{H}_{\ell}^{\mathbf{w}}(s_{0},s_{1},\ldots,s_{h})= \begin{cases} \mathbf{C}\mathbf{\Phi}(s_{0})\mathbf{B},\ \ \ell=0,\ \mathbf{w}=\epsilon,\\ \mathbf{C}\mathbf{\Phi}(s_{0})w_{1}\mathbf{\Gamma}(w_{1},\mathcal{S}_{1}) \cdots w_{\ell}\mathbf{\Gamma}(w_{\ell},\mathcal{S}_{\ell})\mathbf{B},\ \ \ell\geqslant 1.\end{cases}  \tag{27}\]  
The rational functions introduced in (27) are hence divided into subcategories or levels. It follows that a number of \(2^{\ell}\) functions are associated with level \(\ell\geqslant 0\). More precisely, the one corresponding to level 0 is the linear transfer function \(\mathbf{H}_{0}^{\epsilon}(s)=\mathbf{C}\,\mathbf{\Phi}(s)\,\mathbf{B}\), which can also be denoted with \(\mathbf{H}(s)\). Furthermore, the two functions corresponding to level 1 are written as:  
\[\mathbf{H}_{1}^{\mathbf{N}}(s_{0},s_{1})=\mathbf{C}\,\mathbf{\Phi}(s_{0})\, \mathbf{N}\,\mathbf{\Phi}(s_{1})\mathbf{B},\ \ \ \mathbf{H}_{1}^{\mathbf{Q}}(s_{0},s_{1},s_{2})=\mathbf{C}\,\mathbf{\Phi}(s_{0}) \,\mathbf{Q}\,\big{(}\mathbf{\Phi}(s_{1})\mathbf{B}\otimes\mathbf{\Phi}(s_{2} )\mathbf{B}\big{)}.\]  
Note that the rational functions above can indeed be computed by applying the multivariate Laplace transform of time-domain kernels as explained in [38; 39]. In general, a \(\ell\)th level transfer function defined in (27) is a multivariate rational function depending on \(h\) variables \(\{s_{1},\ldots,s_{h}\}\) for \(\ell+1\leqslant h\leqslant 2\ell+1\). Note that, if \(\mathbf{w}=(\mathbf{N},\mathbf{N},\ldots,\mathbf{N})\in\mathrm{Y}^{\ell+1}, \ell>0\) and \(\mathcal{S}_{\ell}=(s_{\ell})\), then the transfer function \(\mathbf{H}_{\ell+1}^{\mathbf{w}}\) exactly coincides to the bilinear transfer functions proposed in [15], i.e.,  
\[\mathbf{H}_{\ell}^{\mathbf{w}}(s_{0},\ldots,s_{\ell})=\mathbf{C}\mathbf{\Phi} (s_{0})\mathbf{N}\mathbf{\Gamma}(\mathbf{N},s_{1})\cdots\mathbf{N}\mathbf{ \Gamma}(\mathbf{N},s_{\ell+1})\mathbf{B}=\mathbf{C}\mathbf{\Phi}(s_{0}) \mathbf{N}\mathbf{\Phi}(s_{1})\cdots\mathbf{N}\mathbf{\Phi}(s_{\ell}) \mathbf{B}.\]