## 3 Deriving Input-Output Mappings for the Two Classes of Systems
### Frequency Domain Mappings  
The unilateral (or one-sided) Laplace transform of a univariate function \(\mathbf{f}:\mathbb{R}\rightarrow\mathbb{R}\) is defined as  
\[\mathbf{F}(s)=\mathcal{L}\{\mathbf{f}(t)\}=\int_{0}^{\infty}\mathbf{f}(t)e^{- st}dt, \tag{23}\]  
while the extension of it directly follows for the multivariate case:  
**Definition 2**. : _Let \(\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}\) be a multivariate function in \(n\) variables. The Laplace transform of \(\mathbf{f}\) is defined as_  
\[\mathbf{F}(s_{1},s_{2},\ldots,s_{n}) =\mathcal{L}\{\mathbf{f}(t_{1},t_{2},\ldots,t_{n})\}\] \[=\int_{0}^{\infty}\cdots\int_{0}^{\infty}f(t_{1},t_{2},\ldots,t_ {n})e^{-s_{1}t_{1}}e^{-s_{2}t_{2}}\ldots e^{-s_{n}t_{n}}dt_{1}\ldots dt_{n}.\]  
**Definition 3**. : _The transfer function \(\mathbf{H}(s)\) of an LTI system \(\mathbf{\Sigma}_{\mathbf{L}}\) (with \(\mathbf{E}=\mathbf{I}_{n}\)) is defined as the Laplace transfer of the impulse response \(h(t)=\mathbf{C}e^{\mathbf{A}t}\mathbf{B}\). Hence, the transfer function can be written in terms of the system matrices as follows \(\mathbf{H}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\). By replacing the identity matrix with \(\mathbf{E}\), we write the more general definition_  
\[\mathbf{H}(s)=\mathbf{C}(s\mathbf{E}-\mathbf{A})^{-1}\mathbf{B}. \tag{24}\]  
The exact derivation follows  
\[\mathbf{H}(s) =\mathcal{L}\{\mathbf{h}(t)\}=\int_{0}^{\infty}\mathbf{h}(t)e^{ -st}dt=\int_{0}^{\infty}\mathbf{C}e^{\mathbf{A}t}\mathbf{B}e^{-st}dt=\int_{0} ^{\infty}\mathbf{C}e^{-(s\mathbf{I}-\mathbf{A})t}\mathbf{B}dt\] \[=\mathbf{C}\Big{(}\int_{0}^{\infty}e^{-(s\mathbf{I}-\mathbf{A})t }dt\Big{)}\mathbf{B}=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}.\]  
**Proposition 1** ([38]). : _If the mapping \(\mathbf{f}:\mathbb{C}^{n}\rightarrow\mathbb{C}\) can be written as a product of two multivariate factors \(\mathbf{x}\) and \(\mathbf{y}\) that do not depend on any common variables, i.e., \(\mathbf{f}(t_{1},\ldots,t_{n})=\mathbf{x}(t_{1},\ldots,t_{j})\mathbf{y}(t_{j+ 1},\ldots,t_{n})\), then by applying the multi-variate Laplace transform to this equality, it follows that \(\mathbf{F}(s_{1},\ldots,s_{n})=\mathbf{X}(s_{1},\ldots,s_{j})\mathbf{Y}(s_{j +1},\ldots,s_{n})\)._  
**Definition 4**. : _The \(\ell\)th (regular) transfer function associated with the system \(\mathbf{\Sigma}_{\mathbf{B}}\) is defined by the Laplace transform of the \(\ell\)th regular kernel. It can be expressed in terms of the system matrices as_  
\[\mathbf{H}_{\ell}^{\mathbf{w}}(s_{0},s_{1},\ldots,s_{\ell})=\mathbf{C}(s_{0} \mathbf{E}-\mathbf{A})^{-1}\mathbf{N}(s_{1}\mathbf{E}-\mathbf{A})^{-1} \mathbf{N}\cdots\mathbf{N}(s_{\ell}\mathbf{E}-\mathbf{A})^{-1}\mathbf{B}, \tag{25}\]  
_where \(\ell\geqslant 1\) and \(\mathbf{w}=(\underbrace{\mathbf{N},\mathbf{N},\ldots,\mathbf{N}}_{(\ell-1) \text{ times}})\), and for \(\ell=0\), we have \(\mathbf{H}_{0}(s_{0})=\mathbf{C}(s_{0}\mathbf{E}-\mathbf{A})^{-1}\mathbf{B}\)._  
In what follows, we introduce a special class of rational functions that can be interpreted as generalized transfer functions of QBTIs and were originally introduced in [23, 39]. The advantage of using these functions is that they are written as one product only, requiring chain multiplication of certain matrices, similar to the case of bilinear systems. Other types of transfer functions have been used for MOR of QBTI systems, such as the class of symmetric ones in [40]. Next, a series of definitions is set up following [23]. **Definition 5**. : _For \(\mathbf{Y}=\{\mathbf{N},\mathbf{Q}\}\), \(b\in\{1,2\}\) and \(\mathcal{S}\in\mathbb{C}^{b}\) consider the function \(\mathbf{\Gamma}:\mathbf{Y}\times\mathbb{C}^{b}\rightarrow\mathbb{C}^{n^{b} \times n}\), defined as follows:_  
\[\mathbf{\Gamma}(w,\mathcal{S})= \begin{cases}\mathbf{\Phi}(s_{1}),\text{ if }w=\mathbf{N},\ \mathcal{S}=(s_{1})\text{ and }b=1,\\ \mathbf{\Phi}(s_{1})\mathbf{B}\otimes\mathbf{\Phi}(s_{2}),\text{ if }w=\mathbf{Q},\ \mathcal{S}=(s_{1},s_{2})\text{ and }b=2.\end{cases}