## 3 Deriving Input-Output Mappings for the Two Classes of Systems
### A Connection between the Transfer Functions of the Two Classes of Systems  
The original quadratic system is transformed into a bilinear system by means of Carleman linearization [32], as originally shown in [41] and later in [42], for particular models from electrical or mechanical engineering. Now, let us revisit the previous derivation by using the original equations of the original quadratic-bilinear system written as  
\[\dot{\mathbf{x}}(t) =\mathbf{A}\mathbf{x}(t)+\mathbf{Q}(\mathbf{x}(t)\otimes\mathbf{ x}(t))+\mathbf{N}\mathbf{x}(t)\mathbf{u}(t)+\mathbf{B}\mathbf{u}(t) \tag{28}\] \[=\mathbf{A}\mathbf{x}(t)+\mathbf{Q}\mathbf{x}^{(2)}(t)+\mathbf{N} \mathbf{x}(t)\mathbf{u}(t)+\mathbf{B}\mathbf{u}(t).\]  
Then, we can write that  
\[\dot{\mathbf{x}}^{(2)}(t) =\frac{d\mathbf{x}^{(2)}(t)}{dt}=\frac{d\big{(}\mathbf{x}(t)\otimes \mathbf{x}(t)\big{)}}{dt}=\dot{\mathbf{x}}(t)\otimes\mathbf{x}(t)+\mathbf{x}( t)\otimes\dot{\mathbf{x}}(t) \tag{29}\] \[=\Big{(}\mathbf{A}\mathbf{x}(t)+\mathbf{Q}\mathbf{x}^{(2)}(t)+ \mathbf{N}\mathbf{x}(t)\mathbf{u}(t)+\mathbf{B}\mathbf{u}(t)\Big{)}\otimes \mathbf{x}(t)\] \[+\mathbf{x}(t)\otimes\Big{(}\mathbf{A}\mathbf{x}(t)+\mathbf{Q} \mathbf{x}^{(2)}(t)+\mathbf{N}\mathbf{x}(t)\mathbf{u}(t)+\mathbf{B}\mathbf{u}( t)\Big{)}\]  
By neglecting the higher-order cubic terms \(\mathbf{x}^{(3)}(t)\) in the equality above, we can write  
\[\dot{\mathbf{x}}^{(2)}(t)\approx(\mathbf{A}\mathbf{x}(t)+\mathbf{N}\mathbf{x} (t)\mathbf{u}(t)+\mathbf{B}\mathbf{u}(t))\otimes\mathbf{x}(t)+\mathbf{x}(t) \otimes(\mathbf{A}\mathbf{x}(t)+\mathbf{N}\mathbf{x}(t)\mathbf{u}(t)+\mathbf{B }\mathbf{u}(t)). \tag{30}\]It follows that the matrices of the BTI system computed from (13) via [PERSON]'s linearization are given as below:  
\[ \begin{split}\mathbf{A}^{\otimes}&=\left[\begin{array}[] {cc}\mathbf{A}&\mathbf{Q}\\ \mathbf{0}&\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A} \end{array}\right]\text{, }\ \mathbf{B}^{\otimes}=\left[\begin{array}{cc}\mathbf{B}\\ \mathbf{0}\end{array}\right]\text{,}\\ \mathbf{N}^{\otimes}&=\left[\begin{array}{cc}\mathbf{N}&\mathbf{0}\\ \mathbf{B}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{B}&\mathbf{N} \otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{N}\end{array}\right]\text{, }\mathbf{C}^{\otimes}=\left[\mathbf{C}\quad\mathbf{0}\right]\text{,}\end{split}  \tag{31}\]  
while the state vector is given by \(\mathbf{x}^{\otimes}(t)=\left[ \begin{array}{cc}\mathbf{x}(t)\\ \mathbf{x}(t)\otimes\mathbf{x}(t)\end{array} \right]=\left[ \begin{array}{cc} \mathbf{x}^{(1)}(t)\\ \mathbf{x}^{(2)}(t)\end{array} \right]\). A first observation is that the poles of the BTI model in (31), derived by applying [PERSON]'s linearization, and the poles of the original QBTI model are clearly connected. By denoting with \(\{\sigma_{1},\ldots,\sigma_{n}\}:=\sigma_{\mathbf{A}}\) the set of eigenvalues of matrix \(\mathbf{A}\in\mathbb{R}^{n\times n}\), since \(\mathbf{A}^{\otimes}=\left[ \begin{array}{cc}\mathbf{A}&\mathbf{Q}\\ \mathbf{0}&\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A} \end{array} \right]\), it follows that the set of eigenvalues of matrix \(\mathbf{A}^{\otimes}\in\mathbb{R}^{N\times N}\) are written as \(\sigma_{\mathbf{A}}\cup\{\sigma_{\mathbf{A}}\oplus\sigma_{\mathbf{A}}\}\); here, the set \(\{\sigma_{\mathbf{A}}\oplus\sigma_{\mathbf{A}}\}\) includes all possible sums between two elements of \(\sigma_{\mathbf{A}}\), by allowing repetitions (poles with multiplicities). For example, when \(n=2\), we have that \(\{\sigma_{\mathbf{A}}\oplus\sigma_{\mathbf{A}}\}=\{2\sigma_{1},\sigma_{1}+ \sigma_{2},\sigma_{2}+\sigma_{1},2\sigma_{2}\}\). Next, we derive explicit formulas involving the transfer functions of the BTI system characterized by the matrices in (31), in terms of the matrices of the original QBTI system. We will be using the following matrix identity based on the Schur complement in order to explicitly compute the inverse of matrix \(\mathrm{s}\mathbf{I}_{N}-\mathbf{A}^{\otimes}\), i.e.,  
\[