## 5 Numerical Examples
### A Nonlinear RC Ladder Circuit  
The next test-case is a nonlinear resistor-capacitor (RC) ladder electrical circuit first introduced by [41]. Various variants of this model were used in many other MOR works, such as [34] or [42]. The nonlinear first-order system under study models a RC network; the nonlinearity is, hence, explained by nonlinear resistors consisting of a parallel connection between classical resistors with a diode as depicted in the schematic in Figure 4.  
Figure 4: Schematic of the RC ladder circuit from [41]. Figure 3: The output of the original large-scale and of the reduced-order BTI systems (**left**) and the absolute error, i.e., the magnitude of the deviation (**right**). Figure 2: The singular value decay of the Loewner matrix + tolerance value (**left**) and the poles of the original QBTI system and of the reduced-order BTI system (**right**). As shown in the original contribution [41], the underlying model of ordinary differential equations forms a SISO nonlinear time-invariant system characterized by:  
\[ \begin{bmatrix}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\\ \vdots\\ \dot{x}_{k}(t)\\ \vdots\\ \dot{x}_{n}(t)\end{bmatrix} = \begin{bmatrix}-g(x_{1}(t))-g(x_{1}(t)-x_{2}(t))\\ g(x_{1}(t)-x_{2}(t))-g(x_{2}(t)-x_{3}(t))\\ \vdots\\ g(x_{k-1}(t)-x_{k}(t))-g(x_{k}(t)-x_{x+1}(t))\\ \vdots\\ g(x_{n-1}(t)-x_{n}(t))\end{bmatrix} + \begin{bmatrix}u(t)\\ 0\\ \vdots\\ 0\\ \vdots\\ 0\end{bmatrix} , \tag{61}\]  
where the mapping \(g\) is defined as \(g:\mathbb{R}\to\mathbb{R}\) defined as \(g(x_{i})=g_{D}(x_{i})+x_{i}\), which combines the effects of a diode and a resistor. The observed output is given by the first state variable \(y(t)=x_{1}(t)\). The non-linearity \(g_{D}\) models a diode as a nonlinear resistor, based on the classical Shockley model:  
\[g_{D}(x_{i})=i_{5}(\exp(u_{P}x_{i})-1), \tag{62}\]  
with material parameters \(i_{5}>0\) and \(u_{P}>0\). For this benchmark, the parameters were selected as follows: \(i_{5}=1\) and \(u_{P}=40\) as in [41]. By substituting these values into (62), we find that \(g_{D}(x_{i})=\exp(40x_{i})-1\), and hence it follows that \(g(x_{i})=\exp(40x_{i})+x_{i}-1\). Analogous to the example in Section 2.2.1, the original nonlinear RC-ladder model can be lifted into an equivalent QBTI model. For more details of the exact procedure, we refer the reader to Section 5.5.1 in [42]. The lifted QBTI model from before, which is equivalent to the original nonlinear system, is then transformed into a bilinear system by means of Carleman linearization as originally shown in [41] and later in [42]. Consequently, the resulting BTI system has dimensions \(n^{2}+n\) with \(n\) denoting the number of circuit blocks of the original system. We follow here the procedures outlined in Sections 2 and 3. We choose \(n=50\), and hence the original QBTI model \(\mathbf{\Sigma}_{\text{GB}}\) derived from (61) is of dimension \(n_{q}=100\) and the bilinear model \(\mathbf{\Sigma}_{\text{B}}^{\otimes}\) derived from (61) is of dimension \(n_{b}=2550\). Next, we choose \(100\) logarithmically spaced points in the interval \([10^{0},10^{2}]_{t}\). Then, by following the extended Loewner approaches, we compute a reduced-order BTI model of dimension \(r_{b}=39\) and a reduced-order QBTI model of dimension \(r_{q}=17\). As before, these values were chosen with respect to the singular value decay of Loewner matrices, as shown in the left plot of Figure 5 (the tolerance value was \(\tau=10^{-14}\)). Additionally, the poles of the original QBTI and BTI systems are displayed in the right plot of Figure 5.  
Next, we perform time-domain simulations by using as the control input, the signal \(u(t)=0.2\cdot\text{sawtooth}(60t)\cdot\sin(100t)\). Here, \(\text{sawtooth}(t)\) generates a sawtooth wave signal with period \(2\pi\) for the elements of the time array \(t\) (using the MATLAB specifications). We choose the end simulation time to be \(T=1\), and a discretization step of \(\Delta t=10^{-3}\).