## 2 Materials and Methods
### Parametrization of the State Responses  
It is an important fact that the sequences of sampled state responses \(N(kT_{s}),L(kT_{s}),\)\(T(kT_{s})\) for \(k=1,2,\ldots,n\) and the sample time \(T_{s}\) cannot be effectively used as inputs to the MLP NN. Although theoretically possible, it is very unpractical due to the dimensions, i.e.,the large number of inputs to the MLP NN. Besides that, it is very likely that the MLP NN would not be able to sufficiently generalize the information comprised in such raw data and thus provide a reliable parameter estimate.  
Therefore, the proposed strategy will be to extract the features of the sampled state responses \(N(kT_{s}),L(kT_{s}),\ T(kT_{s})\) by parametrizing them. The parametrization technique will be based on an approximation of the state responses by a parametric analytical function while determining its parameters in terms of linear regression. After the parametrization, the sampled state responses are described by vectors \(\mathbf{p}_{N}\in\mathbb{R}^{n_{p}},\mathbf{p}_{\mathbf{L}}\in\mathbb{R}^{n_{p}},\mathbf{p}_{ \mathbf{T}}\in\mathbb{R}^{n_{p}}\) of \(n_{p}\) parameters as  
\[\begin{split}& N(kT_{s}),\ k=1,2,\ldots,n^{\text{ parametrization}}\mathbf{p}_{\mathbf{N}}=\begin{pmatrix}p_{N1}&p_{N2}&\cdots&p_{Nn_{p}}\end{pmatrix}^{ \mathsf{T}},\\ & L(kT_{s}),\ k=1,2,\ldots,n^{\text{ parametrization}}\mathbf{p}_{\mathbf{L}}= \begin{pmatrix}p_{L1}&p_{L2}&\cdots&p_{Ln_{p}}\end{pmatrix}^{\mathsf{T}},\\ & T(kT_{s}),\ k=1,2,\ldots,n^{\text{ parametrization}}\mathbf{p}_{\mathbf{T}}= \begin{pmatrix}p_{T1}&p_{T2}&\cdots&p_{Tn_{p}}\end{pmatrix}^{\mathsf{T}}. \end{split} \tag{14}\]  
This approach will lead to a significant reduction in the training complexity and the memory footprint since \(n_{p}\ll n\).  
Consider a parametrizable weighted sum of exponential functions \(\overset{\sim}{X}(t):\mathbb{R}\rightarrow\mathbb{R}\) as  
\[\overset{\sim}{X}(t)=p_{1}e^{a_{1}t}+p_{2}e^{a_{2}t}+p_{3}e^{a_{3}t}\ldots p_{ n_{p}-1}e^{a_{n_{p}-1}t}+p_{n_{p}}, \tag{15}\]  
where \(p_{1}\), \(p_{2}\), \(\ldots\), \(p_{n_{p}}\) are the function parameters to be fitted and \(a_{1},a_{2},\ldots,a_{n_{p}-1}<0\) are appropriately chosen tuning coefficients.  
This function will approximate signal \(X(t)\), which substitutes for the state responses \(N(t)\), \(L(t)\), \(T(t)\) available in the form of finite sampled sequences \(N(kT_{s}),L(kT_{s}),\ T(kT_{s})\) for \(k=1,2,\ldots,n\). The absolute term \(p_{n_{p}}\) in (15) is involved to reflect nonzero steady states of model (1), allowing for a nonzero finite limit \(\lim\limits_{t\rightarrow\infty}\overset{\sim}{X}(t)=p_{n_{p}}\).  
Equation (15) can be noted in linear form  
\[\overset{\sim}{X}(t)=\begin{pmatrix}e^{a_{1}t}&e^{a_{2}t}&e^{a_{3}t}&\cdots&e ^{a_{n_{p}-1}t}&1\end{pmatrix}\begin{pmatrix}p_{1}\\ p_{2}\\ p_{3}\\ \vdots\\ p_{n_{p}-1}\\ p_{n_{p}}\end{pmatrix}=h^{\mathsf{T}}\mathbf{p}, \tag{16}\]  
where \(\mathbf{h}\in\mathbb{R}^{n_{p}}\) is the regression vector.  
Based on (16), we can formulate the linear regression problem  
\[\begin{pmatrix}e^{a_{1}T_{s}}&e^{a_{2}T_{s}}&e^{a_{3}T_{s}}&\cdots&e^{a_{n_{ p}-1}T_{s}}&1\\ e^{a_{1}2T_{s}}&e^{a_{2}2T_{s}}&e^{a_{3}2T_{s}}&\cdots&e^{a_{n_{p}-1}2T_{s}}&1\\ e^{a_{1}3T_{s}}&e^{a_{2}3T_{s}}&e^{a_{3}3T_{s}}&\cdots&e^{a_{n_{p}-1}3T_{s}}&1 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ e^{a_{1}nT_{s}}&e^{a_{2}nT_{s}}&e^{a_{3}nT_{s}}&\cdots&e^{a_{n_{p}-1}nT_{s}}&1 \end{pmatrix}\begin{pmatrix}p_{1}\\ p_{2}\\ p_{3}\\ \vdots\\ p_{n_{p}-1}\\ p_{n_{p}}\end{pmatrix}=\begin{pmatrix}X(T_{s})\\ X(2T_{s})\\ \vdots\\ X(nT_{s})\end{pmatrix},\!H\mathbf{p}=\mathbf{X}, \tag{17}\]  
where \(H\in\mathbb{R}^{n\times n_{p}}\) is the regression matrix and \(\mathbf{X}\in\mathbb{R}^{n}\) is the vector of samples.  
The parameter vector \(\mathbf{p}\) will be obtained in terms of the least squares method analytically as [52]  
\[\mathbf{p}=\left(H^{\mathrm{T}}H\right)^{-1}H^{\mathrm{T}}\mathbf{X}. \tag{18}\]