## 3 Two-Step Parametrization Method
### Parameter Sensitivity Analysis  
The FC model used in this work has numerous unknown parameters, which raises the question of whether each parameter is unambiguously identifiable. The parameter identifiability is, in general, strongly dependent on the model structure and the available measurement data. A parameter sensitivity analysis can aid in answering these questions, and the Fisher information matrix (FIM) \(\mathbf{F}\) is well-established for conducting such analysis [28]. Under the assumption of Gaussian prediction errors with zero mean values and time-independent covariances, the Cramer-Rao inequality holds [29]:  
\[\text{Cov}(\theta)\succeq\mathbf{F}^{-1} \tag{15}\]  
The inequality says that the inverse of \(\mathbf{F}\) is the lower bound of the parameter covariances. The first step for obtaining the FIM is computing the state parameter sensitivities\(\xi_{i}=d\mathbf{x}/d\theta_{i}\), where \(\theta_{i}\) for \(i\in\{1,2,\ldots,n_{\theta}\}\) denotes a parameter, and \(n_{\theta}\) the number of parameters. They are obtainable from solving the following first-order ODE:  
\[ \begin{split}\dot{\xi}_{i}&=\frac{d}{dt}\bigg{(} \frac{d\mathbf{x}}{d\theta_{i}}\bigg{)}=\frac{d}{d\theta_{i}}\bigg{(}\frac{d \mathbf{x}}{dt}\bigg{)}=\frac{d}{d\theta_{i}}\mathbf{f}(\mathbf{x},\mathbf{u},\theta)\\ &=\frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u},\theta)}{ \partial\mathbf{x}}\frac{d\mathbf{x}}{d\theta_{i}}+\frac{\partial\mathbf{f}( \mathbf{x},\mathbf{u},\theta)}{\partial\theta_{i}}\\ &=\frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u},\theta)}{ \partial\mathbf{x}}\xi_{i}+\frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u}, \theta)}{\partial\theta_{i}}\end{split}  \tag{16}\]  
The second step is to calculate the output parameter sensitivities \(\psi_{i}=d\mathbf{y}/d\theta_{i}\) with  
\[ \begin{split}\psi_{i}&=\frac{\partial\mathbf{g}( \mathbf{x},\mathbf{u},\theta)}{\partial\mathbf{x}}\frac{d\mathbf{x}}{d\theta_ {i}}+\frac{\partial\mathbf{g}(\mathbf{x},\mathbf{u},\theta)}{\partial\theta_{ i}}\\ &=\frac{\partial\mathbf{g}(\mathbf{x},\mathbf{u},\theta)}{ \partial\mathbf{x}}\xi_{i}+\frac{\partial\mathbf{g}(\mathbf{x},\mathbf{u}, \theta)}{\partial\theta_{i}}.\end{split}  \tag{17}\]  
The model described by [PERSON] et al. [10] has analytic derivatives available, and this work utilizes MATLAB R2020b's Symbolic Math Toolbox [30] to compute them. All output parameter sensitivities \(\psi_{i}\) merged into one matrix yields the output parameter sensitivity matrix \(\psi(t)=[\psi_{1}(t),\psi_{2}(t),\ldots,\psi_{n_{\theta}}(t)]\). The FIM is determinable with \(\psi(t)\) taken at the sample times \(t_{\mathrm{f}}\) for \(k\in\{0,1,\ldots,n_{k}\}\), where \(n_{k}+1\) is the number of sample instants. Finally, using the sampled output parameter sensitivity matrix \(\psi(t_{k})\), the FIM is computable with  
\[\mathbf{F}=\sum_{k=0}^{n_{k}}\psi^{\mathrm{T}}(t_{k})\Sigma_{\mathbf{e}}^{-1} \psi(t_{k}), \tag{18}\]  
where \(\Sigma_{\mathbf{e}}\) denotes the prediction error covariance matrix, and \(\mathbf{F}\in\mathbb{R}^{n_{\theta}\times n_{\theta}}\) holds. Under the assumption of a perfect model, the prediction error covariance \(\Sigma_{\mathbf{e}}\) is identical to the measurement noise covariance and thus assumed to be known. According to the Cramer-Rao inequality (15), the inverse of the FIM is the lower bound of the absolute parameter variances. Thus directly analyzing \(\mathbf{F}\) would mean that the parameter's physical units and their magnitudes bias the analysis. Using the normalized dimensionless FIM \(\mathbf{F}_{\mathrm{norm}}\) for a physical unit independent analysis resolves this issue [31; 32; 33]:  
\[\mathbf{F}_{\mathrm{norm}}= \begin{bmatrix}\theta_{1}&0&\cdots&0\\ 0&\theta_{2}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\cdots&0&\theta_{n_{\theta}}\end{bmatrix} \mathbf{F} \begin{bmatrix}\theta_{ 1}&0&\cdots&0\\ 0&\theta_{2}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\cdots&0&\theta_{n_{\theta}}\end{bmatrix}