I have an idea on how to integrate uncertainty of the parameters of an ODE in the state error propagation. But I am unsure if my idea is correct. I have a non-linear ODE of the form:
$$ \frac{d}{dt} \mathbf{x} = \mathbf{F}(\mathbf{x}, t, \boldsymbol{\beta}) = \sum_{i=1}^N \phi_i(\mathbf{x}, t) \cdot \beta_i$$
whereby $\phi_i(\mathbf{x}, t)$ are non-linear basis functions of $\mathbf{x}$. When I use this ODE to propagate a state $\mathbf{x}_k = \mathbf{x}(t_k)$ with associated covariance matrix $\Sigma_{x_kx_k}$ I would like to also do error propagation in order to get the updated state $\mathbf{x}_{k+1}$ with associated $\Sigma_{x_{k+1}x_{k+1}}$.
The error propagation can be done using a linearization: $$ \mathbf{T} = \frac{\partial \mathbf{F}(\mathbf{x}, t, \boldsymbol{\beta})}{\partial \mathbf{x}}|_{\mathbf{x}_k, t_k} $$
$$\boldsymbol{\Theta} = e^{\mathbf{T}\Delta t} $$ $$ \Sigma_{x_{k+1}x_{k+1}} = \boldsymbol{\Theta} \Sigma_{x_kx_k} \boldsymbol{\Theta}^T$$
Now to my problem: The coefficients of the ODE, $\boldsymbol{\beta}$, have their own covariance matrix $\Sigma_{\boldsymbol{\beta}\boldsymbol{\beta}}$ which I would also like to incorporate into the error propagation.
My idea for this is the following: Extend the linearized ODE to have $\boldsymbol{\beta}$ as part of the state vector:
$$ \frac{d}{dt}\begin{bmatrix} \mathbf{x} \\ \boldsymbol{\beta} \end{bmatrix} = \begin{bmatrix} \frac{\partial \mathbf{F}}{\partial \mathbf{x}} && \frac{\partial \mathbf{F}}{\partial \boldsymbol{\beta}} \\ 0 && 0\end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \boldsymbol{\beta}\end{bmatrix} = \mathbf{T}^* \mathbf{x}^*$$
This extended system leaves the coefficient $\boldsymbol{\beta}$ unchanged and I can use it for the error propagation as before:
$$ \boldsymbol{\Theta}^* = e^{\mathbf{T}^*\Delta t} $$
Using the power series expansion:
$$ \boldsymbol{\Theta}^* = \mathbf{I} + \begin{bmatrix} \frac{\partial \mathbf{F}}{\partial \mathbf{x}} && \frac{\partial \mathbf{F}}{\partial \boldsymbol{\beta}} \\ 0 && 0\end{bmatrix} t +\begin{bmatrix} \frac{\partial \mathbf{F}}{\partial \mathbf{x}} && \frac{\partial \mathbf{F}}{\partial \boldsymbol{\beta}} \\ 0 && 0\end{bmatrix}^2 \frac{t^2}{2!} + \dots + \begin{bmatrix} \frac{\partial \mathbf{F}}{\partial \mathbf{x}} && \frac{\partial \mathbf{F}}{\partial \boldsymbol{\beta}} \\ 0 && 0\end{bmatrix}^n \frac{t^n}{n!} + \dots $$ $$ = \mathbf{I} + \begin{bmatrix} \boldsymbol{\Theta}^*_1 && \boldsymbol{\Theta}^*_2 \\ 0 && 0\end{bmatrix} $$
$$ \boldsymbol{\Sigma}^* = \boldsymbol{\Theta}^* \begin{bmatrix} \Sigma_{x_{k}x_{k}} && 0 \\ 0 && \Sigma_{\beta\beta} \end{bmatrix} \boldsymbol{\Theta}^{*^T} $$
$$ = \begin{bmatrix} \Sigma_{x_{k+1}x_{k+1}} && \Sigma_{x_{k+1}\beta_{k+1}} \\ \Sigma_{\beta_{k+1}x_{k+1}} && \Sigma_{\beta_{k+1}\beta_{k+1}}\end{bmatrix} = \begin{bmatrix} \Sigma_{x_{k}x_{k}} + 2\boldsymbol{\Theta}^*_1\Sigma_{x_{k}x_{k}} + \boldsymbol{\Theta}^*_1 \Sigma_{x_{k}x_{k}} \boldsymbol{\Theta}^*_1 + \boldsymbol{\Theta}^*_2 \Sigma_{\beta \beta} \boldsymbol{\Theta}^*_2 && \boldsymbol{\Theta}^*_2\Sigma_{\beta \beta} \\ \boldsymbol{\Theta}^*_2\Sigma_{\beta \beta} && \Sigma_{\beta\beta}\end{bmatrix} $$
Is this idea correct?