I have a first order ode given by
$$\dot{x}(t) = \alpha f(x,t) + \beta u(t)$$
where $\left(\alpha,\beta\right)\in \mathbb{R}$ are known constants (i.e. parameters).
Starting from a rich set of input-output noise-free data available, what would be the best procedure to identify (or characterise) the unknown time-varying coefficient $f(x,t)$? I am looking for an approach that shall generalize well to arbitrary excitations (in future simulations).
I am confused if this a grey-box system-id problem or a dynamic optimization issue?
Supposing that the available data is : $$\begin{cases} t_1,t_2,...,t_k,...,t_n \\ u_1,u_2,...,u_k,...,u_n \\ x_1,x_2,...,x_k,...,x_n \end{cases}$$ And with known constants $\alpha$ and $\beta$
Compute $\quad \dot{x}_2,\dot{x}_3,...,\dot{x}_k,...,\dot{x}_{n-1}$ : $$\dot{x}_k\simeq\frac{x_{k+1}-x_{k-1}}{t_{k+1}-t_{k-1}}\quad\text{from}\quad k=2 \quad\text{to}\quad k=n-1$$ Then compute $\quad f_2,f_3,...,f_k,...,f_{n-1}$ : $$f_k\simeq\frac{1}{\alpha}\left(\dot{x}_k-\beta u_k \right) \quad\text{from}\quad k=2 \quad\text{to}\quad k=n-1$$ So, you have the function $f$ defined at discret points. This allows to draw it and see what kind of function it might be. The parameters of the guessed function can be adjusted to an approximate fit thanks to the available methods of regression (least squares fitting).