This question concerns nonlinear control theory. The state estimation is an important issue since the feedback loop relies on states availability. During I read the book "Nonlinear Systems: Analysis, Stability and Control", the author explores the observability of the dynamic system.
\begin{equation} \begin{cases} \dot{x} = f(x) + g(x) \, u \\ y = h(x) \end{cases} \end{equation}
A famous case on dynamic systems is the rolling wheel, given below. Is it possible to find some sensor function $h(q)$ such that it is possible to reconstruct the states $q = [x, \, y, \, \theta, \, \phi, \, \omega_\theta, \, \omega_\phi]^{\intercal}$ based on its reading?
\begin{equation} \begin{aligned} \dot{x} & = R \, \omega_\phi \, \cos{\theta} \\ \dot{y} & = R \, \omega_\phi \, \sin{\theta} \\ \dot{\theta} & = \omega_\theta \\ \dot{\phi} & = \omega_\phi \\ \dot{\omega}_\theta & = a_\theta \\ \dot{\omega}_\phi & = a_\phi \end{aligned} \end{equation}
I really think it is a relevant research problem to solve, there is very little material on even famous repositories.
Thank you