# Nonlinear observer of an unknown variation state

consider following diff equations $$\dot{v} = \frac{f_x}{m} = \mu g\\ \dot{\omega} = \frac{T}{J} - \frac{Rf_x}{J}\\ \dot{R} = \eta$$ where the input is $T$. There is a help equation for slip ratio. The unknown variation is $\dot{R}$.where $\eta$ is a state and variable that I declared. $f_x$, $J$, $m$ and $g$ are known. $R$ is known at the beginning but then it varies. $$\lambda = 1-(\frac{R\omega}{v})^n, \:\: n=\left\{-1,1\right\}$$ What I have done is to define some state variable: $$x = [x_1,x_2,x_3]^T = [\omega, v, R]^T \\ u = T\\ y = [x_1, x_2]$$ When I build the system I end up with a singular matrix. What kind of observation approach does work on such system to observe $$x_3$$. I really want to use a slide mode observer. Otherwise everything else works.

I would appriate it alot if someone gave me a hint.

Its not a complete answer. But I found out that the dynamics need redefining such that a transformation of the state is performed. A system can be generically observable if the systes can be expressed by output and input and finite numbers of their derivatives i.e $$x=X(y, \dot{y},..., y^j , u , \dot{u}, ..., u^i)$$ Where $$j,i \in \mathbb{N}_+$$ Denote the number of derivation. When the dynamics are transformed to a simpler and standard form you can take the unkowns as uncertainties and handle the system using desired observer.