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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.

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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.

More about this can be found in the book Advances and Applications of Nonlinear Control.

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