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Given the system: $$\begin{cases}\dot{x_1}= x_2\\\dot{x_2} = -10x_1+1.8{x_1^2}-0.25x_2 +u,\end{cases}$$ where $$u=-1.8{x_1^2}+v,$$ I get the system: $$\begin{cases}\dot{x_1}= x_2,\\\dot{x_2} = -10x_1-0.25x_2 +v,\end{cases}$$ with the matrix $$\mathbf{A}=\begin{pmatrix}0&1 \\ -10&-0.25\end{pmatrix}$$
Is this a good way to feedback linearize a given system? Is there anything I should be careful about?
Your state and feedback transformation is globally well-defined, so there are no issues. In particular, you've used the state-transformation, $$z := \Phi(x) = x,$$ and feedback transformation, $$v := \Psi(x, u) = 1.8\,x_1^2 + u.$$ to render the dynamics of your original dynamical system to a linear controllable system in state $z$, $$\dot{z} = \begin{pmatrix} 0 & 1 \\ -10 & -0.25 \end{pmatrix}\,z + \begin{pmatrix} 0\\ 1\end{pmatrix}\,v.$$ It is globally valid since the state and feedback transformations have a smooth inverse globally.
