Considering the inverted pendulum equation as follows:
$$ \begin{split} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \sin(x_1) - 0.5x_2 \end{split} $$
Now if I want to apply frozen state methods, I can use factorization to get a pseudo linear system
$$ \dot{\mathbf{x}} = \mathbf{A}(\mathbf{x})\mathbf{x} $$
for example by factorizing out $x_1$ like
$$ \begin{split} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \sin(x_1)\frac{1}{x_1}x_1 - 0.5x_2 \end{split} $$
which gives the state dependent $\mathbf{A}$-matrix as
$$ \mathbf{A}(\mathbf{x}) = \begin{bmatrix} 0 & 1\\ \frac{\sin(x_1)}{x_1} & -0.5 \end{bmatrix}\,. $$
Remark: The singularity can be removed here using L'Hopitals rule.
However: To me it is not clear how to choose such a factorization. In especially I could also have chosen $x_2$ instead of $x_1$ for the factorization - which one should be prefered then? In this special case it seems to be $x_1$ because the term $\sin(x_1)$ depends only on $x_1$ and it allows nicely to remove the singularity.
But: How to do this in general? Assuming a system like
$$ \begin{split} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= f(x_1, x_2) - 0.5x_2 \end{split} $$
with $f$ nonlinear. Should I go for $f(x_1, x_2)\frac{1}{x_1}x_1$, for $f(x_1, x_2)\frac{1}{x_2}x_2$ or maybe even a linear combination of these?
Is there any theory/best practise available how to do this?