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Allright,

I have a system \begin{aligned} \dot{x} &= Ax + Bu \\ y &= C x \end{aligned} I can get it in Canonical Observable form (in this form, it is NOT controllable), AND in Canonical Controllable form (in this form it is NOT observable), is there any form, where I can get it as both Observable and Controllable, OR can I prove, it is not possible ? What is the reasoning behind the latter option ?

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If you use a similarity transform,

$$ \begin{array}{l c r} \bar{A} = T\,A\,T^{-1} & \bar{B} = T\,B & \bar{C} = C\,T^{-1} \end{array} $$

then certain properties, such as the eigenvalues of $A$/$\bar{A}$, observability and controllability should be conserved.

Either you might be dealing with numerical rounding errors when calculating the transformations, or you are considering the dual system,

$$ \left(\bar{A},\bar{B},\bar{C}\right) = \left(A^T,C^T,B^T\right) $$

which does not always have a similarity transformation.

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