Regarding the canonical controllable form question in the post, I learned it from this lecture notes

http://control.ee.ethz.ch/~ifalst/docs/LectureNotes.pdf

In Section 9.2, it is proved that if a single-inupt single-output system $(A,B,C,D)$ satisfies

$[B, AB, \dots, A^{n-1}B]\in \mathbb{R}^{n\times n}$ is of full rank,

then there exists an invertible matrix $T$ such that $(TAT^{-1}, TB, CT^{-1}, D)$ is an equivalent system with the canonical controllable form mentioned in the post. This kind of system is called controllable system (for some other reason).

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