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What does it mean, when my system has controllability matrix that is full rank and identity matrix? Having full rank means that the system is controllable, but does the Identity feature of my controllability matrix say anything specific about my system?

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If you start with a SISO system in controllable canonical form, then you'll end up with a controllabilty matrix that's an identity matrix, or an identity matrix rotated by 90 degrees.

So if you have any controllable SISO system, you can put it into controllable canonical form, and have such a controllability matrix.

I'm sure there's parallels to multi-input systems.

I don't see much deep meaning -- either you were handed a system that was in that form, you described it that way when you put it into state-space, or you stumbled onto an alternative form with the same effect.

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  • $\begingroup$ Thank you, I do think the same. However, with a full rank identity matrix as controllability matrix, can we directly comment about the state vectors (i.e) state vector consists of past inputs without scaling? $\endgroup$
    – Neuling
    May 9, 2023 at 12:31
  • $\begingroup$ First: that's a separate question, if you really want to know, ask it as such. Second: you can assign any meaning you want -- unless there's some utility to that meaning it is, by definition, trivial. $\endgroup$
    – TimWescott
    May 9, 2023 at 19:42

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