Conceptually you are correct, in that a minimal realization is controllable and observable. However, minreal makes no guarantees of that. As per the doc: Pole-zero cancellation is a straightforward search through the poles and zeros looking for matches that are within tolerance.
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I'm using minreal(sys,tol) as a method of simplifying some system matrix transfer functions for my project. I'd like to be able to talk about this function and how it works in my report - especially how the 'tolerance' parameter works.
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sysr = minreal(sys) eliminates uncontrollable or unobservable state in state-space models, or cancels pole-zero pairs in transfer functions or zero-pole-gain models. The output sysr has minimal order and the same response characteristics as the original model sys.