Jonothan How in his lecture notes emphasizes that the controllable system can change, with a proper sequence of u, the states in ANY time to ANY given state. The first ANY, however, cannot be seen in many textbooks including Chen (linear systems) and instead they say "in FINITE time''. In controllability matrix of LTI (sometimes called Kalman matrix), there is no effect of time. Also in the discrete-time proof of it, final time is at least n, where n is the order of system (A is an n-by-n matrix). Does any body have any comment to clarify this?
Can a controllable system be navigated to ANY given state in ANY time ( if the control inputs are not constrained)?!
Just as update: Kalman in its paper" On the general theory of control systems" has defined with a finite T. He has mentioned that T depends on the desired state.
It seems that the definition for continuous-time systems has "FOR ANY T" but for discrete-time systems, using linear algebra, it is not necessarily true. T must be at least equal to n. Still, comments are welcome.
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