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[edit] Controllability vs Reachability

What historically has been named 'controllability' is ambiguous. Let x₀ be the equilibrium state (usually taken to be 0) of a system in the absence of an input then.

1. Reachability of an arbitrary state, x_f, from an arbitrary state x_i is the ability to transfer from the initial state x_i to the final state x_f in some time by applying a suitable input.
2. Controllability of an arbitrary state, x, is the ability to transfer from this state x to the equilibrium state x₀ in some time by applying a suitable input.

Obviously reachability implies controllability. For linear stystems in continuous time, both concepts are equivalent. However, a discrete time system my be controllable without being reachabable. A trivial example is:

where A is nilpotent.

Mastlab 21:29, 10 September 2006 (UTC)