Editing Controllability (section)

== State controllability ==

The [[state space (controls)|state]] of a [[deterministic system]], which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.

''Complete state controllability'' (or simply ''controllability'' if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.<ref name="Ogata97">{{cite book|author=Katsuhiko Ogata|title=Modern Control Engineering|edition=3rd|year=1997|publisher=Prentice-Hall|location=Upper Saddle River, NJ|isbn=978-0-13-227307-7}}</ref>{{rp|737}}

That is, we can informally define controllability as follows:
If for any initial state <math>\mathbf{x_0}</math> and any final state <math>\mathbf{x_f}</math> there exists an input sequence to transfer the system state from <math>\mathbf{x_0}</math> to <math>\mathbf{x_f}</math> in a finite time interval, then the system modeled by the [[state-space representation]] is controllable.  For the simplest example of a continuous, LTI system, the row dimension of the state space expression <math>\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t)</math> determines the interval; each row contributes a vector in the state space of the system.  If there are not enough such vectors to span the state space of <math>\mathbf{x}</math>, then the system cannot achieve controllability. It may be necessary to modify <math>\mathbf{A}</math> and <math>\mathbf{B}</math> to better approximate the underlying differential relationships it estimates to achieve controllability.

Controllability does not mean that a reached state can be maintained, merely that any state can be reached.

Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within the prescribed finite time interval.