Editing Controllability (section)

== Reachable set ==
Let T ∈ '''''Т''''' and x ∈ ''X'' (where X is the set of all possible states and '''''Т''''' is an interval of time). The reachable set from x in time T is defined as:<ref name=":0" />

<math>R^T{(x)} = \left\{ z \in X : x \overset{T}{\rightarrow} z \right\}</math>, where x{{overset|T|→}}z denotes that there exists a state transition from x to z in time T.

For autonomous systems the reachable set is given by :
:<math>\mathrm{Im}(R)=\mathrm{Im}(B)+\mathrm{Im}(AB)+....+\mathrm{Im}(A^{n-1}B)</math>,
where R is the controllability matrix.

In terms of the reachable set, the system is controllable if and only if <math>\mathrm{Im}(R)=\mathbb{R}^n</math>.

'''Proof'''  
We have the following equalities:
:<math>R=[B\ AB ....A^{n-1}B]</math>
:<math>\mathrm{Im}(R)=\mathrm{Im}([B\ AB ....A^{n-1}B])</math>
:<math>\mathrm{dim(Im}(R))=\mathrm{rank}(R)</math>
Considering that the system is controllable, the columns of R should be [[linearly independent]]. So:
:<math>\mathrm{dim(Im}(R))=n</math>
:<math>\mathrm{rank}(R)=n</math>
:<math>\mathrm{Im}(R)=\R^{n}\quad \blacksquare</math>

A related set to the reachable set is the controllable set, defined by:
:<math>C^T{(x)} = \left\{ z \in X : z \overset{T}{\rightarrow} x \right\}</math>.
The relation between reachability and controllability is presented by Sontag:<ref name=":0" />

(a) An n-dimensional discrete linear system is controllable if and only if:
:: <math>R(0)=R^k{(0)=X}</math>   (Where X is the set of all possible values or states of x and k is the time step).
(b) A continuous-time linear system is controllable if and only if:
:: <math>R(0)=R^e{(0)=X}</math> for all e>0.
if and only if <math>C(0)=C^e{(0)=X}</math> for all e>0.

'''Example'''
Let the system be an n dimensional discrete-time-invariant system from the formula:
::<math>\phi(n,0,0,w)=\sum\limits_{i=1}^n A^{i-1}Bw(n-1)</math>  (Where <math>\phi</math>(final time, initial time, state variable, restrictions) is defined as the transition matrix of a state variable x from an initial time 0 to a final time n with some restrictions w).
It follows that the future state is in <math>R^k{(0)}</math> if and only if it is in <math>\mathrm{Im}(R)</math>, the image of the linear map <math>R</math>, defined as:
::<math>R(A,B)\triangleq [B\ AB ....A^{n-1}B]</math>,
which maps,
::<math>u^{n}\mapsto X</math>
When <math>u=K^{m}</math> and <math>X=K^{n}</math> we identify <math>R(A,B)</math> with a <math>n\times nm</math> matrix whose columns are <math>B,\ AB, ....,A^{n-1}B</math> in that order. If the system is controllable the rank of <math>[B\ AB ....A^{n-1}B]</math> is <math>n</math>. If this is true, the image of the linear map <math>R</math> is all of <math>X</math>. Based on that, we have:
::<math>R(0)=R^k{(0)=X}</math> with <math>X\in\R^{n}</math>.