Editing Controllability (section)

===Derivation===

Given the state <math>\textbf{x}(0)</math> at an initial time, arbitrarily denoted as ''k''=0, the state equation gives  <math>\textbf{x}(1) = A\textbf{x}(0) + B\textbf{u}(0),</math> then <math>\textbf{x}(2) = A\textbf{x}(1) + B\textbf{u}(1)= A^2\textbf{x}(0)+AB\textbf{u}(0)+B\textbf{u}(1),</math> and so on with repeated back-substitutions of the state variable, eventually yielding

:<math>\textbf{x}(n)=B\textbf{u}(n-1) + AB\textbf{u}(n-2) + \cdots + A^{n-1}B\textbf{u}(0) + A^n\textbf{x}(0)</math>

or equivalently

:<math>\textbf{x}(n)-A^n\textbf{x}(0)= [B \, \, AB \, \,  \cdots \, \, A^{n-1}B] [\textbf{u}^T(n-1) \, \,  \textbf{u}^T(n-2) \, \, \cdots \,  \, \textbf{u}^T(0)]^T.</math>

Imposing any desired value of the state vector <math>\textbf{x}(n)</math> on the left side, this can always be solved for the stacked vector of control vectors if and only if the matrix of matrices at the beginning of the right side has full row rank.