Editing Controllability (section)

== Controllability via state feedback  ==
When control authority on a linear dynamical system is exerted through a choice of a time-varying feedback gain matrix <math>K(t)</math>, the system

: <math>\dot{\mathbf{x}} = (A - BK(t))\mathbf{x}</math>

is nonlinear, in that products of control parameters and states are present. The accessibility distribution <math>R</math> is, as before,

: <math>R= \begin{bmatrix} B & AB & \cdots & A^{n-1}B \end{bmatrix}. </math>

It is clear that for the system to be controllable, it is necessary that <math>R</math> has full column rank. It turns out that this condition is also sufficient. However, the (optimal) control strategy explained earlier needs to be slightly modified so that the trajectory when applying an optimal input to steer the system between the specified states, does not pass through the origin, else the regulating input cannot be written in feedback form <math>u=-K(t)\mathbf{x}</math>.