Editing Controllability (section)

=== Continuous linear time-invariant (LTI) systems ===

Consider the continuous linear [[time-invariant system]]

: <math>\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>

where 
: <math>\mathbf{x}</math> is the <math>n \times 1</math> "state vector",
: <math>\mathbf{y}</math> is the <math>m \times 1</math> "output vector",
: <math>\mathbf{u}</math> is the <math>r \times 1</math> "input (or control) vector",
: <math>A</math> is the <math>n \times n</math> "state matrix",
: <math>B</math> is the <math>n \times r</math> "input matrix",
: <math>C</math> is the <math>m \times n</math> "output matrix",
: <math>D</math> is the <math>m \times r</math> "feedthrough (or feedforward) matrix".

The <math>n \times nr</math> controllability matrix is given by

:<math>R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}</math>

The system is controllable if the controllability matrix has full row [[Rank (linear algebra)|rank]] (i.e. <math>\operatorname{rank}(R)=n</math>).