The ControllabilityMatrix command computes the controllability matrix of a state-space system.

If the parameter sys is a state-space System, then the A and B Matrices are sys:-a and sys:-b, respectively.

If the parameters Amat and Bmat are Matrices, then they are the A and B Matrices, respectively.

The controllability matrix has dimensions n x n*m, where n is the number of states (dimension of A) and m is the number of inputs (column dimension of B). It has the form << B | A . B | A^2 . B | A^3 . B | ... | A^(n-1) . B >>.

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\&colon;$

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\&colon;$

$\mathrm{sys1}≔\mathrm{StateSpace}\left(\frac{1}{s^2+s+10}\right)\&colon;$

$\mathrm{ControllabilityMatrix}\left(\mathrm{sys1}\right)$

$\left[\begin{array}{cc}0& 1\\ 1& \mathrm{-1}\end{array}\right]$

$\mathrm{sys2}≔\mathrm{StateSpace}\left(⟨⟨-3\left|1\right|0⟩\&comma;⟨-5\left|0\right|1⟩\&comma;⟨-3\left|0\right|0⟩⟩\&comma;⟨⟨1\&comma;2\&comma;3⟩⟩\&comma;⟨⟨1\left|0\right|0⟩⟩\&comma;⟨⟨0⟩⟩\right)\&colon;$

$\mathrm{ControllabilityMatrix}\left(\mathrm{sys2}:-a\&comma;\mathrm{sys2}:-b\right)$

$\left[\begin{array}{ccc}1& \mathrm{-1}& 1\\ 2& \mathrm{-2}& 2\\ 3& \mathrm{-3}& 3\end{array}\right]$

$\mathrm{sys3}≔\mathrm{StateSpace}\left(\mathrm{DiagonalMatrix}\left(\left[a\left[1\right]\&comma;a\left[2\right]\&comma;a\left[3\right]\right]\right)\&comma;⟨⟨0|0⟩\&comma;⟨b\left[1\right]|0⟩\&comma;⟨0|b\left[2\right]⟩⟩\&comma;⟨⟨c\left[1\right]\left|0\right|0⟩\&comma;⟨0\left|0\right|c\left[3\right]⟩⟩\&comma;⟨⟨0|0⟩\&comma;⟨0|0⟩⟩\right)\&colon;$

$\mathrm{sys3}:-a,\mathrm{sys3}:-b$

$\left[\begin{array}{ccc}{a}_1& 0& 0\\ 0& a_2& 0\\ 0& 0& a_3\end{array}\right],\left[\begin{array}{cc}0& 0\\ b_1& 0\\ 0& b_2\end{array}\right]$

$\mathrm{ControllabilityMatrix}\left(\mathrm{sys3}\right)$

$\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ b_1& 0& a_2{b}_1& 0& a_2^2{b}_1& 0\\ 0& b_2& 0& a_3{b}_2& 0& a_3^2{b}_2\end{array}\right]$