The ObservabilityMatrix command computes the observability matrix of a state-space system.

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

If the parameters Amat and Cmat are Matrices, then they are the A and C Matrices, respectively.

The observability matrix has dimensions o*n x n, where n is the number of states (dimension of A) and o is the number of outputs (row dimension of C) It has the form <<C>, <C . A>, <C . A^2>, <C . A^3>, ..., <C . A^(n-1)>>.

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

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

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

$\mathrm{sys1}:-a,\mathrm{sys1}:-c$

$\left[\begin{array}{cc}0& 1\\ \mathrm{-10}& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}1& 0\end{array}\right]$

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

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

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

$\mathrm{sys2}:-a,\mathrm{sys2}:-c$

$\left[\begin{array}{ccc}\mathrm{-3}& 1& 0\\ \mathrm{-5}& 0& 1\\ \mathrm{-3}& 0& 0\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

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

$\left[\begin{array}{ccc}1& 0& 0\\ \mathrm{-3}& 1& 0\\ 4& \mathrm{-3}& 1\end{array}\right]$

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

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

$\left[\begin{array}{ccc}{a}_1& 0& 0\\ 0& a_2& 0\\ 0& 0& a_3\end{array}\right],\left[\begin{array}{ccc}{c}_1& 0& 0\\ 0& 0& c_3\end{array}\right]$

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

$\left[\begin{array}{ccc}{c}_1& 0& 0\\ 0& 0& c_3\\ a_1{c}_1& 0& 0\\ 0& 0& a_3{c}_3\\ a_1^2{c}_1& 0& 0\\ 0& 0& a_3^2{c}_3\end{array}\right]$