The Ackermann command calculates the static (Luenberger) observer gain for the single-output systems to put the observer error dynamics poles in the desired location based on the Ackermann's formula. The (Amat,Cmat) pair (or the sys object) must be observable. The closed-loop observer error system matrix is then $\mathrm{Ac}=\mathrm{Amat}-L·\mathrm{Cmat}$ (or Ac = sys:-a-L.sys:-c) where $L$ is the calculated observer gain.

$\mathrm{with}\left(\mathrm{ControlDesign}\right)\:$

$\mathrm{Amat}≔\mathrm{Matrix}\left(\left[\left[-0.9054\,-39.7500\,22.4100\,0.1891\,-0.7366\right]\,\left[-0.3734\,-77.0100\,41.1600\,4.1900\,-1.0450\right]\,\left[-0.2815\,-120.8000\,63.6500\,4.3650\,-2.1850\right]\,\left[0.4264\,2.2290\,-0.2781\,0.0662\,-0.6509\right]\,\left[-0.5441\,-39.1800\,21.8400\,2.5120\,-0.8066\right]\right]\right)$

$\mathrm{Amat}≔\left[\begin{array}{ccccc}\mathrm{-0.9054}& \mathrm{-39.7500}& 22.4100& 0.1891& \mathrm{-0.7366}\\ \mathrm{-0.3734}& \mathrm{-77.0100}& 41.1600& 4.1900& \mathrm{-1.0450}\\ \mathrm{-0.2815}& \mathrm{-120.8000}& 63.6500& 4.3650& \mathrm{-2.1850}\\ 0.4264& 2.2290& \mathrm{-0.2781}& 0.0662& \mathrm{-0.6509}\\ \mathrm{-0.5441}& \mathrm{-39.1800}& 21.8400& 2.5120& \mathrm{-0.8066}\end{array}\right]$

$\mathrm{Cmat}≔\mathrm{Matrix}\left(\left[3\,9\,5\,4\,6\right]\right)$

$\mathrm{Cmat}≔\left[\begin{array}{ccccc}3& 9& 5& 4& 6\end{array}\right]$

$p≔\left[-3+2\mathrm{I}\,-1\,-2.5\,-3-2\mathrm{I}\,-4\right]$

$p≔\left[\mathrm{-3}+2\mathrm{I}\,\mathrm{-1}\,\mathrm{-2.5}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-4}\right]$

$\mathrm{StateObserver}:-\mathrm{Ackermann}\left(\mathrm{Amat}\,\mathrm{Cmat}\,p\right)$

$\left[\begin{array}{c}1.082871936\\ \mathrm{-0.4535446046}\\ \mathrm{-0.8242844418}\\ 0.2624744304\\ 0.3998350201\end{array}\right]$

[1] T. Kailath, Linear Systems, Prentice-Hall, 1980.

[2] C. T. Chen, Linear System Theory and Design, 3rd Ed., Oxford University Press, 1999.