ControlDesign[StateObserver]

 Ackermann
 calculate the observer gain for single-output systems using Ackermann's formula

 Calling Sequence Ackermann(Amat, Cmat, p) Ackermann(sys, p)

Parameters

 Amat - Matrix; system matrix of a state-space system Cmat - Matrix or Vector; output matrix of a state-space system sys - System; a DynamicSystems system object of state-space format p - list ; list of desired closed-loop poles (real or complex). Complex poles including those containing symbolic parameters must be given in complex conjugate pairs. All symbolic parameters in the list are assumed to be real.

Description

 • 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 $\mathrm{Ac}=\mathrm{Amat}-L\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{Cmat}$ (or Ac = sys:-a-L.sys:-c) where $L$ is the calculated observer gain.

Examples

 > $\mathrm{with}\left(\mathrm{ControlDesign}\right):$
 > $\mathrm{Amat}≔\mathrm{Matrix}\left(\left[\left[-0.9054,-39.7500,22.4100,0.1891,-0.7366\right],\left[-0.3734,-77.0100,41.1600,4.1900,-1.0450\right],\left[-0.2815,-120.8000,63.6500,4.3650,-2.1850\right],\left[0.4264,2.2290,-0.2781,0.0662,-0.6509\right],\left[-0.5441,-39.1800,21.8400,2.5120,-0.8066\right]\right]\right)$
 ${\mathrm{Amat}}{:=}\left[\begin{array}{ccccc}{-}{0.9054}& {-}{39.7500}& {22.4100}& {0.1891}& {-}{0.7366}\\ {-}{0.3734}& {-}{77.0100}& {41.1600}& {4.1900}& {-}{1.0450}\\ {-}{0.2815}& {-}{120.8000}& {63.6500}& {4.3650}& {-}{2.1850}\\ {0.4264}& {2.2290}& {-}{0.2781}& {0.0662}& {-}{0.6509}\\ {-}{0.5441}& {-}{39.1800}& {21.8400}& {2.5120}& {-}{0.8066}\end{array}\right]$ (1)
 > $\mathrm{Cmat}≔\mathrm{Matrix}\left(\left[3,9,5,4,6\right]\right)$
 ${\mathrm{Cmat}}{:=}\left[\begin{array}{rrrrr}{3}& {9}& {5}& {4}& {6}\end{array}\right]$ (2)
 > $p≔\left[-3+2I,-1,-2.5,-3-2I,-4\right]$
 ${p}{:=}\left[{-}{3}{+}{2}{}{I}{,}{-}{1}{,}{-}{2.5}{,}{-}{3}{-}{2}{}{I}{,}{-}{4}\right]$ (3)
 > $\mathrm{StateObserver}:-\mathrm{Ackermann}\left(\mathrm{Amat},\mathrm{Cmat},p\right)$
 $\left[\begin{array}{c}{1.082871936}\\ {-}{0.4535446046}\\ {-}{0.8242844418}\\ {0.2624744304}\\ {0.3998350201}\end{array}\right]$ (4)

References

 [1] T. Kailath, Linear Systems, Prentice-Hall, 1980.
 [2] C. T. Chen, Linear System Theory and Design, 3rd Ed., Oxford University Press, 1999.