condition  : one of relativestability, damping or all

output : one of verify or plot

The RegionPoles command verifies whether the system poles are in a desired region. The desired region is specified by zeta and omegan and is defined based on relative stability and damping conditions as follows:

Relative Stability: The desired region is part of the complex left half plane (LHP) with real part less than $-\mathrm{\zeta }\mathrm{omegan}$. This is equivalent to the relative stability of the closed-loop system with respect to the line $s=j\mathrm{\omega }-\mathrm{\zeta }\mathrm{omegan}$ (rather than the imaginary axis). Clearly, if zeta or omegan are set to zero, the relative stability reduces to the absolute stability with respect to the imaginary axis.

Damping: The desired region is part of the complex left half plane (LHP) inside the angle +/-$\arccos \left(\mathrm{\zeta }\right)$ measured from the negative real axis.

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

$\mathrm{sys}≔\mathrm{DynamicSystems}:-\mathrm{NewSystem}\left(\frac{s+5}{s^3+12s^2+22s+14}\right)$

$\mathrm{sys}≔\left[\begin{array}{c}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\end{array}\right.$

$\mathrm{\ζ1}≔\frac{4}{5}\:$$\mathrm{omegan1}≔2\:$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{verify}\right)$

$\mathrm{false}$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{verify}\,\mathrm{condition}=\mathrm{damping}\right)$

$\mathrm{true}$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{plot}\,\mathrm{condition}=\mathrm{damping}\right)$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{verify}\,\mathrm{condition}=\mathrm{relativestability}\right)$

$\mathrm{false}$

$\mathrm{RegionPoles}\left(\mathrm{sys}\,\mathrm{\ζ1}\,\mathrm{omegan1}\,\mathrm{output}=\mathrm{plot}\,\mathrm{condition}=\mathrm{relativestability}\right)$