strange curve that NyquistPlot gives for delayed first-order system

Question

I've got a strange result from NyquistPlot for a delayed first-order system $\frac{e^{-i\pi /2}}{s + 1}$, using the mathematica code:

NyquistPlot[TransferFunctionModel[E^(-\[ImaginaryJ] \[Pi]/2)/(s + 1), s]
    ,AxesOrigin -> {0, 0}, PlotRange -> Full,PlotLegends -> "Expressions"]

this is the result plot with a strange “drifting” “upside down” portion of arc:

which is expected to be a complete normal circle right focus at (0,-0.5) like this (I drawed it using Graphics and Circle but not NyquistPlot!):

Addionally, for simple first-order system without time-delay $\frac{1}{s + 1}$, the plot result is just right circle focus at (0.5,0):

 NyquistPlot[TransferFunctionModel[1/(s + 1), s], AxesOrigin -> {0, 0},
      PlotRange -> Full, PlotLegends -> "Expressions"]

I check existed questions here by keyword NyquestPlot but nothing relevant.

Even is there some limitation in mathematica? or what's wrong in my code or in my expectation? Is there something I miss when using NyquistPlot?

Answer 1

First, I think this is a nonphysical system rather than a time-delay system. (The output response is imaginary and not delayed.)

I don't think NyquistPlot was designed to automatically handle such systems. As a workaround you can do:

NyquistPlot[TransferFunctionModel[E^(-I \[Pi]/2)/(s + 1), s], {-501, 500}, 
AxesOrigin -> {0, 0}]