The generalized Nyquist stability criterion

Hi I am trying to plot the Nyquist plot of MIMO system. any help please

Answer 1

[Mohamed Belkhayat]

Once you have a MIMO matrix transfer function you can generate the plot by finding the eigenvalues of the matrix as a function of frequency. A 2x2 MIMO matrix transfer function will have 2 eigenvalues at every frequency point. The imaginary vs. the real part of the two eigenvalues will yield two loci that should always connect. This is the generalized Nyquist. I include a sample code for a textbook example in Multivariable Feedback Design by Maciejowski. Example 2.7. This example is limited to 2x2 but can be extended easily to higher dimensions.

[Khaled Abojlala]

Thanks, prof. this is very helpful

[Mohamed Belkhayat]

Note that the 2017 file only plotted one eigen value, which was an oversight. This updated version Gnyquist2 plots both eigen values as it should and it's a bit faster. Note that in some cases the eigenvalues need to be sorted to maintain the continuity of the eign-loci. Otherwise the eign-values will trade places and it shows as a jump in the plot.

Answer 2

[Mitul Saini]

Is it only applicable for square matrices?

[Andrea]

Yes but normally the open loop K*G(s) is always square for state and output feedback, also if G(s) is not.