Make sure Nyquist criterion works for MIMO

[Richard Murray]

On 23 Jan 2019, Karl wrote: One of our visting professor Richard Pates gave me the attached paper which indicates that there may be a good idea to reconsider the Nyquist contour so that it works in the multivariable case, the example from Roy is new to me. Let us ponder it for a while.

Karl also felt that “it may be a good idea to replace (conjugate points) by (antipodal points) antipodal leads you directly to think about spheres. Let us ponder this a while too.”

In the note from Richard Pates, he wrote that “the formula you have is right for
antipodal points on the Riemann sphere too:
https://math.stackexchange.com/questions/480891/projections-on-the-riemann-sphere-are-antipodal”

[Richard Murray]

Karl looked into this and what we have is OK. Some notes:

• Vinnicombe has a good discussion on this topic and the difference in choosing indentations to the right or to the left. His contour is indented to the RHP as in Figure 1.2 on p 15

• Vinnicombe has also an insightful discussion on page 18, 19, our contour is the sama as ii) in Theorem 1.10 which he also prefers, notice that he claims that for multivariable systems it is usually indented into the LHP.

My suggestion is that we stick with what we have we could perhaps consider modifying the contour in Fig 11.4 a to show complex polse more clearly. Personally I dont think that the Nyquist theorem based on det(I-PC) is particularly useful it is much better to use the singular value of the Gang of Four as Vinnicombe does on p 19.