Make sure proof of Bode's integral formula is correct

[Richard Murray]

Noah Olsman reported:

I was looking at the proof in your book of Bode's Integral Formula on page 12-34 and I found a point I couldn't figure out how to justify. When making the point about contour integration around RHP poles, you say

Since S(s) ≈ k/(s− pk) close to the pole, the argument of S(s) decreases by 2π as the contour encircles the pole

However the poles we are looking at aren't poles of S(s), but of L(s). They are zeros of S, which makes them poles of log(S). It seems like to rigorously apply the residue theorem here, you really need the fact that we are using the log here (that is what lets you modify S to be analytic and simply add up residues).

I'm not 100% sure on this, but I don't think you can make the argument for why the contour is equal to the sum of the real parts of the poles in as much generality as is presented in the book, I think you have to use the specific form of log(|S|) to get that the residues work out the way they do. If you take the same argument for an arbitrary holomorphic function, I don't think you get the same type of result (you have to explicitly use the Laurent series of the function I think).

[Richard Murray]

Karl has updated the text for the proof. Need to go through and make sure Noah's issue is addressed.

[Richard Murray]

Read through the new version of the proof and we are much more explicit about the structure of log(S(i w)). Looks ok.