Sort out notation for poles, zeros, winding number in Nyquist, Vinnicombe

[Richard Murray]

From Karl, 18 June 2019:

In wikipedia we have the following definition of winding number

Inmathematics <https://en.wikipedia.org/wiki/Mathematics>, the*winding number*of a closedcurve <https://en.wikipedia.org/wiki/Curve>in theplane <https://en.wikipedia.org/wiki/Plane_(mathematics)>around a givenpoint <https://en.wikipedia.org/wiki/Point_(mathematics)>is aninteger <https://en.wikipedia.org/wiki/Integer>representing the total number of times that curve travels counterclockwise around the point. The winding number depends on theorientation <https://en.wikipedia.org/wiki/Curve_orientation>of the curve, and isnegative <https://en.wikipedia.org/wiki/Negative_number>if the curve travels around the point clockwise.

Theorem 10.2

\begin{equation*}
\winding=\frac{1}{2\pi}\Delta \arg_\Gamma f(z) =\frac{1}{2\pi
i}\int_\Gamma\frac{f'(z)}{f(z)}dz = \nyqZ - \nyqP,
\end{equation*}where $\Delta \arg_\Gamma$ is the net variation in the angle when $z$
traverses the contour $\Gamma$ in the counterclockwise direction, $\nyqZ$
is the number of zeros

The origin of the notiation $Z$ and $P$ comes from math and we implicitely assume that we are talking about right half plane poles and zeros

In Vinnicombe metric Equation 13.7

To define a metric Vinnicombe introduced the set $\mathcal C$ of all
rational transfer functions $P_1$ and $P_2$ such that
\begin{equation}
\aligned
\mathcal C = \bigl\{P_1, P_2: \:&1 + P_1(i\omega) P_2(-i\omega) \neq 0,
\forall\,\omega, \\
& \winding (1 + P_1(s) P_2(-s)) + \vinP(P_1(s))
- \vinP(P_2(-s)) = 0\bigr\},
\endaligned
\label{eq:loopsyn:wnocond}
\end{equation}
where $\winding(P)$ is the winding number,\index{winding number} and
\$\vinP(P)$ is the number of poles of the transfer function $P$ in the
open right half-plane. (Compare with the corresponding conditions in
Nyquist's criterion in Theorem~\ref{thm:loopanal:nyquist}.) The
metric is then defined as follows.

I think it would be nice to replace \nyq Z with $n_{z,D}$ and \nyq P with $n_{p,D}$ locally in Theorem 10.2, because that is the only place we talk about general curve C and general region D, BUT keep \nyqZ and \nyqP because they appear in several other places, but I let you decide.

[Richard Murray]

5 July followup from Karl: [Chapter 10] looks good but we could consider a slight modification of Theorem 10.2. The symbols $Z$ and $P$ comes from the standard math notations but we use P in so many meanings in the book. I have two suggestions,

a) Replace $Z$ and $P$ with $\mathcal Z$ and $\mathcal P$ notice that $P$ also appears in Theorem 10.3!

b) Use the same notation as in Theorem 10.3 and replace $Z$ by $n\text{p,D}(f(z)$ the number of zeros of in $D$ and $P$ by $n_text{p,D}(f(z))$ the number of poles of $f(z)$ in $D$.

I have a slight preference for b) it is a bit more clumsy but it matches what we use in Theorem 10.3.

In any case i suggest that "$Z$ is the number of zeros in $D$" is replaced by "$Z$ is the number of zeros of $f(z)$ in $D$" and that "the number of poles" is replacecd by "the number of poles of $f(z)$"

[Richard Murray]

I’ve made a cut at implementing the requested changes. Here is the current state (in commit 529863a):

• Changed the notation in the Theorem 10.2 (principle of variation of the argument) to use notation from Vinnicombe (n_{p,D}, n_{z,D}). In the main theorem I didn't add the function as an argument (n_{p,D}(f(s))), though this appears in some of the body text below the theorem. The macros \vinP and \vinZ are used to generate this notation.
• I also changed the complex variable used in the principle of the argument from z to s. This is defined using the macro \argz so that it can be changed back if desired. The reason for this change is to avoid confusion with the locations of zeros (which we write as z_i).
• I added a paragraph at the end of the generalized Nyquist subsection re-emphasizing the usual engineering practice of tracing out the Nyquist contour in the clockwise directions and also added back the formula Z = N + P, with appropriate definitions for Z, N, and P.
• I moved the inverted pendulum example back up into the generalized Nyquist subsection, instead of the subsection on conditional stability, and updated the text at the start of the conditional stability subsection. I felt like we needed an example that was clearly focused on the generalized Nyquist and so this flowed better.

Will discuss these changes with Karl on 24 Jul and see if we agree that everything is now OK. If so, we can close out this issue.

[Richard Murray]

24 Jul 2019: Seems OK. Karl to double check.

[Richard Murray]

Final decision was to use \winding\Gamma, which expands to w_{n, \Gamma} for the winding number. Poles and zeros use \vinPD, which expands to n_{p, D}(P(s)) and similarly for zeros. The macros \nyqP and \nyqZ expand to $P$ and $Z$ and are used for the clockwise version of Nyquist (just in one spot + example).

Final version is in commit 4484285.