Controls Contours

Note

The goal here is to test closed-loop stability visually which is helpful for determining the range of a parameter (like gain), for which the system is closed-loop stable.

Definitions

$\Gamma$ is…

• Simple: No self-intersections
• Closed: Starts and ends at the same point

Furthermore, $\Gamma$ is a contour if it is a simple, closed curve with a direction.

Lemma

In general, Let $p$ be in $\mathbb{C}$ and let $\Gamma$ be a contour. Then $\frac{1}{2\pi j}$, ${\int }_{\Gamma }\frac{1}{z-p}dz=\{\begin{array}{ll}1& \text{if }\Gamma \text{ encloses p},\\ 0& \text{otherwise}\end{array}$

This is effectively a check to see if $p$ is inside the contour. If $p$ is on the contour, this would result in an undefined integral ($\infty$)

The contour that we care about here is the unit circle so that we can test stability.

If we have a $G[z]$ which is real, rational, proper, and $\Gamma$ as a contour. Then $G[z]=\frac{{\prod }_{j=1}^m(z-z_j)}{{\prod }_{i=1}^n(z-p_i)}$
$\frac{d}{dz}(\log G[z])=\frac{{\sum }_m^{j=1}1}{z-z_j}-{\sum }_{i=1}^n\frac{1}{z-p_i}$ where now we have a sum
Taking the integral of both sides and using the lemma from class…
This becomes ${\sum }_{j=1}^m\{\begin{array}{ll}1& \text{if }\Gamma \text{ encloses zj},\\ 0& \text{otherwise}\end{array}{\sum }_{i=1}^n\{\begin{array}{ll}1& \text{if }\Gamma \text{ encloses pi},\\ 0& \text{otherwise}\end{array}=z-p$ where each of these variables are the number of zeroes or poles (counting multiplicities) enclosed by $\Gamma$.

Another application

$\frac{1}{2\pi j}{\int }_{\Gamma }\frac{d}{dz}(\log G[z])dz$ also is $\frac{1}{2\pi j}{\int }_{\Gamma }\frac{G^{\prime }[z]}{G[z]}dz=\frac{1}{2\pi j}\int \frac{dw}{w}$

Where we made the change of variables $w=G[z]$, $dw=G^{\prime }[z]dz$

If we make a second change of variables $w=r{e}^{j\theta }$ and $dw=e^{j\theta }dr+jr{e}^{j\theta }d\theta$ = $\frac{1}{2\pi j}{\int }_{G[\Gamma ]}\frac{dr}{r}+jd\theta =\frac{1}{2\pi j}[\log r{|}_{r_0}^{r_1}+[j\theta {|}_{{\theta }_0}^{{\theta }_1}$ where $r_1=r_0$ because $G[\Gamma ]$ is closed, and ${\theta }_1\ne {\theta }_0$ in general.

So our result is $\frac{1}{2\pi }\Delta \theta$ and $\Delta \theta =2\pi \ast N$ where $N$ is the number of times this contour $G[\Gamma ]$ circles the origin (this all evaluates to just $N$).

Here, direction matters, CCW equates to positive integers, and CW equates to negative integers.

The result of this application leads to the following Lemma…

• Argument Principle