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11
4 Qualifying Exam/3 Notes/Feedback Control Theory.md
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@ -74,4 +74,13 @@ Generally speaking:
2. Performance specs that involve u result in weights on S
$$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right] $$
# Chapter 4 - Plant Uncertainty
# Chapter 4 - Plant Uncertainty
>[!important] Multiplicative Disk Perturbation
>$$\tilde{P} = (1+\Delta W_2)P$$
> - P is a nominal plant transfer function
> - $\Delta$ is a variable stable transfer function s.t. $||\Delta||_\infty <1$
> - P and $\tilde P$ have the same unstable poles.
> If $||\Delta||_\infty <1$:
> $$ \left| \frac{\tilde P (j\omega)}{P(j\omega)} - 1 \right| \leq | W_2(j\omega) | \text{ , } \forall \omega$$
$|W_2(j\omega)|$ is the uncertainty profile. This inequality describes a disk in teh complex plane: at each frequency the point P~/P lies in the disk with center 1, radius |W_2|.