From c6ca56802e464ccb0e3e5cfb621830fc6b928ba8 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Tue, 15 Oct 2024 16:49:32 -0400 Subject: [PATCH] vault backup: 2024-10-15 16:49:32 --- 4 Qualifying Exam/3 Notes/Feedback Control Theory.md | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/4 Qualifying Exam/3 Notes/Feedback Control Theory.md b/4 Qualifying Exam/3 Notes/Feedback Control Theory.md index 20552842..efddcf68 100644 --- a/4 Qualifying Exam/3 Notes/Feedback Control Theory.md +++ b/4 Qualifying Exam/3 Notes/Feedback Control Theory.md @@ -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 \ No newline at end of file +# 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|. \ No newline at end of file