From 12c47b9f2d707cecec09c0a84ba91165915216cd Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Sat, 12 Oct 2024 13:54:15 -0400 Subject: [PATCH] vault backup: 2024-10-12 13:54:15 --- .../3 Notes/Feedback Control Theory.md | 28 ++++++++++++++++-- Pasted image 20241012135404.png | Bin 0 -> 314557 bytes 2 files changed, 25 insertions(+), 3 deletions(-) create mode 100644 Pasted image 20241012135404.png diff --git a/4 Qualifying Exam/3 Notes/Feedback Control Theory.md b/4 Qualifying Exam/3 Notes/Feedback Control Theory.md index 353b55d63..8274f418c 100644 --- a/4 Qualifying Exam/3 Notes/Feedback Control Theory.md +++ b/4 Qualifying Exam/3 Notes/Feedback Control Theory.md @@ -9,10 +9,32 @@ Notable signals: - n - sensor # Chapter 2 - Norms >[!note] Signal Norms ->1-Norm +>1-Norm: > $$||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt$$ -> 2-Norm +> 2-Norm: > $$||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}$$ > $\infty$-Norm -> $$||u||_\infty = \text{sup}_t |u(t)|$$ +> $$||u||_\infty = \sup_t |u(t)|$$ +> Power Signals (Not really a norm): +> $$pow(u) = \left( \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T u(t)^2 dt \right)^{1/2}$$ +> If the limit exists, u is called a *power signal* +> > [!caution] Some Implications +> >1. $||u||_2 < \infty \rightarrow pow(u) = 0$ +> >2. u is a power signal and $||u||_\infty < \infty \rightarrow pow(u) \leq ||u||_\infty$ +> >3. There's a third one in the book about the one norm. I'm ignoring it. + +>[!nnote] System Norms +>$\hat G$ means the transfer function $G$ in the frequency domain. +>2-Norm: +>$$||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} $$ +>$\infty$-norm +>$$||\hat G||_\infty = \sup_{\omega} |\hat G(j\omega)|$$ +>>[!hint] Parseval's Theorem +>> If $\hat G$ is stable, then +>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$ + +![[Pasted image 20241012135404.png]] + + + diff --git a/Pasted image 20241012135404.png b/Pasted image 20241012135404.png new file mode 100644 index 0000000000000000000000000000000000000000..21b01c405aa8dada1f447fb6353e25bbe22f719c GIT binary patch literal 314557 zcmd?QWl)?;)HOP|ySsbP;10n81P>70-C=MGgA?3>TOb5Ya3{FC>tMm%ZQz@n^S$TR zt^4o(xOJ<(u9}*fsi(W2-o3kf_gZU5sj0|gppv2j004~l3Nji1075tPbB2rv{Rh+* zDGL39cb9&zg$ynJ$QF^%-y|Ngx*nR&RvuoaZk7ORCuc`Xc6W0(OG_tr8)uI*xK45C 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