vault backup: 2024-10-12 13:54:15

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
+>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$
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+![[Pasted image 20241012135404.png]]
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