41 lines
1.4 KiB
Markdown
41 lines
1.4 KiB
Markdown
# Chapter 1:
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![[Pasted image 20241012132644.png]]
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Notable signals:
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- r - reference or command input
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- e - tracking error
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- u - control signal, controller output
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- d - plant distrurbance
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- y - plant output
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- n - sensor
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# Chapter 2 - Norms
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>[!note] Signal Norms
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>1-Norm:
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> $$||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt$$
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> 2-Norm:
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> $$||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}$$
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> $\infty$-Norm
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> $$||u||_\infty = \sup_t |u(t)|$$
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> Power Signals (Not really a norm):
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> $$pow(u) = \left( \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T u(t)^2 dt \right)^{1/2}$$
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> If the limit exists, u is called a *power signal*
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> > [!caution] Some Implications
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> >1. $||u||_2 < \infty \rightarrow pow(u) = 0$
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> >2. u is a power signal and $||u||_\infty < \infty \rightarrow pow(u) \leq ||u||_\infty$
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> >3. There's a third one in the book about the one norm. I'm ignoring it.
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>[!nnote] System Norms
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>$\hat G$ means the transfer function $G$ in the frequency domain.
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>2-Norm:
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>$$||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} $$
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>$\infty$-norm
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>$$||\hat G||_\infty = \sup_{\omega} |\hat G(j\omega)|$$
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>>[!hint] Parseval's Theorem
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>> If $\hat G$ is stable, then
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>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$
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![[Pasted image 20241012135404.png]]
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