# Chapter 1:
![[Pasted image 20241012132644.png]]
Notable signals:
- r - reference or command input
- e - tracking error
- u - control signal, controller output
- d - plant distrurbance
- y - plant output
- n - sensor
# Chapter 2 - Norms
>[!note] Signal Norms
>1-Norm:
> $$||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt$$
> 2-Norm:
> $$||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}$$
> $\infty$-Norm
> $$||u||_\infty = \sup_t |u(t)|$$
> 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]]