Obsidian/4 Qualifying Exam/3 Notes/Feedback Control Theory.md

Chapter 1:

! 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}

!