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

All info from doyleFeedbackControlTheory2009

Chapter 1 - Introduction

! 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}

!

Chapter 3 - Basic Feedback Loop

! P, C, and F are system transfer functions. For a system to be well-posed, they cannot all be strictly proper. P is almost always strictly proper while the others aren't.

Something interesting - If \frac{1}{1+PCF} is proper, then this suggests that the system output goes to zero when j\omega\rightarrow \infty. This isn't true in reality, because real systems will behave in a not linear way at high frequencies.

[!tip] Nine System Transfer Functions

\left(\matrix{x_1 \\ x_2 \\ x_3}\right) = \frac{1}{1+PCF} \left[\matrix{1 & -PF & -F \\ C & 1 & -CF \\ PC & P & 1}\right] \left(\matrix{r \\ d \\ n}\right)

Notable Properties:

• All 9 transfer functions are strictly proper if 1+PCF is not strictly proper.
• If all 9 transfer functions are stable, then the system is internally stable