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4 Qualifying Exam/3 Notes/Feedback Control Theory.md
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# Chapter 1: All info from [[doyleFeedbackControlTheory2009]]
# Chapter 1 - Introduction
![[Pasted image 20241012132644.png]] ![[Pasted image 20241012132644.png]]
Notable signals: Notable signals:
- r - reference or command input - r - reference or command input
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>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$ >> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$
![[Pasted image 20241012135404.png]] ![[Pasted image 20241012135404.png]]
# Chapter 3 - Basic Feedback Loop
![[Pasted image 20241014145054.png]]
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**

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4 Qualifying Exam/3 Notes/Robust Control.md
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# History # History
## Where did Robust Control come from? ## Where did Robust Control come from?
After the beginnings of modern control and the development of optimal control, John Doyle released a paper in 1978 titled [Guaranteed Margins for LQG regulators](doyleGuaranteedMarginsLQG1978a). This is a less than one page paper that basically gave birth to the robust control field, with a three word abstract: "There are none." I'm working out the kinks in this one ([[Basic Feedback Control]]), but essentially the gaussian part of the LQG is what destroys the guaranteed part of the phase and gain margins. The additional estimator involved can really screw with things. After the beginnings of modern control and the development of optimal control, John Doyle released a paper in 1978 titled [Guaranteed Margins for LQG regulators](doyleGuaranteedMarginsLQG1978a). This is a less than one page paper that basically gave birth to the robust control field, with a three word abstract: "There are none." I'm working out the kinks in this one ([[Feedback Control Theory]]), but essentially the gaussian part of the LQG is what destroys the guaranteed part of the phase and gain margins. The additional estimator involved can really screw with things.
I should add some context: [[4 Qualifying Exam/3 Notes/Feedback Control Theory]]. I should add some context: [[4 Qualifying Exam/3 Notes/Feedback Control Theory]].
# What does Robust Control do? # What does Robust Control do?

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