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

@@ -1,4 +1,5 @@
-# Chapter 1:
+All info from [[doyleFeedbackControlTheory2009]]
+# Chapter 1 - Introduction
 ![[Pasted image 20241012132644.png]]
 Notable signals:
 - r - reference or command input
@@ -34,7 +35,14 @@ Notable signals:
 >> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$
 
 ![[Pasted image 20241012135404.png]]
+# 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**

4 Qualifying Exam/3 Notes/Robust Control.md

@@ -1,6 +1,6 @@
 # History
 ## 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]].
 # What does Robust Control do?