vault backup: 2024-10-17 11:01:49

4 Qualifying Exam/2 Writing/2. QE State of the Art.md

@@ -38,6 +38,4 @@ The other type of uncertainty considered is unstructured uncertainty. This type
 $$ \tilde P = (1+\Delta W_2) P $$
 Where $\Delta$ is a variable stable transfer function with $||\Delta||_\infty < 1$, and $W_2$ is the uncertainty profile.  
 
-(The disk multiplicative perturbation)
-
-(Explain how actually getting to W_2 isn't really trivial).
+The 'disk' part of the multiplicative disk uncertainty comes from analysis in the complex domain, specifically looking at the Nyquist Stability Criterion. Stability according to this criterion is determined when the loop gain of a system does not pass through the point -1 during a sweep of all frequencies on the imaginary access. 

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

@@ -80,7 +80,7 @@ $$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\
 > - P is a nominal plant transfer function
 > - $\Delta$ is a variable stable transfer function s.t. $||\Delta||_\infty <1$
 > - P and $\tilde P$ have the same unstable poles.
-> If $||\Delta||_\infty <1$:
+> If $||\Delta||_\infty <1$, $W_2$ should be chosen s.t.:
 > $$ \left| \frac{\tilde P (j\omega)}{P(j\omega)} - 1 \right| \leq | W_2(j\omega) | \text{ , } \forall \omega$$
 
 $|W_2(j\omega)|$ is the uncertainty profile. This inequality describes a disk in teh complex plane: at each frequency the point P~/P lies in the disk with center 1, radius |W_2|. 
@@ -104,6 +104,6 @@ Something really helpful to think about came to mind as a result of watching a S
 $$ y = \frac{L}{1+L} r $$
 If at a certain frequency $\omega$, L approaches -1, big problems happen. What this means is that the denominator in the above equation gets really small, which means the gain from r to y actually gets really big. If it IS -1, immediate undefined blow up. 
 
-This is where robustness comes from. The distance between L and -1 for all frequencies is what robustness is. Less distance, less room for plant perturbation that could make you unstable. More distance, safer response. 
+This is where robustness comes from. The distance between L and -1 for all frequencies is what robustness is. Less distance, less room for plant perturbation that could make you unstable. More distance, safer response. This gets integrated when you start thinking about $W_2$ and $\Delta$. These two things are how you account for the uncertainty and look at how that gets you closer to -1 or not. 
 
 [^1]: [[stevebruntonControlBootcampSensitivity2017]]