vault backup: 2024-10-17 11:39:29

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

@@ -3,7 +3,7 @@ title: State of the Art
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@@ -40,4 +40,8 @@ Where $\Delta$ is a variable stable transfer function with $||\Delta||_\infty <
 
 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 $L$ of a system does not pass through the point -1 during a sweep of all frequencies on the imaginary access. For robust stability, we examine if a system is still stable when calculating the Nyquist plot of $W_2 L$. If it is, then all perturbed plants $\tilde P = (1+\Delta W_2)P$ are also stable. 
 
-This is useful for us. If we can find an uncertainty transfer function $W_2$ that we are satisfied with, 
+This is useful for us. If we can find an uncertainty transfer function $W_2$ that we are satisfied with, and pair it with a design of a controller that maintains the Nyquist criterion, then we know our system is robust to any perturbations captured by $||\Delta||_\infty <1$. Robust performance can be achieved using a similar process @doyleFeedbackControlTheory2009 . 
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+$\Delta$ is almost always considered a free variable transfer function. Since $||\Delta||_\infty < 1 \text{ } \forall \omega$, $\Delta$ will not decrease the minimum robustness margin. This is fine for developing a controller, but when it comes to actually verifying robustness of a controller implementation, $\Delta$ cannot be a variable. To create a plant to simulate a perturbed plant, $\Delta$ must have an expression. 
+
+**Limitation**: There is no current method for creating examples of random examples $\Delta$.