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

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

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 ## Attempt
 Robust control as a field determines how resilient a control system is to a difference in plant dynamics for a given characteristic. In a real system, there will always be some inaccuracy in the model of plant dynamics, disturbances, or other noise. These unmodeled features will affect plant behavior if they are not anticipated. Robust control gives us tools to design for these perturbations proactively. We can design characteristics such as performance and stability to guarantee as 'robust'. 
 
- Robustness is dependent on two features: the characteristic to be guaranteed, and the set of reasonably possible perturbed plants $\mathcal{P}$. Usually the characteristic we're interested in is internal stability or performance. The possible set of plants, however, is less straightforward. The set $\mathcal{P}$ can be structured or unstructured. A structured set in this instance can be a discrete number of possible perturbed plants, or possibly a parametric study with a finite number of parameters @doyleFeedbackControlTheory2009. Let's consider an example. 
+ Robustness is dependent on two features: the characteristic to be guaranteed, and the set of reasonably possible perturbed plants $\mathcal{P}$. Usually the characteristic we're interested in is internal stability or performance. The possible set of plants, however, is less straightforward. The set $\mathcal{P}$ can be structured or unstructured. A structured set in this instance can be a discrete number of possible perturbed plants, or possibly a parametric study with a finite number of parameters @doyleFeedbackControlTheory2009 . Let's consider an example. 
 
 Suppose a plant representing a spring-mass-damper system is described as follows @controltutorialsformatlab&simulinkInvertedPendulumSystem :
 $$ P = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + bs +k}$$
@@ -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 . 
+
+$\Delta$ 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 $\Delta$.