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4 Qualifying Exam/3 Notes/Feedback Control Theory.md
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@ -74,7 +74,7 @@ Generally speaking:
2. Performance specs that involve u result in weights on S 2. Performance specs that involve u result in weights on S
$$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right] $$ $$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right] $$
# Chapter 4 - Plant Uncertainty # Chapter 4 - Uncertainty and Robustness
>[!important] Multiplicative Disk Perturbation >[!important] Multiplicative Disk Perturbation
>$$\tilde{P} = (1+\Delta W_2)P$$ >$$\tilde{P} = (1+\Delta W_2)P$$
> - P is a nominal plant transfer function > - P is a nominal plant transfer function
@ -86,3 +86,14 @@ $$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\
$|W_2(j\omega)|$ 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|. $|W_2(j\omega)|$ 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|.
W_2 is basically a transfer function that will always be greater in magnitude than that P~/P -1 W_2 is basically a transfer function that will always be greater in magnitude than that P~/P -1
>[!note] Robustness
>A controller $C$ is robust to set of plants $\mathcal{P}$ with respect to a characteristic if this characteristic holds for every plant in $\mathcal{P}$.
>> [!important] Robust Stability
>> A system is robustly stable if it is internally stable for every plant in the set $\mathcal{P}$.
>>