(Solved): Problem 5. Consider the basic servo loop with multiplicative uncertainty, where the nominal plant i ...

Problem 5. Consider the basic servo loop with multiplicative uncertainty, where the nominal plant is $G(s)=1/s^2$. The actual plant is $G^{?}=G\left(1+W_2?\right)$ where $??\mathbb{R}(s)$ is asymptotically stable and satisfies $??{?}_{{\mathrm{H}}_{?}}<1$ but is unknown, and the weight is $W_2(s)=2.1s/(s+10)$. Complete the following: a) Design a controller $G_{\mathrm{c}}?\mathbb{R}(s)$ such that the nominal basic servo loop (i.e., BSL with $?=0)$ is internal stable. b) Does the controller from part a) yield robust stability with respect to $W_2$ ? If not, then redesign the controller such that it achieves robust stability with respect to $W_2$. c) Let $W_1(s)=1/(s+1)$. Does the controller from part b) yield robust performance with respect to $W_1$ and $W_2$ ? d) We define the level of robust performance as ${?}_{\mathrm{RP}}?{\max }_{??{?}_{{\mathrm{H}}_{?}}<1}{?\frac{W_{1}S}{1+W_2?T}?}_{{\mathrm{H}}_{?}}={?\frac{W_{1}S}{1??W_{2}T?}?}_{{\mathrm{H}}_{?}}.$ What is the level of robust performance ${?}_{\mathrm{RP}}$ for the controller from part b)? Note that ${?}_{\mathrm{RP}}<1$ implies that we achieve robust performance. Hint: Use Matlab to complete b)-d).