(Solved): Problem 1. Consider the basic servo loop, where G(s)=1/s and Gc(s)=kR is a proportional contr ...

Problem 1. Consider the basic servo loop, where $G(s)=1/s$ and $G_{\mathrm{c}}(s)=k?\mathbb{R}$ is a proportional controller. The closed-loop system from $r$ to $e$ is $\hat{e}=S\hat{r},$ where $S?1/\left(1+G{G}_{\mathrm{c}}\right)$. We are interested in following sinusoidal commands $r$ with frequencies in the range $[0,1]\mathrm{rad}/\mathrm{s}$. Consider the weight $W_1(s)=10/(s+1)$, which is designed to yield good command-following performance in the range $[0,1]\mathrm{rad}/\mathrm{s}$. Complete the following: a) Find all values of $k?\mathbb{R}$ such that we achieve nominal performance with respect to $W_1$. You must complete the design by hand; however, you may use Matlab to check your answer. b) Assume $k?\mathbb{R}$ is such that we achieve nominal performance with respect to $W_1$, and assume $r(t)=\sin ?t$, where $??[0,1]\mathrm{rad}/\mathrm{s}$. What is the maximum possible amplitude of the steady-state error? c) Assume $r(t)=\sin ?t$, where $??[0,1]\mathrm{rad}/\mathrm{s}$. Provide a weight $W_1?\mathbb{R}(s)$ such that if we achieve nominal performance with respect to $W_1$, then the maximum possible amplitude of the steady-state error is 0.05 (i.e., $5\%$ ).