Problem $2($Ball & Beam$).$ The servo motor from Problem $1$ is used as an actuator in an overall system that controls the position of a ball on a beam. That control system may be approximated as shown below, with a plant model of G$($s$)=0\mathrm{.}419$ s $2.$ It is desired that the ball & beam control system exhibit no more than $15\%$ overshoot for a step input and the bandwidth of the system should not exceed a tenth of the bandwidth from Problem $1,$ i$.$e$.$ c$2<$ c$1/10.$ r C$($s$)$ G$($s$)$