This problem assumes our standard unity (H(s) = 1) negative feedback closed-loop control system. The given open-loop transfer function is 2 ( ) ( 2) G s s s   . Using denominator parameter matching, design a PD (proportional-derivative) controller ( ) G s K Ks C PD   to achieve a closed-loop overdamped second-order response with 1s  2 and 2 s  4 . What is the resulting numerical closed-loop transfer function T(s)? For a unit step input R(s) and assuming 1.0 is the desired steady-state output value, determine the closed-loop correction factor, if one is required. Determine the pre-filter transfer function ( ) G s P , if one is required. Given a unit step input R(s), sketch the open-loop and input-corrected, pre-filtered closed-loop output responses, assuming zero initial conditions for both cases.