For the system shown below do the following: 1. Derive the equation of motion. 2. Find the transfer Function. 3. For the open loop system with unit step input and m = 1 kg, B = 2√3 N.s/m, K = 4 N/m, find the steady state response (x) and the steady state error (ess). 4. Design a proportional controller with = (Sketch the root locus and find x,, and ess). 5. Design a PI controller with = 6. Design a PD controller with locus and find xs, and ess). = 7. Design a PID controller with = √2 (Sketch the root locus and find X., and e..). with half reduction in settling time (Sketch the root with half reduction in settling time and zero steady state error for a step input (Sketch the root locus and find X., and ess). 8. Compare the results and write the conclusion. K 0000 B -x(t) M f(t)