Problem 4 (Average Root Locations and Minimum Possible Settling Times) All of the above analysis ignores the solution of the homogeneous equation. Since the homogeneous equation sets the right hand side of the differential equation to zero, this part of the solution knows nothing about the command or desired output. It only relates to initial conditions1 and we want it to decay to zero in a reasonable amount of time (after which we are left with the errors analyzed above. lConsider a third order polynomial 33 +12.512 +bls+b = [ssl](ss2)[ssg] = s3 (sl + 32 + Sal-92 +(sls1 + s2s3+ sgsds slsls3 = CI One can conclude that in a polynomial the coefcient of the next to the highest power is the negative of the sum of the roots. Or the average location of all roots is minus this coefcient divided by the number of roots. {A} 1|What happens to the average location when you go from a proportional controller to an integral controller? (B) The fastest possible decay would occur when all the roots happened to be at the average location. If all roots happened to be at the average location what would be the setting time for the proportional control system and for the integral control system. Did the steady state errors get better with integral control by the solution of the homogeneous equation take longer to decay