Question

answer ASAP

2. (20 points) Consider the following unity-feedback system. Suppose G(s) sis+45+6s+4) R) Ets) G(s) A. Check the stability of the closed-loop system by using Routh-Hurwitz Criterion. B. Find the steady-state error when the input is a unit step function.

Answer 1

Given G(s) = 1/s(s^3 + 4s^2 + 6s + 4) characteristic equation = denominator of closed loop Transformation function = 0 Transfer function = c(s)/R(s) = G(s)/1 + G(s) = 1/s(s^3 + 4s^2 + 6s + 4/1 + 1/s(s^3 + 4s^2 + 6s + 4) = 1/s(S^3 + 4s^2 + 6s + 4) + 1 = 1/s^4 + 4s^3 + 6s^3 + 4s + 1 characteristic equation = s^4 + 4s^3 + 6s^2 + 4s + 1 = 0 \r\n A) stability from Routh - Hurwitz criteria s^4 1 6 1 s^3 4 4 0 s^2 4 times 6 -1 times 4/4 = 5 4 times 1 - 0/4 = 1 0 s^1 5 times 4 - 4 times 1/5 = 16/5 0 0 s^0 1 0 0 first column of Routh - Hurwitz criteria all the elements are same sign no sign change so system is absolute stable system. \r\n B) error = input - output [steady state] E(s) = R(s) - c(s) from figure [c(s) = E(s) G(s)] put this c(s) in above equation E(s) = R(s) - E(s) G(s) E(s) [1 + G(s)] = R(s)