Question

For each of the four plants below and controller given by C(s) K, find the transfer function of the closed loop sy stem, find the expression for the CL poles (by hand) as a function of K, and sket ch the root locus using the expression for the poles, not Matlab s +2 s +1 s1)21 s1)2 s+ 1

Answer 1

Given four plants p1(s) = 1/s + 1, p_2(s) = s + 2/s + 1, p_3 (s) = 1/(s + 1) (s + 3) p_4 (s) = (s + 1)^2 + 1/(s + 1)^2 therefore for system 1 p_1 (3) = 1/s + 1 therefore closed loop function therefore c(s)/R(s) = T(s) = k 1/s + 1/1 + k/s + 1 = k/s + 1/s + 1 + k/s + 1 = k/(s + 1 + k) \r\n therefore T(s) = k/(s +(1 + k) closed loops poles -(1 + k) therefore Root loop c_1(s) = k/s + 1 starting point (-1) pole ending point infinity p_2(s) = s + 2/s + 1 T(s) = k(s + 2)/s + 1/1 + k (s + 2)/s + 1 = k(s + 2)/s + 1 + ks + 2k = k(s + 2)/s(1 + k) + (1 + 2k) = (k/I + k) (s + 2)/s + (1 + 2k/1 +