Question

For the unity feedback system with forward loop T.F. G(s) =s(s+2)(s+4) Find, using Routh criterion, the value of K for which the system has undamped oscillatory response. A unity feedback control system has the following open-loop T.F k1 s(s+1)00.5s +1) And an input of r(t) 5t b) It is desired that, for a ramp input, Ess 0.1. What minimum value must ki have for this condition to be satisfied? cFor the value of ki determine in part (b) is the system stable or unstable? Check the stability for the following system: Check the stability for the following system: $5+2s4 +3s3 6s2 20s 10

Answer 1

G(s) = k/s(s + 2)(s + 4) characteristics equation can written as - 1 + G(s) = 0 [because H(s) = 1] 1 + k/s(s + 2)(s + 4) = 0 s(s^2 + 6s + 8) + k = 0 s^3 + 6s^2 + 8s + k = 0 Routh table - for oscillatory implies 48 - k/6 = 0 k = 48 At this k value system has oscillatory Response. G(s) = k_1/s(1 + s)(0.5 s + 1) (A) For r(t) input k_V = lim_s rightarrow 0 s middot G(s) H(s) = lim_s rightarrow infinity k_1/(1 + s)(0.5 s + 1) \r\n k_V = k_1 if r(t) = 5t then E_ss = 5/k_V = 5/k_1 E_ss = 5/1.5 (because k_1 = 1.5 sec^-1) E_ss = 3.33 (B) For r(t) = 5t E_ss = 5/k_1 for E_ss < 0.1 implies 5/k_1 < 0.1 5/0.1 < k_1 k_1 > 50 (c) G(s) = k_1/s(s + 1)(0.5 s + 1)