Question

3)er the ety m where wo)-A lNe-2) D CO A) e ks a d the kolowie ples,sp a hea way and beak points heawe ol gh K he keen the watem stable INle etx to yel a stable ystem with second onder complex poles, with a damping ratio of 1-2 FrI 1

Answer 1

G(s) = k(s - 1) (s - 2)/s(s + 1) a) The break point are given by dk/ds = 0 where k = -[s^2 + s]/(s^2 - 3s + 2) dk/ds = -2(2s^2 - 2s - 1)/(s^2 - 3s + 2)^2 = 0 2s^2 - 2s - 1 = 0 The roots are - 0.37, 1.37 since the locus at poles breaks between 0 and - 1 The break away point is - 0.37 and the break in point is 1.37 b)c) The characteristic equation is 1 + K G(s) = 0 1 + K(s^2 - 3s + 2)/s^2 + s = 0 s^2 + s + ks^2 - 3ks + 2k = 0 s^2 (1 + k) + s(1 - 3k) + 2k = 0 equation (1) fn the system to be stable all the co - efficient of 's' mist be positive ie 1 - 3k greaterthanorequalto 0 or k lessthanorequalto 1/3 = 0.33 so fn the system to be stable the gain must be less than 0.33 k lessthanorequalto 0.33 \r\n The