Answer Happy

Q5 (a) A laboratory experiment can be modelled as a system with an open-loop transfer function 1 P(s) (s2 + 4s + 3) with a controller C(s) which has the transfer function K(s +5) C(s) S +8 where K>0. (i) Derive and sketch the root-locus of the corresponding closed loop system. In your answer, show all intermediate steps such as the locations of the open-loop poles and zeros, the relative degree of the open-loop system, the number of asymptotes and their meeting point, and the location(s) of any double point(s) of the root locus. (ii) Comment on the characteristics of this system for small values of gain K and for larger values of K. Does the system become unstable? What is the maximal value of K for which the closed-loop system is overdamped? [4] Hint: The equation x3 + 13.5x2 + 60x + 75.5 = 0 has the solutions x1 = -2.09, X2 = -5.71 +j1.90 and X3 = -5.71 -1.90.