Answer Happy

Problem 3 [25 marks] Consider the generic feedback system shown in Figure 4, where G(s) is the plant's transfer function and C(s) is the controller's transfer function. y C(s) G(s) Figure 4 The plant G(s), which is open-loop unstable, and the Controller C(s) are given as follows: (s+5)(s+10) k C(s) = G(s) (1) (s - 10)(s+20)' where k >O is a proportional gain. Study the root-locus of the feedback system through the following steps (a summary of the rules for root-locus construction is given at the end of the examination paper): (i) state the open-loop poles and zeros. [2.5 marks] (ii) determine the regions of the locus lying on the real axis. [1.5 marks] (iii) draw an approximate root-locus diagram. [4 marks] (iv) determine the point at which the root-locus re-enters the real axis and the corre- sponding value of k. Your answer should be justified. [4 marks] Hint: the re-entry point is one of these three points: -15.15, -7.15,-5 (v) knowing that the root locus intersects the imaginary axis at the point s= +5.5j, find the corresponding value of k. [4 marks] (vi) on the basis of the root-locus obtained, discuss the variation in the properties of the feedback system (e.g., oscillatory behaviour and stability) for different inter- vals of values of k. [9 marks]