Answer Happy

The transfer functions of the control system shown in Figure Q1a are as follows: K Gc(s) = (s +3)2 (s +5) G(S) = (s + 1)(s+4) and H(S) = 1. 7 R(s) E(s). e(t) Gc(s) C(s) c(t) r(t) + Gp(s) H(s) Figure Q1a (b) Sketch the root-locus diagram, including any asymptotes. Indicate the point of instability and the positions of small positive values of K. Comment on the nature of system damping for 0 <K 500. [8 marks) (c) Derive an expression for the open-loop frequency response of the system with respect to K for W - and w → 0 and hence, show a simple Nyquist sketch of the open-loop response of the system. [6 marks) (d) With the aid of the Routh-Hurwitz stability criterion, investigate the range of the gain that would guarantee the stability of the system and hence, find the I frequency of oscillation at the marginal stability point. [8 marks] (e) An input r(t) = 1 Au(t) is applied to the system, where the function u(t) is the unit step function. It is desired that the steady-state error, ess <0.13. Determine the minimum value of K that would satisfy the design requirement. [6 marks]