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The transfer functions of the control system shown in Figure Q1a are as follows: K Gc(s) - (s + 5) (s + 1)(s + 4) Gp(s) and H(s) = 1. = (s + 3)2 ) R(s) r(t) E(S) e(t) Gc(s) Gp(s) C(s). c(t) 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 SOO. [8 marks] (C) Derive an expression for the open-loop frequency response of the system with respect to K for w → 0 and w → 00 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 K that would guarantee the stability of the system and hence, find the frequency of oscillation at the marginal stability point. [8 marks]