Answer Happy

The transfer functions of the control system shown in Figure Q1a are as follows: K (s + 5) G(S) = (S + 3)2 , Gp(S) = (- 4) and H(S) = 1. = s + 1)(s + = R(S) C(s). E(S) e(t) r(t) + Gc(s) Ge(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 <KS. [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] (e) An input r(t) = 1.5u(t) is applied to the system, where the function u(t) is the unit step function. It is desired that the steady-state error, esse < 0.4. Determine the minimum value of that would satisfy the design requirement. [6 marks)

The transfer functions of the control system shown in Figure Q1a are as follows: K (s + 5) Gc(s) = ( Ls + 3)2 , Gp(S) = ( 4) and H(S) = 1. = = s + 1)(s + R(S) E(S) e(t) G(S) C(s). c(t) r(t) + Ge(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 <KSO. [8 marks]