Answer Happy

for question 1
Where A = B = 9 , so r(t) = 1.9u(t) and ess<0.19
please solve in 60 mins i will give thumb up
Q1. Attempt all parts of this question. (a) Control systems engineering and technology have hugely influenced our everyday lives. Write a concise narrative on the future trends and scopes of control systems engineering and technology including potential challenges. [6 marks] The transfer functions of the control system shown in Figure Q1a are as follows: K Gc(S) Gp(s) (s + 5) (s + 1)(s + 4) 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 <0. [8 marks] (c) Derive an expression for the open-loop frequency response of the system with respect to K for w → 0 and w → 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. 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 < 0.1B. Determine the minimum value of K that would satisfy the design requirement. [6 marks]

(f) For the block diagram of a control system shown in Figure Q1b, derive the closed-loop transfer function, C(s)/R(s). [6 marks) G(s) D(S) + R(s) E(s) G2(s) G3(s) C(s) Hi(s) Figure Q1b