Question

please provide a detailed answer to part (e) only

Thanks

A2 2013 2. Consider the feedback system in Figure 3 in which G(8) gain. S-2 and K(8) = K is a $ +2 variable 3. + R(S) E (5) D() K(s) G(3) Figure 3 (a) Find the closed-loop transfer function between R(s) and D(s). [3 marks) (b) Sketch the Nyquist diagram of G(8), carefully indicating the direction in which the origin is encircled. Hence or otherwise, find the range of gains for which the closed- loop is stable; consider both positive and negative values of K. [6 marks) 8 +2 Carefully explain how the Nyquist (c) Now suppose that the plant model is G(8) diagram of part (b) changes. 8-2 [3 marks) (d) In the case of this new plant, find the range of gains for which the closed-loop is stable; again consider positive and negative values of K. [3 marks) (e) Consider the implementation of a controller of the form sta K(8) = 8 + b 8 +2 with G(s) = Show that K(s) must be a lag compensator if it is to stabilize the S-2 closed loop. Select values for a and b so that the closed-loop poles are located at -1=j- (5 marks

Answer 1

atb = 2 = a + b =4 -ean a-b=2 - egna from Eq n and ean G 923 and b=1 Now Kess ما is given by (6+3) (3+1) KIS) = e -3 from pole-zoro plot of kis) it is observed that pole is dominating then zero. Hence kis) is a leg compensator.