Question

1. Consider the following feedback control system Controller Process 1 G(s) R(s) Y(s) $2+5s+6 Below are two potential controllers for this system: 1) Ge(s) K (Proportional controller) 2) Ge(s) K(1 1/s) (Proportional-integral controller) The design specifications are t 3.2s and P. 0. 10% for a unit step input (a) Determine the area on the S-plane where the dominant closed loop poles must be located such that the design requirements are satisfied. (b) Sketch the root locus with each of the given controllers separately (c) Using the results of parts (a) and (b), determine whether it is possible or not to satisfy the design requirements with each of the two controllers. Explain your answer

Answer 1

sal: Gissa - § +5576 han the given specifications ts= 4 = 3:2 za suna 125 and -nelors e = » S=0.6 and Wn 1.25 10 100 = 2.0833 he dominant poles are = {wn & wn sorg²J" fax -1.25 I 1.6675 $ -1.25 1 1667 J & 11600 wa . 1.665 i pwa -1.251 I b) 11) GLD =k. The open loop ök --|-1665 Sawn Gis) Gels) is Teansfer function s²+5576 Rost locus ll There are a poles at s= -2, -3 (11) The locus between the poles break at a - break point and more along the Asymptotes (A) (ii) = 180 (2X+1) K>0,1, na nooy poles (9) nom me no of zeros (0) Mr qó, 9700 (PU) The Centroa to da on Real aux is a -6= {Real Port of Poles - Real part of zeny n- m

2-O - > (2+3] -0 = 2.5 J T -2.5 I W) The break point is de zo Where k=-(5755+6) cm - (25+5320 3 52-2.5 The locus is drawn below ২১ t roty - 7-225 P270 Ta -2 (1) GEBEU = *(1+$) = K () >> k (5+1) s(5²+55+6) Rost locus is poles at S=0,02, 3 and zuo at sa- i pole at são terminate at zao at sa-l and The locus between 2 and 3 break at break point and more along Asymptotes (ii) 0 = 180 (2k +1) K20,1, ZJ 97 98, 2700 O nom RU) break point k is ka-s(5²+55+6) (5+) dk no - 2(3²+ 45²7 55+3) 20 (5+)2 2) S= -2.47, 20.870.85. Since the lokey is between -z and -3 sa-2.47 is the break point dis -

u Go [0+ 2+3] – [13 = 23 cm = -2 The locus is 3-1 chaun below | c) The Real ancix i t swn locus when neither Controlla the point where {wn= 1.25. Controlles Cannot meet The to from the two locus pars through so both Itu Requnements