Question

Need help with 1a and 1b. Please show all work

I (a) The unit step response of a linear control system is shown below: 1.35 1.0 0 0.1 (sec) Find the transfer function of the second-order prototype system to model the system (b) The block diagram of a unity feedback control system is shown below. Find the step-, ramp-, and parabolic-error constants. The error signal is defined to be e(t). Find the steady-state errors in terms of K and Kt when the following inputs are applied. (Assume that the system is stable.) (i)r(t) = u(t) (step), (ii) r(t) = tu(t) [ramp) and (iii) r(t) = (t/2)u(t) (parabolic] R(s) E(S) Y(s) Go(s) = 1 +0.25 205

Answer 1

w @ from from the step response of and order control systems, fC4) peak time (tp) = 0.1 See 1.35 1.0 - tiset wda encertpiaz (wd = 31.416 rad/sec — fook owgobt C) — cc) ce) C Cool 1. Mp 31-35-1 Mp - T Mp = 0.35 . -&T/I-92 7 Mp = e -ET/STER - 0.35 ge ? In (00 35) = - -1005= -&T : : Vię? square on both sides. (105) 2= CET 22 .. (1005) ? T2 (1- - ī The -1 → 0.11172

1 2 - 2 = 0.117 9.952. & 30.1 & = so. I {=0.32 demping ratio . from one, od = 31.416 -&2 = 31.416 .. un on: 31:46 S1 -0.1 {wn = 33 rad se i Natury - Angulae frequency 3. Second order Transfer function .... . wn? CCS). To CCS).. TCS) = s2+2€wns twn? TEST= Reisst (@X0.32133) S +(33) ? 1089 TCS)= POCS= 52+21.185 71089

06 Block diagram:- sis) at 12 » Res) - input function £cs) = Eovor function, => ECS) - E(S) : P1A1 Ris - (4th 2) where P2 forward path gain L Loopgain of L1 . Lay Loop gain of 22 - ECS) Rcs) 1(1) salookt : 1+0.25 100k 205C1+0.28) ELS) L 1) 'a Rep look 100 Kt it to be s + 203C180-23) 205C(+0.25 ito.25

{(s) RCS) . dos(1+0.25) +100 kg 7100k. 205 (1+0.25 ) 205+45 E(S) = RCS) 205 +48 + 2060 kS + look 45²+20s . → 453+(2000 kt t20) S+100k ☺ input rct) zuct) - Step RCS) - Ys 5-domain steady state -Error exal Ecs) ". sto . 8(45+20) x del 8 S70 45+ (2000kr+2015+100kr 20 I en en u bok for step input 100k ( input ret =tuct) - Ramp input, RCS) = Steady State Emor ens: Lt Ecs) 570 $C45+20) - A - x - sro 452+(2000k + + 20)st look lt

ess=lt 45+20 sto 45+ (2000kE+20)5² tlook w ens = 20 Es=oo) for Ramp inpect ... input rot) = t'just) Res) = 2X 333 RCS)= '53 uct) Steady State Error em = ecs). ex L (454204) $_ ? 45+ (goooke +2075tlook ess. = 20 less = d.] for parabolic input