Question

please help to solve this. Thank you

B1. Consider the second order system where damping ratio 3-0.6 and natural angular frequency Ww=5 rad/sec. find the rise time tr, peak time tp, maximum overshoot Mp, and settling time ts (2%) when the system is subjected to a unit-step input. I B2. Find the steady-state errors for inputs of 5 u(t), 5t u(t), and 5t.u(t) to the system shown in the following figure. The function u(t) is the unit step. R(S) + E(S) C(s) 120(3 + 2) (s + 3)(s + 4)

Answer 1

From basics of response of a second order system to unit step response the solution is as follows

given damping ratio , & = 0.6 Natural angular frequency, wn = 5 l/h for andt step input response is 2 wer CCS) = 3² + 2quenstun Rise time, tra Time taken by response to reach from 0 to 100-f- a-d 128-qr won. Wd = /1- 0= cos(&) = tr= wa c00927 tr = 0-55359 peak time: Teme taken to reach mare value T wd wed=wn rasa tp tp 00785, 0-64 3014 ) Manimum overshoot mp: (PT-O-BE el & / Mp = e mp = 0.09478 & fol time (2% tolerence) settling 1.331 ts = 48 = 4/gwn & (0.675

2.from basics of Fourier transform and response of an unity feed back system the steady state error of system to applied input is as follows

The given system is in the form, oltor RIS) t EW (GB) y cool tere, G16) = 120 18+2) (3+3) (8+42 state enor, steady from Baskes of control system, ess S s20 1+ GIO) R16)= sult) =) 5.15 (1) sf input is ess : ㅋ S. 5 (4) 120(8+2) (373) (674) 1 + ess : 0238 21 12, If input is 5t ult) R)= 5 (1152) -) ess It S 0 S. 1 + 120 (8+2) (3+3) (3+4) S-70 S + 120 (8725) 87 76 +12 ) Not :- 120 BY L- Hospital Rule. It S+0 120 (8²+28) 34 75+12 000416 ess - 120

5tut → R16)- 5/83 input applied is s -) egy - sto It 1201872) (5+3)(3+4) It sto 5% so 120 [3+2) + +32 (574) Nole ! 420 1 3² 25 By - Hospital Rule sto 8t78+ 12 ess sto S+0