Question

Given the following differential equation for some plant, dy +7.+ 15y = 2x(t) dt dt a. Find the steady-state output for a unit-step input. b. Find the step response of the plant; that is, solve for the output if the input is a step function, x(t) = u(t).

Answer 1

Sol! Guven differential equation dy + + dy + 154 = 2x(4) - TD dt2 taking dablace toon foora of equatron sy() + 7 Sy(3) +15 y(3) - 2 x (5) Where y (4) L'T » Ils x16 is input X (s) YU) 13 output act) 2 Y(s) X (3) FH($) s²75115 Impulse response Transfer function of the Plant Y(3) = H(S) X (s) is given as a) steady sterte out but Yss (4) = lim y(t) = lim Sto sy(s) to Guver input x0) = u(1) X (3) = Ś [u uit) um syu) Yss (t) = sto

Yes (4) - Jim SHS) (3) 5-> - lim of 2 S-> $775+ 15 lim 370 s²+ 75+15 2 87.0115 2 15 steady state outful lyss(4) - 2 I this 15 b output y(+) = ? When input *14) = u(d) => x () -- From port @ H(S) = 2 s+75 +15 response Impulse output y(t) for step-response is calculated asz- 7/3) = H(S) XL) ys s2+ s +15 Solving equation using borttal fraction Bstc s (s'4 7515) 5² + 7st15 12 S 2 A S +

2 = (n.) + (70+e)s + 15 A comparing the coefficient of equal powers A+B=0B2-A => Bis 7+2=0 → 02-7A--- 14- 15 151 = 2 A= 15 putting values of A, B and c in equation © and then equation © becomes - y() = 5 + 14/15 32+ 7st 15 2. 15 2 15 s 5²775+15 14 15 s²+75+15 To 6 + 73+15= 2425 1+5)* 15-2)}=(3+ 3) + (m) S 15 y(s) = 2.1-2. (s+J+15 (547 2 15 7.2 VII (st 7 / 4 (W 15 VTi (st za

Taking Anverse Laplace Transform -24 7(4) = 2 Sin you do 1.LT - at t e caswot steb Response of blant g (sta)²t wo 1.67 at se sinwot (stapt est I.LT Inverse doplace Tondom