Question

Problem 1a (6 points): Find the differential equation associated with the transfer function: C(S) = R(S) 53 +52 + 1 e-ST

Answer 1

Date Page 7.4 The given transfer function is -XT C(s) R(8) s3+2+1 Cross multi bly ul get (&?+ 3² + 1) ((A) = é STR(8) A3((0) + ²((o)+((o)= ést RCO) - take in ursie laplay both side we get [ { 13() + 12C68) + (68) 42 (Pest R08)} the linear property of inverne day Hlaul trans hern LGGW+CF, Cos}= CEEAcolyt alff (a} abone becames [143 c«)] L7 { g²((a)} + Lt {((0)} [Yest R(87} by wing the - we know the when Fm (t) meam formula Lt {gh fol} = frict the nth derivative of F(t) where L&f gly Elt) is the fca) FW 4 f (t) = do fu) den abone using Formula becames

C (t) C (Alt (lt) 1" (e ATR (s] but second shifting the oren inverse ني laplace is if {f(x)}=F(t) then Le frazy z j Flta) ; t>a o tea alf L" f R(AI) - Rit) Lth é torsi} then R(t-T) E>T t<T to using beemas using this d²CH) dt3 (lt) t d 2014) dta R (A-T) if t>T if tCT differential equation of abone This is required heat equation (1)