Question

Question 4 Suppose we are given a system which is initially at rest with input u(t) and output x(1) described by the differential equation x - 6x + 13x = 2ủ + u 41 Write down the transfer function G(p) for the system and say with adequate justification if this system is stable or not 42 Use the initial value theorem to find g(0+) where g(t) = 1-'[G(p)] (3) 43 Can we, in this case, trust the final value theorem to give lim where g is as before? Justify your answer (3) 1-00 44 Determine the response of the system if a forcing function of u(t) = e 2 is applied momentarily at time (Hint Let p? – 6p+13 = (p – 3)+4) (5) [16]

Answer 1

- 6x + 13 x = zutu Apply laplace transform on both sides t{i-64 +13x} = L{ zutu} 52 x(s) – 65*(s) +13 *CS) = 25 U(S) +06) x(s) (3-15 +13) = (25+1) UCS) GCS) = 2541 52-65+13 TF G(P = 2P+1 s²6p+13 Open Loop zeros :-P =-½ left hay of p plane Open loop poles :- 123721, 3-2i =) right hay g p plane poles lies in the right holy of p plans, the system is (31 intial value theorem X0) - Gm x (C). 960+) = lin : ( 24+1 i 2 (2+ Yol P700- p2-68+03 Pa pa (-+13)p3 got) = lim 2 + / - 2to za you l-G+!? l-oto L. 9cot) = 2]

(3) Find value theoren X las = lim sx(s) 9600) = lim P (2p+1) = 0 (0+0 p70 p2 88 +13 2-30 +13 0 -0 olm 1 9Coo) = 0 (4) U(t) = e E12 L?u(t)= U48) = 2 + '/2 X($) = 28+1 uip s2_8p+13 = x(P) = (20 +1) Coton sec 6p+13 X (P) = apti = (P+%) (.p²_0p +13) 2 (P+Y2) (P+2) (62-68 +13)

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