Question

1. Consider a continuous system whose input x(t) and output y(t) are related by dy(t) + ay(t) = x(t) dt where a is a constant. (a) Find y(t) with the condition y(0) = yo and x(t) = Ke-bu(t) (b) Express y(t) in terms of the zero-input and zero-state responses.

Answer 1

+ igit dy It) (t)=710) D dt a is constant if we take the Laplace toons for m of a boup egin (ů with initial condition y = yo and 21+)- ke 417) 915) - Ķ stb bt of LOT 210) S*18) – 210). da we know that time. diffesentintion property Laplace a tansform x 15) 21t) Low above propesty in sen SYIS) – y ro) + aY !s) = TIS) y 10) = yo given Y18 of 579} - yo = 815) from een ① apply 1918) &st tag yo= K (s+b) K YIS) of staza + 8°. Stb t yo 4.182 sta к sta) (stb) for y It) we take in nesse Laplace of abone egn.

A) sta Eaplautif e-at ult) By using partine fraction Concept A of B (s+a) (s+ b) (S+9) (s+b) Sara for fou put I K__ A b-q -atb B = k. for B for saab -bta B -k K к k (b-99 (s+ 9) (6-9) (stb) yo Yis) = S+9) stb) (5+9) (6-9) ㅗ u(t) sta inverse b-q (5+9) (+) + к we know that LT e-at = ) -at ult ult) e-bt ult) e +yoyat yits ķ yit it) -kestult) eraturtilyo tk a } (69) (6) zero input and reso state response means i's relaxed. ystem init itally yo

if we take the caplace Transform of een @ without initial conditions . SY's) to a Y15) = res) X (8) (Sub) K YIS) & stay (sth) YU)= k Stay cS+b) {as in 1) K u + 6-99 (sto) (6-9). (549) past a} yet): t)- (6-a engt -erbe enbt} 4et

YIS) = + 115) sta 410) + e.at -gt o eat u (t) =) othes wire zou (a) if y(o)= yo po from past 1(a). SYIS) - Y10) + 9 YIS) = 818) Y15) 15+9) = yo +*11) yo (sta) 81+)= yo enotuit) 41t) (t) yit) = ult) alt) + Yoe 410) for to yit): enq talt) + yo e at R At) inint y(t) Outbud from the linear and non-linear. defination System system is said to be linear if it satisfy lië additive and the super position principle homogenossina ons, nature, for linear easily filt), alt) are input and Yılt) and Yelt) are outfont, then the system need to satisfy transform of combined signal must be equal to addition of transform of individual signals TELE) +21+] - 9,7 ( 9/1t) +92 T [421t)] A

lystem system I= e.at 9,811t) + yo ce engt at and yo erat is called nom lineas tom otherwise the -at ayılt) & treated 92Yz (t) as constant Oh It) & Yo erat it is not input or not nat %p. e t 2 et if we dise scaled a, a 9, and a in at 9,y,lt) + 92Yz1t) - at seh an 2 71 directly Ech et (974 (t) + 927214) +40. et highlighted) in een a system is is non Joenat

( 3 if A B alt) 3 paat 4 9, 7210) and ® erat c CB rat yo 20 Then in abone past put yo=o.in. neph ylt) = erat 91411t). enat 9,714) 92 Yz1t) = Add eph 6 ar gult) +92 Y2(+) [4,4(+)+9+at+)] Now of me dire direcly scaled egn quyilt) +92 Y2lt) - e (9,7,10) + 97214) compasing eph ☺ and B same asult. is linear in nature system dylt) ay (t) = 81t) we want to find impulse response ht) remember that impulse response find le linear time only is varient system le system tem initially relaxed, and ® meeg get so the w + dt for LTI system

reen with zeso ine Now we take the laplay pans from of abouy Egh initial conditions, SYIS) faYIS) = x(s) * 13) & stay tis) Y 11 #(s)- 705) +18) (sta abone Een -at istas Now the innerre Caplace of