Question

4. (12 points) LTI systems. * = [ 11, lo -2* +11 y = (1 1]x. (a) Calculate the unit step response of the system with x(0) = 0. (b) Calculate the response of the system to x(0) = (1 1]', with u(t) = 0. (c) Calculate the unit step response of the system with (0) = (1 1]. HINT: Is there an easy way to do this?

Answer 1

Solution! n ri= Ant Bu y = [1 1] n + Cogu Y= ant ou do A=[8 ss XIA [25-A3 = (* :]-[ (31-A) = P -] < zt dit LSI-A] = 868+2) = $18+2) to fetorn (1593 = [ st2 nej127 12) = ( ]

:: (81-A]!= Adjl8I-A] det (SZ-AJ = I r 8+2 37 $18+2), o sta Los e As we know that: C [ SI-A][X10)+ BUIS)] Y(8)= Y)= take [SI-A] -X10) + CESI-A] Ř Uls) in verse laplace trane form on bom-lice ylt) = LIC (SI-A] X10)] ++[C [SI-AIR VIT Total Zeno input relponse zero state (ZIR) respon ise (ZSR) Reapone a: Input - ulticul) = unit - Step USIE I Mio=0 So Reiponee so Reeponie is zno state es teno initial conclition is z ero r eeponee, because :. 4181= C[SI-AI BUIS) YVR)= (2 t) [21] [2]

rs+2 ] sis+2) s+2 St1 ) (842 343 [1] +2) sti 8218+2) Yout= 0,25+ best to take enverie laplace tranform on bom-dich, ylt) = (0.25 +0.504 -0-25e-2t) ult) here b>: Response és zero ultico input reeponce because .: 448= c [SI-AJT XIO) = [1 IJJ L T &+2 27rez

Y(31= 3+2+ 3+1 s18+2) Y101= as+ sis+2) s YISI 1.50+ 0.50 3+2 take enverie laplace frane form on bom- siche 940).=11050 +.50 e 2t)ult e>: Now Totat response yle) = ZIR + ZSR 1 Responee in reepone in part (6) past I a) get) = (0.25 +0.50 +-0.25e-2t) ult) + (150 +0.50 e-2t Jult) =11075to.sot +0.25 e 2t) ult)