Question

1. The system S = {A, 5.e, where A = [1 2] 1-() c=[1 -1.Jis excited by the 1. The system S = {A, b,c}, where A = is excited by the input u(t) = (5e 21 cost).1(t) where l(t) is the unit step function. Use the Caley-Hamilton Theorem to find the complete response of the system. Identify a. The zero state response to the given input; b. The zero input response to the initial state x(0) = (x2(0) C. The steady state and transient responses to the given excitation and initial conditions.

Answer 1

I Jäählt Gtate Gpace equations x = A x +Bee. X (0) = ,00) 7 y=CX +DU. LOC200 . at docet) = 6*69 - 20603 A = F-4 37 B = 517 cala-17 ucts = 5e2 Complete re@ponse of System - zero input reoponge + Zero Gtater response (C). zero input respon Ge: het ult)=0 (Input =0 ), = Agc and y=co Apply Laplace transform : SxS) - 20) = A x2 x S) CGI-A) = 20), Gtate transition matrica < (s) = ST-A5zco) = 0s), co ) Caley hamilton theorem: Every Square matriac Gatisfies its con Characteristie Equation (A-GI-0. 3 7 [S+, -37.. Td-b7 2] Le seg (ST-)^= 1 6-9 +37 (6+4) CS-2) + 9 L3 6+4 L SI-A L -40 ad-bcl-cal ICS Scanned with CamScanner * * ** *

(GI-A3-1- DE S (GT-A-). 3 7 C+4 F 1,7r6S-2 3.7i de a se LIS+1)%|| -3. Gre 1 0-2 +9Q 8+9 L-3 G-2 (6+1) 2 Gti 2 G+ 4 Gtu C+1) PA4 BB B + A +1 152 ( + 3? + Scanned with CamScanner B E a B G+1) + . A +1 L G + (5+12

* SH 1S. -#1 + OH)2 - AGT AT8 A+B=-9' and AG=G => A=2 B=-9-1--6. CS * B (6+ DE AGUA +B اورمی) Stijl Comparing 'G' and conGtant terme. AG=G4 A=1 and A+B = 4 and B=G. (61-452 - Gta + 1)2 + &#1377_bcom L ije GI te tijd ILT CSI-A3-2-rét-atét ! L-ate- t ē tt 3tē Otet - State transition matreat X6S) = piss. & Co) A e rote Ste 2,60)7--0th Oct 2x1 '2x3 2x1 7-etet éttatet Locaco99 xey or cēt-3tét, 20160) + Btēt coc.com) Celestēt ) 2010 tlēt + 3te394204. Jun ts ys) = C X(S) 1x9 2x1 4 yes =cēt tēt, 32 603 +(8t e tjx200) + (tötet) 5610) - 1. let+ötét) 2002 scom Ocicos cét - atétzötét)+ Sco, cēt taxét_ 34ét) sam sancoslēt) +90260(-et) = ZIB

T zero State response or Indeal conditions CX (S) = A OCCS) + Bus), E XCS) COI-A) Bucs) eines yok.CX6)+D uc). 4 65) = C (C1-A)?B uc) +D U(S). = c bs B u(s) +D UCS) =(cos).B + DJU(S). yos uld - 50 2t cost 5.8+237 1 499=[61-2200. [1]+ :o] 46). 06[1] - [ Gotija 1927777 1 X 2 S 2x2 G-2 (St12 - 3 0-9+3 (6+1) - 3+6+4 (6+12 ( + ) (s+4 (6+1) (Štaja 999 = [[1,-27 p! C+2 +1] [5 (S + 0+! 1x2 LS+1) 9X 1 I 1921 Le corps 469 zero Initial conditeons = Z6B =o. Complete responde = ZIB + ZOB, = Tranbientresponse + Steady State reoponge gitt = et a,o- etx.co Transient responge =é tocaco - ētoceros. Trabstent response es a response cobich der is dier ocet after certain teme, means et ils le dependent on tema Steady State response to Independent of time to cs Search Centerta) = teat