Question

Question#1 Given the following block diagram of a continuous-time system with Ki 10 and K2=5 Rs) Yo) a) Using the forward rectangular rule equivalent discrete-time system (FRR), find the state- variable model (SVM) of the above system in matrix form. b) If the overall z-trans fer function of Y(z/R(2) is found to be 10z2 Y(z) R(2) 6622-57z+1 What discrete-time equivalent is used to reach to the above transfer function? Show work c) Using Y(z)/R(z) of part (b), if r(k)-a unit-step sequence and y(0)-5, y(1)-8, find y(3)

Answer 1

Given yis) RCS) transfer function mason's the obtaining gain fomula (-forward rectorngular Yule) forward Paths (fom R to y) fouard path gain Loops mo of loop gain kik 1+S loop gan K1 SCSH Transer Functon 1 Kik k. SCstt 19en ta SCS) RCS) KI t SCSH

k 1 RO) SCI+Kk)) k, =to 2-5 Givon RLD s+ SC1+50) Res 51S Tran ster funchòn in Topresent thi Ctate vanable model we chose pv F Cphaa vartabk fom 51 S +12 = 10RCs) yCS) Laplate apply d 51 ay dt ct =inpat output Let 4ct) sub

~メ」 -S1 X2 + 100 Svm moda AXBU OX X2 ou -x,-5)x tiou 3- matax fomm Rl2 662-57z t1 ya)G6-572 1) = 102 Ree from スtransfomm Z-T 66 yR

both ztrantom invete des get we 1o Cn- h-23-yn) yng 66 yl Hen ca 10 n-2 direte tme eualent of abova trans fer function nC)unit sep Cn utn - yり =8 put 5 -66 *nl-2] = 0 5= 3) n2 put ye 100 -66 403 +57

yе2)- 46(5) t57 (8)410 () и Со]-я го) 3) у С2) - -330 + 456 t10 у f2) 36 put in у с5) -6G 4 1 t$7 ut1- я0) Ч Е3)- -66 (8) 5(136) + 10C) E-528 10 +452 ус5) 7234 Henca