Question

VAL REALIZATIONS AND CUPRIME FRACTIONS 7.4 Consider the circuit shown in Fig. 2.21(a). Is its two-dimensional state-space equation developed in Problem 2.11 controllable and observable? Is the circuit completely charac terized by its transfer function? Find a simpler circuit which is completely characterized by the transfer function.

Answer 1

masi dt dt 14 Yeca ficat Here outpeet y =ícs. ou af trica IF vcs We can take 2 Gpace variable's here 902 ic and 90g = (cs. Electrical Circuit Vca - Codica = 1.0 = x; Vcg = ca dica 2.062 = 2002 - apple kve in Loop: (coe know from ckt ie, voi = vc ->-47 ix1 + VCL = >> VCA - u-e 4-ici-icg. - 001 - 4- X1 - X,. - (1). | => • Yc a = - = 4 (cu Lich -> 9.35 - 4-901 - 2025 2 - - - - . From Equation 1 and 2 F -17042712] y = ica = x2 y=0x61 +X, Scanned with -44sche Eo o] L.

D - as AX + BU. y = coc tou To check controllability C-1B AB -..RA?-?B7 rank of Ic) should be i.e., Icito bere D=9 theen Co I a. L0.5 F-1-1 7 17 -1-0.5.7 -1.57 -0.5 -0. 5 -0.5-0.5 -0.75 [c]=11 -1.57 Scanned wit0.5 -0.751 1C1 = 1XC-0.75) + 1.5*0.500 30 atne oystem Co not controllable Q-27 A-B- - 2x 2 2X1

To check observability [6] = [CT AT CT ... (AT 2 (T) Rank of [b] Ghould be nl.es, 1 bl to coon Gingular matrices ATCT-1-1-0.575 .7 -0. I lo27 -0.5.7 1-1 -0.5 L -0.5 gx92x1 [b] - [og-0.57 1b1 = ad-bc = L2] -0.51 0 +0.5 20.5 #o Co the Gystem to obGervable Scanned with CS Scann CamScanner

Innan - . * To find transfer function from Gtate Gpace model: TO) --40) - c 61-4540B + D. Cocoa]: B= 6.6 4(S) 17 GI-* - [88]-04] - [615 Gio.] [add] - ad-bc Escada ] 0.3333 +0.5 GCG11.5) B G+ 1.5 -1.5 - 1.5 1.5t +05 I G+1.5 . - = 0.6666 (GI-1) 4 +0.5 -17. 16+0.5-17 (C+DCG+0.5) -0.5 -05 G+1 5+1.55 -0.5. 6+1 - Get 5,5) = det ; +16=0 = 0.3333. B16-1.5 = -1.5+05- =0.6666 649775) 4 (0.3333 + 0.6666 71.97 juct. o GST1.5) = B 6 60.66666 +0.6666 ē 1.55) ucts. Alo-o = -0.6666 818 -55 = 0.6666. -0.5 05.016 ( +2.5 15 +1.64>L-0.3333 + 466-0 - -06 -0.3333 : B|6--15 = = =0.3333 0 64642-9) - 4+ $72.6. 1945 (0.6666 +0.3333 15tjues. A = hs - 0.6666 8 = -1.542 - .5 = +0.333 DA -1.5 = -0.66 66 1.5 Is=-1.5 B -0.3333 +0.3333 e 5 +1.5 - 15+1 -0.5 40 33 - =0.6666 - 1.5 -1.5 1 TS = 405) 45) G+0. 55 - 1 CSC+15) CCS +1.5) -0.5 CS+2 GC6+15) 2X9 0.5 182 Lei) +G+ 9 SCO+15) US) horas I S+0/5 -05 0 || G6+1:59. Try uડ). -0.5 +0.55 +05Ľ GCU+1.5) G+1.5 0.5 C+1.5 A GILD.SI 0.5 G+1.5 1S) Camscanner 20+ 6.

TE = y ) = 1- UCS) 26+3 het Gystem is lenite feedback system, HS)=1 TE = GOD 2+GS) = 96+3. GS).90 + GS) 3 = 1+ GS Gos) (26+3-1) = 1 = GOS) = = 0.5 6+ 95+2 1 405) 0. 5LF45.' 6+2 465) = 2 Blockeddiagram CamScanner