Question

1. Controllability and Observability L = 100 m R1 = 10 Ohms Mm R2 = 100 Ohms R4 = 100 Ohms ( = 100 microfarads ult) 1V R3 = 100 Ohms R5 = 100 Ohms Xı = i(t) y = valt) vi(t) = 1 di:(t) i(t) = c dvelt) x2 = valt) A) For the schematic above find the state-space equations that define this system. B) Using the controllability rank test determine if this system is controllable. C) Using the observability rank test determine if this system is observable.

A) For the schematic above find the state-space equations that define this system.

B) Using the controllability rank test determine if this system is controllable.

C) Using the observability rank test determine if this system is observable.

Answer 1

n L = loomH VL (t)=40 = Rolc Dis 3 R4 = loon Etica 100 mp tris ii-is R3= loon St 2+5 Tato Lig - R5 = kou at ☺ * = ict) VL(+) = Lail | Xp= V 4) . . ic (I) = collect y= Vect) usiha KUL in loop. ☺ UC) - iRi + V2 (4). Hi - ) R2 +ii- Uc+) - il CRI+ Re + R3) - Is Me - is R3. Vylt using KCL at node A : iz = ictis id = -ig - 0 using lane in coob-3 - 0

aid R&- + Sicat +(ii- iQ) P2 =0 ii Rp – (84+Re] 10 - +sicolt = 0 - 0 using KUL in look-3 ť şicat - is as +(-) R3 = 0 lipa - (P3+ Ps) iz + ť sic dt=0 - ① from ②. id=-Ć sicdt tii Rp (P4+B) from 13 = 1 R3 + < şi colti ī R3+Rs)

putting this value of is and is in ean and eqno Ut)= i Crot R2 + R3) -1 -1 siedt tiiRo IR - Essicatti R3 ) R2 I R34 RG R2 + R4 7 thech U (+) = 11 [(R+R+P3 ) - 1 Be \_/ R₃² tesiedt [R? e+R+) Cotes. J T . +VLCt). L> 5 LR1 R4 Ret RS ean® ic = ip-ig. ie = - + Sic. c + iR2 -1 183 + tsicch) (RtRs)

"c= is lehetne VA TRAIRE te + Sicht [fie et le tras) Paths L o Putting vawes in 6 and is) i cil=24. 6 < srecht - Vect) = x2. . ULELE 1 = col ni C CH ic= cave solla

th poo 200 + Qolai = lonit oolai - - 110021 – Youet).- Oct) 20 9n ® 2 = 0+ 2[3] x 10. B = - +2. - © Aurranging A and B in Matrix. Til -1100 01727+ Fioluet). LW LO -009] P], matrix -A Output y = V(CZ) = 2 · y = [0, 13[] b.) . Conteronitoring matrix Qc = [B Lyc [ ] AB] fove ordet op ABOS [ sytem. + ]-[•]- [home] Oc = -10 110007 Ocla o [0] 0 Lence Syutem four controllable Syetem is not controlladce

-) Observability matrix Oo= [cT AT CT Į coulder & system) CT = to JT - []. ..^= [-100 cm] =>ACT [Ano Qo- [ • omn] 100/= 0 deso fool Observable sys Since observable is not sg system ile [ool=0