Question

We want to analyze the following electrical circuit R + + C C R y The output is the voltage across the resistor R,. We assign the state variables like this voltage across C xcurrent through L x3 = voltage across C, Then the state vector is x 1) Obtain the state-space representation of the system 2) Determine the observability of the state equation obtained in (1) 3) Let x(0) [x (0)x,(0) x,(0) =[1 0 o] Derive a control input uf), 0 t2, such that x(2)[0 0 1. The control input u(t) should be a piecewise constant function 2

Answer 1

given Ciuit RI12, Rr 14 C1EIF C IF 3 R L 114 Sate variables 21 3 KCL at -Ve Nlode@ fe,ナiレ-iR -!RI C dvci u-Ve C. Ne Nei RI C Ri C -L Ney RIC RiC Subs titute Values in ea. a - Ve-iLt u No de 2) kcL at + 7) tie co fL-iR :fR,= Ves O-12A C. dve C C R

C Noco kVL fn betoeen node D Node - VetVLA VC o VeiVe L diu Vei-Ver Vei om . and ejuations State space Repre sentatio n of the Syster i NC out put 9-EO Vt for obseivabitty A7CT (ACT C. CA t0. ecao) z eCA), then Sy stemis Observable

-1 X 3 3 x3 1 E:AC9 -1 -1 AY 1 I0t0 C +1+0 1-1+D -2 11 -1+0-1 -1+0+0 D+1 0-I+0+0 0-1 t CCA) O Now と0-D+ (a. = l 0 -1 D ,- Con] a110)- ecA) ニー2キ0, = -| -| So Sysfem fCbser vable PCA) 3 )RCo) PCA) C

<sub>Given aLD): J' uct)204tt We know $1 10331 r-1 2()=し O S r-1-D o tucs) 1:C rS41 1 D S+1 2 Ct S+1 t -ts+1) 1 1 6t1)s1) (S11)C32) 1 +C5t1) 1+S S 1 stst Cst1) 1 +1) st1) S1 T+5 TStS (St)) CStt) +1) (s41) Cs4t 2) (にち)っ 54 1) +511) S)C42)

IHSPS (S1) Cs 2) St1) (ss+ C cC S+t1) St)s+2) S+1) Ct 2) Given xctに 02) c Ctst1 So 2 Ct) 1 S 2 C 5うt (っ 2 St1 2 s4s 2- 2 て+J七んS て+Stas -t Cos 1 +5 NoW )ニ t Cos T- 2 C= -0.729 N1 t2 u ct) 0. 7 29 0Lt2 Now</sub>