Question

8. Write down the state space equation for the system shown below US) + 2 y(s) $+3 2 s(s+1) 9. Derive the state space equation for the system shown where the coefficients of the system matrix are in diagonal form and the elements of the control matrix are unity. U(S) 1 X2 $+2 X 3+1 X = y $+3 $+4 S

Answer 1

Dung for the Write dowen state system shown below. ва гарақ ол 2 Se 5+1) 1 + 2.5 CH) 2 wo 2 st3 s(+1) (nh $($+ 2 -- S(5+1)+25 c2f35 vy) finding Transfer function to (2) yu 4 (กด Ets 5E 75 (5+ 3) (32435) +4 2 1 + 2 4 $3+ 352+ 352+95+4 : Yes) 7 (sooh + (noha ou chy 3 yu) ( 33 +682 + 95+y] s3yes) +652 yu) + 9 syu) fyyu) taking Donnerse laplar ☺ W) + 6 ý' Lt) + 2 y +) { y ư) : Ý tư ) w yu) = x, f(t)= x; -X2 ý (t) = x, = 2 ýt) : 33 -X3

in the eq. we get aboue putting 4 iz +683 +9x2 + 4x, =Yult) X= 48, 942-623 +4 it). 1 UU) 31105161 • 17. and yet) = x :.900= [1 vo] ( X2 Deine where Ous @ state space ego for the system the coeffécents of the ku- inafix are in diagonal form and the elements of the control mafix are unity, X= sta sty $ SF 2 sts finding Transfer function (sto) (sty) 1 + $. sta Y es) ves) TIF S+2 (5+1)(sty) (25453

442 $ +2 VU) (sti) (sty) (2573) partial fraction. By using K3 K2 + Yil 04) t KI (s+i) 28+3) (M+S) Се ky : 5+2 isht +2) (1+y)(-2+3) cyricstas sty) (2+3) s=1 be 3*1 KI E K2 (styt. (3+2) (5+49 (sze) (28+3) Sary 4+2 15 -3x - 5 K. (~4 +1) (2x-4+3) (2343) (Ste) (st)) (sty) (2/6) 3=-312 215 12+2) (34+) (-3) +4) 1/28/2 215 40 /3 yes) Uu) is Tery (5+1) (25+3) 215 ys ind inn 93 (sti) (sty) (5+312)

for Diagonal canonical form. hesla bne sny bistbo son + an-1 S 1-1 + 9252+9,5+q. Gus= bru S7 + + bis f bo RI t (s+p,) (stPz) 15tPn) By using partial fraction Gula R2 (Stp) (s fP2), (sten define reperate for each partial fraction term taking inverse laplan dai da2 ot independent & state -P, x, tall = --P2X2 + I. dt and y = Rit, f 22 X2 0 0 0 O 0 u 10 10 Yo (Ri Ry Ro By vising denivaben, we can find diagonal maksx.

(کلا ves + (sti) of (-4,5) (5+4) (1/5) (5+312) dau setu ct dar at --442+4) das =~3/13 till dt 5 đ) = - (Ys -2/15 -\s] X2 X3 [*] ( - 4 Ult). JCJ:(}. o'-312 Ane