EEL5173exam1sol - 3— O I O 0 0 O/— V[L O A UCF SEECS...

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Unformatted text preview: / 3— O '- I O 0 0/ O * // . /— V [L O A UCF SEECS EEL 5173 LINEAR SYSTEM THEORY EXAM ONE Spring 2003 PLEASE READ AND FOLLOW ALL INSTRUCTIONS. PRINT NAME: [ID NO. Viewing Location: The exam is closed book and closed notes. A note sheet (two sides) is permitted. Do your work on this test paper. Use the back necessary. The problems have equal weight. Answers given without sufficient justification may result in diminished credit? Good luck! (1) (a) For the block diagram shown, using state variables x1 and x2 as state variables, find the state space representation (state equation and output equation). 1 (b) Assume that the state space representation is given by A e t 4’ Ole" . [ 0 1] H ‘I l‘ 13* X: X+ LI —2 —3 1 q y:[1 1]x+ u. ~~ *7) Find any other valid stcate space representation. INT 1 LC T X ' P W! WHE‘ZC' Jon-37,06 H -‘Pw WmP:[’ l Lqi fi:[ J W J IQ Ye ) d ‘ mfimnc O l —! -\ >5 ‘5 : w T V M s2: Nu:/4?w4 EM w ’I’M7 \— i \ 'i i J _ l l _ i \ \ 0 '- \ 1 pa: K l V"T['I(1' 0‘ F 0‘0! 4. ‘ i ' r * - + ‘ . Ana-mg, flP/loficfl 75 a) . VD - 5—1;?— “ "L f 375; :1 y N" #3“ [Mr-y) é WWW I L———— 5 ’7 sx,+5x,= X2 "'7 => )2, = ~39 WWW-7 “ ml? WWW” y: Xl'f ZX’L - -, 'r— g-X 4r~X~2>CL => XL ~ 361,1” )’ l I \fL: \Xl'3XL+r (2) ‘ Given the discrete time state space representation x(k+1)=[" 0'2 0 :lx(k)+[l:lu(k) — 1 0.8 0 M) = [ —1 1]x(k) + 1 we a (a) Determine whether the system is controllable. fr ‘ (b) Determine Whether the system is observable. ' (c) What can you conclude about the controllability and observability of the dual 3373mm? 603 (on, Qumomtu‘c’y ' ' i - ‘ r) ‘ , é 4.1 Pv’fisfl UWCE 'fltog’u{ J n [on '_ ) IE ‘3 ' 09555 $11,157,?! / i in «W xao 7 n (5) Fort oégvaILc—ry m/(r “02:6,” ) 0 rl Ell: “ ‘ C“ l" “l” “1 Du m gysfim: 7 JUL): 37 WW + 9414) GNTMLL/WE /fi/3 Y[ CT ATCT] : n (:1) 49.7, ; T ._ C , [CT Na K? - rEC/J : I (mm (5)) : ‘4 ‘O‘E r I l 0.3 ,1 r[CT kTCT3< “(32) sax; (“WSW WE: w 7w flwsflw 6:; f"; 756mg (2 N0 A470 J;— . m %w’ : mm“)? mm ‘wme’ nan/:2) r -—— ‘ - CHM/fa 57W“ 4w am my; Wm.) 75 7/7; 0mm It? fiWU/ILL WLE wag—:0. (3) Given a linear vector space consisting of polynomials of degree less than three, having real coefficients, OVer the field of real numbers. Also given the bases 2 e125 ,e2 :5, e3 =1 f1 :52 —25, f2 =s—2, f3 =1. (a) (i) Represent the polynomial 332 + 25 +1 inthe {ei} basis. (ii) Find the change of basis matrix B and use it to find the representation of the above polynomial in terms of { basis. (b) Given an operator that takes polynomial v(s) and returns a new polynomial w(s) 2 313(5) + 2v(s) . (i) Find the matrix representation of this operation in the {ei } basis. (ii) Find the matrix representation of this operation in the } basis. i ‘ _ a 2mm hS’rCU) . (m @3314 um ._ are, +5624 C 3 7% Q; ’9: Z C;l ‘ p 3 a m Veam/ flawew’: I - $3 % '0; $47Mme 3g *25 , (5;) C ‘107"7—IHE flamed: 6\ - 3L ‘ 051”: 1‘ £00? 4 IF; .c‘ ‘ v - C ‘ 8”“ mnrmloiw)" “Wat 2 Q.C $71- 25) 4 5‘3 1W ,0 0046/ . 2.7 @wmwm; 5% : Ml _ 2&9 H” izgfic. Mu VEcrmye;,c:/,z,3 l/u ‘ , , . 713nm; 0L New £570F Adm :5 ®\‘ : \ \ ’20“; (so) ._Zb|+¢,vo Vac-rams) ,1}, 2:45;) [B a, I 9i: It“: g. F t "r I ' l” 91:”! _ Cy (3 c3 ; as him g: :3 9:33.) bkmlo :5 big—2: 0 Cu Q 6 0.1 (See 8 N 2.19 l 3 { ‘ 7 Q 6‘ Jaw) Z l ALTZn/Jm'e ,427/7nc!,¢§#: Zflrzsw: gal/Ls) flak—L) +c0) ‘-> a:3/ b:S) c: (7 T A 6 AL ; 3101+ 247' : g(15~z)+z(skzs) : 65 ‘LA z<zsus = 2‘91 *‘73‘4’ — 04(32,153+b(9‘2)+ CO) 5 : 1M1, tea; 4 53 ’Zb+c a m: z kzexzz C‘Z‘F'é 7%,: H. a MKQ fit»: 3?» 42‘“. : g(‘\ ‘L 1(34’) ...
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