EEL5173exam1sol

# 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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