EEL5173exam1spring2004 - UCF ECE DEPT EEL 5173 LINEAR...

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Unformatted text preview: UCF ECE DEPT EEL 5173 LINEAR SYSTEM THEORY EXAM ONE Spring 2004 PLEASE READ AND FOLLOW ALL INSTRUCTIONS. PRINT YOUR NAME CLEARLY: ID NO. (if known) The exam is closed book and closed notes. A note sheet (two sides) with no SOIVed problems is permitted. Do your work on this test paper. Use the back if necessary. The problems have equal weight. Answers given without sufficient justification will result in diminished credit. Good luck! (1) (a) For the given circuit, using the state variables and output shown Find the state space representation, that is, find A, B, C, and D in the following: it = Ax + Bu x1 where x = y = Cx + Du x2 — 2 1 0 )2? = x + u (b) Assuming that 1 - 2 1 , find the transfer function, U (S) y = [1 1 ]x + u. (03 KOL out noJeCQ: 1%, + x,— L l “L "3‘ V‘°&€@11‘§<24 Viz/2‘14 “was; 52,: 1x ~2xL+xA l l WLI 'KM/Vl—le a» )1: 4xpr Letbhfi X’Y XL #3 X Y! ~L]X+ 1}“ ~\ CM :{Lg’} : cars/vi 8 +5 U “7 - ti +13 [tr :2] [91+ 1 u H] aimlle Azs‘ziwsg _ it; (2) (a) if} 373)” [7],, (a) (b) \I: For For the state space representation of (l-b) above, find a new state space representation w = 21w + Eu y = aw + Du inwhich 1:1 is diagonal. That is, find 1:1, 3, 6', Determine the controllability and observability of the following state space representation 0 l 0 x(k +1 = [—1 _ 2:|x(k) +[1]u(k) y(k)= [1 1]x(k)+ u(k> [fix-Al=0>> WWW?“ 7% 7“:~() 7,2:'- t [3, 1‘][¥2]>[31 [‘ GX-ifi/u W gatisrizr anagram: "g A“ 1' : ii jjwmrm :‘igxmtmma ,1 7., “I I. G I. , Z A [ [IL 17w : ~m m l \ 4 F\|L \Ew 3 m "2 U931] U f l”{ F_LJ uz (3) (3.) Given the underdetermined set of equations, x1 4 0 O 4 x = O 5 0 5 , x3 find, the shortest solution vector , i.e. that x for which the Euclidean norm is smallest. Given the quadratic function V(x) = x12 + 3(x12 — 2x22) . Determine which it is: pos. def, pos. semidef., neg. def, neg. semidef., or indefinite Justify your answer. (b) (M X: ATM/(W . ‘40 .L o . Y»:[ 3’1 [6 'X if _ L G ‘ i go [a trig/1’ 2;“;1“: l O O O Q35 vm: >634 mini) 2 : (Hag. '1 ~— Li 0 X! I 6K2 QM ¥L§[ o ~6]<¥1) ...
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This note was uploaded on 01/14/2012 for the course EEL 5173 taught by Professor Staff during the Fall '11 term at University of Central Florida.

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EEL5173exam1spring2004 - UCF ECE DEPT EEL 5173 LINEAR...

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