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EEL5173exam1spring2004

# 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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EEL5173exam1spring2004 - UCF ECE DEPT EEL 5173 LINEAR...

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