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examI_f07 - EE 428 Last Name(Print First Name(Print ID...

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Unformatted text preview: EE 428 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO EXAM I SolJE i 005 Problem Weight Score 1 25 2 l 25 3 l 25 4 25 Total 100 INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. This is a closed book exam. You may use one 8.5” X 11” note sheet. 3. Calculators are allowed. 3 October 2007 4. Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey What you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1. (9 points) The ODE (1% (Pg dy d2u du — 4 —— 2 — 8 t : 4 -— 2 — 9 t . (1153+ dt2+ dt+ y” (1252+ dt+ “(l represents a certain LTI system with input u(t) and output f (t). Represent the system using an all—integrator block diagram that uses the smallest number of integrators. 2. (9 points) Another LTI system with input u(t) and output y(t) has the statespace representation r — _2 4 1' + 2 ’ ’ 2 0 0 u y : ( 2 2 )$. Find the transfer function representation Y(3) / U (s), and place your answer in the standard form Y(s) bmsm + bm_18m’l + - - - + bls + be -\ = H- (SI'F) 6 11(5) 3n 4—0171—1871—1 + ‘ ‘ ‘ +018 +(L0 \ 5 ‘l _ 5+?- —*1 _ -\= _.._.___..-——— 31' 'p "' (-7. 5 ) (S: F) 52+ 25 -8 L 5+2.- .. - 2 5 <5:-r=>‘6= (ST’Fb‘io = ”L..— 2 (54")(5-1) L! H(sr~—P)“6. = (z mas-FTC» 3. (7 points) A fourth—order system with input u(t) and output y(t) has the transfer function representation Y(8) 352 + 65 + 4 U(8) s4 + 353 + 852 + 43 + 2' Write a MATLAB m—file that: o (2 points) Determines the poles and zeros of the transfer function. 0 (3 points) Determines the F , G, H , and J matrices of a state—space representation of the system. 0 (2 points) Calculates the eigenvalues of the system matrix. 70 iefnnz nvnzrwlzor (0) am-Q— flewomlflw‘bvf‘ («9.) POE/1017113.“ n = E 3) 6.) H]; L: L" 1, 3J 8,?) 2.]; 70 'qul learns roots ('0 70 'chfl- poles T‘oo‘lii') Cab) ’70 Construct sta'énerquco. mmsenfiu‘éton CF, 6, H13] : t-S-zss (AJJL) 070 'qul- Qxaemuam % sashem Muir 1X 946 C F) Problem 2: (25 Points) 1. (12 points) A LTI system with input u(t ) and output f(t) )has the state— space representation 11’" _ 1 3 ’ _ 2 0 y : (3 (1 )5r, where a is a real-valued constant. 0 (4 points) Is the system BIBS stable? Justify your answer. 1. +. —3 _ = 7x + 7‘ — e = o AI—F = (fl auto: F3 j“: +2. )21, 1—6 2: (2 +3)(7\'2.) = O )2. .3 =+Z Because the, JdJ‘Lem Maid)». F has ._ Fagin/n. 2.17.3”wa 7! the. Sasffim .5 ”at 5183 Mable o (8 points) For what value of a, if any. is the system BIBO stable? If you cannot choose a to obtain a BIBO system7 explain why in one or two sentences. If you can choose a to obtain a BIBO system, what is the corresponding transfer function representation. 2'95. = M5: — F)“(,. {1(3) ‘ s 3 -3 -‘ _________.. (st—F3 = (5:7: 3) (st-F) : s"+$ ‘6 7— 5+1 _\ _ _ ,1 z : l 25 (SI—F) 6 _ (5:: F) (o) 51+5‘6 ‘1 .. - -‘ — 6.5 + ‘109 -F‘&= (3%)(‘11'355‘ ”“1 ) (”sacs—2.) rm = 6(S+ -73=~o=-2 ”‘3‘ (mm—a) Ba’ chaos”; GC. :-3) fie, eran‘s‘kr- "Fx—hc‘blon IJ' Ym 6 (5-2) g [A Smile, pole we -——-—r um (5 +3) ($1,) 5 +3 5 g ‘3 u Thsds-Lem 1‘5 6/80 s‘énue. {r CC: -3. 2. (13 points) Once again consider the LTI system with input u(t ) and output y(1‘ ) that has the state— space representation ,. v —1 3 , + 2 u I — 2 0 ‘5 0 y = ( 3 a )1 o (8 points) Compute the state transition matrix (Mt) for t 2 0. 3 S -l -l ..._————-— = _ ‘I = (s +2004.) (s+3)(,s~2.) it“) ; lCSI F3] 3? 2. 5+1 (Hus-L) (“capo-2.3 ‘ = 41-— 4— i- 9: (::3XS‘L) Ln“? 1 ‘ Cl? __. 5+ 5-2 " (H330 7-3 3 1 5-2. : i:- B (usxs-l} s“- A : §($+3) ;: 3—. ________.S = 49—- + .L cmxs-a 5:4 5 (“336—1) “'3 ‘5‘” B :. 5(5—z) l 2 A cs+s>ts~2> 3=L 5 3t z-L- i-ll ‘5 —-e. «[494» ?e 0.9%?) (51.3361) at J“ St _ 3-3;;3fiom + 3‘56 “0(5) —2 (51-333 (.5 ) -31: 2. 2th“) i~\i _-_- -3_2—'.e_ “.0093" '37-‘6- 3r 3 (5:3)Cs- 2- 33;- _3‘_ + (5., ....L Cat-“0H0 IAE S-H = (%-1¢:)€. the”) J‘ 5' o (5 points) Write a MATLAB In-file that calculates and displays Mt) as a function of t. \ ‘53-’25: S J103j F- — “a 3’ Phi. :. L\c&folu.(.e. ( MW (5 to: 9512.0.) .. p)) Problem 3: (25 Points) 1. (12 points) Consider the LTI system that has the sFtate—space representation r-_'_‘ ~. _ 1 2 2 , : 5 ,7 y ( a)1 o (6 points) For what values of a, if any, is the system unobserverable? The, 86¢me 75 unobsO—Nw‘AQa Men 902562 :0. 01-21: (a) = (63073 - ((1.3(64'300) -= o 72 '60:. -3cc.7" ——.=o «’2 +' 2-“ ' l7 :0 (cc. -‘!)COC+63 :0 1 «Q a, 710. 6d {tam 7.5 vnobje NOLLIL 'Q/Y‘ (X; ‘4 q . (13 page) Consider the linear equation Where (1 is a constant parameter. 0 (5 points) For what value of the parameter a is the vector y not in the range space of A? a. Caflndk 50’ efi/oreJreQ. «a 0.. wzyhfcco. Svm %A ‘ ' “ cue _ the. (wk 4 90 drug. {0 g .5 056? I" “Q. 0(an %_ aaQ/‘ECBX: 0"”! 1-4-0) mug. 9 ML! vCu“ rank, m tim. mndcjflare% ”- (”Hem 6C— 1") when océl) . Veda/r 75 In thus Cause.) my 1’“ alt o (8 points) For the value of a chosen above, specify the rank, range space, nullity, and null Space of the matrix A. For 01’ :(J A = (II ll)- 00 Problem 4: (25 Points) With Halloween closely approaching it is prudent to consider the dynamics of vampirism. Consider the Lotka—Volterra nonlinear system 1'1 : ia1i(t)+dv(t)h(t) (1) i2, : nh(t)—de(t)h(t) (2) where h is the stock of humans in an isolated Pennsylvania community; ’17 is the number of vampires, (1, is the death rate of vampires due to contact with sunlight, crucifixes, garlic, and vampire hunters, n is the growth rate of the human population and d is the contact coefficient. While h and ’U are functions of time, a, n, and d are constant positive parameters. The term dii(t)h(t) appearing in the state equation indicates that each time a vampire meets a human being, the former extracts blood from the latter and by doing so, turns the human into a vampire. For this model define a state vector 17(15) as 1:05): < Mt) > . U(t) 1. (6 points) Find a_ll possible static equilibrium states m. “/J‘ r- l‘iQ, '=' .— 0: -a-\Je,+69a\k,hg_ = V€.(-—a. +0le) V2.- 0 0: Abe ’szhe. 1:. haCfl—o‘x/a) her—o hez “/3- X :. (he): (0) anl— Xez 7' <V€p> = ( 2‘ V2» 0 IV}. 2. (10 points) For each of the equilibrium states found in part 1, find a linearized model of the form , 6i: : F6117. 3‘) H) :: < 5‘15) xco = W van >°< - n x. — i xle ~ “9 WW“) I Sq: -axz + JL,><.xL = 41.09,)0 \ Din/Ox, afi/axL n-vfi-xze. “me— 5-6/9 X\ 8.5/3 XL “n u u 8 #2 For xe,:(€ For hem; (not) ,____,_______, n o O ’0‘“ F= 0 ”L F“ n o 9 3. (9 points) For each linearized model Obtained in part 2, state whether or not the corresponding equilibrium state is stable or unstable. If the stability of the equilibrium state (cannot be determined, state Why this is the case. For Xe‘ : (3) F = <2 .6“) £21263 IT'F> =- (96E (:‘nflcij :o (fi’ NOR-HQ =0 s V1 7] S '0» 7“ + J k 11"“?— eéuflbviwn Sew-(Q Xe! r 9. V5 [4.5 UL 4.19. n “V‘Q PoS.{3\_/Q_. {70. am I l.) unsfiLl-e- beam/Se, 7‘] >0- a-I&_ Fflw X9»: (AM, a 0‘) ~"o .. — {3 r F: (a K) oae'lz<7\1"F)’ 04:, —r\ 7‘ 10 ...
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