examI_s08 - EE 482 EXAM I 28 February 2008 Last Name...

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Unformatted text preview: EE 482 EXAM I 28 February 2008 Last Name (Print): SOL/£30 0.5 First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Weight Score INSTRUCTIONS 1. You have 2 hours to complete this exam. This is a closed book exam. You may use one 8.5” x 11” note sheet. Calculators are allowed. 6‘93!" 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. 7. For your convenience, a transform table appears on page 12. Problem 1: (25 Points) 1. (12 points) Figure 1 shows the all—delay block diagram representation of a discrete—time system where the parameters a, fl, 6, A, n, and 0 are constants. Using the state-vector $06) = < $206) > , (133(k) Where the variables 301(k), 302(k), {103(k) are defined in Figure 1, determine the state—space representation w(k +1) : Am(k) + Bu(k) we) = owe) + Duos) by specifying the matrices A, B, C, and D. Figure 1: Discrete—time system with input u(k) and output x\(K+D = " @«X,(K7 + 6 X309 4- ecu-.09 ch K-H) = X)(k.) _ ’8“ *0 = X24 n - “X3 Cb) In) 8.00 = 7. Now) +x,,cn) = —é7\X‘(K) + xzm + fiexswb + 00““ 2. (13 points) Another discrete-time system has the state-space representation _% ‘fi 16 mac) + ( )uUc) < 0 > 0 y = (5 —5)a:(k). Determine the transfer function Y(z)/U(z) and express your answer in the standard form m(k+ 1) NH Y(z) bmzm + - - - + 120 [1(3) 2" + an-1z"_1 + ' ' ' + a0. p\ z+-‘- +4“ We} = Cal-4943 = (s —s) V '6 (IL?) -.L. a q u, : l (5 '5') Z '7}: a .L. “Wang +44 an? ° has Problem 2: (25 Points) 1. (10 points) Consider a 8180 LTI discrete-time system represented by the transfer function Y(z) 322 z _ 1 . 4 o (1 point) Without determining for all k 2 0, determine the initial value y(0) of the zero-state response for a unit-step input. 375‘ 23 YCE)’: 6(33211‘0‘33: 2.1-4“. .2—’ Use the Inltial Vaulue theorem 7%»: mm l“) = 3 2-900 o (2 points) Without determining for all k 2 0, if possible, determine the final value limk._.oO of the zero-state response for a unit-step input. (a- JiX£+ Jaine-I‘) “"5"” the vm‘é Gn‘de.’ the, ‘Fmo/ Val-{92. Mao-mm is ale/(aye: 45 all pol€$ % (2-!) W15) am, 2-4,; Jul-J5; 2"] : -gi‘ + J- “? + ‘l—L‘ J- 2. 2-42: 7- 23+:- a—-t a" LCM <———> 43—» , 2. (15 points) Consider a discrete-time system with the state—space representation x(k+1) : ;)x(k)+<(l)>u(k) y = (1 o (3 points) Suppose that and u(k) = 0 for all k 2 0. Determine the values of 31(0) and Q: (:4) 3 o (3 points) Can you find a different initial state vector, w(o)¢(_21 that generates the values obtained for 31(0) and y(1) above? If so, what value(s) of $(0) yield the values of 31(0) and y(1) determined above, if not, explain, why. geCau/Je 4— OJ 'U'e 00" Spam. fig & 34.5 GQIMQILS/on fig, "71%! “Q .5 water Adm fie {‘0 drug, wz can .50 {hi =o_ This means thqré c0) ,. I 7 X(b) ftp who vielooa. (gun ~- (‘1); WLerz, oz; .5 am} Sadatr Amy» 59 tkm are cm Mka numb-1r S'Eu‘beé yum”? = -f‘ank C6?) = :5 New!)qu (o/wmu% 6? r, I vanrbr (6?): n « *dnlC—(Q) = I :9 Nu" Sync, v By Insien‘blofl Q? :. (e?) -For F = 31)) null space 9 spam 4% Im'huQ s‘bulae. 75(0) '-= 1(0) + OCCJI) ammo 627(0): 517‘“) 4‘ 00620,) ‘— o (9 points) Determine the state transition matrix quc) for the statevspaee system mm = (f ;)m<k>+(3)u<k> y 2 ( 1 1 Verify your answer by determining qb(0). ¢<I<>= 02"iz(aJ:-19Y’3 = fl l? 2133‘} 0(a) {p3 —I ~' _ —-Z 5-3 I = ( bus) > 1'5 (_‘ Z_> " Zz_6a+ 3 \ E_3 0(1) DCi) b(%) 2—3 i— +..L .- ~fl 1‘ (a 4962-?) Problem 3: (25 Points) The Laplace transform of a signal y(t) is W3): 6s+16 = 65+” 32 + 53 + 6 (s+2)($+3) 1. (8 points) For the sample period T = ln(2) determine Y*(s). Express your answer in the form a: 01 C2 Y s : —— + _____7 ( ) 1 — 6—3Td1 1 — 6—3Td2 and specify the numeric value of the constants ci and dz». 2 im‘l’esetwyofcs-m} , c, + Ca. SlfiaJN fonts i (73:?) (QT-'3) vol Y(7l)J -2 -3 Y‘C-‘D = 62+ 16 l ———-—‘-l--—'“ ’ ’25—: 0|: (774.) “T5” = Parse-m2) Pa '1 (2/2)(2+3) l e 9— my: 1 ,____7:_..—-—-— = ._._3—-——— c - (2/3) "LL [6 -T5 T7\ = [-5.1323ch I~ Fri-137 7“ (2 +¢DC7~/3) [-6. e. 7mg (no‘l-A. thud: 82"“), €942”) :- 27‘), 2. (2 points) Using Y“(s)7 determine Y(z) and express your answer in the standard form y(z):_7l_fl.zm_:__'_l;+_b°__. z +an—lz +"‘+ao 2 hi): 7*“) = v—l—T 4- #3“; 2=¢57 1—2."? 1“ s7 a .3 l'l'z(Z"/¢3) 4‘ 2% (lad/y - “1 EA? 4' Z {TIE ‘ (z—Jq)(-Z’Ji) 3. (6 points) Determine the causal signal y(t) from the Laplace transform = 63 + 16 = _ A + 6 _. 32+53+6 (5+z)cs+3) — “7-— 5+3 65+”, 65+]6 9:- S-I# 1 _ '1 6:. #3 —-—'——'—'"l —; Z ' +2 (.3 (Lt/36+» J31 (.5 ) $5 S=_3 4. (4 points) Using an ideal sampler with sample period T : ln(2), find an expression for y(k), where : y(t)lt:kT' Express your answer in the form 3406) = 61(T1)k1(k) + C2(7“2)'°1(k) and specify the numeric values of the constants Cr and n. Reflux. i: with ‘l: = KT = 14,0“,(2), then 3.06”: “(629141”? 1-0“) + 2(ee3flvvC'Z-D k) 106,-) 5. (5 points) Verify your answer in part 2, by computing the Z transform of the sequence obtained in part 4, and express your answer in the standard form bmzm+---+bo Y z 2 W. ( ) zrz,+an_1zn—l+_n+ao Y(2) = ‘1 .2 4.. Z .__a'.___. 1 '59 eboa‘élcn Problem 4: (25 Points) Consider a continuous-time system with state-space representation m) = [1])x(t)+<2>u(t) -.-. Ax +6..» y (1 0)x(t). 1— C; 1. (7 points) Using the state-space matrices, determine the transfer function representation of the system and place your answer in the standard form Y(s) bmsm + m+ be G6 = : . (S) U(s) {Wan—ls"—1 +~~+ao 6¢c9= ¢Csr-»)"8= <\ o)(5 —'>-‘(f‘)= 5'7 H 07C — ‘< 3‘ ._.‘,z|ca.s 2. (8 points) Determine the state transition matrix ¢(t) of the continuous—time system7 and check your answer by evaluating ¢(0)- ¢Cifl = i“ i (S I nyq] .1. —L- -. . s! - s a: (SI-I9)-| : (S '1‘) : 52' ° 5 0 S 05 10 3. (10 points) For the sample period T, use the result from part 2 to determine the zero-order hold discrete—time equivalent state—representation m(k+ 1) WC) @3609) + FuUc) 0:8(16) + Du(k) l | by specifying the value of the matrices <1), I‘, C, and D in terms of T. ll /'\ 3. “l1 -) a"; ll ‘-l phi, :2 ll ~\ v\ 11 TRANSFORM PAIRS Time Function Laplace Transform 6(75) 13(8) 1(t) % t1(t) ;% g 1(75) $3 6“” 1(t) Si te-m 1(t) (Sid? Time Function z—Transform 6(k) W?) :31 k 1(k) (zfm W M) 32:32 ak 1(k) fa kak 1(k) W 12 ...
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This note was uploaded on 07/23/2008 for the course EE 482 taught by Professor Schiano during the Spring '08 term at Pennsylvania State University, University Park.

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examI_s08 - EE 482 EXAM I 28 February 2008 Last Name...

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