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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 statevector $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 KH) = 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 discretetime system has the statespace 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" + an1z"_1 + ' ' ' + a0. p\ z+‘ +4“ We} = Cal4943 = (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 discretetime 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 zerostate response for a
unitstep input.
375‘ 23 YCE)’: 6(33211‘0‘33: 2.14“. .2—’
Use the Inltial Vaulue theorem 7%»: mm l“) = 3 2900 o (2 points) Without determining for all k 2 0, if possible, determine the ﬁnal value limk._.oO of the
zerostate response for a unitstep input. (a JiX£+ JaineI‘) “"5"” the vm‘é Gn‘de.’ the, ‘Fmo/ Val{92. Maomm is ale/(aye: 45 all pol€$ % (2!) W15) am, 24,; JulJ5; 2"]
: gi‘ + J “? + ‘l—L‘
J
2. 242: 7 23+: a—t a" LCM <———> 43—» , 2. (15 points) Consider a discretetime 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 $(0) yield the values of 31(0)
and y(1) determined above, if not, explain, why. geCau/Je 4— OJ 'U'e 00" Spam. ﬁg & 34.5 GQIMQILS/on ﬁg, "71%! “Q .5 water Adm ﬁe {‘0 drug, wz can .50 {hi =o_ This means thqré
c0) ,. I 7
X(b) ftp who vielooa. (gun ~ (‘1); WLerz, oz; .5 am} Sadatr Amy» 59 tkm are cm Mka numb1r S'Eu‘beé yum”? = f‘ank C6?) = :5 New!)qu (o/wmu% 6? r, I
vanrbr (6?): n « *dnlC—(Q) = I :9 Nu" Sync, v By Insien‘bloﬂ Q? :. (e?) For F = 31)) null space 9 spam 4% Im'huQ s‘bulae. 75(0) '= 1(0) + OCCJI) ammo 627(0): 517‘“) 4‘ 00620,) ‘— o (9 points) Determine the state transition matrix quc) for the statevspaee system mm = (f ;)m<k>+(3)u<k> y 2 ( 1 1 Verify your answer by determining qb(0). ¢<I<>= 02"iz(aJ:19Y’3 = fl l? 2133‘} 0(a)
{p3 —I ~' _ —Z 53 I = ( bus) >
1'5 (_‘ Z_> " Zz_6a+ 3 \ E_3 0(1) DCi) b(%) 2—3 i— +..L .
~ﬂ 1‘ (a 4962?) Problem 3: (25 Points)
The Laplace transform of a signal y(t) is W3): 6s+16 = 65+”
32 + 53 + 6 (s+2)($+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’esetwyofcsm} , 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” = Parsem2) Pa '1
(2/2)(2+3) l e 9— my:
1 ,____7:_..——— = ._._3————
c  (2/3) "LL [6 T5 T7\ = [5.1323ch I~ Fri137
7“ (2 +¢DC7~/3) [6. e. 7mg (no‘lA. 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: SI# 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”? 10“) + 2(ee3ﬂvvC'ZD 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 continuoustime system with statespace representation m) = [1])x(t)+<2>u(t) .. Ax +6..»
y (1 0)x(t). 1— C; 1. (7 points) Using the statespace 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‘
._.‘,zca.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: (SII9) : (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 zeroorder 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 tem 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.
 Spring '08
 SCHIANO

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