examII_s07

examII_s07 - EE 429 EXAM II 18 April 2007 Last Name(Print...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE 429 EXAM II 18 April 2007 Last Name (Print): Sal. [£3 9 Q S First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO INSTRUCTIONS 1. You have 2 hours to complete this exam. This is a closed book exam. You may use two 8.5” x 11” note sheets. Calculators are allowed. P9330 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 example7 Problem 1.2 Continued. N 0 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. (13 points) Figure 1 shows the block diagram of a data-sampled control system. ms) m9 Figure 1: Sampled-data system with command input R(s) and controlled output Y(s). o (3 points) Which dynamic block(s) in Figure 1 implicitly contain a ZOH DAC? 0 (10 points) Without performing any calculations, explain in a single sentence whether or not is possible to find the transfer function Y(z)/R(z). If the answer is yes, compute Y(z)/R(z). If the answer is no, find an expression for Y(z) that involves R(s). . Ms) an». 6(9 contain a... 20H 06c. because. Hw— MM: 5" 1"“ “wk; are. SumPiQfl signals. o Because. this} ‘6 sqmphl bgcvm, if .3 El£er=ee0J we. Can Comp/LE- \fl‘vmfi). Fn-sf *FmL er-prcuiom ‘Far' the. Sampler Mir-{:5 au-Q dash"? “P915 Ms): RC5) ~ am «(gang :7 n" = R“- are? =% M3 =R® ‘ “‘93” 3(5) = M5) 9‘15 — «(gang z; 3*: o“ 4“—c.’8*==9 3(a) -— Ma) 91;) - era/5(a) chi = so) may 1; Tm) = am» 5‘5) :; yaw-,ma 13(2) El'mowte— BL 53 hum}. 8620 =- 7(2)/6(%) : % = K _ _ ‘ H- ate) = "6”) we) A c 3 (13 6: Via) Am wow 61%) DH, 6m Ehmonuk't ( _______,_______ a7; (.9 I + GL1) WE) =3 IE)... :1 H- 6(a) 3TH?) 23-- Y(%) ‘-'-' N. 63) out) 66%) Ni) breasts) 1’ en.) ‘ ____________________._. “a 1 + 6(1-.\ + Na 6H“) 2. (12 points) Consider a continuous-time plant with the transfer function representation 012(5) : where Td = 1.4 ln(4) sec. Using the method of residues, compute the zero-order hold discrete-time equivalent transfer function representation using the sample period T = ln(4) sec. 1-41.: LHT' = nT+AT “:9 Z rnT's 2| lZe'MS} e - ’- _ ~n _ = ——a— so“) = g—E—L E z—‘ 235(w-M] Z} 50*” I 2e. :- C *C\ ZfimT‘J l* 2- re$\&s2$% Mac)?” arty?» 0 ______~, = - i 3(S'I'I') Po‘es+ Zemfi M— N244) Co, I‘aSIuo-JQ— “5506|wb¢&4 Wu‘b“ 1:0 l 2. 2am» = ~1-s - ( a ’———_’—' ‘flrfl ‘—¢ (1° " [KCMO he r0 —mT‘ c VeSlo‘uQ, assocm'lzflr “4th 7‘" ’11! - 18- ‘J = ‘T )s Problem 2: (25 points) 1. (13 points) A discrete-time system has the characteristic equation Q(z) = z“ —1.223 + 0.07;:2 + 0.32 — 0.08 = 0. Using the Jury stability test, determine whether or not the discrete-time system is BIBO stable. Show the steps used to construct the Jury stability table, as well as the conditions examined for determining stability. '3 . 2° -z' is" 23 2‘ 2° 5' 2" Z 2“ Go a1 qt. 03 01 “0.03 0.3 0.07 “L?— I l "'2 0.07 0.3 -o.03 be b| b!— 173 -o_9736 LI7L -0.075'6 'O.207 b3 51- bu be -O.2.o|1 —o.075'6 "‘75 vo.¢r73e 00 GI C1. o'cmgelg 4473375 0.315‘0202 ac q 49 a. “o L : 5°: ac C.“ L) = as [32: i l’3 \GH 43 Chi “0 a“ q) q\ q?— bo b3 b0 bl c : be bl O - = 2- b by b: b0 b} l». J Therm are, n+1 = 5' (gng'trmn‘fio £0 fe5-t GU] -.-. l-I.Z+ 0.074(13-0-0? : L‘l)“ 90'!) = I-H-‘L +0.07 ‘0-3 '0.03 :: [.870 > 0 laol<an => 0.03 4.1 \/ lbol3 lb“! =5 0.9V36 v 0.2.07 \/ leol> [Co-2., =7 0.74:“: > 031502.07, / aféf'fia‘é W syfkm I“ A) #‘ue ConSfirq/h'éj arO/ S 61(50 séaiaie. 2. (12 points) The natural response of a closed-loop system is observed to have the form > o (3 points) Identify the mode in the natural response that dominates the transient response characteristics explain your reasoning in one or two short sentences. 3k ynatUc) = c1(0.1)'° + 02 (0.5)k cos (3 where the coefficients c1 and 62 are real-valued. o (9 points) Specify the characteristic equation of the closed—loop system in the form Q(Z)=Zn+awn—1zn'1+~-+a1z+ao :0 by specifying the value of n and the coeflicients 04’. - ’39, mole- ,and ab It tfl-h—G) ’00»er 153 relax to taro than the. flick ' 719% sari—em .5 thing amoer- all-U» po‘e) rou‘iéeg 45$ 17 (gin/3 ,‘ “i /3 1P1: OJ} 2?; ': 0.52, ’ 2P3 :. fir; -.= O.r2_ . fl as In feo‘LaHinmr’ 4:me pom; 2.pz any 2?; am, (gr—[arena I17. 5/1, t 5 2P1“ 05°9‘7g'yg)*05'/5\(1E)= - J..- ['3' 2P3 ' 39’; ‘I a"? UNIV. thQ. Dad: 32% a"; YaSU/fis -L’ E 2-1174— 1— am: (2—ap.3(z—z,;)ce-%,3\ = (330.0(2 , ; 7)( j —, \l (2—0.0(22' “12-:- + _. - ..L. 7- _. - a 2% + {12. “ME-.1 +0445?- -0-‘ H1) +03% — 0.02.5” Problem 3: (25 points) Consider the closed-loop data-sampled system in Figure 2 where the controller and plant transfer functions are S Gp(s) : s + 1 and the sample rate is T = ln(2) sec. Figure 2: Sampled-data system using cascade compensation. 1. (9 points) Determine the loop transfer function B(z)/E(z) and specify the system type. 2. (8 points) For a unit-step—input r(k) = 110(k), estimate the time, in seconds7 for the system to reach steady-state as four times the dominant time constant of the closed—loop system. 3. (8 points) Without calculating e(k), determine the steady—state error for the unit-step input. I. Kczu “ E“) am YC-z) {gm/5(a) caves we “w “a 2. W2) _ arrange) , 52-7.5— r<c a = 2: '+6¢'*75p(*) Z-o.s'+$-a-Y.r =- - 1'11"“ fl pola 65%» 676 6% -- 5.0 F 739m. nwtvr-JZ, response. #— the Sachem ’us the. firm. c4676]. I’QQ. Gofre$ponj~uy 5-,glunq, pale, “as a. fume, cooI-Ecw't‘ t = " I?" = ‘ LN“) = 3.802 Sec.) ‘- £954) + all So the vstem wll rem}, SEQ/"$430669 I” “- °‘“ VI 1:. lf-L sec. In response, '50 a. unit’sbpmflfé. Eu) __ 1 Mt) I+Go69 ._ 2- 6.5 6% - 3.0 swam is Sim“ ‘“ .3 v uQ. FTO" {wart 2. we, hm.» find: "the. doSOQ—pavf ‘6 {he po‘e, ,afas mulL 1L; umfi. clrck' ‘fihe. '9 theorem, 2 Qf (-2- E( = Lag) 91%) e“ 297 4) 4") 31/21 (*4) W55) = ,Q.:n(4)_i'2_'°_-5L i = ‘~°-5 f, 1. a" 6%-J‘.o 3/? 6-5“ 2 e55 =Ji Problem 4: (25 points) 1. (13 points) The forward difference rule approximates the area f m) = 'e(T)dT under the curve e(t) using samples e(kT) as z(k + 1) : TZeaT), 1120 where k = 0, 1, 2, . . . is an integer and T is the sample time. o (7 points) Show that X(z) » T E(z) _ z — 17 and hence the integration transfer function can be approximated as 1 . T g = . ( m o (6 points) Provide a sketch that shows where the forward difference rule maps the left-half-plane in the s—domain into the z—domain. K'l ><(|<+:) = 1- Zeccr) + T'e(KT) = x(K3+T2(¥-) L’o The. 2—trqns'RJf-m e‘F‘ the, «QU'E' OJ-frafidrm y‘alflo 2 X(%‘) = x112) + T 5(a) :3) Ma) From abduction CD) aalbove. 51-33-, 0y. E=ST+I Doha that 5=0 maps to 2:0 9.) dXL) 4',- the, SvPIQnQ, maps gn‘éo 6U point 4/0? QM royEegQ. .25 (29. 223 =1. % «.49. any pom'b “JV-"W the- Ir. (a) IMCS) . (5.6%) Recs) ’9 1. 9 “a «M Le? Poles cm W fivplaflom Outslbe- the, um'l: arc/e. I.” flu. 2. (12 points) In a system identification experiment the measured plant response y(k) is estimated as we) = «WM, where the 1—by-n regressor vector ¢T(k) contains known measurement data and the n-by-l parameter estimate 9 is chosen to minimize the performance index J = Dye) we»? ( 0) Suppose that n = 2, N = 4, and W2) = (1 0) W3) = (0 1) $00 = (1 1) while 11(2) = -1 31(3) = 2 21(4) = 0 Q4 points) Using the given data set, and without solving for é, how can you determine if a least—squares estimate 0 exists, and whether or not this estimate is unique? 0 (8 points) Determine the least-squares estimate éLs. The. flararmpter‘ eatm'l'g, é; th-GE Mlmrmzflé T saints-Le; 4.‘ '1 A A ( 2 Mafia») 9 = 2 (Magus) => ,9 e z b K=Z K:]_ A Sb; 1-? the. (hurtful. A 75 nonb'flaulurj A. ufllbve, évL-‘b’m fix 6 ext ' I-F- F} is slrvaJuy-J by": lo 75 7a ‘U'Q- Calumn {pace 2! '41 than an m'l'nmi‘e 0vm‘3-QJ‘ 04:— so(u‘broo.s' elus'é3 othermw—J 0... L5 Saki-w” does not axis-b. In ‘U‘n‘s Prdble": 9 = ¢’(7—D¢TCZD + ¢(3)¢Th) +—¢M¢Tm = l (I 03 o (o I) ‘ ( l l\ (‘3) + (l) 'l (I) .. l 0 a l l = (7.. 1 fl 0 :3 4’ <0 I) + (I I) I z 10 Because 49:4: (9) : :_ (290)—1600) = 3 75C; the, muffir\x )9 ZS mmsrryulay any WQ, Can 09 ‘1!“va 5 ob -| gzfib, We dlreu‘py' have, 4‘) w‘n'g E 7.5 a ‘2. .1 ) ( ( J b fifty“) aaacz) +¢c3 5M3) + 05 ‘1); ‘I = mm mm +< = ( 11 TRANSFORM PAIRS Laplace Transform Time Function E (s) e(t) % 710(t) g1? tuo(t) S—i—a 6““u0(t) swim (1 — 6"”) U0“) 32(sa+a) (t — Lift“) U0“) 12 z—Transform z (aT—1+e‘“T)z+(1—e_aT—aTe““T) a(z—-1)2(z——e—a ) ...
View Full Document

This note was uploaded on 07/23/2008 for the course EE 482 taught by Professor Schiano during the Spring '08 term at Penn State.

Page1 / 12

examII_s07 - EE 429 EXAM II 18 April 2007 Last Name(Print...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online