HWCE3_Sol - HW/CE# 3 SOLUTION 1. Problems from Franklin’s...

Info iconThis preview shows pages 1–10. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: HW/CE# 3 SOLUTION 1. Problems from Franklin’s text: 4.7 part a 2 15% (0e pay, 2 180 Ciro/c of Facial: lé‘l’ (2&2 She/{(1 fiver/277.? Wee/ff Spec 5, ( I (Jé/E/fz (‘er affrta-nvnerrf: 5%,)” (3th W? > E“) = / /+ 6- s7- ; LII] .5 (S) 7" A» “if.” if: , fir 5 0,005 5—90 5—70 3+5? _ 5&0 56— ~ we ‘m—yaxr/ gamma. 6 Z J/ Sfloos') WCM é z: 01 «A; "a .‘ ‘ I. , ;_ \ , : z, --23 o r“ .2, 2-0 Map "’ "‘ “r \0 IC. 0/61 L} > i233 {D 4.23 part a; Use the simpler approximating realtionships on the first page of the reference sheets and do your design for two different values of C , 0.5 and 0.7 3:51" __ 45+ 5,; __ £0§+».Ll la -—* "’ "”"” 1- 52+4(9’“2_ <‘+ 27'“)? “MM Arm; ‘ I e 0 5” ’5 * ’13 < a / w) a)”; F" ’3 /" ,) I“ “w - ‘n z; i 0' 5’ -~ :3: 6.)]? E? ’15: Qawrl'n J/f'5> g”)? Jew 5.5, (’f‘fo /” Ker 5+} (,2 PC? cv/Iv'é 12/ '3 cf: “: 50,7 3‘ §§8, é a H a = 22%) a Riff“ ‘1 w x z . w a! 5,5, {erI‘ 327‘ flaw-74:7 wmgr at “ ‘9'3‘0/ J a“ «My 53— . \\ — E:€”Y'(/—(é$+é/£n jK Sin/‘6?! S .2 S/SJ'tcé' 2,, DOS, {(4 z?" r (5,” O " 5—7 0 “*N W“ _ ...,__.,.{_,, Ma .a “>0 52—3.“, 5 '44. (a? .1 til-- g e ,a/e 04, g :1 4 —o.o/a — /5a/ 0 z o / z 64M \e/fmfz 3cm 69 ‘ fl, = ~22.§é 50 5/ = 22“, __ 72.9 1 2514 < 94 fun 7'7" Jo we W/OeV‘IL—V‘xc-esf rat/$400+. Z? we flaw-“we, f a 0:? Mag-172(4)” %m- fir6>6¢cflwf£ v A)?! a IE (6r Kym/1514,7495 ) we j¢t A: 325‘ A=2/7Z 4=2§z 51:5pr / ) o J I z A, z 1%?!) 05" M73?” 1 I. /7— £0 29).? ’ th" ‘ Viper" . K 2. Con31der a system where: KL(s) = 7% H , S(s‘ +4s+68) w 2‘ a . M - ] Make a hand sketch showing the complete positive and negative root loci. Find the angles of the asymptotes for roots approaching zeroes at infinity and identify 0t (the intersection of the asymptotes). Find the range of values of K for stable performance. Find the value of positive K at which roots cross over into the right half plane and the values of the roots on the imaginary axis for this value of K. i ’7 J7 “I, ’2‘“ if?» if, '1: [4'5 > v” (I l I; {:3 I, J (r /~ '5 7 g r t' "A t I ‘ Q a ‘ ,1, x L t C Q ‘1) p -‘ p I; . C; / “13"; 1 _, ' ~‘\:V\ l/ i 9' >1, 5/ / WWW." V ‘ [2/ A, ' “ r ’ A: I } V / V_ “(1.93.” , - f' no r . A ’3‘ _ r i. I, . 4; A l f ;i ’L 7 p; ~ " s t g l i J'/ _ l '3 )‘ l’! “7 ‘a E t v L \T ‘22“. i ‘( ’ ,, 7' I K (‘rk’v l - a “x , '1‘ 2 A i ‘7 ‘ " 3:) V. , i; K r, u 0 _'/ '1 g I :5 . o , 'r“ _, ’ 5 ‘9‘. 3. Turn in a MATLAB generated plot for the root locus. (This should be consistent with your hand sketch. You may submit one plot for the positive and a second plot for the negative root loci.) >> num=[1]; >> den=[l 4 68 0]; >> sysL=tf(num,den) Transfer function: '1‘" kg».- Roo‘t Locus ii)‘: .ww WW .y” z i 3 WE} 2 s E ; g i ‘ : E - M. l,“ ; _/ i—g ‘ I l ! \Véméw‘ :1» g : '5‘ E E i_ = n , n a (5 M‘ ———_——'——3§ E : r51 _ E ‘ _ g \t I : Xxx x ‘ \\ i “X. 9 s 'k ; ‘x t \L,_ x"- x .K i \ w ~ iii} - i L; Real Axis >> r10cus(-sysL) Root Locus WWW,‘YwmmvWW,.A.».,_..V_V_“W “Wm-..” _. inary films 3 IlTlaI Real Axis K 4. Turn in a MATLAB generated Bode plot for KL(s) = 2—— s(s + 4s + 68) where K is set to the positive value determined in problem 2. at which roots cross over into the right half plane, i.e., roots on the imaginary axis. Estimate from your Bode plot the frequency at which the magnitude of the response is 1.0 (0 dB) and the angle of the response is -180°. >> Bode(272*sysL) Bode Diagram - ...... .wwwwwuwywwr...‘ . _._ S n I .x 'J . a _ _ ..._ :1 a 1 w ,3- -..L...,... A u-..\..w.. x M~-a,9w--_,._..c. -.w...,-....-r...- f System: untitlecH " Frequency (radlsecjz 8.28 K ‘ ” ’ “ ” ‘ ” ' ‘ ‘ ’ ‘ ‘ ; Magnitude (dB): 43.0829 3 ‘ Magnitude [d5] 1 ) a a r a a a 2 x a x a 2 L---— .Jgt a CD 3 3% E 3 System: unt'rtled1 ; I a -225 Frequency (radrsec): 3.23 . " Phase(deg)z-1S1 ‘ i t i i i in; ‘E 2- is” < a: ‘ "58‘" Frequency (radtsecj >> margin(272*sysL) See next page. Note that gain and phase margins are approx. zero at 8.25 rad/sec. Magn‘rtude (GE!) 6 (deg) Phas Bode Diagram Gm = BEBE—0‘1 5 dB [at 8.25 radisec) , Pm = 5093-014 deg fiat 8.25 radisecj Frequency [radisecj 1 -:. 0.49113- 5. A certain system has a controller: DC (s) = 8 s + a) Assume the sampling rate T is 10% of the controller time constant. Find approximate expressions for Dc[z] for a digital realization of the controller using: Tustin’s method DCZ3J= 366:) ,3 , S‘? E :4) = /éo(§:nl \‘ /Aa/z:l>+8 7’ 5f: 4-: Euler’s method EC [3 y = "DC a; z- x (573;) :?G{3—I) b) Find a difference equation relating controller output to controller input for the Dc[z] you found using Tustin’s method. 4V3] “[3] r 7 flab] 7 513’ = 4353: fit" ,, . 515] mg; dun— /é8’-;3:‘ y a )Elg'] Cg" 0”“ 15f j) N; 3 = C” /* X? a , /68 17.11:] - 1’5"}: graffiti}: €[t7 + cit—1:] toe #1 k] = f} mfh’ij'f‘ ,L. [cla'lfle 24—17] (68 M8 // ...
View Full Document

This note was uploaded on 11/05/2010 for the course EE 362K taught by Professor Friedrich during the Fall '08 term at University of Texas at Austin.

Page1 / 10

HWCE3_Sol - HW/CE# 3 SOLUTION 1. Problems from Franklin’s...

This preview shows document pages 1 - 10. Sign up to view the full document.

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