quiz1solutions

quiz1solutions - 16.060 QUIZ 1 October 15, 2003 Each...

Info iconThis preview shows pages 1–11. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 16.060 QUIZ 1 October 15, 2003 Each question is worth an equal number of points. 1 . Consider a pure second-order, underdamped system (quadratic lag) with poles at (-a ijb). Consider the effect on the system step response of the following changes. \ You should discuss the qualitative effect on percentage overshoot, settling time, ' frequency of oscillation, peak time, and system stability. Mafia 0 45M) heel/my of Mel/o.er 35 CJQ} , {10-} C0,, (‘ T [+91 (landed; Ea”; go 7; Sq” a. lnereasingh *4 ” _!‘~ 1|. 4 “N51 i J/ @ Puts: Ear—e CAI-lqu {film I“ CHIS, “cl/r f) 773 So S‘laLi‘fr'ltf SGF‘C, ecreasing 0 (Le. moving the poles to the right) i ‘1') @ T5 7 Uflcfi‘qfl fol G? Uficficwjbj (9 SJOILI'IIJ')! J/ ,[3€(Gv£€ PUB ([016? 75 In; GIVES ' c. Addingazcro at --a (990- 7i @Ts 5W6 690/ game (,9 Tie ii (9 $499,511)! 50/“?! d. Adding a pole at —a .flea/ WOLF MU doman) g‘o {Psfou/‘ie UH [wk Poo-EH)! MW ‘3 i we; . if? 0* J/ TS S4”! [L4 'iooVL any GASL/r'r) (.JJ qu6 “0+ (96",)! $71GLI/tlx/ fCrp Wok} _._,_____._--"" 2. Consider the following system (D know Adv 710 aliqCIQ/gnyf,‘ale amine” a. new or mam" U55. Ii I“ (f) (r?) C(s) a) What is the closed-loop transfer function? b) The nominal plant is K=1, 1:1. Calculate the static sensitivity of the closed- loop system to variations in K. Sketch a plot of this static sensitivity versus K; for Kc>0. c) Consider the closed—loop system response to a unit step input with a controller gain of K5]. i. What is egg, the steady-state output value? ii. - If the value ofK varies by 10% to K=l.1, to first order what will be the new value of css? iii. How could this steady-state output sensitivity to variation in K be reduced? 9 - IVE :Tfs) (a) R " 1:me __._.——.._._——...-. (L) i QT K {IMHWJMZ{MUM}. T541+W6h "CWT. “3 574' *1?” '{Z’SflaL 1414‘}; Va — “(54”!ka SKI-J. git/5:0 SK}. : l . .50 to retry». [my We? Va (a) {E} 1L5 SyS‘IPN is lype O TL: jam :5 “C K I / {if 0:” (“w/0f We I SO €09” 16/ 05430 {fr/mt f5 5% {Fl/T 6rd 05) 3 TM- W5) (q) 3. Consider the following system, which has a unity standard gain. 1- 2.; 42:05” [8,, m 0.3 0 J Re Im 0.4; NOT TO SCALE a) What is the root locus gain of the system? b) Consider a unity step input. Write down the equation that describes the output response C(t). You do not have to calculate the residues, but you should fill in all other numerical values and define clearly any notation that you introduce. 0) Identify the dominant mode(s) and justify your answer. . _ .. 9.10.; a: .—, Cm” 5:225. 92: ,M 0 ~96! I > 0105' 540,04 5 2 +0.6; 4— 0. 25' :_ 0.2. {3+ 005') " (54L 0.04%; 2 $06; 4-0.25’) :> “(L “:0 2 1.-.“ .W L“ _,,_ fl“__Mw_,_fi_,_m_ .. 4. / f flail ofmosio’ve Oniyif 0‘; FPJiJvc. a'} (99' due ' ‘f F re JV 6 5:0 pole q+ 51204 a+ viper C0 (“93' UH?” Eompbx J] F0 p” C pole U6 know :11 Mud‘ Ea. i £1?wa +‘c jar. (5 A 50 "He (ta—I‘m Of a“: C(Gyg (a S)({th9"}[q£ 094Fu+ 40 "He (n S‘lf’fiJ/ "S'Jajf {of}; My”! 4f 1 . (C) gé’rn/x, 00(7119 ZFFO C(ofic 'Aa [vote a)" $3,.04/ “hi (UHFQ’K (030/2407) (Hob/{n Lu” Anne 0 4y)"- fesxdve qnb/ m” dommaé‘: ear/2! on. gt’mum [706 mL {3 304 fins 0; flog” 70W: mm, m mm ul/ dam-«.4 Mar. Woks ’4 Zero de’In’jt LCM, a {kw/{Cf fie (7) I" fulfil/fr. [or Lg of fpgidve’ gr (27572” (“OM/99*" pale. —_.._.——._.. I'IL l5 > \ LJ/ntga lbw VL/oni fl,» 1. Complex rail/3?“; POL”) US‘C; 52 J. 2;.(4/‘3 + CJnL I 0—!" NJ”!be 0v!— {f- upfr; page) (Iv My” fate); LJ {En-«en. Lgr Aow 710 I‘M/H7067 (-ku flMAP/s , 3.5. {5 Log-agflwawag) A: Show; +0.!é :2 52+ 06; +[0./€¢0.0?) 151;» M; +0.25‘ 4. For the following system . _3_ I where 6(3) 2 ————1—— and G609) = 3 +3 l 3542 — 2 S 2 .— [5] +fl+1 S 5 4 4 50 [1:2 a) What is the steady state error (855) when r0) is a unit step input and d (t) = 0 ? 635:0 _ b) What is the steady state error (en) when r(r) is a unit ramp input and d(r)=0? 4ch 1 s) 955 s. 7)? s :1; c) What is the steady state output (CH) when do) is a unit step and r(t) = O ? Fig-"traitor Murat”! (Q Gal D “*9 raft} Sitp Gibb/Lame 1-") Cs; 3 0 d) What is the steady state output (cu) when d (t) is a unit step and r(t) is a unit step input? fqu/‘flofi‘ifon i at” (6') GM, fr) In (c) J «655:0 .1) [55:] - go \[ur (CUT (I; 3 C556?” (“(C) I “O :m ' i/Uu (an also Solve (c) L), (mics/airy 1%: CW! UM“? (q; you (a LONG Work)- Problem 5 5a) Find a state space model and A, B, C and D matrices for the system represented by the following block diagram. 5.b) Given a state Space model with A, B, C and D matrices A z 0 1 B = 0 —1 0 1 C = [2 _ 2] D=l find the system transfer function. V - -’ 492 t. (W :. SZt-‘£§_-.:’5‘ (q) U, ‘” 51" 5+3 5' (unfit-5) ' 594547 WTflmw X. y u] ‘—> 53114513! - 5 5E: 4‘ 4'3}: 4'31! ‘3 y : 4—4)?! 4 fl“! Let - Jr; :22, 3> it} : )1; z u, -31, ~41; . 9 Lay: [2 0.722 + DJ“ C. D Problem 6 6a) Given a system represented by the following differential equation, where y is the system output and u is the system input y+y2a+3u 6.a. 1) Determine the system transfer function 6.3-2) Create a state space model of the system by determining A, B, C and D matrices 6b) Given a system represented by the following matrix transfer function 5+3 s(s+l) 32+2 s(s+1) 6(3): 6b. 1) Find a state space model of the system by determining A, B, C and D matrices Note: Be sure that the state vector for your model is of smallest possible dimension. Hint: Your results from part 6.3.2) should be useful here 6.b.2) Check that your model is correct by applying your A, B, C and D matrices to the formula 6(3) 2 Ch] — A]*‘B+D Proum 6- (9) G) 525/4 Syrsu 4}“ 2) I _. 343 M $245 f7— }, M u. 9 ‘_ 1 l PM“ LEE?" W“; *3 W 34411»; CV12? 5M We gig-z _______“___‘m__fl_ 50 be 5M! Lam: E: [0 ']1. 4/67 "” 0 fl 5 L ’J Ndu W quo Lo.va y; :2 J’le l: [jg-X2, 4'2JC, (51m (6”) and y( {5 IL +3)” {CU ...
View Full Document

This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

Page1 / 11

quiz1solutions - 16.060 QUIZ 1 October 15, 2003 Each...

This preview shows document pages 1 - 11. Sign up to view the full document.

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