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quiz1_practice_sol

quiz1_practice_sol - NAME 16.060 QUIZ All questions are...

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Unformatted text preview: NAME: 16.060 QUIZ October 24, 2002 All questions are worth 10 points each. 1. Answer TRUE or FALSE (you may include a sentence of explanation if you wish) a. A linear system with feedback may be characterized as stable for certain inputs and as unstable for other inputs. FM. 93 I Siabluiy {S a F®F1°rtgf of IQ $753k"?! r1071 7th {apt/13 b. Increasing the root-locus gain of a first-order, linear feedback system always tends to make the system more stable. {WW4 ‘7’”! LI”? 57(de (boo! {oft/.5 Mel‘s/J) 0. Good tracking and good disturbance rejection are typically the goal of control system -. design, but cannot be achieved simultaneously with the use of feedback control. ' HUB: we can 3:! Lo” 97 {74ny “130'" 44W“ More I? a”! 0 entn- fie 90f. d. The open-loop zeroes of a linear system can be moved with feedback. FALSF' (Cm 0N; Molfc‘. 0L {point Lita“- flea/£454 e. The root-locus gain is important because it determines steady-state errors. FALIE; ch Siaw/a/r/l (yarn 4b denier-rake €55 NAME: 2. Consider the system represented by the following polefzero diagram, where the pole at the origin represents a step input. The system has unity gain. not to scale firm: : 4i? ‘— w—> -O.1 £24283? \ _ 4 M - step input a. Write down the form of the step response C(t). No numbers are necessary for the residue magnitudes and angles, but define your notation. b. What is the root locus gain of the system? c. Calculate the magnitude of each of the residues. d. What is the decay time of each exponential mode? e. Is there a dominant mode or modes? Explain the rationale for your answer in a few words. A (H f 315 (n) (M): 1m Ice 4 2V; 6" rust-+9) (L) .SfSL’“ Ln; Vail}; S'L’o- jib-:1 jfih I I E? )3“ £19} 3.fi,__,__._ .555 It». 05!? (.5 4 W “ i ' s 0.} ~- m: or}; ._- . .._. : > ( 0-D f. 5’ 0.} S 24-(54‘ ’2? ‘3 Km : )S‘ ....___ .. k; w _ . L3 (C) CM' é” 53?, “L 5::ng 4,5434%» W= -{_(§J_/9-J_2_L : (0.0mm my: (Wow) : 0.273 {0- WWW?) W3]: M .: 0.33:? {WWX 6) (J) gnarl; POL! @ 5: B0.) .' T1 /0 C”"f"*P°“>@ Sr wig: T": 73 ((71 683%(05061‘9) 'IdBNIMJ-(y erg/J}, 01"} £6ch :1' [c5 élijr/f’fi'duca 8-0.” CIOMI‘dU laL/p 6"), LPrav-hc 1+ [965 [Griff 917M? (M471). ,0.) I“! decays SIOUEF NAME: 3. Consider the following system with controller 66, reference input R, and disturbance input D: a. If Gc is a proportional controller, GC= KC, what value of Kc will cause the steady-state error for a unit ramp reference input to be 0.1? ' b. If we want a zero steady-state error for a unit ramp reference input, what form should the controller Gc take? - ' . c. For a unit step disturbance input, what form of controller GE should we choose to make the steady-state error zero? - (1. Assume that the reference input is a unit step, the disturbance input is zero and we are using a proportional controller. What value (or values) of Kc will yield an underdaniped system with a settling time of 4 seconds? 6. Considering the same case as part d, what effect does increasing KC have on the damping ratio? (a) CC:{(° ‘3" 71pr (1er .21} (/ “Jymfir 1,, 6(3)) :) €53 For {‘6er Imp} 3 l/Kax 55/57/ij J Fonda/J {Ml/ix 7%139‘ [wagon (5: 66'): 5%» '2 5%} $54.? [1’ 1 :> 1/ 3 “'19 U8 Md €55 : i:0.) :2) K3“) a> K (L) To go)" €55 :0 (of unfit ramp (ops-1.} (1255/ 2 ”6&9le ,9, 6(3) ‘ 1') Cc (5) SLO'uiaP! [\GLC (AI-«9 A”! [wiry {Q'A'Jr‘ (C) Ti) 94} Zfi’O SS Paar L/g (wet/110 (OFFICIr/{y Pulsar 4Q (/qung :) new 655“) w D WMMJCF 2/5 $90 5~>0 “(1* 2946(1) -— m .L F 5190 65(5) “4) ija-J' Css'o L18 {My} Love 0? VI 'A’KM [-4 Cg (-5“) (0') Med 40 94 g .. clam—100p C We _ V J H— : _{.}5 ~— C/ -> 0 40'“ 3 iii}: M Slflséh I? [[563 2rl/F) 5(9)) : fl i J??? To yo;— an Unclrrc/aqflcf/ 5);”!er {Mg Am] Cuffed-m C(«pa’éi 2) [Va/Z (flak: KL), 12> [30sz 51+ §:«/i(# .9 T:II:.) T554T5'4ECM) (e) Incrwl-J {to m” ’7v3L VH4: FOL; away gm 7% réa/ am 17> (Jar-«PIT; (“mire J/ 1gflcrrw£yfic x X ¢ NAME: 4. For the system shown below, sketch the root locus for K>0 and K<0 on separate plots. Where appropriate, calculate pg and the angles of departure of the complex poles. NAME: 5. The following plot shows the response of 5 different systems to a unit step input. The dashed lines indicate the response of the baseline system. Step Response '1 Tifine (secii 10 Consider the baseline pole-zero diagram shown below and the four modified pole-zero diagrams shown on the next page. Each diagram corresponds to one of the responses shown above. For each of the four modified systems, match the step response to the pole- zero diagram. Justify your choice with a few words. Baseline: NAME: Pole-zero diagram 1: Pole diagram 2: )'( ------------------- - —j 213 Which response? @ Which response? @ Why? [AU [“43 yo”? (lawn, Why? 9 56'”? 5! Lax/he g. [was 301-4! VIP “:5 56""? P0. (35 £41365”? ‘5) [Omar P0./ 10!! r;- T? CJJ Ln) flat? (.2er I») 5101(ka Pole-zero diagram 3: . . Pole diagram 4: Which response? 3 Which response? / Why? Saw; r85 Flick, 0:5 Why? (“WI/7 Zero +0 4% Law/w, £66au}c- Zea I) “3“" Car/ks 20. 7'} TI}; J/ “in 9,» may 4» Law: Wei 6550! NAME: 6. The following plot shows the response of a second-order system to a unity step input applied at t=0. At t=0, the system was in its equilibrium position. Step Response 3.6 I I I I I I I "" l 1. r ! ' Amplitude 2 :_ , i l I I _ LW__J. I i- ........ J- i 0 1 2 3 4 5 6 7 8 9 10 ‘11 12 13 14 15 Two (sec) a. What is the 5% settling time (T5)? 2: 3- 7 56604115 b. What is the percentage overshoot (P.O.)? RU, :5 9—1"? 3 (3 Z), c. What is the peak time (2;)? T!” :3, i. 6 Run!) d. What is the riSe time (T)? fr 2 O“? 51¢“ ,1, e. What is the damped natural frequency (an)? :5 32. 51701); Mire:- [796-19 z) D 5.1:; 2i: ...,_ _ f. What are the locations of the poles? J 3'? N 2 {Gal/g T 2 T5 (5%} z (.2 seem! = i 3 J W” [M95 «J— 631-)” :t. 0J6; : v0.57 1 2% Practice State-Space Problem for Quiz 1 16.060: Principles of Automatic Control October 8, 2003 For the following differential equation: 1 tfi+5w+6w=f+r+2r . Derive a state—space model in the form: a = AE+B€E g“ = oe+oe Take r to he the input and w to be the output. . Draw a block diagram for the system, and clearly label the input, the state variables, and the output. _ (Hint: look at the diagram on page 4 of the Lecture 13 notes.) Using your state—space model, determine the transfer function 0(3) from the input to the output. Calculate the state—transition matrix @(t). . If the initial state is £(0) = E] , and there are no inputs, what is the state vector :E'(t) at time t > 0'? !" Ww'wfi ‘i 914 t J - (2-->l L 152+ zeal! _iwlb o} Ga-erosr >L_ $4.; ' l u 2:) X; +SJI*(];JP L3 ‘1. )(i + I; L21! ' ' s N , .. Skits". I; 2 :2) X’ 3 Mfg}, tier“ ‘ij )l22h-J - AI“ - _ ; (54;)(542) §(fl* 6’ 9* aM1/ 0*- L 2.6—. 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