141_1_Chapter6_problems

141_1_Chapter6_problems - 390 Chapter 6 The...

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Unformatted text preview: 390 Chapter 6 The Frequency-Response Design Method 6.3 6.4 e Bode plot magnitude and phase for each of the following s. After completing the hand sketches, verify your result MATLAB results on the same Sketch the asymptotes of th open—loop transfer function using MATLAB. Turn in your hand sketches and the scales. (3) L(s) : mag-(3%)? (b) L(s) : MEG-11.1“) (c) L(s) : W73 W MS) 3 (s : 0%; 2s 4) (e) us) = W? (t) L(S) : KW (g) L(S) = W (h) L(s) z 4s(s~ 10) (s + 100)(4s2 + 55 + 4) ‘ S (l) L(S) — (S + ”(S +10)(32 + 25 + 2500) ch the asymptotes of the Bode plot magnitude and phase oop transfer functions. After completing the hand sketches, hes and the MATLAB results Real poles and zeros. Sket for each of the listed open—l verify your result using MATLAB. Turn in your hand Sketc (d) L(s) 2 (S S(S 6.6 Multiplepoles at the origin. Sketch the asymptotes for each of the listed open—loop transfer functions verify your result with MAT LAB. Turn in your hand s W +2s+ 10) on the same scales. (a) L(s) : 2 l S (S+ 8) .3" (s + 8) (c) us) 2 4 1 __ S (S + 8) (d) [1(5) : «W S (S + 8) (9) LC?) r. £5 + 3)_ S (3 ~— 4) (f) L(s) : (XLL r)2 S‘ (s _- 4) Problems 391 of the Bode plot magnitude and phase ter completing the hand sketches. ketches and the MATLAB results on the same scales. l (a)L<s>=S(S 1)(s 5)<s 10) fl ( +2) (mm—as 1x? 5>s 10> (s n 2X5 " 6) (c) L“) : s(s+ l)(s+5)(s+ 10) (S 2)( s4) ' (d) L“) 2 5(3 + l)(s + 6.5 Complex poles and zeros. Sketc for each of the listed open—loop transfer function second—order break point, based on the value 0 the hand sketches, verify your result using MATL the MATLAB results on 5)(s + l0) the same scales. _ 1 (3)1’6)‘ s +3s+ 10 d 1 (b) L“) fl 5(52 33 10) a (52 + 25 ~- 8) (0 MS) " 5(52 2s 10) h the asymptotes of the Bode plot magnitude and phase 5. and approximate the transition at the f the damping ratio. After completing AB. Turn in your hand sketches and 6.7 (g) L(s) : 43;}: S" (s + 10)2 MXCCI Cl [’1 L0 [7 p [e . 1; 61C 1‘ l6 5/ “We 0 l e ”()(le 1) 3g 1 “It 62 t l 16‘ l a (I I77, [61 0 S k '19 S 0 c mp S f h 10 m plla e l l eaCh ()1 [he hSth Open-100p l a nSfe] u] Ct 0113. E] ll)€lll$h the asympt t6 pl {9 g r Ina h t I 11013 [0 ea p l 0 p 0 g 0 WI a ]()U l 6“ le ()1 6 (NH! Cl bleak 0 Mt. ANSI C l l Curl the hand sketches verif ' F i, y your result With MATL ' MAFLAB resting on the same scales. AB. Turn in your hand sketches and the (8) Lo) = “MW 5(5+10)(S2 + 25 + 2) (b) LC?) : 2 (S + 2) s (s + 10x92 + 65 + 25) (c) 14(5) 2: “2*“ (M2)2 5 (5+ 10W +63--25) (d) Lo) : TQM S (S "10)(52 + 4s w 85) (6:) Lo) 2 W 5 (S ~ 2>(s+ 3) 392 Chapter 6 The Freque Figure 6987 Magnitude portion of Bode plot for Problem 6.9 ncy—Response Design Method ‘ ‘tude , totes of the Bode plot magni . and zeros. Sketch the asymp ‘ . ( that the phase 6.8 nght half’Plane [flatlifSthe listed openvloop transfer functions. Mithiig the complex and phfsiiirzgerly take the REF singularity into acgqun: 33:0 s After completing the asymp O ' ' oes from o . changes as s g . ‘ and th P1 an: t1: :6: :Oyeiilf: fiftirsiesult with MATLAB. Turn in your hand sketches e han s e c e , MATLAB results on the same scales. S + 2 1 (the model for a case of magnetic levitation with lead (a) L“) = 5+1032d1 compensatISOE)2 1 (The magnetic levitation system with integral control (1)) MS) =3 5(3 --1()) (52 #1) and lead compensation) (aLm=S;‘ 32+2t§+1 @LW3EjfiHKZEE (5+2) (e) L<s> = m 1 (f) 145): W 1 ~ F‘ 687 . '— .~ , shown in 1g. . . . (9 is represented by the asymptotic Bode diagram ssuming zero initial 6.9 A Gerald] Slilsttefinthe response of this system to a unit—step mpUt a Find an S e C ‘ conditions). 6 0 () 6 11131113. [HHSSUJEU '— 13” (16 plOtC 113 p “" Odb d ad g 1 1 0 0 S Onds t0 2 p61 CC 5 .1 PT V t d i u ’ y p 4 ~— . n U 6 11 A nonnahzed SeCOUd 01 del S Stein Wltil 3 d3“! 111g 121th 0 5 and a addl ()lldl 2610 IS glVen by C(S) ' 2 1 ' s + S + ’— 0 01 for a —- . , ATLAB to compare the Mp from the step response of the system Use M ' ‘ se. ls therea ‘ 10 d 100 with the M r from the frequency response of each ca 0.1, l, , an C011elat1013 between Mr and 1W 0? I g y 6.12 A 1101 3.1123(1 SeCOn ‘(Kdel Syste Il Wlth 2/ ~-—- 0 5 and all addltlonal p016 IS 1V6“ b m d G(§) [(5/17) 1](3 " 1) W h -— ( i l i i and i H i [1 con i can () draw 1 t it 1 1 1, 10, ‘ l . W at 0111810115 y u _ D13 Bode p O S W p . , . , h . ' ndWldth fm [hi5 h ff t Of an eXtra p016 0n the bandWIdth Compared Wlth thC bd. aboutt C 6 6C ' Second'order System Wlth n0 6X“ 3 p016 . llUULCHI) JyJ 6.13 For the closed—loop transfer function 2 T(s) = 0)” sz+2§wns+wg7 derive the following expression for the bandwidth wBW of T(s) in terms of (on and g“: wBW = com/l — 244 + ,/2 +454 — 4;? Assuming that a)” = 1, plot wBW forO 5 g 5 1. 6.14 Consider the system whose transfer function is Aowos G(s) : ~2————~—2. Qs + cogs + wOQ This is a model of a tuned circuit with quality factor Q. (3) Compute the magnitude and phase of the transfer function analytically, and plot them for Q = 0.5, l, 2, and 5 as a function of the normalized frequency w/wo. (b) Define the bandwidth as the distance between the frequencies on either side of (00 where the magnitude drops to 3 db below its value at £00, and show that the bandwidth is given by 1 (00 BW = _ _ ‘ 2n Q (c) What is the relation between Q and g“ ? 6.15 A DC voltmeter schematic is shown in Fig. 6.88. The pointer is damped so that its maximum overshoot to a step input is 10%. (a) What is the undamped natural frequency of the system? (b) What is the damped natural frequency of the system? (c) Plot the frequency response using MATLAB to determine what input frequency will produce the largest magnitude output? (d) Suppose this meter is now used to measure a l-V AC input with a frequency of 2 rad/sec. What amplitude will the meter indicate after initial transients have died out? What is the phase lag of the output with respect to the input? Use a Bode plot analysis to answer these questions. Use the [Sim command in MATLAB to verify your answer in part (d). Figure 63% Voltmeter schematic [2 40 ><10’6kg~nn2 k = 4 X 10—6 kg ‘ mZ/sec2 T = input torque = Kmv v 2 input voltage Km = 1 N ~ m/V 394 Chapter 6 The Frequency—Response Design Method ' a n, 6.2: Neutral Stability 4' . 6.18) are stable for ’ . (1-1 0 s stems (see Fig . . hmhthe 010% £0313: l and imagining the magnitude ts. Verify your answers by using a very Problems for Sectlr) ' ' f K for w .16 Determine the range 0 ' 6 each of the cases below by making a Bode plo plot sliding up or down until instability resul 6.2] rough sketch of a root—locus plot. K +2) (a) mo: liq-tr K . (b) KG(S) = W Figure 689 Control system for K(S+10)(£ill Problem 6.21 s ‘ :___ Mflww 3 (L) KW) (s + 100)(s + 5) f 6 17 Determine the range of K for which each 0 . plot for K r: l and imagining the magnitu fl h gk results. Verify your answers by usmg a very ioug i K(s+1l the listed systems is stable by making a Bode de plot sliding up or down until instability etch of a root-locus plot. 6.22 leG“*:se+5) w fits + 1) (me””¥n+m) K WW (c) KG(s) :: (S + 2X92 + 9) 2 . _§§ilL (d) KG“) . s3 (S + 10) ‘ ’ lemon Problemi‘for Section 6 3 The Nyquist Stability Cri f t. 1/52. that k ‘fer unc ion . , ' ‘ —loop system With trans t plot fOi an open 6.18 (3) Sketch the Nyquis is, sketch 1 1 m2, ‘ , S 5:61 ‘ ' ' 'r. 6.17. (Him: h C is a contour enclosing the entire RHP, as shgw: 301;: in Fig. 6.27.) W Cline/E takes a small detour around the poles at s z , a. t‘ n is Gm : 28811 t palrt (a) for an open—loop system whose transfer funCio ea , 00) GP; (02) _ V S and 6.23 ”(A + 0 . r each of the followmg system , lot based on the Bode plots f o ‘ ‘ n uist: ult with that obtained by using the MATLAB command yq Sketch the Nyquist p then compare your res Ks+m (a) KG(s) : 7(le— 6.19 K flfiflf (s -_ 10) (S + 2) K(s + lO)(s + l; (s + 100)(s + 2)‘ s estimate the range lt by usin (h) K 0(5) :2: (C) K015) I .‘ ‘ I d of K for which each system is stable, an (d) USlUg YOUY le g a rough sketch of a root—locus plot. qualitatively verily your resu 6.20 Draw a Nyquist plot for K“ + 1) (677) mflflzso+3y rioutems 333 choosing the contour to be to the right of the singularity on the jw-axis. Next, using the Nyquist criterion, determine the range of K for which the system is stable. Then redo the Nyquist plot, this time choosing the contour to be to the left of the singularity on the imaginary axis. Again. using the Nyquist criterion, check the range of K for which the system is stable. Are the answers the same? Should they be? Draw the Nyquist plot for the system in Fig. 6.89. Using the N yquist stability criterion, determine the range of K for which the system is stable. Consider both positive and negative values of K. (a) For a) z 0.1 to 100 rad/sec, sketch the phase of the minimum—phase system . l GCY) 2 s + s + 10 SZJ-w and the nonminimum—phase system G( ) S — l I s 2 ~ , S + 10 s=jco noting that 4(ja) — 1) decreases with a) rather than increasing. (b) Does an RHP zero affect the relationship between the —l encirclements on a polar plot and the number of unstable closed—loop roots in Eq, (6.28)? (c) Sketch the phase of the following unstable system for a) = 0.1 to lOO rad/sec: 8+1 5—10 0(5) : Szja) (d) Check the stability ofthe systems in (a) and (c) using the Nyquist criterion on KG(s). Determine the range of K for which the closed—loop system is stable, and check your results qualitatively by using a rough root—locus sketch. Nyquist plots and the classical plane curves: Determine the Nyquist plot, using MAT— LAB, for the systems given below, with K = l, and verify that the beginning point and end point for the jw > 0 portion have the correct magnitude and phase: (a) The classical curve called Cayley’s Sextic, discovered by Maclaurin in 1718: l o+nT (b) The classical curve called the Cissoid, meaning ivy»shaped: l s(s + l) (c) The classical curve called the Folium of Kepler, studied by Kepler in 1609: l (sw—l)(si—l)2' KG(S) : K KG(s) : K ...
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