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ps5fpe - 394 Chapter 6 The Frequency—Response Design...

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Unformatted text preview: 394 Chapter 6 The Frequency—Response Design Method Problems for Section 6.2: Neutral Stability 6.16 Determine the range of K for which the closed-loop systems (see Fig. 6. 18) are stable for each of the cases below by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify your answers by using a very rough sketch of a root-locus plot. (a) KG(s) = 53%? b KG = K ( ) (S) (s+1om (C) KG(s) = W (s +100)(s + 5)3 6.17 Determine the range of K for which each of the listed systems is stable by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify your answers by using a very rough sketch of a root—locus plot. (a) KG(s) = %S+t51_)) (b) KG(s) = 5%(5J-rt—110L) (e) KG(s) = m (d) KG(s) = fig—:3?) Problemsfor Section 6.3: The Nyquist Stability Criteriorz 6.18 (a) Sketch the Nyquist plot for an open-100p system with transfer function 1/52; that is, sketch 1 ' S2 S=C1 , where C1 is a contour enclosing the entire RHP, as shown in Fig. 6.17. (Hint: Assume C1 takes a small detour around the poles at s = 0, as shown in Fig. 6.27.) (b) Repeat part (a) for an open-loop system whose transfer function is G(s) = 1/(s2 + mg). 6.19 Sketch the Nyquist plot based on the Bode plots for each of the following systems, and then compare your result with that obtained by using the MATLAB command nyq uist: (a) KG(s) = 5%? . b G = __K.__ ()K (S) (s+10)(s+2)2 (c) Kan) = W11 (5 + lOO)(s + 2)3 ((1) Using your plots7 estimate the range of K for which each system is stable, and qualitatively verify your result by using a rough sketch of a root—locus plot. 6.20 Draw a Nyquist plot for ' ’ K (s + l) ———-—, (6.77) s(s + 3) KG(S) = Figure 6.89 Control syste Problem 6.2'. :Fig. 6.18) are stable for magining the magnitude nswers by using a very stable by making a Bode )r down until instability :oot-locus plot. ! 1sfer function 1 /s2; that wn in Fig. 6.17. (Him: as shown in Fig. 6.27.) er function is G(s) = : following systems, and AAB command nyquist: :h system is stable, and a root-locus plot. (6.77) Figure 6.89 Control system for Problem 6.21 Problems 3 choosing the contour to be to [the right of the singularity on the jw-axis. Next, using Nyquist criterion, determine the range of K for which the system is stable. Then the N yquist plot, this time choosing the contour to be to the left of the singularity 01 imaginary axis. Again, using the Nyquist criterion, check the range of K for whicl system is stable. Are the answers the same? Should they be? 6.21 ‘Draw the N yquist plot for the system in Fig. 6.89. Using the Nyquist stability criter determine the range of K for which the system is stable. Consider both positive negative values of K. and the nonminimum-phase system 571 G =— (S) s+1O 7 s=jw noting that 4(ja) — 1) decreases with w rather than increasing. (b) Does an RHP zero affect the relationship between the 71 encirclements on a p( 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 100 rad/sec: s+1 5—10 s=jw G(s) : (d) Check the stability of the systems in (a) and (c) using the Nyquist criterion on KGt Determine the range of K for which the closed—loop system is stable, and check yr results qualitatively by using a rough root-locus sketch. 6.23 Nyquist plots and the classical plane curves: Determine the Nyquist plot, using M2 LAB, for the systems given below, with K = 1, and verify that the beginning point 2 end point for the ja) > 0 portion have the correct magnitude and phase: (a) The classical curve called Cayley’s Sextic, discovered by Maclaurin in 1718: KG(s) = K (Si—D3. (b) The classical curve called the Cissoid, meaning ivy—shaped: KG(s) = K s(s + 1) ' s (c) The classical curve called the Folium of Kepler, studied by Kepler in 1609: 1 W = Km 396 Chapter 6 The Frequency—Response Design Method Figure 6.90 Nyquist plot for Problem 6.24 (d) The classical curve called the Folium (not Kepler’s): l W = Km (e) The classical curve called the Nefihroid, meaning kidney-shaped: 2(s +1)(s2 — 45 +1) KG(s) = K (s _1)3 (f) The classical curve called Nephroid of Freeth, named after the English mathemati- cian 'l‘. J. Freeth: 2 l KG(s) = Kw 4(s , 1)3 (g) A shifted Nephroid of Freeth: (52 +1) KG(S) = K(5 _1)3. Problems for Section 6.4: Stability Margins 6.24 The Nyquist plot for some actual control systems resembles the one shown in Fig. 6.90. What are the gain and phase margin(s) for the system of Fig. 6.90, given that or = 0.4, ,8 = 1.3, and <l> = 40°. Describe what happens to the stability of the system as the gain goes from zero to a very large value. Sketch what the corresponding root locus must look like for such a system. Also, sketch what the corresponding Bode plots would look like for the system. ‘ Im[G(s)] 6.25 The Bode plot for 100[(.v/10)+ 1] G“) : sr<s/1>—1][<s/100)+ 1] is shown in Fig. 6.91. Figure 6.91 Bode plot for Problem 6.25 Figure 6.92 Control systr Problem 6.2 Problems 401 6.38 The Nyquist diagrams for two stablejopen—loop systems are sketched in Fig. 6.100. The proposed operating gain is indicated» as K0, and arrows indicate increasing frequency. In each case give a rough estimate of the following quantities for the closed—loop (unity feedback) system: (a) Phase margin (1)) Damping ratio (c) Range of gain for stability (if any) (d) System type (0, l, or 2) Figure 6.100 Nyquist plots for .38 A Re[G(S)l ‘ K for which the s sketch. )ply the Nyquist 'hich the system (a) (b) Y for WhiCh the 6.39 The steering dynamics of a ship are represented by the transfer function sketch. V(s) _ G“) _ K[—(s/0.142) +1] 8r(s) _ _ S(s/O.325 + l)(s/0.0362 + 1)’ where V is the ship’s lateral velocity in meters per second, and 8, is the rudder angle in radians. (3) Use the MATLAB command bode to plot the log magnitude and phase of G(ja)) for K : 0.2. (b) On your plot, indicate the crossover frequency, PM, and GM. (c) Is the ship steering system stable with K = 0.2? (d) What value of K would yield a PM of 30°, and what would the crossover frequency be? ly the Nyquist ich the s t ys em 6.40 For the open—loop system K (s + 1) s2 (S + 10)? determine the value for K at the stability boundary and the values of K at the points where PM = 30°. for which the KGCY) = .ketch. Problems for Section 6.5: Bode ’S Gain—Phase Relationship 6.41 The frequency response of a plant in a unity feedback configuration is sketched in Fig. 6.l01. Assume that the plant is open—loop stable and minimum-phase. (a) What is the velocity constant K, for the system as drawn? (b) What is the damping ratio of the complex poles at w = 100? ...
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