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Unformatted text preview: ‘igure 6.100 yquist plots for roblem 6.38 Problems 401 6.38 The Nyquist diagrams for two stable, open—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 (b) Damping ratio (c) Range of gain for stability (if any) ((1) System type (0, 1, or 2) EA (a) (b) 6.39 The steering dynamics of a ship are represented by the transfer function we G( ) _ K[—(s/O.l42) + 1] are) “ s — S(S/0.325 + 1)(s/0.0362+ 1)’ where V is the ship’s lateral velocity in meters per second, and (Sr is the rudder angle in radians. (a) 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? 6.40 For the open-loop system K (s + 1) s2(s + 10)2’ determine the value for K at the stability boundary and the values of K at the points where PM = 300. KG(S) = Problems for Section 6.5: Bude’s Gain—Phase Relationship 6.41 The frequency response of a plant in.a unity feedback configuration is sketched in Fig. 6.101. 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 a) = 100? 402 Chapter 6 The Frequency-Response Design Method 6.42 For the system i 3 it Figure 6.101 ‘ i \ Magnitude frequency 100 “l— - 40 r 1‘; response for ' e W l Problem 6.41 We“ 6 4 10 20 1‘ t - in. r |G| 1 > 0 db § 0.1 20 }, ~ we ~—- Jr—w ‘ 0'01 10 20 100 \ J.1300 40 i I - w (rad/sec) r l . 1: i (c) Approximately what is the system error in tracking (following) a sinusoidal input ‘ ofcu = 3 rad/sec? V i ' (d) What is the PM of the system as drawn? (Estimate to within ::10°.) i r 100(s/a + l) s(s + l)(s/b +1)“ where b = 10a, find the approximate value of a that will yield the best PM by sketching only candidate values of the frequency-response magnitude. G(s) : Problem for Section 6. 6: Closed—Loop Frequency Response 6.43 For the open-loop system K (s + 1) s2(s + 10)2’ determine the value for K that will yield PM 3 30° and the maximum possible closed loop bandwidth. Use MATLAB to find the bandwidth. KG(S) = Problems for Section 6.7: Compensation Design 6.44 For the lead compensator ' D“) = aTs + 1 whereot < l, (a) Show that the phase of the lead compensator is given by r) = tan—10w) — tan—1(aTw). (b) Show that the frequency where the phase is maximum is given by 1 wmax — m Problems 403 and that the maximum phase corresponds to l — a l + or i (c) Rewrite your expression for wmax to show that the maximum-phase frequency occurs at the geometric mean of the two corner frequencies on a logarithmic scale: Sin ¢max = l l " 1 legwmaX = 5 (log ? + log W). (d) To derive the same results in terms of the pole—zero locations, rewrite D(s) as D(s)=s+z, s+p and then show that the phase is given by ¢> =.tan_1 ~— tan_1 (3), IZI lPl such that wmax = lzl Hence the frequency at which the phase is maximum is the square root of the product of the pole and zero locations, 6.45 For the third—order servo system 50,000 a s(s +10)(s + 50)’ use Bode plot sketches to design a lead compensator so that PM 2 50° and wBW 2 20 rad/sec. Then verify and refine your design by using MATLAB. G(s) = 6.46 For the system shown in Fig. 6.102, suppose that ‘ _ 5 i _ s<s+1>(s/5 +1)’ ‘ Use Bode plot sketches to design a lead compensation D(s) with unity DC gain so that PM 2 40°. Then verify and refine your design by using MATLAB. What is the , ‘> approximate bandwidth of the system? ’ ' G(s) Figure 6.102 Control system for Problem 6.46 6.47 Derive the transfer function from Td to 6 for the system in Fig. 6.70. Then apply the Final Value Theorem (assuming Td = constant) to determine whether 6(00) is nonzero for the following two cases: (a) When D(s) has no integral term: lims_,0 D(s) : constant; (b) When D(s) has an integral term: D/ D(s) = (S) . S In this case, lims_>0 D/ (s) = constant. 404 Chapter 6 The Frequency—Response Design Method 6.48 The inverted pendulum has a transfer function given by Eq. (2.31), which is similar to l s2—1‘ G(s) : (a) Use Bode plot sketches to design a lead compensator to achieve a PM of 30°. Then verify and refine your design by using MATLAB. (b) Sketch a root locus and correlate it with the Bode plot of the system. (c) Could you obtain the frequency response of this system experimentally? 6.49 The open—loop transfer function of a unity feedback system is K G“) : 5(5/5 +1)(s/50 + 1)‘ (a) Use Bode plot sketches to design a lag compensator for G(s) so that the closed-100p system satisfies the following specifications: (i) The steady-state error to a unit—ramp reference input is less than 0.01. (ii) PM 2 40°. (b) Verify and refine your design by using MATLAB. 6.50 The open—loop transfer function of a unity-feedback system is K s(s/5 +1)(s/200 + l)‘ C(S) = (3) Use Bode plot sketches to design a lead compensator for G(S) so that the closed—100p system satisfies the following specifications: (i) The steady—state error to a unit-ramp reference input is less than 0.01. (ii) For the dominant closed—loop poles, the damping ratio 4“ 2 0.4. (b) Verify and refine your design using MATLAB, including a direct computation of the damping of the dominant closed—loop poles. 6.51 A DC motor with negligible armature inductance is to be used in a position control system. Its open—loop transfer function is given by 50 s(s/5 +1)‘ G(s) = (3) Use Bode plot sketches to design a compensator for the motor so that the closed-loop. system satisfies the following specifications: (i) The steady—state error to a unit-ramp input is less than 1/200. (ii) The unit—step response has an overshoot of less than 20%. (iii) The bandwidth of the compensated system is no less than that of the uncompensated system. (b) Verify and/or refine your design using MATLAB, including a direct computation of the step—response overshoot. 6.52 The open-loop transfer function of a unity-feedback system is _._,..£____ s(l + s/5)(l + 3/20)~ (3) Sketch the system block diagram, including input reference commands and sensor noise. 6(3) : ...
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