ch09 - N I N E Design Using the Graphical Tool SOLUTIONS TO...

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Unformatted text preview: N I N E Design Using the Graphical Tool SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: Lag-Lead Compensation a. Uncompensated: From the Chapter 8 Case Study Challenge, 76.39 7194.23 150 1.32 150 1.32 K G s s s s s s s with the dominant poles at 0.5 6.9 j . Hence, 1 6.9 cos tan 0.0723 0.5 , or %OS 79.63% and 4 4 8 0.5 s n T seconds. Also, 7194.23 36.33 150 1.32 v K . b. Lead-Compensated: Reducing the percent overshoot by a factor of 4 yields, 79.63 %OS 19.91% 4 , or 0.457 . Reducing the settling time by a factor of 2 yields, 8 4 2 s T . Improving K v by 2 yields 72.66 v K . Using 4 4 s n T , 1 n , from which 2.188 n rad/s. Thus, the design point equals 2 1 1 1.946 n n j j . Using the systems original poles and assuming a lead compensator zero at 1.5 , the summation of the systems poles and the lead compensator zero to the design point is 123.017 . Thus, the compensator pole must contribute 123.017 180 56.98 . Using the geometry below, 1.946 tan56.98 1 c p , or 2.26 c p . 9-2 Chapter 9: Design Using the Graphical Tool Adding this pole to the system poles and the compensator zero yields 76.39 741.88 K at 1 1.946 j . Hence the lead-compensated open-loop transfer function is Lead-comp 741.88 1.5 150 1.32 2.26 s G s s s s s . Searching the real axis segments of the root locus yields higher-order poles at greater than 150 and at 1.55 . The response should be simulated since there may not be pole/zero cancellation. The lead-compensated step response is shown below. Since the settling time and percent overshoot meet the transient requirements, proceed with the lag compensator. The lead-compensated system has 741.88 1.5 2.487 150 1.32 2.26 v K . Since we want 72.66 v K , an improvement of 72.66 29.22 2.487 is required. Select 0.002922 0.0001 Lag s G s s to improve the steady-state error by 29.22. A simulation of the lag-lead compensated system, Solutions to Problems 9-3 Lag-Lead-comp 741.88 1.5 0.002922 150 1.32 2.26 0.0001 s s G s s s s s s is shown below. UFSS Vehicle: Lead and Feedback Compensation Minor loop: Open-loop transfer function 2 0.25 0.437 2 1.29 0.193 K s G s H s s s s ; Closed-loop transfer function: 2 3 0.25 0.437 ( ) ......
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ch09 - N I N E Design Using the Graphical Tool SOLUTIONS TO...

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