Control system appendix by Benjamin kuo, Essays (university) of Electrical Engineering

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Answers to Selected Problems

JOHN WILEY & SONS, INC.

TO ACCOMPANY

AUTOMATIC CONTROL SYSTEMS

EIGHTH EDITION

BY

BENJAMIN C. KUO

FARID GOLNARAGHI

Copyright © 2003 John Wiley & Sons, Inc.

All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508)750-8400, fax (508)750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM. To order books or for customer service please call 1-800-CALL WILEY (225-5945).

ISBN 0-471-13476-

Ans-2  Answers to Selected Problems

4-16. (c) Forward-path transfer function: 4-18. (c) Characteristic equation: 4-21. (b) State equations: ia(t) as input,

4-26. (b) Forward-path transfer function:

CHAPTER 5 State Variable Analysis 5-3. (a) Eigenvalues: (c) State transition matrix: (g) Characteristic equation: 5-6. (a) Eigenvalues of A : 2.325, (b) (3) Output transfer function:

5-8. (c)

(e)

5-11. (c) S is singular. 5-13. (a)

5-16. (a) Forward-path transfer function:

Closed-loop transfer function:

5-19. (b) State transition matrix:

5-28. (b) State equations:

5-32. (b) Characteristic equation: 5-42. The system is controllable. 5-43. (a) Transfer function:

5-45. (b) For controllability, for observability,

CHAPTER 6 Stability of Linear Control Systems 6-1. (b) Poles are at Two poles on imaginary axis. Marginally stable. (d) Poles are at All poles in the left-half s-plane. Stable. 6-2. (b) No roots in RHP. (f) Two roots in RHP. 6-3. (e) Conditions for stability: and 6-5. (b) Condition for stability: 6-9. Stability requirement: 6-12. (a) There is one root in the region to the right of s  1 in the s-plane.

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Answers to Selected Problems  Ans-

CHAPTER 7 Time-Domain Analysis of Control Systems 7-3. (c) 7-5. (a) Unit-step input: unit-ramp input: ess   (d) Unit-step input: ess 0; unit-ramp input: ess  0.05; unit-parabolic input: ess   7-6. (c) Unit-step input: ess  2.4; unit-ramp and unit-parabolic inputs: ess   7-10. (a) Stability requirement: and 7-13. Rise time tr  0.2 sec;  n  10.82 rad/sec; K  4.68; Kt  0. 7-15. System transfer function: ymax  1.2 (20% overshoot) 7-24. (b) (c) 7-27. (a) Stability requirement: (c) Stability requirement: 7-28. (f)

7-32. (a) (e)

7-35. (a)

CHAPTER 8 Root-Locus Technique 8-2. (a) K 0:  1  135 K 0:  1   45 (e)  1  108. 8-4. (a) Breakaway-point equation: Breakaway points: 0.7272, 2. 8-5. (h) Intersect of asymptotes: breakaway points: 0, 4,  8 (l) Breakaway points: 2.07, 2.07, j1.47, j1. 8-7. (d)   0.707, K  8. 8-16. (a) breakaway points: ( ) 0, 3. 8-21. (a) Breakaway-point equation: For no breakaway point other than at s  0,

CHAPTER 9 Frequency-Domain Analysis 9-1. (c) For K  100,  n  10 rad/sec,   0.327, Mr  1.618,  r  9.45 rad/sec 9-2. (b)  r  4 rad/sec, BW  6.223 rad/sec (e)  r  0.82 rad/sec, BW  1.12 rad/sec 9-5. Maximum minimum BW  1.4106 rad/sec 9-9. (a) The Nyquist plot encloses the 1 point. The closed-loop system is unstable. The characteristic equation has two roots in the right-half s-plane. (b) Nyquist plot intersects the negative real axis at 0.8333. Thus the closed-loop system is stable. 9-10. (a) The system is stable for 9-11. (a) The system is stable for 9-12. The system is stable for except at K  1. 9-14. (a) The system is stable for 9-15. (a) For stability, (N has to be an integer). 9-18. (a) For stability, 9-21. (b) Maximum D  15.7 in. 9-25. (e) GM  6.82 dB, PM  50. (h) GM  infinite, PM  13. 9-26. (c) K can be increased by 28.71 dB. 9-27. (d) K must be decreased by 2.92 dB. 9-33. (b) PM  2.65 , GM  10.51 dB, Mr  17.72,  r  5.75 rad/sec, BW  9.53 rad/sec 9-35. (a) The gain-crossover frequency is 10 rad/sec. (d) (g) BW  30 rad/sec 9-36. (b) Td  0.1244 second 9-40. (b) GM  30.72 dB, PM  infinite

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