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Appendix A

Complex Variable Theory

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-

A-2  Appendix A Complex-Variable Theory

as single-valued (Fig. A-2). If the mapping from the G ( s )-plane to the s -plane is also single-valued, the mapping is called one-to-one. However, there are many functions for which the mapping from the function plane to the complex-variable plane is not single- valued. For instance, given the function

(A-2)

it is apparent that for each value of s , there is only one unique corresponding value for G ( s ). However, the inverse mapping is not true; for instance, the point G ( s )   is mapped onto two points, s  0 and s  1, in the s -plane.

A-1-3 Analytic Function

A function G ( s ) of the complex variable s is called an analytic function in a region of the s-plane if the function and all its derivatives exist in the region. For instance, the func- tion given in Eq. (A-2) is analytic at every point in the s -plane except at the point s  0 and s  1. At these two points, the value of the function is infinite. As another exam- ple, the function G ( s )  s  2 is analytic at every point in the finite s -plane.

A-1-4 Singularities and Poles of a Function

The singularities of a function are the points in the s -plane at which the function or its derivatives does not exist. A pole is the most common type of singularity and plays a very important role in the studies of classical control theory. The definition of a pole can be stated as: If a function G ( s ) analytic and single-valued in the neighborhood of s (^) i , it is said to have a pole of order r at s  si if the limit

has a finite, nonzero value. In other words, the denominator of G ( s ) must include the factor ( s  s (^) i ) r , so when s  s (^) i , the function becomes infinite. If r  1, the pole at s  s (^) i is called a simple pole. As an example, the function

(A-3)

has a pole of order 2 at s  3 and simple poles at s  0 and s  1. It can also be said that the function G ( s ) is analytic in the s -plane except at these poles.

G 1 s 2 

101 s  22 s 1 s  121 s  322

lim s S s i

S 1 s  si 2 r^ G 1 s 2 T

G 1 s 2 

s 1 s  12

s 1 s

v 1

0 Re G

G ( s )-plane

G ( s 1 )

j Im G

0

s -plane

s 1 = s 1 + j v 1

j v

Figure A-2 Single- valued mapping from the s -plane to the G ( s )-plane.

A-1 Complex-Variable Concept  A-

A-1-5 Zeros of a Function

The definition of a zero of a function can be stated as: If the function G ( s ) is analytic at s  si , it is said to have a zero of order r at s  si if the limit

has a finite, nonzero value. Or, simply, G ( s ) has a zero of order r at s  si if 1  G ( s ) has an rth-order pole at s  si. For example, the function in Eq. (2-3) has a simple zero at s  2. If the function under consideration is a rational function of s , that is, a quotient of two polynomials of s , the total number of poles equals the total number of zeros, count- ing the multiple-order poles and zeros, and taking into account of the poles and zeros at infinity. The function in Eq. (A-3) has four finite poles at s  0, 1, 3, and 3; there is one finite zero at s  2, but there are three zeros at infinity, since

(A-4)

Therefore, the function has a total of four poles and four zeros in the entire s -plane, in- cluding infinity.

 REFERENCES

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed., Prentice Hall, Englewood Cliffs, NJ. 1965.

lim s Sq G 1 s 2  lim s Sq

s^3

lim s S si S 1 s  si 2  r^ G 1 s 2 T

- The total numbers of poles and zeros of a rational function is the same, counting the ones at infinity