Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems

Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M.Thoma 12 lan Postlethwaite Alistair...

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Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M.Thoma

12 lan Postlethwaite Alistair G. J. MacFarlane

A Complex Variable Approach to the Analysis of Linear

Multivariable Feedback Systems

Springer-Verlag Berlin Heidelberg New York 1979

Series Editors A.V. Balakrishnan • M. Thoma

Advisory Board A. G. J. MacFarlane • H. Kwakernaak • Ya. Z. Tsypkin

Authors Dr. I. Postlethwaite, Research Fellow, Trinity Hall, Cambridge, and SRC Postdoctoral Research Fellow, Engineering Department, University of Cambridge. Professor A. G..I. MacFarlane, Engineering Department, University of Cambridge, Control and Management Systems Division, Mill Laqe, Cambridge C B 2 1RX.

ISBN 3-540-09340-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09340-0 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-VerlagBerlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2061/3020-543210

Contents

CHAPTER

1

Introducti0n References

CHAPTER

2 2.1 2.2 2.3 2.3-1 2.3-2 2.4

Preliminaries System description Feedback configuration Stability Free systems Forced systems Relationship between open- and closed-loop characteristic polynomials for the general feedback configuration References

CHAPTER

3 3.1 3.2 3.3 3.3-1 3.3-2 3.3-3

3.3-4 3.3-5 3.3-6 3.4 3.4-1 3.4-2

CHAPTER

4 4.1 4.2 4.3

CHAPTER

5 5.1 5.2

1 6 8 8 ii 13 13 16 17 20

Characteristic gain functions and characteristic f r e q u e n c y 'functions 22 Duality between open-loop gain and closed-loop frequency 22 Algebraic functions: characteristic gain functions and characteristic frequency functions 25 Characteristic gain functions 27 Poles and zeros of a characteristic gain function 29 Algebraic definition of poles and zeros for a transfer function matrix 32 Relationship between algebraically defined poles/zeros of the open-loop gain matrix G(s) and the poles/zeros of the corresponding set of characteristic gain functions 36 Riemann surface of a characteristic gain function 39 Generalized root locus diagrams 46 Example of frequency surface and characteristic frequency loci 47 Characteristic frequency functions 49 Generalized Nyquist diagram 50 Example of gain surface and characteristic gain loci 51 References 57 A ~eneralized Nyquist stability criterion Generai izea Nyquist stability crfterion Proof of the generalized Nyquist stability criterion Example References A generalized inverse Nyquist stability criterion Inverse characteristic gain functions Pole-zero relationships

58 58 60 72 75 77 77 78

N 5.3 5.4 5.5 5.6 CHAPTER

6 6.1 6.2 6.2-1 6.3 6.4 6.5 6.6

CHAPTER

7 7.1 7.2 7.3 7.4

Inverse characteristic gain loci generalized inverse Nyquist diagrams Generalized inverse Nyquist criterion Proof of generalized inverse Nyquist stability criterion Example References Multivariable root loci Theoretical background Asymptotic behaviour Butterworth patterns Angles of departure and approach Example 1 Asymptotic behaviour of optimal closedloop poles Example 2 References On parametricstability and future research Characteristic frequency and characteristic parameter functions Gain and phase margins Example Future research References

81 81 86 97 99 iOO IOO 104 109 113 115 123 126 130 132 132 135 135 138 140

Appendix

1

Definition

Appendix

2

A reduction to the irreducible rational canonical form ..........................

143

of an algebraic

functipn

143

Appendix

3

The discriminant

147

Appendix

4

.method for constructing the Riemann surface domains of the algebraic functions correspondins~to an open-lo.op gaip matrix G(s)

150 157

Appendix

5

Extended

Appendix

6

Multivariable pivots from the ch'arac£eristic equation A'('g,s).=O

166

Association between branch, point.s and stat.ionary points on the gain and frequency surf.aces

168

References

169

Bibliography

170

Index

174

Appendix

7

Principle

of the. Argume.nt

1.

Introduction

The great success of the optimal filtering techniques

developed

for aerospace work during the

late 1950's and early 1960's n a t u r a l l y apply these techniques variable

led to attempts

to

to a wide range of e a r t h - b o u n d multi-

industrial processes.

less than immediately

control and optimal

In many situations

successful,

particularly

this was

in cases where

the available plant models were not s u f f i c i e n t l y accurate or where the p e r f o r m a n c e plant

indices

required to stipulate

b e h a v i o u r were m u c h less obvious

space context.

Moreover,

in form than in the aero-

the controller w h i c h results

direct a p p l i c a t i o n of optimal control and optimal synthesis

techniques

if it incorporates dynamical

in fact,

filter it has a

since the filter e s s e n t i a l l y

model w i t h feedback around it. for many m u l t i v a r i a b l e

process

consists of a plant

In contrast,

what was needed

control problems was a relatively

simple controller w h i c h w o u l d both stabilize,

about an operating

a plant for w h i c h only a very a p p r o x i m a t e model might be

available,

and also mitigate

disturbances

by i n c o r p o r a t i n g

the effect of low-frequency integral

action.

engineers brought up on f r e q u e n c y - r e s p o n s e optimal control methods

seemed difficult

forward techniques,

To industrial

ideas the s o p h i s t i c a t e d

to use;

e s s e n t i a l l y relied on a m i x t u r e of physical

action,

filtering

is in general a c o m p l i c a t e d one;

a full K a l m a n - B u c y

from a

complexity equal to that of the plant which it is

controlling,

point,

the controlled

these engineers

insight and straight-

such as the use of derivative and integral

to solve their problems.

huge gap in techniques loop f r e q u e n c y - r e s p o n s e

It became obvious

existed between the classical methods,

that a single-

based on the work of Nyquist

[i],

2

Bode

[2] and Evans

industrial variable

[3],which

applications,

time-response

were still in use for many

and the elegant

methods

and powerful

developed

multi-

for aerospace

applications. For these reasons methods

first step towards

an optimal approach

control

approach

closing the yawning and the classical

was taken by Kalman

characterization

[4]

,

and design

theory

pioneering

paper by Rosenbrock

of increasing approach.

control

A systematic

attacks

problem.

transfer

function

diagonal

form.

controller

[5] which ushered

controller.

a cascaded

frequency-response

compensator

If such a compensator

such that the overall system had a

off using standard compensating

from such a procedure objection

to this approach

to go to such drastic

to reduce

A natural

approach

to multivariable

achieved

by way of standard matrix papers

studying

further

single-

matrix which

is necessarily

that it is not essential interaction.

consisted

could be found then the

The required

and the most succinct

matrices;

some fairly

Their procedure

matrix of the compensated

techniques.

results

in a decade

and Hood [6] put forward the

design could be finished

loop design

analysis

had been made on the multivariable

Boksenbom

simply of choosing

attack

systems was begun in a

in a rejuvenated

idea of a non-interacting

usually

frequency-response

Prior to this new point-of-departure

straightforward

gap between

a frequency-response

for multivariable

interest

An

who studied the frequency-

of optimality.

on the whole problem of developing

one,

in frequency-response

slowly began to revive during the mid-1960's.

important

domain

an interest

a complicated is simply

lengths merely

step in this initial

control was to see what could be calculations

the problem

using rational

in this way were produced

3

by Golomb and Usdin and Freeman

[12]

a completely

[7]

,

Raymond

, [13]

Rosenbrock

sophisticated way.

[14],[15],

to classical

[9] , [iO],[ii] however,

enable single-loop

techniques

completely.

investigators

to be employed,

interaction

Return Difference

approach of Mayne

In the non-interacting, approach to m u l t i v a r i a b l e deployment of classical

control the m o t i v a t i o n was the eventual

single-loop

is to investigate

frequency-response

the t r a n s f e r - f u n c t i o n

be suitably extended?

single-loop

diagram and root locus diagram?

:

how can the key basic

concepts of pole, It is to questions

complex-variable

ideas have an important

of m u l t i v a r i a b l e

feedback

approach

systems.

A generalization

was put forward by MacFarlane complex-variable

to the

zero, Nyquist of this sort

and it is shown that

role to play in the study

An early attempt to extend

Nyquist diagram ideas to the m u l t i v a r i a b l e

[20]

matrix representation

frequency-response

that the work p r e s e n t e d here is addressed,

and K a t z e n e l s o n

approach,

What are the relevant g e n e r a l i z a t i o n s

case of the specific

, [18]

techniques

An alternative

as a single object in its own right and to ask concepts of the classical

as in the

or partially non-interacting,

during the final stages of a design study.

generalization

the

to develop ways of seeking to reduce a m u l t i v a r i a b l e

[163.

treatment,

-

The success of this m e t h o d led other

Sequential

multivariable

, [15]

approach was based upon a

criterion of partial

concept.

in a more

rather than to eliminate

problems,

[17]

opened up

[14]

control p r o b l e m to a succession of single-loop

however,

,

to an amount which would then

The Rosenbrock

careful use of a specific dominance

techniques

In his Inverse Nyquist Array Method

the aim was to reduce interaction

diagonal

Kavanagh

new line of d e v e l o p m e n t by seeking to reduce a multi-

variable p r o b l e m to one amenable

interaction

[8] ;

p r o b l e m was made by Bohn

of the Nyquist

[19] and,

stability

following

criterion

that heuristic

based proofs were supplied by Barman

and MacFarlane

of the Nyquist

situation was soon followed by

and Postlethwaite

stability

[21]

.

This

criterion to the m u l t i v a r i a b l e

complementary

generalizations

of the

root locus technique

E21],

E22],

~23],

The aim of the work p r e s e n t e d the concepts

underlying

to m u l t i v a r i a b l e linear feedback

in this text is to extend

the techniques

systems.

of Nyquist,

In the two classical

system design the N y q u i s t - B o d e

gain as a function of frequency frequency

E24].

and the Evans'

as a function of gain.

In Chapter

Bode and Evans

approaches

to

approach studies approach studies

3 it is shown how

the ideas of studying complex gain as a function of complex frequency

and complex frequency

as a function of complex gain can

be extended to the m u l t i v a r i a b l e function matrices pair of analytic characteristic

case by a s s o c i a t i n g with transfer

(having the same number of rows and columns) functions

frequency

:

a characteristic

function.

These are algebraic

[253 and each is defined on an a p p r o p r i a t e Chapter

considered; theorems;

w i t h basic definitions and w i t h a fundamental

c l o s e d - l o o p behaviour

to an algebraic In Chapter

of stability and related

a comprehensive

systems w h i c h is p r e s e n t e d

this criterion

such as

system being

based on the r e t u r n - d i f f e r e n c e

3 also contains

[26].

r e l a t i o n s h i p between open- and

to the g e n e r a l i z e d Nyquist stability feedback

feedback

and a

functions

Riemann surface

2 deals With a number of essential p r e l i m i n a r i e s

a d e s c r i p t i o n of the type of m u l t i v a r i a b l e

Chapter

gain function

a

operator.

discussion of the b a c k g r o u n d

criterion

for m u l t i v a r i a b l e

in Chapter 4.

The proof of

is based on the Principle of the Argument applied function defined on an appropriate

5 a generalization

Riemann surface.

of the inverse Nyquist

stability

criterion to the multivariable case is d e v e l o p e d which is complementary to the exposition of the g e n e r a l i z e d Nyquist criterion the previous the Evans'

chapter.

Using the material

root locus approach

developed

given in

in Chapter

is extended to m u l t i v a r i a b l e

3,

systems

in Chapter function

6;

this uses well established

theory.

based approach

It is also shown how an algebraic-function

poles of a multivariable

linear regulator

index approaches

In Chapter

optimal

of the concepts

Nyquist

loci.

of

developed

and suggestions

that

of the are not

but to any parameter

system is considered 'parametric'

This chapter

evident

as a parameter

7 the effect of parameter

feedback

duction

Information

it becomes

that the techniques

to gain and frequency

on a multivariable

proposals

progresses

used can be considered

and consequently

frequency.

time-invariant

of

zero.

As the work presented

only applicable

behaviour

as the weight on the input terms of a quadratic

the gain variable system,

in algebraic

can be used to find the asymptotic

the closed-loop

performance

results

concludes

and

variations by the intro-

root loci and

'parametric'

with a few tentative

for future research.

of secondary

importance

which would unnecessar-

ily break the flow of the text has been placed

in appendices.

References

are listed at the end of each chapter

in which they

are cited,

and also at the end of the text where a bibliography

is provided. References [i]

H.Nyquist, "Regeneration ii, 126-147, 1932.

theory",

Bell Syst.

[23

H.W.Bode, "Network analysis and feedback Van Nostrand, Princeton, N.J., 1945.

amplifier

design",

[3]

W.R.Evans, "Graphical analysis AIEE, 67, 547-551, 1948.

systems",

Trans.

[4]

R.E.Kalman, "When is a linear control system optimal?", Trans. ASME J.Basic Eng., Series D., 86, 51-60, 1964.

[5]

H.H.Rosenbrock, "On the design of linear multivariable control systems", Proc. Third IFAC Congress London, I, 1-16, 1966.

of control

Tech. J.,

[6]

A.S.Boksenbom and R.Hood, "General algebraic method applied to control analysis of complex engine types", National Advisory Committee for Aeronautics, Report NCA-TR-980, Washington D.C., 1949.

[7]

M.Golomb and E.Usdin, "A theory of multidimensional servo systems", J.Franklin Inst., 253(1), 28-57, 1952.

[8]

F.H.Raymond, "Introduction a l'~tude des asservissements multiples simultanes", Bull. Soc. Fran. des Mecaniciens, 7, 18-25, 1953.

[9]

R.J.Kavanagh, "Noninteraction in linear multivariable systems", Trans. AIEE, 76, 95-100, 1957.

[i0 ]

R.J.Kavanagh, "The application of matrix methods to multivariable control systems", J.Franklin Inst., 262, 349-367, 1957.

[li

R.J.Kavanagh, "Multivariable control system synthesis", Trans. AIEE, Part 2, 77, 425-429, 1958.

]

[12 ]

H.Freeman, "A synthesis method for multipole control systems", Trans. AIEE, 76, 28-31, 1957.

[13 ]

H.Freeman, "Stability and physical realizability considerations in the synthesis of multipole control systems", Trans.AIEE, Part 2, 77, 1-15, 1958.

[14 ]

H.H.Rosenbrock, "Design of multivariable control systems using the inverse Nyquist array, Proc. IEE, 116, 1929-1936, 1969.

[15

]

H.H.Rosenbrock, "Computer-aided control system design", Academic Press, London, 1974.

[16

]

D.Q.Mayne, "The design of linear multivariable systems", Automatica, 9, 201-207, 1973.

[17

]

E.V.Bohn, "Design and synthesis methods for a class of multivariable feedback control systems based on single variable methods", Trans.AIEE, 81, Part 2, 109--115, 1962.

[18 ]

E.V.Bohn and T.Kasvand, "Use of matrix transformations and system eigenvalues in the design of linear multivariable control systems", Proc. IEE, iiO, 989-997, 1963.

[19 ]

A.G.J.MacFarlane, "Return-difference and return-ratio matrices and their use in the analysis and design of multivariable feedback control systems", Proc. IEE, 117, 2037-2049, 1970.

[20 ]

J.F.Barman and J.Katzenelson, "A generalized Nyquisttype stability criterion for multivariable feedback systems", Int.J.Control, 20, 593-622, 1974.

[21]

A.G.J.MacFarlane and I. Postlethwaite,° "The generalized Nyquist stability criterion and multivariable root loci", Int. J. Control, 25, 81-127, 1977.

[22]

B.Kouvaritakis and U.Shaked, "Asymptotic behaviour of root loci of linear multivariable systems", Int. J. Control, 23, 297-340, 1976.

[23]

I. Postlethwaite, " The asymptotic behaviour, the angles of departure, and the angles of approach of the characteristic frequency loci", Int. J. Control, 25, 677-695, 1977.

[24]

A.G.J.MacFarlane, B.Kouvaritakis and ~ E d m u n d s , "Complex variable methods for multivariable feedback systems analysis and design", Alternatives for Linear Multivariable Control, National Engineering Consortium, Chicago, 189-228, 1977.

[25]

G.A.Bliss, "Algebraic functions", 1966 (Reprint of 1933 original).

[26]

G. Springer, "Introduction to Riemann surfaces", Addison-Wesley, Reading, Mass., 1957.

Dover, New York,

2.

Preliminaries

This text considers techniques dynamical

the g e n e r a l i z a t i o n

of the classical

of Nyquist and Evans to a linear time-invariant feedback system which consists of several multi-

input, m u l t i - o u t p u t

subsystems

connected

in series.

chapter a d e s c r i p t i o n of the m u l t i v a r i a b l e under c o n s i d e r a t i o n

feedback

In this system

is given.

The chapter also includes

basic definitions

of stability,

some associated theorems,

and a fundamental

r e l a t i o n s h i p between open- and closed-loop

b e h a v i o u r based on the return-difference 2.1

operator.

System description

The basic description

of a linear t i m e - i n v a r i a n t

dynamical

system is taken to be the state-space model x(t)

=

Ax(t)

+ Bu(t)

y(t)

=

Cx(t)

+ Du(t)

(2.1.1) where x(t)

is the state vector,

y(t)

the output vector,

i

u(t)

the input vector;

x(t)

x(t) w i t h respect to time; matrices.

denotes the derivative of A,B,C,

and D are constant real

For c o n v e n i e n c e the model will be d e n o t e d by

S(A,B,C,D)

or S when the m e a n i n g

diagramatically In general state-space

is obvious,

and r e p r e s e n t e d

as shown in figure i. S(A,B,C,D)

representation

will be considered of several

subsystems

Si (Ai,Bi,Ci,Di) :

xi(t)

= A . x . (t) + B.u. (t) 1 i i i

i=1,2, ...... h

Yi(t)

= C.x.l l(t)

+ Diui(t)

as i l l u s t r a t e d

in figure

connected

in series,

example,S

consists of two subsystems

as being the

(2.1.2)

2.

If, for

S 1 and S 2 then the

"I D ,I +1

t)

u(t '

g''~~

I 1

J I

fA

'c

I

I I

S J

Figure 1. Stote-spctce model

r"

u{t}=u~(t}

"~

I

1

llYh(t)= y(t) I

' I

I I

S

J

Figure 2. Series connection of subsystems

.~

10 state-space x(t)

description

= Ix l(t)] !

[x2(t)

of S is g i v e n by e q u a t i o n s

,

the c o m b i n e d

]

=

ul(t)

,

the input to S 1 ,

y(t)

=

Y2(t)

,

the o u t p u t

B2c I =

subsystems

A2

[ D2C 1

The s t a t e - s p a c e

with

states of both s u b s y s t e m s ,

u(t)

C

(2.1.1)

(2.1.3)

tB2DlJ

C2 ] ,

model

of S 2,

and D

=

D2D 1.

for an i n t e r c o n n e c t i o n

can be d e r i v e d

from the above

of s e v e r a l

formula

by s u c c e s s i v e

application. The s t a t e - s p a c e to as an i n t e r n a l

model

of a s y s t e m is o f t e n

description

of the s y s t e m ' s

internal

or i n p u t - o u t p u t

description

(2.1.1)

single-sided

s~(s)

- x(o)

since it r e t a i n s

dynamical

Laplace

structure.

is o b t a i n e d

+ B~(s)

~(s) = c~(s)

+ D~(s)

a knowledge An e x t e r n a l

if in e q u a t i o n s

transforms

= A~(s)

referred

[i] are taken,

to give

(2.1.4)

where

~(s)

initial

denotes

conditions

then the input

the L a p l a c e

transform

of x(t).

at time t=O are all zero

and o u t p u t

transform

vectors

If the

so that x(o)=O, are r e l a t e d

~(s) = G(s) ~(s)

by

(2.1.5)

where G(s) I

n

= C(SIn-A)-IB

is a unit m a t r i x

of a matrix.

G(s)

+ D

of o r d e r n and

(2.1.6) ( [idenotes

is a m a t r i x - v a l u e d

rational

the i n v e r s e function

11 of the c o m p l e x

variable

function m a t r i x the o p e n - l o o p G(s)

for the set of i n p u t - o u t p u t

~ain matrix.

can be r e g a r d e d

an e x p o n e n t i a l the c o m p l e x frequency When

s, and is c a l l e d the t r a n s f e r

The t r a n s f e r

as d e s c r i b i n g

variable

each s u b s y s t e m

can be c o n s i d e r e d

consists

Gi(s ) = Ci(SIn.-Ai)

to

a complex single-output

case.

of h s u b s y s t e m s

i=i,2, .... ,h}

has a t r a n s f e r

response

and t h e r e f o r e

as in the s i n g l e - i n p u t ,

S(A,B,C,D)

{Si(Ai,Bi,Ci,Di):

s

s [2],

or

function matrix

a system's

input w i t h e x p o n e n t

variable

transforms,

, as shown in figure

2,

function matrix

-1Bi +

Di

(2.1.7)

1

and the i n p u t - o u t p u t

transform

vectors

~(s) =Sh (s) Gh_l (s) .... Gl(S) with

the o b v i o u s G(s)

relationship

of S are r e l a t e d

~(s)

(2.1.8)

for the o p e n - l o o p

= G h ( S ) G h _ l ( S ) .... Gl(S)

For the p u r p o s e

of c o n n e c t i n g

form a f e e d b a c k

loop G(s)

by

gain m a t r i x

(2.1.9)

outputs

is a s s u m e d

back to inputs

to

to be a square m a t r i x

of o r d e r m. 2.2

Feedback

confi@uration

The g e n e r a l

feedback

is shown in figure

3.

configuration

The output

of the f e e d b a c k

is shown as that of the hth s u b s y s t e m m a y be the o u t p u t the

later

subsystems

compensators. variable

from an e a r l i e r

system

but in p r a c t i c e

subsystem

can be t h o u g h t

The p a r a m e t e r

that w i l l be c o n s i d e r e d

in w h i c h

of as b e i n g

it case

feedback

k is a real gain c o n t r o l

c o m m o n to all the loops.

The s y s t e m ' s

input and

12 output

are r e l a t e d

e(t)

=

r(t)

u(t)

=

ke(t)

to the r e f e r e n c e -

input

r(t)

by the e q u a t i o n s

y(t) (2.2.1)

and c o m b i n i n g closed-loop

these w i t h

equations

state-space

equations Bcr (t)

x(t)

=

AcX (t)

y(t)

=

CcX (t) +

+

(2.1.1)

the f o l l o w i n g

are obtained:

(2.2.2) D c r (t)

where

A-B(k-IIm+D)-Ic

Ac =

B c = k B - k B (k-iIm+D)-iD Cc

=

(Im+kD) -i c

Dc

=

(k-iim+D) -I D

~l 7"'ml -1°1 1, "7"hi Figure 3. Feedback configuration

y(t)

13

2.3

Stability Stability

of a feedback

is the most important system and for general

single requirement time-dependent

nonlinear

systems

it poses very complex problems.

stability

problem

for linear time-invariant

systems,

however,

The

dynamical

is much simpler than in the general

case.

This is because: (i)

all stability

to time, (ii)

properties

are constant

properties

are global,

with respect

and

all stability

since any solution

for the state of the system is proportional at time zero; There

see equation

(2.3.2).

are many definitions

and broadly

speaking

these

i.e.

those

ioe. those

types of stability associated

concerns

following

[3]

2.3-1

Free systems

below,

of free

the second

of forced input.

Both

the definitions

and

very closely those given by

.

Let us consider figure 3 described

the closed-loop

by equations

Then the stability

x(t)

stability

the behaviour

are discussed

theorems

considering

into two classes.

in which there is a given

Willems

Cc=I.

concerns

in the literature

in which there is no input;

class of definitions systems

of stability

can be divided

The first class of definitions systems

to the state

dynamical

(2.2.2)

problem

system of

with r(t)=O and

reduces

to that of

the free system

= AcX(t)

The equilibrium

state for equation

(2.3.1) (2.3.1)

is clearly

the

14

origin

(assuming A

solution

of

is non-singular),

c

(2.3.1) which passes

time remains

through the origin

there for all subsequent

is called the null solution. equilibrium

and therefore

times;

The stability

state is characterized

a at some

this solution of the origin

using the following

definitions• Definition called

i.

The origin of the free system

stable

if when the system is perturbed

origin all subsequent small neighbourhood Definition

2.

remain

is

from the

in a correspondingly

of the origin•

The origin of the free system

called asymptotically slightly

motions

(2.3.1)

stable

(2.3.1)

is

if when the system is perturbed

from the origin all subsequent

motions

return to

the origin. Definition

3.

The origin of the free system

called asymptotically asymptotically motion

stable,

converges

which stable

in the large,

if it is stable,

to the origin

The general x(t

stable

solution

;x(t O),tO)

(2.3.1)

is

or globally

and if every

as t+~.

of equation

(2.3.1)

= exp [ Ac(t-t O) ].x(t O)

is

[3]

(2.3.2)

shows clearly that if the free system is asymptotically it is also asymptotically

is the Jordan

canonical A

c

stable

form [4] of A c such that

=

TJT -I

=

Jl

(2.3.3)

with J

in the large.

J2 •

where each Jordan block J

l

Jk

has the form

If J

IS J. l

=

I. l

1 ~i ". hi

1

and li is an eigenvalue of Ac, then it can be shown exp[ Ac(t-to) ] =

T exp[J(t-to)].T-i

[ 3] , that

(2.3.4)

with exp [ J(t-t O) ] =

exp[J 1 (t-to) ] exp [ J2 (t-to)] exp[ Jk (t-to)]

and exp [ Ji (t-to)] = "i t t2/2 ' .... tr-I/(r-l) : 0 1 t

exp[~ i (t-to) ]

tr-2/(r-2)'

000

1

where ~ = (t-t O ) and r is the order of the Jordan block Ji"

The general

solution of the free system can therefore be expressed as x(t; x(t O),t O ) = T exp[J(t-to)].T-iX(to )

(2.3.5)

and from this the following theorems can be derived;

see

[31 for proofs. Theorem i.

The null solution of system (2.3.1) is asymptotically

stable if and only if all eigenvalues of the matrix A c have negative real parts. Theorem 2.

The null solution of system (2.3.1) is stable

if and only if the matrix A c has no eigenvalues with positive real parts, and if the eigenvalues with zero real parts correspond to Jordan blocks of order i.

16 2.3-2

Forced s~stems Let us consider the closed-loop dynamical system

(2.2.2) which has the general solution [3] , x(t; x(t o),t o ) = exp (Act).X(to)+/texp[Ac(t-T)].Bcr o (2.3.6)

(T)dT

To study the stability properties of this system we need to introduce the concept of input-output stability. Definition 4.

A dynamical system is called input-output

stable if for any bounded input a bounded output results regardless of the initial state. By theorem i, asymptotic stability of the unforced system (2.3.1) implies that all the eigenvalues of A c have negative real parts, in which case there exist positive numbers P and a such that llexp(Act) ll < P exp (-at) where

~t~O

(2.3.7)

II. II denotes the Euclidean norm of a matrix or From equations

vector [3].

(2.2.2),

(2.3.6) and (2.3.7)

we then have

II y(t)ll -< 4

H Ccx(t)II

+ II Dcr(t)II

II Dcr(t) ll + II Ccexp(Act)'x(to)II +Cfotllexp[Ac (t- T)] IIllBcr(t)lld~

d +c llx(to)li+

cbMP/a

where b=llBcll, c=llCcl], d=llDcl], and ]]r(t)l]4M ~

t >~ O.

This result is summarized in the following theorem. Theorem 3.

If the null solution of the unforced system

(2.3.1) is asymptotically stable, then the forced system (2.2.2) is input-output stable. Note that input -output stability implies asymptotic stability

17 of the equilibrium state at the origin only if the system (2.2.2) is state controllable and state observable; all unobservable real parts.

or if

and/or uncontrollable modes have negative

In the remainder of this

book

system stabilit ~

is understood as meaning input-output stability coupled with asymptotic stability of the equilibrium state at the origin. Theorem 3 is important because it tells us that the stability of a linear time-invariant

system can be determined

solely from a knowledge of the eigenvalues of the system "A" matrix.

The stability conscious eigenvalues corresponding

to the closed-loop dynamical system

(2.2.2)

are values of 1

which satisfy the equation det[IIn-Ac] =

(2.3.8)

O

The left-hand side of equation

(2.3.8)

closed-loop characteristi ~ polynomial,

is called the abbreviated as CLCP(1)

so that CLCP(1)~ det [ IIn-Ac]

(2.3.9)

Similarly for the open-loop system S(A,B,C,D) characteristic polynomial, OLCP(1)

OLCP(1),

an open-loop

is defined as

~ det[IIn-A ] = detflInl-Al]det [ IIn2-A 2] . . . .

.... det[iInh-Ah]

(2.3.10)

In the next section it is shown how the open- and closed-loop characteristic polynomials are related via the return-difference 2.4.

operator [ 5 ] .

Relationship between open- and closed-loop characteristic

polynomials

for the general feedback configuration

Let us suppose that all the feedback loops of the general

18

~(s)

(a)

P(s)

9(s)

6(s) r--'q ~(S) r--q

- ~ k i

[m~----~Gl(S)k ..........-,,pl~S)1 0

0

....

(b) Figure 4. Feedback configuration (a) dosed-loop (b) open-toop

19 closed-loop

configuration

are represented figure

4.

are broken

by their transfer

The corresponding

and that the subsystems

function matrices;

return-difference

see

matrix

[5] for this break point is

F(s)

I

+ L(s)

m

(2.4.1)

where L(s)

=

kGh(S)Gh_l(S) .... Gl(S)

=

kG(s)

(2.4.2)

is called the system return-ratio difference

operator

generates

the difference

and returned

signal transforms

transform.

It plays

the essence of signals between

matrix

equal,

them identically rational

feedback

detF(s)

partitioned detF(s)

between

= det[sI

which is equivalent

to

open- and

configuration. of equation

model,

formula

determinants

and

which is now derived

(2.4.1)

[6]

B

(2.4.3)

for the evaluation

can be rewritten

-A i

and represent

we obtain

= det[Im+kC(SIn-A)-iB+kD]

which using Schur's

are

of the return-difference

polynomials

If we take determinants G(s) by its state-space

and L(s)

around the properties

in the relationship

characteristic

for the general

revolve

since

two sets

of a complex variable

The importance

is emphasized

closed-loop

text

signal

the difference

Both F(s)

functions

injected

theory

link is making

thus making

zero.

in this

of such matrices.

between

from the injected

a feedback

identically

the key concepts

A return-

a major role in feedback

of forging

matrix-valued

matrix [5]

of

as

l+det[SIn-A ]

(2.4.4)

20 detF(s) = det -I

:

% letlSn l+etSn Is

L--~C--J, I S k

= det IsIn-A+B (k-IIm+D) -IC :--im+~ 0 ]-- det[SIn-A ]

[----ic ...... = det

-,

[SIn-A+B(k-iIm+D)-iC]det[Im+k~

(2.4.5)

det [SIn-A ] Now from equations

(2.2.2) we have

A c = A-B(k-IIm+D)-Ic and it is obvious from equation

(2.4.3) that

detF(~) = det[Im+k ~ and therefore under the assumption that det F(~)~O we have from equation

(2.4.5) the following relationship

detF(s) =

det[SIn-Ac] =

d e t [,,,,s, In-, ,A,,,,,,,,c,, ]

~ CLCP(s)

detF (~)

det [Sin_ A ]

det [SInh_Ah] . . ..det ..... [slnl_A~

OLCP (s)

(2.4.6) The zeros of the open- and closed-loop characteristic polynomials, OLCP(s) and CLCP(s), are known as the openand closed-loop poles or characteristic frequencies respectively. Relationship

(2.4.6) shows how the matrix-valued

rational transfer functions F(s) and G(s) are intimately related to the stability of a dynamical feedback system. The study of such matrices and their eigenvalues opens the way to suitable extensions of the classical techniques of Nyquist [7] and Evans [8;9] ;

the results of such a study are given

in Chapter 3. References

[I]

R. Bracewell, "The Fourier Transform and Its Applications", McGraw-Hill, New York, 1965.

21

[2]

A.G.J. MacFarlane and N. Karcanias, "Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex variable theory", Int. J. Control, 24, 33-74, 1976.

[3]

J.L. Willems, "Stability Theory of Dynamical Systems", Nelson, London, 1970.

[4] 5]

P.M. Cohn,

"Algebra", Vol. i, Wiley, London, 1974.

A.G.J. MacFarlane, '~eturn-difference and return-ratio matrices and their use in analysis and design of multivariable feedback control systems", Proc. IEE, 117, 2037-2049, 1970.

[6]

F.R. Gantmacher, York, 1959.

"Theory of M a t r i c e ~

Vol. I, Chelsea, New

[7]

H. Nyquist, "The Regeneration Theory", Bell System Tech. J., ii, 126-147, 1932.

[8]

W.R. Evans, "Graphical Analysis of Control Systems", Trans. AIEE, 67, 547-551, 1948.

[9]

W.R. Evans, "Control System Synthesis by Root Locus Method", Trans. AIEE, 69, 1-4, 1950.

3.

Characteristic gain functions and characteristic frequency functions

In the analysis and design of linear single-loop feedback systems the two classical approaches use complex functions to study open-loop gain as a function of imposed frequency (theNyquist-Bode approach), and to study closedloop frequency as a function of imposed gain (the Evans root locus approach).

The primary purpose of this chapter

is to show how these techniques can be extended to the multivariable case by associating with appropriate matrixvalued rational functions of a complex variable characteristic gain functions and characteristic frequency function s. 3.1

Duality between open-loop vai n and closed-loop frequency For the general feedback configuration of figure 4 we

have from section 2.4 the fundamental relationship detF (s) detF(-~

=

det [Sln-Ac ] det[s~n, A ]

(3.1.1)

where the return-difference matrix F(s) is given as F(s)

=

I + kG(s) (3.1.2) m If we substitute for F(s) in equation (3.1.1) we obtain det[ SIn-Ac] _ det[ SIn-A ]

det[ Im+kG(s) ] det[ Im+kD ] det[k-iIm+G(s~

=

(3.1.3)

det[ k-iIm+D] and substituting for the gain variable k using the expression g= -I where g is allowed to be complex i.e. g e ~ plane), we have

(3.1.4) (the complex

23

det[ SIn-Ac ] det[ Sin-A ]

det[gIm-G (S)] (3.1 .5)

det[ qIm-D ]

The closed-loop system matrix A c is given in equations (2.2.2) as A c = A - B(k-IIm+D)-Ic and substituting for k from equation

(3.1.6) (3.1.4) we have

A c = A + B(gIm-D)-Ic S(g) The expression

(3.1.7) (3.1.5) can therefore be rewritten as

det[ SIn-S (g)] det[ Sin-A ]

det[gIm-G(s) ] det[gIm-D ]

or

(3.1.8) det[ SIn-S (g)] det[ SIn-S (~ ~

det[ gIm-G (s)] det[ gIm-G (~)]

The form of this relationship shows a striking 'duality' between the complex frequency variable s and the complex gain variable g via their 'parent' matrices S(g) and G(s) respectively.

This duality between the roles of frequency

and gain forms the basis on which the classical complex variable methods are generalized to the multivariable case. S(g) is called the closed-loop frequency matrix;

its eigen-

values are the closed-loop characteristic frequencies and are clearly dependent on the gain variable g.

The eigenvalues

of the open-loop 9ain matrix G(s) are called open-loop characteristic ~ains and are clearly dependent on the frequency variable s.

The similarity between G(s) and S(g) is

stressed if one examines their state-space structures: G(s) = C(SIn-A)-IB + D

(3.1.9)

S(g) = B(gIm-D)-Ic + A

(3.1.10)

24 In figure 5 the feedback configuration of figure 4a is redrawn with zero reference input, the state-space representation for G(s),and the substitution

(3.1.4)

for k in order to

illustrate explicity the duality between the closed-loop characteristic frequency variable s and the open-loop characteristic gain variable g.

~ I n _

] ......

!

Figure 5. Feedback configuration illustrating the duality between s and g The importance of relationship

(3.1.8)

is that it shows,

for values of s ~ ~ (A) and values of g ~ ~(D)

(this condition

is equivalent to det F(~)~O which has already been assumed), where a(A) denotes the spectrum of A , that I det[SIn-S(g)]

= O ~

det[gIm-G(s)]=O~

(3.1.11)

25

This tells us that a knowledge gain as a function of closed-loop gain.

of frequency

knowledge

is equivalent

to a knowledge

from this is that it ought to be possible

the stability

of a feedback

of the characteristic

Note that from equation CLCP(s)

characteristic

frequency as a function of

characteristic

The inference

to determine

of the open-loop

from a

gain spectrum of G(s).

(3.1.8)

= det[gIm-G(s)]

system

we have that

. OLCP(s) (3.1.12)

det[gI m -D] and such an expression makes

it intuitively

there should be a generalization to loci of the characteristic

obvious

of Nyquist's

gains of G(s)

that

stability

theorem

as a function

of

frequency. 3.2

A19ebraic

functions:

characteristic ' frequency The characteristic

characteristic

9ain functions

and

function s. equations

for G(s)

and S(g)

i.e.

d(g,s)

~ det[gI m - G(s)]

= 0

(3.2.1)

?(s,g)

~ det[sI n - S(g)]

= 0

(3.2.2)

and

are algebraic

equations

Each equation

can be considered

with coefficients respectively, functions [i; (i)

which

defines

v a r i a b l ~ s and g.

as a polynomial functions

in g or s in s or g

over the field of rational a pair of algebraic

functions

i]:

a characteristic

loop characteristic (ii)

the complex

are rational

and if irreducible

each equation

appendix

relating

9 a i n function

gain as a function

a characteristic

frequency

g(s) which gives openof frequency,

function

and

s(9) which gives

26 closed-loopcharacteristic

frequency as a function of gain.

In general equations irreducible

(3.2.1)

and

(3.2.2) will not be

and each equation will define a set of characteri-

stic gain and characteristic

frequency

simplicity of exposition

and because

usual situation

and S(g)

for G(s)

functions.

For

this is in any case the

arising

from p r a c t i c a l

situations,

it will normally be assumed that equations

(3.2.1)

(3.2.2)

and

are irreducible over the field of rational

functions. A l t h o u g h b o t h equation

(3.2.1)

define the same functions g(s)

and equation

and s(g),

will in general contain more information

(3.2.2)

equation

(3.2.2)

about the system.

It is possible under certain c i r c u m s t a n c e s

that ?(s,g)

will

contain factors of s independent of g w h i c h are not present in (i)

~(g,s).

These factors occur in the following situations:-

When the A - m a t r i x of the open-loop

system S(A,B,C,D)

has eigenvalues which correspond to modes of the system which are u n o b s e r v a b l e

and/or u n c o n t r o l l a b l e

from the point

of view of considering the input as that of the first subsystem and the output as that of the hth sybsystem. that if output m e a s u r e m e n t s available

then in practice

of S(A,B,C,D) (2) G(s)

for earlier

subsystems

Note are

some of the u n o b s e r v a b l e modes

m a y in fact be observable.

When the poles and zeros of the o p e n - l o o p gain matrix are different

gain function g(s);

from the poles and zeros of the characteristic see section

These two conditions, (3.2.2) differ,

3.3-3.

under w h i c h equations

clearly p r e s e n t problems

(3.2.1)

and

to the d e v e l o p m e n t

27 of a Nyquist-like

stability

the characteristic

gains of G(s).

poles and zeros of g(s)

modes these problems criterion

can be overcome; is developed

is given which results

of the characteristic

3.3

Nyquist

frequency

@ain

A(g,s)

4.

3.4 a similar results

to the form ~£(g,s)

factors

are polynomials

Ai(g,s)

and the coefficients

(3.3.3)

functions

have the form

+ ...... +ait. (s)=O 1 (3.3.3)

where t i is the degree of the ith irreducible

equation

in

ti-i +ail(s)gi

common denominator

(3.3.2)

over the field of rational

ti

functions

gain functiong(s)

(3.3.1)

......

Let the irreducible

rational

in a

the characteristic

{Ai(g,s) :i=1,2 .... ,£}

= gi

study

equation

= Al(g,s)~2(g,s)

g which are irreducible

Ai(g's)

Nyquist

in a generalization

function

will be reducible

where the factors

in s.

and uncontrollable

~ det[gIm-G(s) ] = O ~(g,s)

and

functions

via the characteristic

In general

the

diagram.

Characteristic

A(g,s)

by relating

study of the characteristic

In section

The natural way to define is

loci of

a generalized

in chapter

a detailed

of the root locus diagram.

generalized

However,

of the unobservable

In the next section gain function

in terms-of

to the poles and zeros of G(s),

by careful consideration

stability

criterion

polynomial

{aij(s) :i=l,2,...,£;j=l,2,...,t i} are

in s.

Then if b

lO

(s) is the least

of the coefficients

can be put in the form

{aij(s) :j=l,2,...,t i}

28

t. t.-i bio(S)g i l+bil(s)g i i + ...... +biti(s)

= O

(3.3.4)

i=i,2,...,£ where the coefficients are polynomials gi(s)

{bij(s) li=l,2,...,£;j=l,2, .... ,t i}

in s.

The function

defined by equation

function

[i; appendix

loop gain matrix {gi(s):

eigenvalues of G(s)

i].

G(s)

i=l,2,...,Z}

(3.3.4)

of a complex variable

is called an algebraic

Thus associated

is a set of algebraic which are directly

of G(s).

with an openfunctions

related

The characteristic

to the

gain functions

are defined to be the set of algebraic

functions

{gi(s):i=l,2,...,~}. The problem of finding {A i(g,s):i=l,2,...,£} functions

are defined

finding an appropriate was reducible

is closely canonical linear

form [2].

case and a suitable

polynomials

from which the characteristic

to factors

put into Jordan

the irreducible

linked

canonical

to the problem of

form of G(s). in g then G(s)

In general

If A(g,s) could be

this will not be the

form is defined as follows.

Let O O

.

. O -aiti(s)

10

. . . O -ai,ti_l(S (3.3.5)

C(Ai)~

O 1 . . . O -ai,ti_2(s)

O

1 -ail (s)

for t.>l with 1

c (A i)

=A - a l l

(s)

gain

if ti=l

(3.3.6)

29 then a transformation G(s)

= E(s)

where Q(s)

matrix E(s)

Q(s)

E(s)

exists

-i

(3.3.7)

is a unique block diagonal

called the irreducible

such that

rational

matrix,

canonical

which

is

form of G(s)

and is given by Q(s)

~ diag[C(A I),C(A 2) ...... ,C(Az) ]

It is clear that given Q(s) can easily be obtained. for any given G(s)

the irreducible

A proposed

is presented

method

in appendix

3.3-1 Poles and zeros of a characteristic Consider function

the defining

g(s)

(3.3.8)

equation

factors

Ai(g,s)

for finding Q(s) 2.

vain function

for a characteristic

gain

:

~(g,s)~bo(s)g t + bl(S)g t-I + ... + bt(s)

= O (3.3.9)

We will take both bo(S) since,

~ O

and

bt(s)

~ 0

if either or both of these polynomial

were to vanish,

coefficients

we could

find a reduced-order equation such and that both the coefficients of the highest/zeroth powers of g(s) were non-zero;

this reduced-order

be taken as defining

an appropriate

for whose defining

equation

bo(S)

factor and thus both vanish

set of values of

s .

consider

the situation

a common

factor.

Before when

new algebraic

the supposition

It m a y happen however that common

equation would then

together

looking

bo(S)

The algebraic

and

and

function

would be true. bt(s ) share a at some specific

at the effect bt(s)

of this,

do not share

function will obviously

zero when bt(s)

=

O

(3.3.10)

be

30

and w i l l t e n d to i n f i n i t y bo(S)

+ O

(3.3.11)

For this r e a s o n

those v a l u e s

(3.3.10) are d e f i n e d g(s),

bo(S)

=

are d e f i n e d

of

s

which

satisfy equation

s

which

satisfy

(3.3.12)

to be the p o l e s of the a l g e b r a i c

should be t a k e n

the t e r m i n o l o g y

as r e f e r r i n g

s =~ r e q u i r e s

at the end of this

general bo(s)

o n l y to finite poles

special

attention

case, we m u s t share

the zeros

a common

factor.

share a c o m m o n

factor by saying that

simply be d i v i d e d

and bt(s)

empty

set of c o e f f i c i e n t s

have a c o m m o n

not share

this c o m m o n

left-hand

side of e q u a t i o n

t

in the

{hi(s)

such a common

function. factor,

Suppose

by bo(S)

bl(S) gt-i b (s) + b----~ + "'" + u gt-U+, bo(S ) -o" "

factor

t h e n that

but that some non-

Then d i v i d i n g

(3.3.9)

: i=O,2,...,t}

equation

{bu(S),bu+l(S),...,bv(S)}

factor.

when

Let us first d i s p o s e

out to get a n e w d e f i n i n g

algebraic

and

appropriate

all the c o e f f i c i e n t s

bo(S)

g

(3.3.1o)

show that they r e m a i n

case w h e n

for an a p p r o p r i a t e

and zeros.

and is dealt w i t h

and p o l e s of g(s)

of the t r i v i a l

would

g(s).

sub-section.

as d e f i n i n g

and bt(s)

function

'poles and zeros'

In o r d e r to be able to take e q u a t i o n s (3.3.12)

function

the e q u a t i o n

O

stated otherwise

The p o i n t

of

to be the zeros of the a l g e b r a i c

and those v a l u e s

Unless

as

do

through

the

we get

bv(S)

t-v+

"'+bo---~

bt(s)

"''+bo (s) = O (3.3.13)

Then,

as s + s w h e r e

and bt(s),

the m o d u l i

s is a zero of the c o m m o n of the c o e f f i c i e n t

set

factor of bo(S)

31

bu(S)

bv(S)

all become arbitrarily

large, and it is obvious that g(s)

will have a pole at s = s . Again,

suppose that bo(S)

factor but that some non-empty {bj(s), .... bm(S)} do not.

and bt(s) have a common set of coefficients

Then as

s ~ s where s is

a zero of the common factor, the algebraic equation

(3.3.9)

may be replaced by bj(s)gt-j(s)+

... + bm(~)gt-m(~)

= O

(3.3.14)

where bj(s) ~ O . . . . .

bm(S) ¢ O

so that we must have g(s)

= 0

showing that s is indeed a zero of the algebraic function g(s). We thus conclude that equations

(3.3.10)

and

(3.3.12)

may be taken as defining the finite zeros and finite poles of the algebraic function g(s), and that use of these definitions enables us to cope with the existence of coincident poles and zeros.

The pole and zero polynomials of g(s),

denoted by p (s) and z (s), are defined as g g pg(S) ~ bo/ (s) and

(3.3.15) Zg(S) ~ btl (s)

where b~(s) and bt[ (s) are the monic polynomials obtained from bo(S) and bt(s) respectively, by its leading coefficient.

by dividing each polynomial

32

For the purpose

of considering

g(s)

at the point

s=~

we put s=z

-i

(3.3.16)

so that ~ (g,s) =~ (g,z-l)=z -q ~ (g,z) where q is the number neighbourhood excluded

(3.3.17)

of finite poles of g(s).

of the value

z=O

from it) the equation

(the point ~(g,s)=O

In any

z=O itself being

is equivalent

to the

equation

~(g,z)=O. Therefore if we consider the equation & ~(g,z)=Co(Z)gt+cl(z)gt-l+ .... +ct(z)=O (3.3.18)

it follows (i)

that:

s =~ is a pole of the characteristic

gain function

g(s)

if and only if Co(O)=O (ii)

s =~ is a zero of the characteristic

gain function

g(s)

if and only if ct(o)=O For an open-loop realizable

system,

are considering

gain matrix

here,

for g(s)

2.1) we

to have

In fact it is easy to show that for s=~

the eigenvalues A!~ebraic

a physically

(see section

it is not possible

the values of the characteristic

3.3-2

describing

which by definition

poles at infinity.

simply

G(s)

gain function

g(s)

are

of D.

definition

of poles

and zeros

for a transfer

function matrix Let T(s)

be an mx£ rational m a t r i x - v a l u e d

the complex variable form for T(s),

s.

function

Then there exists a canonical

the Smith-McMillan

form

[3 ]

MCs)

, such

that T(s)

= H(s)M(s)J(s)

of

(3.3.19)

33

where the m×m matrix H(s) and the £×£ matrix J(s) are both unimodular

(that is having a constant value for their determinants,

independent of s ).

If r is the normal rank of T(s)

(that is T(s) has rank r for almost all values of s ) then M(s) has the form M(s) = [ M*(s)rr

] I

Or, Z-r

(3.3.20)

I r,r

Om-r,m-rJ

with

M* (S)

-- diag

Cl(S ) ~--~-~ ,

e2(s) ~2(s ) , . . . .

er(S) l ~--~-~j(3.3.21)

where: (i)

each ei(s) divides all ei+ j (s) and

(ii)

each ~i(s) divides all ~i-j (s).

With an appropriate partitioning

of H(s),M(s)

and J(s) we

therefore have

=

HI (s)M* (S) Jl (s)

(3.3.22)

where M*(s) is as defined in equation

(3.3.21).

Thus T(S) may be expressed in the form T(s)

Idiag{ ei (s) 7 ~i-~-s~}j Jl (s)

= Hi(s)

r

=

i=l

ci(s) t h i(s) ~ 3i(s)

(3.3.23)

where : (i)

{hi(s)

matrix H l(s)

;

: i = 1,2 .... ,r}

are the columns of the

34

{j~(s)

(ii)

: i = 1,2,...,r}

are the rows of the

matrix Jl(S) We know that r ~ min(£,m) and that

H(s)

and

J(s)

are unimodular

matrices

of full

rank m and Z respectively

for all s.

transfer

for a system with input transform

function matrix

vector %(s)

and output

transform

Suppose T(s)

vector ~(s).

is the

Then any A

input vector ~(s)

is turned into an output vector y(s)

by

(3.3.24) For the single-input y(s)

=

single-output

ke(s)

case where

~(s)

~(s) with

k

a constant,

g(s)

=

is defined vanishes

~(s)

as having

zeros at those values of s where

and poles at those values ~(s)

s is a zero of g(s),

is a pole of g(s).

II ~(s)II

of ~(s)

and becomes

~(s)

£ (s)

vanishes.

vanishes

arbitrarily

A natural way therefore

large when s

to characterize

is in terms of those values

becomes

zero for non-zero

and arbitrarily large for finite denotes

of s where

the modulus

the zeros and poles of T(s) s for which

function

ks(s)

Thus for a non-zero when

the transfer

the standard vector norm.

II ~(s) II

II ~(s)ll

' where

9his natural

of '

II "II extension

35

of scalar case ideas leads directly to definitions of zeros and poles of T(s) in terms of the Smith-McMillan

form

quantities

E(s)

because of the following pair of simple results. Zero lemma:

II 9 (s)II

if and only if some Pole lemma: some

vanishes for II ~ (s)II ~ O and s finite e. (s) is zero. l

II 9 (s)II+ ~

for II~(s)II < ~

if and only if

~i (s) + O. These considerations

definitions

lead naturally to the following

[3].

Poles of T(s):

The poIes of T(s) are defined to be the set

of all zeros of the set of polynomials

{~i(s)

: i = 1,2,...,r}.

In what follows we will usually denote the poles of T(s) by {pl,P2,...,p n}

and put

PT(S)

(s-Pl) (s-P2)

where PT(S)

=

...

(S-Pn)

(3.3.25)

is conveniently referred to as the pole polynomial

of T(s) and is given by r PT(S) = ~ ~i(s) i=l

(3.3.26)

Zeros of T(s): The zeros of T(s) are defined to be the set of all zeros of the set of polynomials

{si(s)

: i = 1,2,...,r}.

We will normally denote the zeros of T(s) by {Zl,Z 2 .... ,z } and put ZT(S) where ZT(S)

=

(S-Zl) (s-z 2) ...

(s-z)

(3.3.27)

is conveniently referred to as the zero polynomial

38

of T(s)

and is given by

ZT(S)

=

r ~ i=l

It is important necessarily

g. (s) 1

(3.3.28)

to remember

relatively

that ZT(S)

prime;

for this reason

to simply define

ZT(S)

as the numerator

and demominator

Rules for

and PT(S)

calculatin9 pole

polynomials

polyDomials

for the determination

T(s), particularly The following

are not

it is wrong

for a square matrix T(s)

The route via the Smith-McMillan convenient

and PT(S)

of det T(s).

and z e r o p o l y n o m i a l s form is not always

of the poles and zeros of

if the calculation

is being done by hand.

rules [4] can be shown to give the same results

as the Smith-McMillan Pole polynomial

rule:

definitions. PT(S)

is the monic polynomial

from the least common denominator

obtained

of all non-zero minors

of

all orders of T(s). Zero polynomial

rule:

from the greatest minors

of T(s)

ZT(S)

is the monic polynomial

common divisor

of the numerators

of order r (r being the normal

which minors hav e a l l been adjusted

obtained

of all

rank of T(s))

to have PT(S)

as thei r

common denominator 3.3-3

Relationship

of the open-loop

between

9ain matrix

algebraically G(s)

and the p~es/zer0 s of the

correspondin 9 set of characteristic

@ain functions

As a key step in the establishment Nyquist

stability

criterion,

defined poles/zeros

of a generalized

it is crucially

important

relate the poles

and zeros defined by algebraic

complex variable

theory,

to

means to

and thus to the poles and zeros of

37 the set of characteristic gain functions. The coefficients ai(s) det [gIm-G(s) ] =

in the expansion

gm + al(s)gm-i + a2(s)gm-2 + ... +am(S) (3.3.29)

are all appropriate sums of minors of Q(s)

since it is well

known that: det [gIm-G (s) ] =

g

m

-

[trace G(s)]g m-1 + [~principal minors of G(s) of order 2]g m-2 -

... + (-l)mdet G(s)

(3.3.30)

and thus the pcle polynomial bo;(S) is the monic polynomial obtained from the least common denominator of all non-zero principal minors of all orders of G(s). Now the pole polynomial p~s) of a square matrix G(s) is the monic polynomial obtained from the least common denominator of all non-zero minors of all orders of G(s). Therefore,

if eG(s)

is the monic polynomial obtained from the

least common denominator of all non-zero non-principa! minors, with all factors common to bo;(S) removed, we have that (3.3.31)

PG (s) = eG(S)bo/(S) Furthermore since det G(s)

=

am(S)

=

bm(S) 5--(s) o

and since from the Smith-McMillan det G(s) =

~. ZG(S )

(3.3.32)

form for G(s) (3.3.33)

PG(S) where a is a scalar quantity independent of s, we must have that

38

ZG(S )

=

eG (s) bml(S)

(3.3.34)

In many cases the least common denominator zero non-principal

of the non-

of G(s) will divide bo(S)

minors

,

in which case eG(s) will be unity and the p01e and zero polynomials

for G(s) will be b~(s)

In general a square-matrix-valued

and b i(s) respectively. m function of a complex

variable G(s) will have a set of £ irreducible gain functions the general

characteristic

in the form specified by equation

(3.3.3) and

form for thepole and zero polynomials

can be

written as PG(S)

=

eG(s)

H b/lo(s) i=l

(3.3.35)

ZG(S)

=

eG(s)

~ b i'ti(s) I i=l

(3.3.36)

and

where the pole and zero polynomials gain function gj(s)

are b!o(S) 3

for the jth characteristic

and b/ (s) respectively. 3,tj

Example demonstrating th 9 p o l e V z e r o r e l a t i o n s h i p s Let

G(s)

o]

s+l -i s-i

The pole polynomial =

and consequently ZG(S)

O

(s+l) (s+2)

(s-l) (s+l)

(s+l) (s+2) (s-l)

i PG(S)

(s-l) (s+2) 1

z

=

1 s+2 for G(s)

(s+l)(s+2)

is obviously

(s-l)

the zero polynomial (s-l) •

is

39

The c h a r a c t e r i s t i c det

equation

LgI-G(s)J

=

for G(s)

(g

so that the i r r e d u c i b l e

is

i 1 s+l ) (g - s - ~ ) =

characteristic

Al(g,s)

=

1 g - ~-~ =

A 2 (g,s)

=

1 g - s+---2 =

O

O

0

equations

are

and

which may be w r i t t e n

as

(s+l)g - 1

=

O

(s+2) g - 1

=

0

and

Therefore

the p o ~ and zero p o l y n o m i a l s

gain functions pgl(S)

=

gl(S)

blo(s)

and g2(s)

=

for the c h a r a c t e r i s t i c

are

(s+l)

Zg I

(s)

=

(s)

=

1

= b21(s) -i

=

1

bll

-1 Pg2 (s)

=

Now for common

b 2 0 (s)

G(s)

=

the m o n i c p o ~ n o m i a l

denominator

all factors e G (s)

common =

which v e r i f i e s PG(S)

=

Zg 2(s)

(s+2)

of all n o n - z e r o

bJ(s)~

to

obtained

from the

non-principal

I I (=b~'o(S)b20(s))a

least

minors

removed

with

is given

(s-l) the r e l a t i o n s h i p s 2 eG(s) H b.; (s) i= 1 lO

and ZG(S) 3.3-4

=

Riemann

eG(s)

surface

A characteristic irreducible

2 i=iH bilJ (s)

equation

of a c h a r a c t e r i s t i c gain

of the

function form

g(s)

gain

function

is d e f i n e d

by an

by

40

bo(S)g t + bl(S)g t-I + ... + bt(s) having

in general

occurs

only if

(a)

bo(S)

t distinct

= O, because

lowered,

and as b

infinite;

or if

o

=

O

finite roots.

(3.3.37) An exception

the degree of the equation

(s)÷O one or more of the roots becomes

(b)

the equation

has multiple

This

last situation

can occur

roots.

for finite values

only if, an exPression , called the discriminant equation,

vanishes.

The discriminant

function

of the equation

by Dg(S)

, and is discussed

Ordinary

points

function

g(s)

that bo(S) Critical

b

o

~

O

plane

or both,

and

Dg(S)

O

or

D

plus the point

Solutions

of

D

O

are called

=

g

(s)

discriminant

gain

plane

such

gain function is any point of the

=

O,

s =~.

finite branch points

function.

function

at which either

=

(s)

gain

~ O.

point [i; 6] of g(s)

Branch points of the characteristic

g

3.

of the characteristic

points of t h e characteristic

(s)

it will be denoted

in appendix

[1;6]

of the

is any finite point of the complex

A critical complex

point

of s if, and

[5] is an entire rational

coefficients;

of the characteristic

An ordinary

is then

The point Dg(Z)

of the characteristic

at infinity

of equation

At every ordinary p o i n t t h e

function

is a branch point

(3.3.18) equation

satisfies (3.3.37)

gain

if the

Dg(O)

= O

defining

the

41 characteristic

gain function has

the discriminant functions

t

does not vanish.

distinct roots,

since

The theory of algebraic

[i] then shows that in a simply connected

region

of the complex plane punctured by the exclusion of the critical points the values of the characteristic form a set of analytic functions

functions;

Arguments

analytic continuation, algebraic equations,

together w i t h the properties

regular function g(s) Functions

(3.3.37)

an irreducible

functions of a complex variable.

function of a complex variable has the set of as both its domain and its range.

function has the complex number set C

but has a new and a p p r o p r i a t e l y Riemann Surface

surface of an algebraic

defined domain R

[~

as its range which is

function plays a crucial role in this

its definition

now briefly considered.

An

Since the Riemann

it is important to have an intuitive

underlying

[7 ] .

defined in this w a y are called algebraic

complex numbers C

work

algebraic

and can be regarded as natural g e n e r a l i z a t i o n s

An elementary

called its

algebraic

defines precisely one t-valued

in the punctured plane

of the familiar e l e m e n t a r y

algebraic

can be

in the following basic theorem

function theory:

equation of the form

the corresponding

of

of

show that the various branches

This is summarized

of algebraic

gain

based on standard techniques

into a single entity:

function.

functions,

each of these analytic

is called a branch of the c h a r a c t e r i s t i c

function g(s).

organized

gain function g(s)

and formation,

grasp of the ideas

which is therefore

42

Figure 6. Anotytic continuation

S u p p o s e we h a v e a r e p r e s e n t a t i o n an a l g e b r a i c

function

a representation

of p a r t of one b r a n c h

in the f o r m of a p o w e r

series;

of

such

is u s u a l l y c a l l e d a f u n c t i o n a l e l e m e n t .

I m a g i n e its c i r c l e of c o n v e r g e n c e

to be cut out of p a p e r and

t h a t the i n d i v i d u a l p o i n t s of t h e p a p e r d i s c are m a d e b e a r e r s of the u n i q u e

f u n c t i o n a l v a l u e s of the e l e m e n t s .

initital element second power

is a n a l y t i c a l l y

series,

another

If n o w this

c o n t i n u e d by m e a n s of a

c i r c l e of c o n v e r g e n c e

can be

t h o u g h t of as b e i n g cut out and p a s t e d p a r t l y o v e r the first, as i l l u s t r a t e d by f i g u r e

6.

The p a r t s p a s t e d t o g e t h e r

m a d e b e a r e r s of the same f u n c t i o n a l v a l u e s t r e a t e d as a s i n g l e r e g i o n

and are a c c o r d i n g l y

c o v e r e d o n c e w i t h values.

further analytic continuation

is c a r r i e d out,

is s i m i l a r l y p a s t e d on to the p r e c e d i n g one. that,

after repeated analytic

are

continuations,

If a

a further disc Now suppose one of the d i s c s

lies o v e r a n o t h e r disc, not a s s o c i a t e d w i t h an i m m e d i a t e l y

43 preceding analytic continuation,

as shown in figure 7.

Such an o v e r l a p p i n g disc is pasted together with the one it overlaps

if and only if both are bearers of the same

functional values.

If,however,they

bear different

functional

values they are allowed to overlap but remain disconnected. Thus two sheets, values, become

which are bearers of different

functional

superimposed on this part of the complex plane.

1 Figure 7 Repeated anatytic continuation Continuing surface-like

this process

configuration

for as long as possible,

is obtained covering

a

t "sheets"

of the complex plane, where t is the degree of the algebraic function.

To form the Riemann

joined together

surface these sheets can be

in the m o s t varied of ways.

This m a y

involve connecting together two sheets which are separated by several other sheets lying between them. a construction

such

cannot be carried out in a t h r e e - d i m e n s i o n a l

space it is not difficult

to give a p e r f e c t l y

topological d e s c r i p t i o n of the process surface-like

Although

configuration

of the m u l t i p l e - v a l u e d

satisfactory

required.

is called the Riemann

algebraic

function.

This surface

On the Riemann

44 surface the entire domain of values of the algebraic function is spread out in a completely

single-valued manner

so that, on every one of the t copies of the complex plane involved,

every point is the bearer of one and only one

value of the function. A m e t h o d for building Riemann surfaces appendix

4.

This involves the use of cuts in the complex

plane and it may be helpful point.

is given in

Let an algebraic

points { a l , a 2 , . . , a r } .

to say a w o r d about them at this

function g(s)

have r critical

Suppose them to be joined to one

another and then to the point at infinity by a line L . Any line joining critical points will be called a cut. L

denote the set of complex numbers defined by the line L.

We then have that the solutions of equation a set of t "distinct" in the cut plane analytically cut

Let

L .

C

analytic

-i

.

continued,

functions

(3.3.37)

define

{gl(s),~2(s) .... ,gt(s)}

Each of these functions

by standard procedures,

can be

across the

NOW it fellows from the fundamental principles

of analytic c o n t i n u a t i o n satisfies

an algebraic

of definition,

that if an analytic

equation

in one part of its domain

it must satisfy that equation

into w h i c h it is a n a l y t i c a l l y

function

continued.

in every region

We must therefore

have that: (i)

there are only t "distinct"

which satisfy the defining algebraic

analytic

equation

functions

in the cut plane

C-i, (ii) analytic

each analytic continuation functions

{~i(s)

of any of these

: i = 1,2, .... t) gives rise to an

45 analytic function which also satisfies equation.

It follows

the defining algebraic

from this that the set of analytic

functions associated w i t h one side of the cut simple p e r m u t a t i o n of the set of analytic

L

must be a

functions

associated with the other side of the cut.

Therefore by

identifying and suitably matching up c o r r e s p o n d i n g functions

(via their sets of computed values)

sides of the cut

L

, one can produce

on which a single analytic defines a continuous

algebraic function,

function may be specified which

This function

from this domain

is of course the

conceived of as a single entity,

the domain so constructed It is sufficient book

on opposite

an appropriate domain

single-valued m a p p i n g

into the complex plane.

analytic

is its Riemann

and

surface.

for the purposes of understanding

this

for the reader to know that a Riemann surface can be

constructed

for any given algebraic

values form a s i n g l e - v a l u e d standard relationships theory generalize,

and properties

using the Riemann

characteristic

of the Argument surface which

surface;

function to the

the Principle

of

an extension

of

in appendix

5.

is the domain of the

gain function g(s) will be called the When the o p e n - l o o p gain

is m x m and has a corresponding

equation which is irreducible the frequency

Many

surface concept,

is developed

frequency surface or s-surface. matrix G(s)

of analytic

in p a r t i c u l a r

the Argument holds on the Riemann

The Riemann

on which its

function of positio n.

algebraic function case and,

the Principle

function,

characteristic

(i.e. the usual case in practice)

surface is formed out of m copies of the complex

46 frequency plane or s-plane. 3.3-5

G e n e r a l i z e d root locus diagrams The characteristic

gain function g(s)

is a function of

a complex variable whose poles and zeros are located on the frequency surface domain. nature of g(s) magnitude

It is convenient to exhibit the

by drawing constant phase and constant

contours

of g(s)

on the frequency

surface.

the computational method outlined in appendix construct

the surface then the superposition

phase and m a g n i t u d e The frequency possible

contours

If

4 is used to of constant

is clearly a simple process.

surface can be thought of as the set of all

closed-loop

characteristic

frequencies

associated

with all possible values of the complex gain parameter g. When the surface magnitude

is c h a r a c t e r i z e d

by constant phase and

contours of g(s) we have a direct correspondence

between a closed-loop loop gain,

characteristic

and since the surface

frequency

is constructed

copies of the complex frequency plane, there are m corresponding From equation g(s)

from m

for each value of s

characteristic

gains.

(3.1.4) we have

= -1

(3.3.38)

so that the variation frequencies)

and an open-

of the c l o s e d - l o o p poles

(characteristic

with the real control variable k traces out

loci which are equivalent

to the 180 ° phase contours

Equation

(3.3.38)

equation

for the single-loop

phase contours

is a direct generalization

of g(s)

of the defining

root locus diagram.

are the m u l t i v a r i a b l e

of g(s).

The 180 °

root loci i.e.

the variation

of the closed-loop poles with the gain control

variable k.

The fact that m u l t i v a r i a b l e

root loci

'live'

47 on a Riemann

surface explains

their c o m p l i c a t e d b e h a v i o u r

[9] as compared with the single-input,

single-output

where the root loci lie on a simple complex plane i.e. one sheeted,

Riemann surface).

case

(a trivial,

The m u l t i v a r i a b l e

root loci will sometimes be referred

to as the c h a r a c t e r i s t i c

frequency loci. Recall that in section characteristic

equations

3.2 it was pointed out that the

for G(s)

and S(g)

are in general

different in that the equation for S(g) may contain of s which are independent of g.

These

factors

factors therefore

correspond to closed-loop poles which are independent of g, or equivalently

independent of the gain control variable k;

and, from the root locus point of view, correspond to degenerate point.

The degenerate

loci each consisting of a single loci are therefore not p i c k e d out by

the 180 ° phase contours of g(s) In practice

these factors

the c h a r a c t e r i s t i c

on the frequency frequency

surface.

loci are g e n e r a t e d

as the set of loci in a single copy of the complex frequency plane traced out by the eigenvalues the negative

of S(g)

real axis in the gain plane.

automatically picks out the d e g e n e r a t e with the classical

control variable k=-g

In common

in terms of the gain

-i

Example of frg~uency

frequency

This approach

root locus approach of Evans the characteristic

frequency loci are usually calibrated

3.3-6

loci.

as g traverses

surface and characteristic

loci

As an i l l u s t r a t i v e variable feedback

example

configuration

open-loop gain m a t r i x

consider the general multiof figure 3 with a corresponding

48

~,.':':~--

Root Cut

Figure 8. Sheet 1 of the frequency surface

....

Root Cut

-I -3

Figure 9. Sheet 2 of the frequency surface

loci

Ioc

49

G(s)

=

The matrix

1 1.25(s+i) (s+2)

The two sheets

constant phase 8 and 9.

are shown characterized

and magnitude

are represented

characteristic

frequency

contours of g(s),

contours

of g(s)

by

in figures

by discontinuities

by thick black

lines;

in the

and the

loci, which are the 180 ° phase

are identified

The characteristic

the appropriate

from two sheets of the complex

The cuts,identifiable

contours,

s] s-2

is of order two and therefore

surface will be constructed s-plane.

[s-i -6

frequency

by a diamond

loci indicate

of the gain control parameter

k, upwards

symbol.

that variation

from zero,

causes

the system to experience

stability,

instability

again.

is clearly

linked with the presence

This phenomenon

of a branch point in the right half-plane Note that since we have completely feedback configuration are no unobservable 3.4

Characteristic

frequency

The natural way to define function s(g) V(s,g)

It is an algebraic characteristic

function

gain matrix

there

functions the characteristic

(3.4.1)

and the detailed

directly

frequency

equation

0

gain function presented

section can be applied

the

modes.

is via the characteristic

~ det[SIn-S(g) ] =

(at s=~4).

characterized

by its open-loop

or uncontrollable

and stability

study of the

in the previous

to it with the roles of

s

and g reversed. The Riemann frequency @-surface.

surface which is the domain of the characteristic

function will be called the ~ain surface or It is formed out of n copies of the complex

50 gain plane or g-plane loop characteristic value of g.

frequency

(closed-loop

The gain surface

set of all possible open-loop

since there are n values

open-loop

gain matrix

closed-loop

G(s)

characteristic

characteristic

to the gain function g(s)

exhibit

the behaviour

for every

of as the gains of the

with all possible

frequencies.

fashion

superimposing

poles)

can be thought

associated

of closed-

In a similar

it is convenient

to

of s(g) on the gain surface by

constant

phase and magnitude Like g(s)

contours

s(g)

onto the surface.

the frequency

s(g)

has poles and zeros but their significance

of

function is quite

different. 3.4-1

Generalized

Each phase

'sheet'

Nyquist

of a gain surface

and magnitude

corresponding

contours

g

surface

(or equivalently

poles.

contours

Therefore

loop Nyquist

k) correspond between

are a natural

diagram

In practice

into regions

given

closed

such a

one can see at a glance which values to stable

of s(g).

generalization

regions

The ~90 ° phase of the single-

and are called characteristic

the characteristic

of

closed-loop

stable and unstable

the ~90 ° phase contours

of s(g)

by constant

and right half-plane

frequencies.

The boundary

is clearly

characterized

of s(g)is divided

to left half-plane

-loop characteristic calibrated

diagram

gain loci.

gain loci are generated

as the loci in the complex gain plane traced out by the eigenvalues D-contour portion

of G(s)

as s traverses

in the s-plane.

of the imaginary

set of loci corresponding ( where

in this context

the so called Nyquist

Suppose axis.

that we consider

a

We can then compute

a

to the eigenvalues j = /--2i-- )

gl(j~),...,gm(j~)

51

in the following way: (i)

Select a value of angular

frequency,

(ii)

Compute the complex matrix G(j~ a)

say

a

(iii) Use a standard computer algorithm to compute the eigenvalues

of G(j~ a)

, which are a set of complex

numbers denoted by {gi(J~a)} (iv)

Plot the numbers

(v)

Repeat with further values of angular frequency ~b,~c, ... etc., continuous

{gi(J~a)}

.

in the complex plane.

and join the resulting plots up into loci using a sorting routine based on the

continuity of the various branches of the characteristic functions

involved.

For the purpose of developing stability criterion

in chapter

traversed in the standard 3.4-2

a g e n e r a l i z e d Nyquist

4 the Nyquist D - c o n t o u r

clockwise

direction.

Example of ~ain surface and characteristic

As an illustrative gain matrix considered minimal state-space

is

~ain loci

example consider the open-loop in subsection

3.3-5 w h i c h has a

realization

=-

0.6 1

O.5

The system has two states and therefore gain surface will be c o n s t r u c t e d complex g-plane.

the appropriate

from two sheets of the

The two sheets are shown c h a r a c t e r i z e d

by constant phase and m a g n i t u d e

contours

i0 and ii.

gain loci, w h i c h are the

~9~phase crosses.

The characteristic contours of s(g),

of s(g)

in figures

are denoted by a series of

52

....

Charocteristic gain loci

mmmm

Cut

Figure 10. Sheet 1 of the gain surface

Characteristic gain loci b C~%

Figure11. Sheet 2 of the gain surface

53

Right. half plane region

~

Left. half plane region .

.

.

.

.

.

.

.....

.

Choract,erbtic

gain loci Cut. between branch points.

"l

-0"8~.533

19-2

Figure12. Sketch of figure 10 emphasizing right half and left hatf-ptane regions

L

•533

1<3,2

Figure13. Sketch of figure 11 emphasizing right half and left half-ptane regions

54 In figures

12 and 13 sketches

made to emphasize closed-loop

From these

sketches

value of g moves

regions

of

it is easy to

on the gain control parameter

Since g=-k -I, as we increase critical

iO and ii are

the right and left half-plane

poles.

infer bounds

of figures

k positively

k for stability. from zero the

from -~ along the real axis towards

the origin on each sheet.

On sheet

plane region

while on sheet 2, for positive

for 1.25
k, g never moves the closed-loop

into a right half-plane system is stable

If k is increased of g

moves

region.

for 0~k<1.25

negatively

Therefore

and 2.5
from zero the critical

from ~ along the real axis towards

on each sheet. half-plane

i, g is in a right half-

the origin

On both sheets the value of g is in right

regions

for -~
k, which corresponds

to positive

loop system is stable

Therefore

feedback,

for negative

the closed-

for -1.875
If the calibrated

gain surface

is projected

complex gain plane ~ , we have the normal

onto the

representation

of the characteristic

gain loci, plus the superposition

contours

both right half-plane

plane

representing

regions.

Stability

the right half-plane point

1 (-~+30).

be difficult contours.

regions

However,

to comprehend Therefore,

because

although

is not fundamental

to system

by considering

to a single

this presentation

Nyquist

[i0~,

in relation

stability

it does afford the simplest

of

of encirclements

criterion

stability

critical

will in general

of the overlapping

the counting

of

and left half-

can now be predicted

in the generalized

Saeks

value

[chapter

4]

as pointed out by method of predicting

55

closed-loop stability

in the gain plane ~ .

From a gain surface plot it is possible to determine the closed-loop poles a n d hence the relative a closed-loop

system.

stability of

This is now i l l u s t r a t e d by finding

the dominant c l o s e d - l o o p poles for the example under consideration with

unity

k.

It is convenient

purpose to have the gain surface real and imaginary

=

by constant

contours as shown in figures

Let the dominant sd

characterized

for this

14 and 15.

closed-loop poles be

ezj8

Then ~ is the smallest

(in magnitude)

negative

that passes through any one of the critical, g, and 8 is the c o r r e s p o n d i n g

imaginary

real contour

-I, values of

contour.

From

figures 14 and 15 we have ~-

-0.05 and 8 = O

so that sd ~

-0.05

By hand calculation

the dominant c l o s e d - l o o p pole is

sd = -0.0528 Also from a gain surface characteristic

frequency

loci.

it is possible to determine Apart

from possible

the

single-

point loci, the root loci are simply the values of s at which the characteristic

gain loci has a phase of 180 ° .

Therefore

from a gain surface plot the root loci are determined by the values of the constant contours real axis on each sheet. frequency surface,

as they cross the negative

Similarly,

from a calibrated

it is possible to determine

gain loci from the values of constant contours the imaginary axes.

the c h a r a c t e r i s t i c as they cross

r~

v

c~

6

U}

~

E

.Q

L

cO

m.

cs O~

0

~ql

v

QJ E

U~

L

LL

57 I ~ a l contours lab¢ll~:l thus : Irnoginary contours labelled thus : 0.5

-0.9 -I'O -I'l -1'2 -#'3 -1'4 -f'5

-oe

1.5

Characteristic gain |oci

-0.7

....... ........

Cut

I'0-

-06 -0.5

-0,4

0-5-

-0.3

°-i.s ~

-,.o

I

-o:s..~f-o'

,!o

4 1.5

:2/ (lll : ,o

0.9

,'0

I"11~2 ~:3 ~.415

-I.5-

Figure 15. Sheet 2 of the gain surface References

Ill

[2] [3] [4] [2]

[0] [7] [8] [9] iO]

G.A. Bliss, "Algebraic Functions", (reprint of 1933 original). P.M. Cohn,

Dover, New York,

1966

"Algebra", vol. l, Wiley, London 1974.

H.H. Rosenbrock, "State Space and Multivariable Theory", Nelson, London, 1970. T. Kontakos,

Ph.D. Thesis, University of Manchester,

1973.

S. Barnett, "Matrices in Control Theory", Van NostrandReinhold, London, 1971. E. Hille, "Analytic Function Theory", Vol. 2, Ginn and Co., U.S.A., 1962. K. Knopp "Theory of Functions", 1947.

Part 2, Dover, New York,

G. Springer, "Introduction to Riemann Surfaces", AddisonWesley, Reading, Mass., 1957. B. Kouvaritakis and U° Shaked, "Asymptotic behaviour of root-loci of multivariable systems", Int. J. Control, 23, 297-340, 1977. R. Saeks, "On the Encirclement Condition and Its Generalization", IEEE Trans. on Circuits and Systems, 22, 780-785, 1975.

58

4.

A generalized Nyquist criterion

The Nyquist fundamental

stability

results

to the multivariable

Locus Method [3]

and design,

by Barman

ignored

quantities

Nyquist

a rigorous criterion

on the

for the general

an appropriate

the essential

of this chapter

feedback

applied Riemann

surface stability

the usefulness

of each depending

are given

4.1 and proved

Generalized

Nyquist

If the subsystems of figure

stability

statement

defined

on

on how the subsystems of the criterion

in section

4.2.

criterion

of the general

3 are each characterized

then the following

function

the Principle

tests are in fact stated and

The two statements

4.1

theory:

based on

[ appendix 5].

are characterized. in section

stability

configuration

to an algebraic

simplicity

is to give

Nyquist-like

in complex variable

Two Nyquist-like proved,

this made their treatment

and obscured

proof of a generalized

result

but their

it also leaned very heavily

The purpose

of the Argument

[4;5]

theorem was

of the

complicated,

a fundamental

systems

properties

use of cuts in the complex plane;

of the result.

called

proof was supplied.

stability

certain key algebraic

involved;

technically

case is of

for feedback

and Katzenelson

systems

was put forward

but no satisfactory

The proof of a generalized

approach

feedback

and u s e d as part of a technique

the Characteristic

undertaken

[i] is one of the most

Such a generalization

by MacFarlane[2]

analysis

criterion

in the theory of linear

and its generalization great interest.

stability

feedback

configuration

by a state-space

of the criterion

model

is applicable.

59

Statement

I.

feedback

configuration

is closed-

the net sum of anti-clockwise

encirclements

of the

loop stable (i)

The general if and only if:

critical point

(-~+jO)

by the set

of characteristic

loci is equal to the number of right half-plane (2)

the characteristic

critical point (3)

the

(-~+jO);

infinity

of G(s) on the imaginary (4)

poles of G(s);

gain loci do not pass through

the number of branches

passing through

gain

the eigenvalues

system S(A,B,C,D),

of the characteristic

gain loci

is equal to the number of poles axis;

and

of the A-matrix,

which correspond

of the open-loop

to modes of the system

which are unobservable

and/or uncontrollable

of view of considering

the input as that of the first sub-

system and the output

from the point

as that of the hth subsystem,

are

all in the left half-plane. If the subsystems their transfer

function matrices,

for each subsystem uncontrollable imaginary

axis,

are completely

characterized

or if it is known

there are no unobservable

modes

in the right half-plane

then the following

statement

by that

and/or including

the

of the criterion

applies. Statement

2.

feedback

configuration

is closed-

bhe net sum of anti-clockwise

encirclements

of the

loop stable (i)

The general if and only if:

critical point

(-~+jO)

by the set of characteristic

gain

loci is equal to the total number of right half-plane of Gl(S),G2(s) .... , and Gh(S) ;

poles

60

(2)

the c h a r a c t e r i s t i c

critical (3)

point

gain loci do not pass through the

(-kl--+jO); and

t h e number of branches

of the characteristic

loci passing through infinity poles of G(s)

on the imaginary

Note that if condition hold the closed-loop

gain

is equal to the number of axis.

(2) and/or condition

(3) do not

system has one or more poles on the

imaginary axis and is therefore not input-output

stable

although the e q u i l i b r i u m state at the origin may be stable. 4.2

Proof of the ~ e n e r a l i z e d Nyquist

stability criterion

In section 2.4 it was shown how the return-difference operator corresponding

to the break point shoWn in figure 4

is related to the open- and c l o s e d - l o o p

characteristic

polynomials by the following expression. detF(s) detF (~)

=

CLCP(s) OLCP (s-sT

(4.2.1)

This fundamental r e l a t i o n s h i p

is the foundation on which

the proof of the g e n e r a l i z e d Nyquist

stability

criterion

is based. The first stage in the proof is to consider the eigenvalue

equation of the r e t u r n - d i f f e r e n c e

matrix

F(s),

that is det[fI m - F(s)] which in general,

=

O

(4.2.2)

as for the characteristic

G(s), can be expressed

equation of

as a product of irreducible

of the form t. t.-i dio (s)f i i+dil(s)f'i i + .... + dit

algebraic

equations

(s) i i=i,2,...,~

defining a set of algebraic

functions

{fi(s)

=

O

(4.2.3) : i=l,2,,..,i}

61 Therefore,

as in sub-section

3.3-3 where the p o l ~ and zeros

of G(s) are related to the poles and zeros of the characteristic gain functions,

the pole and zero polynomials

be related to the Ixle and zero polynomials functions {fi(s):i=l,2,..°,Z} PF(S)

=

eF(s)

of F(s) can

of the algebraic

as follows

£ ~ d io ! (s) i=l

and

(4.2.4)

ZF(S)

=

eF(s)

By definition

K di;tl (s) i=l

the open-loop

characteristic

of the general feedback configuration

polynomial

is given by

OLCP(s)~det[SIn-A ] =det[Sin~Al]det[sin2-A2]

... det[sIL n. -Ah ]

(4~2.5) and it is easily shown [6] that det[SIn[Ai] = where pG

PG. (S)Pd. (s) l l (s) is the pole polynomial

l matrix Gi(s)

(4.2.6) for the transfer

and the monic polynomial

zeros the decoupling

function

Pdi(S) has as its

zeros of the ith subsystem associated

with that set of characteristic

frequencies

(eigenvalues)

of A i which correspond to modes of the ith subsystem which are uncontrollable characteristic OLCP(s)

and/or unobservable.

polynomial

The open-loop

can therefore be expressed

as

= pG l(s)pG 2 ~s)...PGh(S)Pdl(S)Pd2(S)...pd (s)~, (4.2.7)

The pole polynomial PG(S)

for the open-loop gain matrix

G(s) is related to the pole polynomials transfer functions

of the subsystem

through the relationship

62

PG(s)Px(S)

=

where Px(S)

has as its zeros those poles of GI(S),G 2 (s), ....

•d

PGl(S)PG2(s)...p~(s)

(4.2.8)

Gh(S) which are lost when G(s) is formed [ 7] .

zeros of Px(S)

The

are in fact a subset of the unobservable

uncontrollable modes of the system S(A,B,C,D)

and

where the

input is that of the first subsystem and the output is that of the hth subsystem.

The complete set of unobservable

and uncontrollable modes of S(A,B,C,D) of the polynomial Pd(S)

=

is the set of zeros

Pd(S) where

Px(s)Pdl (S)Pd2 (s)...Pdh (s)

(4.2.9)

The open-loop characteristic polynomial can therefore be rewritten as (4.2. iO)

OIX2P(s)

= PG(S)Pd(S)

OLCP(s)

= P G ( S ) P x (s)pd I ( s ) p d 2 ( s ) ' ' ' p ~ ( s ) (4.2.11)

or

and if we combine relationship CLCP (s)

expression

(4.2.10 with the fundamental

(4.2.1) we obtain =

pd (s) pG (s) detF (s) detF (~)

N o w from the Smith-McMillan

(4.2.12)

canonical form for F(s) we

have that detF(s)

= ~. ZF(S) PF (s)

(4.2.13)

where 8 is a scalar quantity independent of s; the structure of F(s), equation the monic polynomials

and from

(3.1.2), it is clear that

ZF(S) and PF(S) will be of the same

order and hence that detF(~)

=

8

(4.2.14)

Therefore combining equations characteristic polynomial

(4.2.12-14)

is given by

the closed-loop

83 ZF(S) CLCP

(s)

=

Pd(S)PG(S)

(4.2.15)

P F (s)

(4.2.4) and (3.3.35)

If we now substitute from equations

this expression can be rewritten as follows CLCP(s)

=

Pd(S)eG(s)

H b/lo(S) H i=l i=l

d / (s) iti

(4.2.16)

~d~o (s) i=l shift theorem

But using the eigenvalue

[ 8] , on equation

(3.1.2), we have that the characteristic {fi(s) :

of F(s) are related to the characteristic

i=i,2,... ,£}

gain functions fi(s) =

functions

i=1,2•...,£}

{gi(s) :

•

l+kg i(s)

by

i=1,2 .... ,£

(4.2.17)

and hence that the p o ~ p o l y n o m i a l s

for both sets of

algebraic functions are identical•

that is

£ dio/(s)

£ ~ i=l

=

i=l and therefore from

CLCP (s)

=

b~o(S)

(4.2.18)

(4.2.16) we have

Pd(S)eG(s)

Note that the polynomial

£ H dit.l (s) i=l

(4.2.19)

H d I (s) is dependent on the i=l it.I

gain control variable k. The relationship

(4.2.19) implies

[ see section 2.3]

that the following conditions are necessary and sufficient for closed-loop stability: (a)

eG(s)

(b)

H dilt (s) i=l i

(c) Pd(S)

= O

~

O has only left half-plane roots; =

O

has only left half-plane roots; and

has only left half-plane roots.

64

The next step in the proof is to show how condition (b) can be replaced by an encirclement

condition similar

to that in the classical Nyquist criterion. For the set of irreducible characteristic equations associated with the return-difference

operator F(s) there

is a corresponding set of Riemann surfaces on which the appropriate characteristic algebraic become single-valued, to a corresponding

functions

{fi(s) : i=i,2,...

and mappings from these surfaces on

fi - plane are one-to-one and continuous.

Let us consider the jth equation of the set defined by equations

(4.2.3).

The degree of the equation is tj and

therefore the corresponding Riemann surface ~ f . is formed 3 by piecing together t. copies of the complex s-plane, C • 3 Suppose now that a Nyquist D-contour, as shown in figure 16, is drawn on each of the tj copies of C before they are pieced

together to f o r m ~ f

. Then when the surface is 3 formed the set of Nyquist D-contours combine to form a set

of closed Jordan contours [ 9] enclosing right half-plane regions o f ~ f . . The extended Principle of the Argument 3 [appendix 5] can then be applied to each right half-plane region on ~ f j .

Therefore

for a particular right half-

plane region, not necessarily

simply connected but with a

boundary made up from Nyquist D-contours, we have that the difference between the number of zeros and poles of the algebraic function fi(s)

in the region,

is equal to the

number of clockwise encirclements of the origin in ¢ complex f-plane) fj(s)

(the

by the image of the boundary curves, under

for that particular region.

If we therefore consider

65 all the right half-plane regions o n ~ f

and apply the

3 extended Argument Principle to each, we have that N(fj , O)

=

Zf.

-

Pf.

3

(4.2.20)

3

where: (i)

N(f@,O) is the net sum of clockwise encirclements

of the origin in C by the set of image curves, under fi (s) I of the set of curves formed on ~ f . by piecing together 3 the appropriate set of Nyquist D-contours when f o r m i n g ~ f 3.; (ii)

Zf. is the number of right half-plane zeros of fj (s); and 3 (iii) Pf. is the number of right half-plane poles of fj (s). 3

Trn (s)

)

S -plane

')///4/

Radius r where

..,......~ r-,-oobut#00 / _

f

Re (s) x

Poles

O

Zeros

z~

Branch points

Figure 16. Theoretical Nyquist D-contour

66

Note that the indentatiorson necessary

only from a theoretical

of the Extended in practice

the D-contour viewpoint

are

in the development

Principle

of the Argument [ appendix

5] and

the D-contour

is taken as the imaginary

axis.

Condition

(b) for closed-loop

stability

is equivalent

to saying that there are no zeros of {fi(s) :

i=1,2,...,~}

in the right half-plane therefore

be replaced

Zf. = 0 l

(4.2.21)

or on the imaginary

axis,

and can

by

i=1,2,...,~

or

(4.2.22) N(fi,O)

= -Pf. l

plus the condition

i=1,2,...,£ that the algebraic

have no zeros on the imaginary

functions

axis.

(4.2.21)

Conditions

(4.2.22)

imply and are implied by N(fi,O)

=

-

i=l so that the necessary loop stability

~ i=l

P

(4.2.23)

and sufficient

can be rewritten

conditions

as

(a')

eG(S)=O has no right half-plane

(b')

e (s)=O has no roots on the imaginary £G

(c')

i=iZN(fi'O)

(d °)

{fi(s):

axis;

and

(e')

Pd(S) Now,as

=-i~l P %

for G(s).

roots; axis;

;

i=l,2,..,Z } have no zeros on the imaginary

has only left half-plane zeros. shown by equations

(3.3.35)

together with the pole polynomials functions

for closed-

{fi(s):

i=1,2,...,£}

and

(4.2.18),e(s)

for the set of characteristic

make up the pole polynomial

This leads us to consider

combining

conditions

67

(a') and

(c r) into the single

i=l

equivalent

condition

N ( f i ' O ) = -PG

where PG is the number The n e c e s s i t y closed-loop

(4.2.24) of right

and s u f f i c i e n c y

stability

are satisfied

when

is p r o v e d

poles

of c o n d i t i o n

conditions

of G(s).

(4.2.24)

for

(b'),(d I) and

(e')

as follows.

From e q u a t i o n s (3.3.35) £ PG = e + E Pfi i=l where e is the n u m b e r

half-plane

and

(4.2.18)

we have

that

(4.2.25)

of right

half-plane

zeros

of eG(s),

and we also know that Z N(fi,O) i=l

=

so that c o m b i n i n g £

Z Zf - Z Pf i i=l ii=l these £

(4.2.26)

two e v p r e s s i o n s

Z N(fi,O) = ~iZfi + e -- PG i=l i £ where i=ZlZfi'e and PG are all p o s i t i v e To e s t a b l i s h suppose that £ N(fi'O) i=l

~

the n e c e s s i t y

(4.2.27) integers,

of c o n d i t i o n

and hence we conclude

that

implies

(4.2.24)

condition

that

(4.2.24)

stability.

For s u f f i c i e n c y £ Z N(fi,O) = i=l then from e q u a t i o n £

or zero.

-PG

then from e q u a t i o n (4.2.27) this £ ~Zf ~O and/or e ~ 0 i=l i

for closed-loop

we have

suppose

that

-PG (4.2.27)

iZl= Zf i = e = O

we m u s t

have that

is n e c e s s a r y

68

and hence the system is closed-loop sufficiency

of condition

We have therefore are necessary (a") (b")

(4.2.24)

stable.

Thus the

is established.

shown that the following

and sufficient

for closed-loop

conditions

stability:

Z N(fi'O) = - PG ; i=l [fi(s) : i=1,2 .... ,£} have no zeros on the imaginary

axis;

(c")

eG(s)

has no zeros on the imaginary

(d")

Pd(S)

has only left half-plane

From equation the Nyquist

D-contour

simply related condition Z

(4.2.17)

the image curve

set mapped under

(a") can consequently =

and

zeros. sets in ~ of

fi(s)

by a unit shift in C •

N (kgi, -i)

axis;

and kgi(s)

are

The stability

be replaced by

-PG

(4.2.28)

i=l where N(kgi,-l)

is the net sum of clockwise

encirclements

of the point (-l+j~ in C by the characteristic scaled by k. scaling by

As in the classical

k is avoided by counting

that the characteristic

gain

(b") is equivalent gi(Jw)

~-~

and in practice gain

1

the

the number of encirclements

condition

(4.2.28)

point

by

(4.2.29)

we also have that condition

to

,

i=i,2, .... i

this corresponds

loci not passing Condition

(4.2.17)

criterion

loci make of the critical

1 (-~ + jO), and hence replacing £ 1 N(gi' -k) = -PG i=l From equations,

Nyquist

gain loci

through

(4.2.30) to the characteristic

the critical

(c") can also be replaced

point

(-~+jO).

by a more practical

69

The p o l e p o l y n o m i a l

test.

(3.3.35),

PG(S) where

for G(s)

equation

as

H b/ (s) i=l Io

= eG(s)

(4.2.31)

b I (s) is the pole p o l y m o m i a l ]o

gain f u n c t i o n

gj(s).

will have no zeros

Therefore

loci is equal Note

branches

to the n u m b e r

axis

if, and only

of the c h a r a c t e r i s t i c

of poles of G(s)

that for a b r a n c h

infinity there will

for the jth c h a r a c t e r i s t i c

it is clear that eG(s)

on the i m a g i n a r y

the number of i n f i n i t e

axis.

is given,

if, gain

on the i m a g i n a r y

of the loci going off to

also be a b r a n c h

returning

from i n f i n i t y

and care should be taken not to count this as two i n f i n i t e branches. The n e c e s s a r y loop s t a b i l i t y £

and s u f f i c i e n t

can t h e r e f o r e

Z N(gi' i=l

(2)

the c h a r a c t e r i s t i c

critical p o i n t the n u m b e r

passing t h r o u g h

Pd(S)

to:

gain

loci do not pass

through

the

(-kl---+jO); of b r a n c h e s infinity

G(s) on the i m a g i n a r y (4)

be r e d u c e d

for c l o s e d -

1 -k ) = - PG;

(i)

(3)

conditions

has only

This c o m p l e t e s Nyquist s t a b i l i t y

of the c h a r a c t e r i s t i c

is equal

axis;

to the n u m b e r

criterion.

loci

of poles of

and

left h a l f - p l a n e the p r o o f

gain

zeros.

of s t a t e m e n t

1 of the g e n e r a l i z e d •

70 If the s u b s y s t e m s their t r a n s f e r descriptions

are c o m p l e t e l y

function matrices

will

characterized

then t h e i r

each be o b s e r v a b l e

by

state-space

and c o n t r o l l a b l e

so

that p d I (s) = p d°~ (s)

=

.... = p d h(s)

=

i

(4.2.32)

and h e n c e Pd(S)

= Px(S)

Condition

(4) of s t a t e m e n t

then e q u i v a l e n t This

situation

by their

(4.2.32)

to Px(S)

information

modes

axis.

information

about

Pd(S)

zeros.

are c h a r a c t e r i z e d

has no u n o b s e r v a b l e

situations

decoupling

is

and we have the a d d i t i o n a l or

or on the

conditions

to give a c r i t e r i o n

To s h o w this we w i l l (i)

subsystems

in the right h a l f - p l a n e

In these

(4) can be combined

if the

subsystem

criterion

only left h a l f - p l a n e

descriptions

that each

uncontrollable imaginary

having

also arises

state-space

1 of the N y q u i s t

(i) and

which needs

no

zeros.

first of all p r o v e

that when

= Px(S)

(4.2.34)

or

(ii) Pd(S) where

the

=

Px(S)P~ (S)Pd 2( s )

zeros

of

{pd

.... p d h (s) (4.2.35)

(s) : i=1,2, .... h}

are

all in the left h a l f - p l a n e , conditions

(a'),

£ N(fi,O) i=l where

PG~

(c I) and h = -Z PG i=l

is the n u m b e r

The n e c e s s i t y

(e I ) are e q u i v a l e n t

to

(4.2.36) i of r i g h t h a l f - p l a n e

and s u f f i c i e n c y

of c o n d i t i o n

p o l e s of Gi(s). (4.2.3 6 ) for

7~ closed-loop stability when conditions

(b') and

(d') are

satisfied is proved as follows. From equations

(4.2.9),(4.2.18)

and

(4.2.31) we have

that h i=l

PG

=

~

e

£ + Z Pfi + P i=l x

(4.2.37)

where Px is the number of right half-plane and combining this with equation £ £ i=iZN(fi,O)

zeros of Px(S),

(4.2.26) we have that h

= i~l Zfi + e + Px - i~iPGi

(4.2.38)

Note also that Pd(S)

= Px(S)Pd] (s) .... Pdh(S)

so that if Pd denotes the number of right half-plane zeros of Pd(S) we have that (4.2.39)

Pd = Px and equation (4.2.38) becomes £ £ h Z N ( f i , O ) = ~ l Z f i + e + Pd - 2 PGi i=l i i=l To establish the necessity of condition

(4.2.40) (4.2.36)

suppose that h 7 N(fi,O ) ~ i=l

-

il~--1PGi

then from equation (4.2.40) this implies that £ Z Zfi ~ O, or e ~ O, or Pd ~ 0 i=l or any combination of these and thus that the system will be closed-loop unstable. (4.2.36)

Hence we conclude that condition

is necessary for closed-loop

For sufficiency suppose that £ h Z N(fi,O ) = - __ZIPGi i=l i

stability.

72 then from e q u a t i o n i~iZfi

(4.2.40)

= e = Pd = O

and hence the s y s t e m sufficiency

is c l o s e d - l o o p

of c o n d i t i o n

We have t h e r e f o r e (4.2.34)

we m u s t have that

or c o n d i t i o n

(4.2.36)

stable.

Thus the

is e s t a b l i s h e d .

shown that w h e n e i t h e r (4.2.35)

is s a t i s f i e d

the f o l l o w i n g

are n e c e s s a r y and s u f f i c i e n t for c l o s e d - l o o p £ h (a"~ Z N(f i O) = - Z PG ; i=l ' i=l (b"')

stability.

have no zeros on the i m a g i n a r y

{fi(s) : i = 1 , 2 , . . . , £ }

axis;

condition

and

(c i'') eG(s)

has no zeros on the i m a g i n a r y

But we have

already

shown,

in p r o v i n g

axis.

statement

1 of the

s t a b i l i t y criterion, that £ £ Z N(fi,O) ~ Z N(gi,-!) k i=l i=l that c o n d i t i o n gain

(b"')

is e q u i v a l e n t

loci not p a s s i n g

and that c o n d i t i o n infinite

branches

(c"')

generalized 4.3

the c r i t i c a l

is e q u i v a l e n t

of the c h a r a c t e r i s t i c

equal to the n u m b e r This t h e r e f o r e

through

to the c h a r a c t e r i s t i c

of poles of G(s)

completes

Nyquist

point

to the n u m b e r gain

of

loci b e i n g

on the i m a g i n a r y

the p r o o f of s t a t m e n t

stability

(-~+jO);

axis.

2 of the

criterion.

Example To i l l u s t r a t e

example

already

the g e n e r a l open-loop

the s t a b i l i t y

criterion

u s e d in s u b - s e c t i o n s

feedback

gain m a t r i x

configuration

consider

the

3.3-5 and 3.4-2,

is c h a r a c t e r i z e d

where

by its

73

0s

l

By d e f i n i t i o n

the c o n f i g u r a t i o n Therefore e i t h e r

of the p r o b l e m

as c o n s i s t i n g statement

is d i r e c t l y

=

statement

of just one

1 or s t a t e m e n t

for G(s)

closed-loop

net sum of a n t i - c l o c k w i s e

gain

critical point and have gain loci are shown

encirclements

no

gain

or on the i m a g i n a r y

is e n s u r e d

infinite

loci is zero, through

branches.

17 from w h i c h

For

is stable

point

the f o l l o w i n g

for O ~k<1.25.

and t h e r e f o r e

the

gain

system

loci pass t h r o u g h is c l o s e d -

for k = 1 . 2 5

(iii) For - O . 8 < - ~ < - O . 4 of the c r i t i c a l loop u n s t a b l e

of, or

and thus the c l o s e d -

1 -~ = -0.8 the c h a r a c t e r i s t i c

loop u n s t a b l e

the

are obtained.

the c r i t i c a l

the c r i t i c a l p o i n t

and

The c h a r a c t e r i s t i c

1 For -~< -~ < -0.8 there are no e n c i r c l e m e n t s

passage through,

if the

of the c r i t i c a l

loci do not pass

in figure

stability c o n d i t i o n s

(ii)

stability

by the c h a r a c t e r i s t i c

if the c h a r a c t e r i s t i c

loop system

Gl(S)=G(s).

(s+l) (s+2)

axis and t h e r e f o r e

(i)

subsystem

is

in the right h a l f - p l a n e

(-~+jO)

we can c o n s i d e r

2 of the s t a b i l i t y

which has no zeros

point

or u n c o n t r o l l a b l e

applicable.

The pole p o l y n o m i a l PG(S)

1

s-2

the s y s t e m has no u n o b s e r v a b l e

modes and by v i r t u e

criterion

i s i-6s

1.25(s+i) (s+2)

point

there

is one c l o c k w i s e

and t h e r e f o r e

for 1 . 2 5 < k < 2 . 5

the s y s t e m

encirclement is c l o s e d -

74 Im

60 ",=0-5

__

jl.O0.

~=1-0

]o.5o ~=2.0.

S ---,,-.- 00

f

~=0.05.

60

-I.00 ~=-0.05

-0-50

60 IRe

~= 20'0

60

~=1-0 ~ =-2.,

-]o.5o

~ j l O=_o.s " 0 Figure17. C h a r a c t e r i s t i c

= -I.0

gain loci

75

(iv)

1 For -~ = -0.4 the characteristic gain loci pass through

the critical point and therefore the system is closed-loop unstable for k=2.5. (v)

For -0.4<-~<0 there are no encirclements of, or passage

through, the critical point and thus the closed-loop system is stable for 2.5
1 For -~=O the characteristic gain loci pass through

the critical point and therefore the system is closed-loop unstable for k =~. (vii)

For 0<-~
of the critical point and therefore the system is closedloop unstable for -~
For O . 5 3 3 < - ~

passage through,

there are no encirclements of, or

the critical point and thus the closed-

loop system is stable for -1.875
References i]

H. Nyquist, "The Regeneration Theory", Bell System Tech. J., ii, 126-147, 1932.

2]

A.G.J. MacFarlane, "Return-difference and return-ratio matrices and their use in analysis and design of multivariable feedback control systems", Proc. IEE, 117, 2037-2049, 1970.

~3]

A.G.J. MacFarlane and J.J. Belletrutti, "The characteristic Locus Design Method", Automatica, 9, 575-588, 1973.

76 4]

J.F. Barman and J. Katzenelson, Memorandum ERL-383, Electronics Research Laboratory, College of Engineering, Univ. of California r Berkeley, 1973.

5]

J.F. Barman and J. Katzenelson, "A generalized Nyquist-type stability criterion for multivariable feedback systems", Int. J. Control, 20, 593-622, 1974.

[6]

A.G.J. MacFarlane and N. Karcanias, "Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex variable theory", Int. J. Control, 24, 33-74, 1976.

7]

C.A. Desoer and W.S. Chan, "The Feedback Interconnection of Lumped Linear Time-invariant Systems", J. Franklin Inst., 300, 335-351, 1975.

8]

A.G.J. MacFarlane, London, 1970.

~9]

"Dynamical System Models", Harrap,

E. Hille, "Analytic Function Theory", Vol. i, Ginn and Co., U.S.A., 1959.

77

5.

A generalized

inverse Nyquis t stability

In this chapter a g e n e r a l i z a t i o n stability feedback

criterion

of the inverse Nyquist

[i] for single-input

systems is developed

criterion

single-output

for the general

c o n f i g u r a t i o n which is complementary

feedback

to the exposition

of the g e n e r a l i z e d Nyquist stability criterion p r e s e n t e d in chapter

4.

The development

is based on an association

of the o p e n - l o o p gain m a t r i x G(s) with a set of inverse characteristic

gain functions,

and a corresponding

set of

inverse Nyquist diagrams which will be termed the inverse characteristic 5.1

gain loci.

Inverse c h a r a c t e r i s t i c , ~ain functions A

main

stability

feature of the g e n e r a l i z a t i o n

criterion,

of a set of algebraic

given in chapter functions

A(g,s)

4, is the association

- the characteristic

functions - with a square transfer by means of the characteristic

of N y q u i s t ' s

gain

function m a t r i x G(s)

equation

= d e t [ gl m - G(s) ]

=

O

(5.1.1)

If G(s)

has normal rank m then its inverse

exists,

and has a corresponding

function G(s)

characteristic

-i

equation

given by A(g,s) If A(g,s)

= det[gI m

G(s) -I] =

O

is regarded as a polynomial

which are rational

(5.1.2) in g with coefficients

functions of s then equation

defines a set of algebraic

functions

{gi(s) : i=1,2,...,£}

w h i c h will be called the inverse characteristic If A(g,s)

is also irreducible

(5.1.2)

gain functions.

over the field of rational

78 functions g(s),

in s, then the inverse

like g(s),has

surface

characteristic

as its domain

an appropriate

formed out of m copies of the complex

suitably

joined together.

of a matrix

In fact,

are reciprocal

gain function Riemann

s-plane

because the eigenvalues

to the eigenvalues

of the matrix

inverse g(s)

=

1 g(s)

and therefore

(5.1.3)

g(s)

and g(s)

hence the same Riemann The inverse

have the same branch points,and

surface

domains.

characteristic

on which the generalized

gain function

inverse Nyquist

is the foundation

stability

criterion

is based. 5.2

Pole-zero

relationships

In this section

a number

of pole-zero

relationships

are derived which will be used in the proof of the generalized inverse Nyquist Because

stability

criterion,

the eigenvalues

of the eigenvalues set of inverse

of a matrix

of the matrix

characteristic

inverse

by pg*(s) and pg.* (s)

Zg*(s), =

the poles of the

gain function

The pole and zero polynomials are therefore

5.4.

are the reciprocal

gain functions

the zeros of the characteristic versa.

section

are simply set, and vice

of gi(s),

expressible

as

z~=.(s)

1

denoted

(5.1.4)

1

and ~

(s)

=

pg

1

(s)

(5.1.5)

1

i=l,2,...,i From sub-section -loop gain matrix

G(s)

3.3-2

it is clear that for an open

there exists

a canonical

form,

the

79

Smith-McMillan

form [2 ] , such that

G(s) = HG(S)MG(S)JG(S) where HG(S)

and JG(S)

the Smith-McMillan

MG(S)

Consequently PG(S)

=

diag

(5.1.6)

are both mxm unimodular matrices

form MG(S)

[ ~l (s)£1(s) •

and

is given by

~2 (s)e2(s)-'

~m--~ )em(s) ]

(5.1.7)

the poles and zeros of G(s) are defined as m = H ~i(s) (5.1.8) i=l

and ZG(S)

m H i=l

=

ei(s)

(5.1.9)

Now if G(s) is of normal rank m equation

(5.1.6)

can be

inverted to give G(s) -1 = JG(S)-IMG(S)-IHG(S) -I which using the elementary

(5. i. lO)

transformation

matrix E, of order

m, and given by

E

=

O

O

...

0

1

O

O

...

1

O

•

:

•

.

O

1

...

O

O

1

O

...

O

0

=

E

-i

can be rewritten as G(s)

-i

=

JG (s) -IEEM G (s) -IEEHG (s)

=

H~(s)M6(s)J~(s)

-i

(5.1.12)

or G(s) -I

(5.1.13)

(5.1.11)

80

where H~(s) = JG(S)-iE J~Cs) = EHG(S)

(5.1.14)

-i

(5.1.15)

and M~(s) = EMG(S)-IE

= diag

[ ~m (s) e~-~) '

which is the Smith-McMillan zero definitions

~m-i (s) . ~I (s) ] em_l(S) ' " "' el(S ) form for G(s) -I.

(5.1.16)

The pole-

for a transfer function matrix [sub-

section 3.3-2] can now be applied to G-I(s) with the result that p~(s)

m -- i=iH s.1(s)

z~(s)

=

=

zG(s)

(5.1.17)

=

PG(S)

(5.1.18)

and m

where p*(s)G

H ~i(s) i=l

and z~(s) denote the pole and zero polynomials

of G(s) -i respectively. In sub-section zero polymomials

3.3-3 it is shown that the pole and

of G(s) are related to the poles and zeros

of the characteristic

gain function set

PG(S) =

eG(s)i=l K b ti°(s)

ZG(S) =

eG(s)

{gi(s) :i=1,2,...,£}

= eG(s)i~iPgi (s)

by

(5.1.19)

and K b ! (s) = eG(s) ~ z (s) i=l iti i=l gi

(5.1.20)

If the pole and zero polynomials of the complete characteristic gain function set are denoted by pg(S) and Zg(S) respectively, and similarly for the inverse characteristic

gain function set

81

using pg*(s)

and z*(s), g

then relationships

(5.1.19)

(5.1.20)

can be combined with relationships

(5.1.18)

to give

PG(S)

= z~(s)

ZG(S)

= P*(s)=eG(S)Zg(S)G

and

(5.1.17)

= eG(S)pg(S)=eO(s)z~(s)

and

(5.1.21)

and

These pole-zero

= eG(s)p~(s)

relationships

the inverse Nyquist

stability

5.3 Inverse characteristic inverse Nyquist diagrams The inverse

the eigenvalues traverses

D-contour

generalized

Statement

i.

loop stable

The general

(clockwise)

the loci follows in sub-section

(iv) the reciprocal

to state and prove

stability

the

criterion.

stability

criterion

feedback

configuration

by a state-space

of the criterion feedback

model

is applicable.

configuration

is closed-

if and only if:

the net sum of anti-clockwise

critical

as s

in the complex plane.

of the general

statement

G(s)

gain loci,

3 are each characterized

then the following

of each of

in the standard

Generalized i n v e r s ~ Nyquist

of figure

(la)

gain matrix

are plotted

inverse Nyquist

If the subsystems

loci are the loci in

that at step

We are now in a position

5.4

gain

for computing

3.4-1, with the modification gi(jwa)

later.

~ain loci - generalized

for the characteristic

of the numbers

presented

out by the reciprocal

The algorithm

that given

criterion

of the open-loop

the Nyquist

direction.

will be used in the proof of

characteristic

the complex plane traced

(5.1.22)

point

(-k+jo),

encirclements,

by the set of inverse

of the

characteristic

82

gain loci, minus the net sum of anti-clockwise of the origin,

by the set of inverse

encirclements,

characteristic

gain loci,

is equal to the number of right half-plane poles of G(s); (2)

the inverse c h a r a c t e r i s t i c gain loci do not pass

through the critical point (3a)

(-k+jo) ;

and

the number of branches of the inverse characteristic

gain loci that pass t h r o u g h the origin is equal to the number of poles of G ( s ) o n (4)

the imaginary

the eigenvalues

system S(A,B,C,D),

axis;

and

of the A- matrix,

of the open-loop

which correspond to modes of the system

which are unobservable

and/or uncontrollable

from the point

of v i e w of considering the input as that of the first subsystem and the output as that of the hth subsystem,

are all in the

left half-plane. Alternatively

conditions

(la) and

(3a) m a y be replaced

by: (ib)

the net sum of anti-clockwise

encirclements

point

(-k+jo) by the set of inverse

characteristic

is equal to the n u m b e r (3b)

of right half-plane

the number of branches

is equal to the number of

are completely

function matrices,

c h a r a c t e r i z e d by their

or if it is known that for each

subsystem there are no unobservable

and/or uncontrollable

in the right h a l f - p l a n e

including

the following

statement

of the criterion

Statement

The general

2

zeros of G(s);

on the imaginary axis.

If the subsystems transfer

gain loci

of the inverse c h a r a c t e r i s t i c

gain loci passing through infinity zeros of G(s)

of the critical

feedback

the imaginary

modes

axis, then

applies.

configuration

is closed-loop

stable

83

if and only if: (la)

the net sum of anti-clockwise

critical point

encirclements

(-k+jo), by the set of inverse

charactezistic

gain loci, minus the net sum of anti-clockwise of the origin,

of the

encirclements,

by the set of inverse c h a r a c t e r i s t i c

gain loci,

is equal to the total number of right half-plane poles of Gl(S),G2(s), .... , and Gh(S); (2)

the inverse

characteristic

the critical point (3a)

(-k+jo);

gain loci do not pass through

and

the number of branches of the inverse characteristic

gain

loci that pass through the origin is equal to the number of poles of G(s) on the imaginary Alternatively

condition

(3b), as in statement are square,

(3a) m a y be replaced by condition

1 of the criterion,

and if the subsystems

i.e. have the same number of outputs

then condition (ib)

axis.

(la) m a y be replaced by:

the net sum of anti-clockwise

critical point

as inputs,

encirclements

(-k+jo) by the set of inverse

of the

characteristic

gain loci is equal to the total number of right half-plane zeros of Gl(S),

G2(s) . . . . ,

Note that if condition

and Gh(S). (2) and/or condition

not hold then the closed-loop on the imaginary

(3a)/(3b)

do

system has one or m o r e poles

axis and is therefore not input-output

stable although the e q u i l i b r i u m

state at the origin may be

stable,

For strictly proper s y s t e m ~ t h a t system D-matrix

is, ones in which the

is zero, the inverse c h a r a c t e r i s t i c

gain loci

84

approach infinity as

s approaches

a pole of the inverse

function.

infinity,

and s=~ is

Therefore,

in practice,

is n e c e s s a r y to traverse the whole of the D-contour,

it

not

just the imaginary axis,

in order to obtain closed curves.

In this way the net sum

of encirclements

point and the origin by the inverse can be obtained.

As an example,

of the critical

characteristic

gain

loci

the inverse characteristic

gain loci for G(s)

=

1

(5.4.1)

(s+l)

3

are shown in figure 18. The traversal however, than

of the D - c o n t o u r off the imaginary

is not n e c e s s a r y

if we use conditions

(ib), in both statements

This is because the extra part of the D-contour,

A useful rule, when using the to join up the corresponding

m a y actually

and thereby cancel each other.

(la) conditions,

is therefore

loose ends of the loci via a that

W h e t h e r the ends are joined via the

or left half-plane

the mapping under g(s), on the Riemann

to the circular

and to forget any extra encirclements

exist.

right half-plane

criterion.

encircle the origin and the critical

point the same number of times,

large semi circle,

(la), rather

of the stability

loci, corresponding

axis,

is fixed by the fact that

from the set of Nyquist

surface of g(s)

curves formed

(by piecing together an appropriate w

set of Nyquist D-contours)

to the g-plane,

means that if we imagine two people, the domain curve, curve,

This

X and Y, X walking

along

along the corresponding

image

such that at each step X defines the corresponding

image point, right.

and Y walking

is conformal.

then if x turns to his right, Y will turn to hi__~

85

*®I~h 20~

~j_ plane

- 3;0

"~x~=.2. 4

-

I0 J

2~0

$=0

-20.

(o)

I.O

/

°-~

,o( o,:: f.,~ = 6 o

~ I ~ ,0°3 -0.5 0.5

l.~l

(b)

Figure18. Inverse charact.eris!,ic gain Loci for G(s)= 1 / (s.l) 3 ((:1)small area around origin (b) large a r e a

86

5.5

Proof of gene!alize d inverse Nyquist stabilit Y criterion For the general

closed-loop transfer

feedback configuration

of figure 3 the

function matrix R(s) is given by the

relationship R(s) = kG(s) [ Im4kG(s)]-i

(5.5.1)

which after inverting becomes -i -1 R(s)

= 1 G(s)

(5.5.2)

+ I

m

If we now denote the set of algebraic eigenvalues

of R(s) -I by

the eigenvalue

functions defining the

{~i(s) :i=l,2,...,Z}

shift theorem[3]

to equation

we can apply (5.5.2) with

the result that ~. (s)

=

1 gi(s)+ 1

(5.5.3)

i=I,2, . . . ,£ Then if the poleand zero polynomials algebraic functions monic polynomials ~r(S)

=

{ri(s):i=l,2,...,Z}

Pr* (s) and

MR (s)-lG (s)

set of

are denoted by the

* Zr(S) respectively,

~g(s)

Now post-multiplying

for the complete

we have that (5.5.4)

equation

=

Im+kG (s)

=

r(s)

(5.5.2) by kG(s) we have (5.5.5)

the return difference matrix

[4] , and taking determinants

this we obtain det F(s)

= k m det[

s

-l]

det[G(s) -I] =y z~(s)

*

g( )

(5.5.6)

of

87

where y is a scalar constant independent det F(s)

= S ~(s)

=

8 zf(s)

PF{S) where

pf(s) and

of s.

But (5.5.7)

pf ('s)

zf(s) are respectively

the monic pole and

zero polynor~als for the set of algebraic

functions

{fi(s):i=1,2,...,£}

and therefore (5.5.8)

and =

z f(s)

Now from equations pf(s)

@

z*(s) r

=

p*(s) )

(4.2.18)

pg(S)

=

pf(s)

and

(5.5.9)

(5.1.21)

z*(S)g

(5.5.10)

and hence combining this with equations

(5.5.4)

and

(5.5.9)

we obtain

Izf(s)

*(s)I

=

In chapter closed-loop

(5.5.11)

zr

4, equation

(4.2.19),

characteristic

polynomial £ = Pd(S)e~s) R d ! (s) i=l iti

CLCP(s)

or equivalently, u s i n g CLCP (s)

=

and therefore CLCP(s)

it was shown that the can be given by

the notation presented

pd (s) eG(s) zf(s)

combining this with equation =

Pd(s)es(S) Zr(S) ]

This implies that the following conditions sufficient

(5.5.12)

for closed-loop

in this chapter, (5.5.13)

(5.5.11) we have (5.5.14) are necessary

stability:

(a)

eG(s) = 0

has only left half-plane

roots;

(b)

Zr(S)

= 0

has only left half-plane

roots;

(c)

Pd(S)

= 0

has only left half-plane

roots.

and

and

88 Suppose now that a Nyquist D-contour,

as shown in figure

16, is drawn on m copies of the complex plane before they are

pieced

together

t o f o r m t h e Riemann s u r f a c e s

on which the {ri(s):i=l,2,...,~} the jth s u r f a c e ~ j

{I~.:i=1,2,...,£} 1

are defined.

corresponding to ~j(s).

Let us consider When the surface

is formed the set of Nyquist D-contours combine to form a set of closed Jordan contours [5] enclosing right half-plane regions o n e * [appendix On~r* • 3

r.]

.

The extended Principle of the Argument

5] can be applied to each right half-plane region Therefore for a particular right half-plane region,

not necessarily simply connected but with a boundary made up from Nyquist D-contours, we have that the difference between the number of zeros and poles of the algebraic function rj(s) in the region,

of the origin

is equal to the number of clockwise encirclements

in

~

(the complex r-plane)

boundary curves, under rj(s),

I f we t h e r e f o r e

consider

all

by t h e image o f t h e

for that particular region.

the right

half-plane

regions

and

apply the extended Argument Principle to each, we have that N(rj, O)

=

Z~j

-

P* rj

(5.5.15)

where: (i)

N(r~,O) is the net sum of clockwise encirclements of the J origin in C by the set of image curves, under ~j(s~ of the set of curves formed on ~ *

rj

by p i e c i n g

together

the appropriate

set of Nyquist D-contours w h e n f o r m i n g ~ . ~ (ii)

3 Z~. is the number of right half-plane 3

w

zeros of rj (s);

89

and p* r.

(iii)

is the number of right half-plane

poles of r.] (s).

3

If we now consider the corresponding gain function gj(s) N(gj

, O)

=

inverse characteristic

in the same way, we find that Z~j

-

P~j

(5.5.16)

where: (i)

N(gj,O)

the o r i g i n

is the net sum

in C by the inverse

corresponding

(ii)

~j

(iii) ~ j

of clockwise encirelements

characteristic

gain

loci

to gj(s);

is the number of right

half-plane

zeros

of gj(s);

is the number of right half-plane poles of gj(s).

Equations N(rj,O)-N(gj,O)

(5.5.15)

N(rj,O)

and

= Z, - P, rj rj

which using equation

(5.5.16) -

Z, gj

can be combined to give + P, gj

(5.5.17)

(5.5.4) becomes

- N(gj,O)

(5.5.18)

= Z~j - Z~j

and if we consider the complete *

{ri(s):i=l,2,...,£}

set of algebraic functions {* and the set gi(s) :i=l,2,...,Z}

we have Z , £ , Z N(ri,O)- Z N(gi,O) i=l i=l Now condition

=

£ Z Z* i=l ri

(b) for closed-loop

- Z Z* i=l gi

(5.5.19)

stability is equivalent

to saying that there are no zeros of {ri(s):i=l,2,...,£} in the right half-plane

or on the imaginary axis, and can

therefore be replaced by N(ri,O) i=l

of

-

N(gi,O) i=l

= -

~ Z* i=l gi

(5.5.20)

and

90

plus the c o n d i t i o n t h a t on t h e

imaginary

conditions rewritten

{ri(s):i=l,2,...,£}

axis.

The

for c l o s e d - l o o p

necessary

have no zeros

and

stability

sufficient

can t h e r e f o r e

be

as

(ai )

eG(s)

(b! )

eG(s)

= O has no right h a l f - p l a n e

= O has no roots on the i m a g i n a r y .

Z

(c ! )

Z N(ri,O) i=l

(d I )

{r*

-

,

axis;

and

(e I )

Pd(S)

= -

Z Z* i=l gi

;

have no zeros on the i m a g i n a r y

has only left h a l f - p l a n e

The p o l e - z e r o

axis;

£

Z N(gi,O) i=l

(s):i=i,2,...,£}

i

roots;

relationship

zeros.

(5.1.21)

n o w leads us to c o n s i d e r

combining

c o n d i t i o n s (a')and(c') into the s i n g l e e q u i v a l e n t . £ , N(ri,O) Z N(gi,O) = -PG (5.5.21) i=l i=l

where

PG is the n u m b e r of right h a l f - p l a n e

The n e c e s s i t y closed-loop

and s u f f i c i e n c y

stability

are s a t i s f i e d

is p r o v e d

From relationship PG = e + where

of c o n d i t i o n

conditions

(bl),

(5.1.21)

s u p p o s e that £ , N(ri,O) i=l

-

and

_

(e I )

we have that (5.5.22)

of right h a l f - p l a n e

Z . - Z N(gi,O) i=l

To e s t a b l i s h

(d')

for

as follows

this w i t h e q u a t i o n

£ . Z N(ri,O) i=l

of G(s).

(5.5.21)

Z Z* i=l gi

e is the n u m b e r

combining

when

poles

condition

(5.5.19)

=

poles

gives

£ Z Z* + e-P G i=l ri

the n e c e s s i t y

of c o n d i t i o n

£ , 2 N(gi,O) i=l

-PG

fl

of eG(s),

(5.5.23)

(5.5.21)

and

91

then

from e q u a t i o n £ 7. Z* ~ O i=l r i

(5.5.23) and/or

and h e n c e we c o n c l u d e for c l o s e d - l o o p

this

that c o n d i t i o n

from e q u a t i o n £ 7. Z, = e i=l r. l

(5.5.23) =

the s y s t e m

of c o n d i t i o n

that =-PG this

implies

is c l o s e d - l o o p

(5.5.21)

stable.

for c l o s e d - l o o p

(b") (c")

eG(S)=O

(d")

Pd(S)

has no roots

Also have

(a")

(-k+jO);

hence

in s t a t e m e n t

axis

to c o n d i t i o n

(la)

criterion. we have that if, and only

{ri(s):

i=i,2,...,£}

if the inverse

the c r i t i c a l

(b") is e q u i v a l e n t

1 of the s t a b i l i t y

F r o m the p o l e - z e r o

and

zeros.

is e q u i v a l e n t

loci pass t h r o u g h

condition

axis;

axis;

(5.5.24)

(5.5.3)

zeros on the i m a g i n a r y gain

stability:

-k)

conditon

from e q u a t i o n

conditions

it is clear that

1 of the s t a b i l i t y

characteristic

sufficiency

on the i m a g i n a r y

on the i m a g i n a r y

left h a l f - p l a n e

F r o m e q u a t i o n (5.5.3) £ , ~ , 7. N ( r i , O ) = 7 N(gi, i=l i=l and c o n s e q u e n t l y

Thus the

shown that the f o l l o w i n g

~ N(ri'O) 7. N(gi'O) = -PG ; i=l i=l , {ri(s): i=i,2, .... £} have no zeros

has only

that

is e s t a b l i s h e d .

are n e c e s s a r y and s u f f i c i e n t £ , ~ ,

in s t a t e m e n t

is n e c e s s a r y

0

We have t h e r e f o r e

(a")

(5.5.21)

stability.

s u f f i c i e n c y suppose , £ , N(ri'O) - 7. N(gi'O) i=l i=l

and h e n c e

that

e ~ 0

For £

then

implies

point

to c o n d i t i o n

2

criterion.

relationships,

(5.1.21)

and

(5.1.22),

92 condition

(c")

is c l e a r l y

and also c o n d i t i o n

equivalent

(3b) in e i t h e r

to c o n d i t i o n

statement

(3a)

of the s t a b i l i t y

criterion. Therefore stability (ib)

to c o m p l e t e

criterion

is e q u i v a l e n t

the p r o o f of s t a t e m e n t

all that is left is to show that c o n d i t i o n to c o n d i t i o n

s h o w that w h e n c o n d i t i o n s the c o n d i t i o n £ , E N(r i, O) i=l where

=

ZG = e

To do this we will

(c") and

(d")

zeros of G(s),

for c l o s e d - l o o p

(5.1.22)

are s a t i s f i e d ,

(5.5.25)

= - Z G,

of r i g h t h a l f - p l a n e

and s u f f i c i e n t

From relationship

(a").

(b"),

£ , Z N(gi,-k) i=l

Z G is the n u m b e r

is n e c e s s a r y

1 of the

stability.

we have that

+ Z P, i=l gi

(5.5.26)

where

P*~ is the n u m b e r of right h a l f - p l a n e p o l e s of gj(s), gj , , but from e q u a t i o n (5.5.3) the p o l e s of ri(s) and gi(s) are the same, ZG

and h e n c e =

e

Z Z P*r. i=l l

+

If we

now

combine

which

is v a l i d

(5.5.27)

equation

(5.5.27)

for { j = 1 , 2 , . . . , £ } ,

Z N(ri,O)~. = Z Z*r. i=l i=l 1 To e s t a b l i s h

+

e

the n e c e s s i t y

with equation

we have (5.5.28)

- ZG of c o n d i t i o n

that £ , F N(ri,O ) ~ i=l then from e q u a t i o n i iZ#l~

~ i

O

-Z G (5.5.28) and/or

this e ~

implies O

(5.5.15),

that

(5.5.25)

suppose

93

and hence we conclude for c l o s e d - l o o p

that c o n d i t i o n

from e q u a t i o n £ Z* = e i=l ri

and hence

(b"),

sufficiency This inverse

(5.5.28) =

(c")

Nyquist

Statement

we have

and

(d")

the p r o o f

providing Thus

the

is established.

of s t a t e m e n t

1 of the g e n e r a l i z e d

criterion.

2 of the s t a b i l i t y in w h i c h

stable,

are satisfied.

(5.5.25)

stability

see chapter 4]

that

is c l o s e d - l o o p

of c o n d i t i o n

completes

that

O

that the system

conditions

is n e c e s s a r y

stability.

For s u f f i c i e n c y suppose £ , N(ri,O) = -Z G i=l then

(5.5.28)

•

criterion

applies

to systems

either

(i)

Pd(S)

= Px(S)

(5.5.29)

(ii)

Pd(S)

= Px(S)

where

the zeros

or

all In these and

in the

situations

Pd I

(s)

Pd 2

(s).

"''Pd h

(s)

(5.5.30)

of {Pdi(S) :i=i,2,...,£}

are

left half-plane.

it is possible

to combine

conditions

(la)

(4), of s t a t e m e n t i, into the single e q u i v a l e n t condition i h * w N(g i, -k) - Z N(gi,O) = - Z PG (5.5.31) i=l i=l i=l i

94 The n e c e s s i t y

and s u f f i c i e n c y

loop s t a b i l i t y is p r o v e d

when

of c o n d i t i o n

conditions

(2) and

(5.5.31)

(3) are s a t i s f i e d

as follows.

F r o m e q u a t i o n s (4.2.37) and h £ PG = e + Z P + i=l i i=l f i But the poles

of {fi(s):

poles of {gi(s): the zeros of

(4.2.39) Pd

(5.5.32)

i~l,2,...,Z}

i=1,2,...,£}

we have that

which

{gi(s) : i = 1 , 2 , . . . , £ } ,

are the same as the

in turn are the same as and t h e r e f o r e

(5.5.32) can be r e w r i t t e n as h ~iPGi = e + Z Z* + Pd i i=l gi Equation

for c l o s e d -

(5.5.33)

to give £ , Z N(ri,O) i=l

equation

(5.5.33)

can now be c o m b i n e d w i t h e q u a t i o n £ , Z N(gi,O) i=l

-

(5.5.19)

£ h = Z Z* + e + Pd - Z=IPGi i=l ri i

(5.5.34) w h i c h u s i n g e q u a t i o n (5.5.24) b e c o m e s , i , £ Z N(gf-k) - 2 N(gi,O) = Z Z* + e +Pd i=l i=l i=l ri

h - __ZIPGi i (5.5.35)

To e s t a b l i s h s u p p o s e that £ , N(gi,-k) i=l then

the n e c e s s i t y £ , - Z N(g,O) i=l

of c o n d i t i o n

(5.5.31)

h ~ -Z PG. i=l l

from e q u a t i o n (5.5.35) this implies £ Z Z* ~ O, or e ~ O, or Pd ~ O

that

i=l rl

or any c o m b i n a t i o n closed-loop is n e c e s s a r y

of these,

unstable.

and thus

that the s y s t e m w i l l be

H e n c e we c o n c l u d e

for c l o s e d - l o o p

stability.

that c o n d i t i o n

(5.5.31)

95

For s u f f i c i e n c y s u p p o s e t h a t , £ , h N(gi,-k) Z N(gi,O) = Z PG i=l i=l i=l i then f r o m e q u a t i o n (5.5.35) we m u s t h a v e t h a t £ Z Z* = e = Pd = 0 i=l r i and h e n c e the s y s t e m is s t a b l e p r o v i d i n g and

(3) are s a t i s f i e d .

(5.5.31)

2 of the

c r i t e r i o n all t h a t is left is to s h o w the e q u i v a l e n c e (la) a n d

that w h e n c o n d i t i o n s i.e. i , Z N(ri,O) i=l

where

T h u s the s u f f i c i e n c y of c o n d i t i o n

to c o m p l e t e the p r o o f of s t a t e m e n t

between conditions

(ib)

(2)

is e s t a b l i s h e d .

Therefore stability

conditions

(ib).

(2) and

(3) are s a t i s f i e d ,

£ , ~ N(gi,-k) i=l

=

To do t h i s we w i l l

h = - ~ ZG. i=l l

show

condition

(5.5.36)

zG

is the n u m b e r of r i g h t h a l f - p l a n e p o l e s of G. (s), is i 1 n e c e s s a r y and s u f f i c i e n t for c l o s e d - l o o p s t a b i l i t y . From equation Z N(ri,O) i=l

(5.5.15), w h i c h

=

Z N(gi,-k) i=l

=

is v a l i d for Z Z* i=l ri

{j=1,2,...,£},

- Z P* i=l ri (5.5.37)

and since the p o l e s the p o l e s of rewritten

of

{gi(s):

{r. (s) : i=1,2,...,£} 1

i=1,2,...,~}

are

equation

(5.5.37)

=

Z Z* i=l ri

If the s u b s y s t e m s

- Z P* i=l gi

as

can be

{Gi(s):

(5.5.38)

i=l,2,...,h}

are e a c h s q u a r e

t h e n P x ( = P d ) is the n u m b e r of r i g h t h a l f - p l a n e

G(s)

same

as

Z N(gi,-k) i=l

poles)

the

w h i c h are lost t h r o u g h p o l e - z e r o

is formed.

Therefore

zeros

cancellations

(or when

if the n u m b e r of r i g h t h a l f - p l a n e

96

zeros of G(s) ZG

=

is g i v e n [ e q u a t i o n 9,

e

+

Z

(3.3.36)] by (5.5.39)

Z

i=l gi t h e n we m u s t h a v e h ZG. = e i=l l or,

+

~ Z i=l 9i

since the p o l e s of gi(s)

zeros,

+

Pd

(5.5.40)

are i d e n t i c a l l y

e q u a l to the

of gi(s), h ZG

i=l

=

e

+

Z P* i=l g i

i

Consequently

equations

to g i v e £ , Z N(gi,-k) i=l

=

+

(5.5.38)

Z ~ Z* i=l ri

+

Pd

(5.5.41)

and

e

(5.5.41)

+

Pd

can be c o m b i n e d

h - ~iZGi i (5.5.42)

To e s t a b l i s h the n e c e s s i t y of c o n d i t i o n

(5.5.36)

suppose

that Z . Z N(gi,-k) i=l then

h

~

-

~ ZG i=l i

f r o m e q u a t i o n (5.5.42) t h i s i m p l i e s £ E Z* ~ O, or e ~ O, or Pd ~ O i=l r i

that

or any c o m b i n a t i o n of t h e s e and t h u s that the s y s t e m w i l l be closed-loop (5.5.42)

unstable.

is n e c e s s a r y

H e n c e we c o n c l u d e t h a t c o n d i t i o n for c l o s e d - l o o p

stability.

For s u f f i c i e n c y s u p p o s e that £ , h N(gi,-k) = - Z ZG i=l i=l l then f r o m e q u a t i o n (5.5.42) we m u s t h a v e t h a t £ Z* = e = Pd = O i=l r i and h e n c e the s y s t e m is c l o s e d - l o o p conditions

(2) and

stable,

(3) are s a t i s f i e d .

providing

T h u s the s u f f i c i e n c y

97 of c o n d i t i o n

(5.5.36)

This c o m p l e t e s inverse N y q u i s t 5.6

the p r o o f of s t a t e m e n t

stability

2 of the g e n e r a l i z e d

criterion.

Example To i l l u s t r a t e

example

the

stability

used in c h a p t e r s

G(s) The p o l e G(s)

is e s t a b l i s h e d .

=

Gl(S)=

criteria

consider

the

system

3 and 4 w h e r e

1 1.25(s+i) (s+2)

and zero p o l y n o m i a l s

[ s-i -6

s ] s-2

for the o p e n - l o o p

gain m a t r i x

are PG(S)

=

(s+l) (s+2)

and

zG(s)

= 1

gain

loci are shown in figure

so that PG = O

and

The i n v e r s e There

Z G = O.

characteristic

are no u n o b s e r v a b l e

and t h e r e f o r e is d i r e c t l y

either

statement

applicable.

only one subsystem,

and/or

uncontrollable

1 or s t a t e m e n t

19.

modes

2 of the c r i t e r i o n

In fact, b e c a u s e we have e f f e c t i v e l y

there

is no d i f f e r e n c e

between

the two

statements. Testing generalized

the c o n d i t i o n s inverse N y q u i s t

the c l o s e d - l o o p -1.875 O 2.5

These

bounds

stability

stability

is stable

as o u t l i n e d

criterion

by the

we find that

for

< k ~ 0 < k<

and

obtained

system

for s t a b i l i t y

1.25

< k < on the g a i n c o n t r o l

in s e c t i o n

4.3

using

variable

k agree w i t h

the g e n e r a l i z e d

those

Nyquist

criterion.

It is i n t e r e s t i n g

to n o t e that

an i n v e r s e N y q u i s t - l i k e

98

g-plane

ca.~ 5.o~

~.,, "%.

\ ..... . /

Figure 19. Inverse characteristic gain loci

99 stability criterion has recently been used by Mees and Rapp [6] to establish stability criteria for multiple-loop nonlinear feedback systems.

References

[1]

[2] [3]

A.L. Whiteley, "Fundamental Principles of Automatic Regulators and Servo Mechanisms", J. IEE, 5-22, 1947. H.H. Rosenbrock, "State Space and Multivariable Theory", Nelson, London, 1970. A.G.J.

MacFarlane,

"Dynamical System Models", Harrap, London,

1970.

[4]

Is] [6]

A.G.J. MacFarlane, "Return-difference and return-ratio matrices and their use in analysis and design of multivariable feedback control systems", Proc. IEE, 117, 2037-2049, 1970. E. Hille, "Analytic Function Theory", Vol. i, Ginn and Go., U.S.A., 1959. A.I. Mees and P.E. Rapp, "Stability criteria for multipleloop nonlinear feedback systems", Proc. IFAC Fourth Multivariable Technological Systems Symposium, Fredericton, Canada, 1977.

6.

Multivariable The Evans'

established design

root loci

root locus approach

graphical

technique

systems

as a function

single-input

for estimating

steps towards

the system closed-loop

case has caused

characteristic

frequency

loci

[6]

point of view.

In this chapter

well established

results

to be determined

and

behaviour

of the

is developed,

function

theory

using

[7;8;9],

and the angles of

of the characteristic

from a Laplace

loci

transfer

frequency

loci

function matrix

of the system.

the asymptotic

behaviour

a multivariable on the input

small numbers

time-invariant

closed-loop

linear regulator criterion

seems particularly

of inputs

can be utilised

of the optimal

in the performance

The method

6.1

frequency

and approach

a method

It is also shown how the method

enough

interest

; both from a state-space

in algebraic

the asymptotic

and approach

description

Its generaliza%ion

considerable

of the characteristic

[4; 5] , and the angles of departure

departure

poles

this end have been made with regard to the

behaviour

which allows

and

single-output

of the gain control variable.

to the multivariable

asymptotic

is a well

used in the analysis

of linear time-invariant

feedback

[1;2;3]

useful

to find poles of

as the weight

approaches

zero [ io].

for systems with

since the calculations

are then simple

to be carried out by hand. Theoretical

background

For the general

feedback

has been shown in sub-section

configuration

of figure

3 it

3.3-4 that the characteristic

I01 frequency contours

loci

(multivariable

root loci)

of the characteristic

are the 180 ° phase

gain function

g(s), where g(s)

is defined via the equation ~(g,s)

~

det [gI m - G ( s ) ]

O

=

(6.1.1)

or the equation ~(g,s)

= bo(S)~(g,s)

where we have multiplied of the rational For simplicity

(6.1.2)

through by the least common denominator

coefficients

in s.

of exposition,

the usual situation from practical irreducible

= O

and because

for transfer

situations,

function matrices

gain function

control variable

g(s)

functions

and solutions

for g in equation

= k -m F(k,s)

= O

we have (6.1.4)

= O

(6.1.5)

of the closed-loop apart

real k, determine

the dependence

poles on the gain control variable

from possible

of this dependence

loci of the given

no single-point equivalent

(6.1.2)

of

for s in terms of positive

frequency

The

(6.1.3)

~(-k-l,s)

description

in s.

k via the expression

so that substituting

Therefore,

is

is related to the gain

1

F(k,s)

arising

it is assumed that ~(g,s)

over the field of rational

characteristic

this is in any case

single-point constitutes

system.

loci then equation

loci,

the graphical

the characteristic

Note that if there are (6.1.5)

is directly

to

CLCP(s)~det[SIn-Ac]=

k.

det[SIn-S(-k-l)]

= O (6.1.6)

I02 and the c h a r a c t e r i s t i c

frequency

E a c h of the n b r a n c h e s

loci have n branches.

of the c h a r a c t e r i s t i c

loci can in t h e o r y be r e p r e s e n t e d si(k) where

j= ~

branches. point

= ui(k)

The t a n g e n t

s I approaches

i are labels

as the l i m i t i n g

the branch,

that

position

20),

(Sl-So)./~k, w h e r e 8 k > O ,

of the

Sl=Sr(ko+~k)

is as gk+O.

Sl-S ° can be r e p r e s e n t e d

from s O to s I (figure

for the v a r i o u s

of the loci at a

s o and a n o t h e r p o i n t

s o along

the c o m p l e x n u m b e r

(6.1.7)

to the rth b r a n c h

is d e f i n e d

line t h r o u g h

in the form

i=l,2, .... n

and the s u b s c r i p t s

So=Sr(ko)

straight

+ Jvi(k)

frequency

and the v e c t o r

as

NOW

by the v e c t o r

corresponding

has the same d i r e c t i o n

to

as that vector.

Si= 5 (ko+6k)

i\soo o Figure 20. To derive a brmuta for the tangent to the characteristic frequency loci

It follows

therefore

that the v e c t o r

st(k°) = dkdSr(k)Jk o

=

~k÷Olim

corresponding

sl~_k-so

to

103

=

lim 6k÷O

Sr(ko+~k) Sk

is tangent to the branch at s vector and the positive %

=

argument

Formula

o

(6.1.8)

and the angle

e between this

real axis is given by

{Sr(ko) }

(6.1.9)

(6.1.9)

is fundamental

the asymptotic behaviour

to the determination

of the c h a r a c t e r i s t i c

loci, and the angles of departure For k=O, equation

- Sr(k O)

(6.1.3)

implies that the c h a r a c t e r i s t i c

loci start at poles of the open-loop

therefore

the angle of departure

a pole is given by formula (6.1.3)

loci terminate

implies

frequency

and approach of the loci.

frequency

equation

of

system,

and

of a branch of the loci from

(6.1.9) w i t h ko=O.

For k=~,

that the c h a r a c t e r i s t i c

at system zeros.

Therefore

of the loci t e r m i n a t i n g at a finite

zero,

frequency

for a branch

formula

(6.1.9),

with ko=~, gives the angle of approach of the branch to the zero.

If a branch terminates

at an infinite

zero then formula

(6.1.9), w i t h k =~, gives the angle that the asymptote o the branch makes with the positive Formula

(6.1.9)

because expressions frequency

for the separate branches

function

for the p r a c t i c a l

theory

[7;8;9],

construction

of a given point.

of

there exists

However,

a method

series representations function

in the n e i g h b o u r h o o d

The m e t h o d consists of repeatedly using

a "Newton diagram"

once,

of the characteristic

in general be found explicitly.

for the branches of an algebraic

in the series.

real axis.

does not at first seem to be useful,

loci cannot

in algebraic

to

to find the next most significant

Therefore,

approximations

terms

by using the Newton diagram just

can be obtained

for the branches of the

104

characteristic or zero s where

frequency

loci in the v i c i n i t y of a p o l e

(finite or i n f i n i t e ) ,

r

of the f o r m

(k) ~ a + bk ~

(6.1.10)

a and b are c o m p l e x n u m b e r s ,

number,

and w h e n e

understood.

~ is a r a t i o n a l r e a l

is f r a c t i o n a l the p r i n c i p a l

For the n e g a t i v e

feedback configuration

c o n s i d e r a t i o n k is real and p o s i t i v e , real,

and t h e r e f o r e

(6.1.10)

applying

root of k is under

~k e-I w i l l a l w a y s be

formula

(6.1.9)

to the a p p r o x i m a t i o n

we h a v e

0 =

argument

{b}

+ ~180 °

(6.1.11)

where [

=

O

when

e >

O

=

1

when

~ <

0

T

as a r e s u l t of the d i f f e r e n t i a t i o n In the f o l l o w i n g approximations

sections

of the forn% shown in e q u a t i o n

b e h a v i o u r of the c h a r a c t e r i s t i c

in

frequency

the

loci

and a p p r o a c h of the loci.

Asymptoti q behaviour The N e w t o n d i a g r a m is a g r a p h i c a l

can be u s e d to find e a c h of the m o s t series

representations

algebraic which

(6.1.10)

and h e n c e to d e t e r m i n e

and a l s o the a n g l e s of d e p a r t u r e 6.2

to k.

it is s h o w n h o w to o b t a i n

the v i c i n i t y of a p o l e or zero, asymptotic

with respect

significant

which

t e r m s in the

for the p a r t i c u l a r b r a n c h e s

f u n c t i o n q(v),

approach

construction

of an

in the v i c i n i t y of the origin,

zero as v a p p r o a c h e s

zero.

Therefore,

whether

f i n d i n g the a s y m p t o t i c b e h a v i o u r or the a n g l e s of d e p a r t u r e and a p p r o a c h of the the p r o b l e m ,

loci,

our f i r s t aim is a l w a y s

by a c h a n g e of v a r i a b l e s

to r e d u c e

in the c h a r a c t e r i s t i c

105 equation,

to one of f i n d i n g a p p r o x i m a t i o n s

an a l g e b r a i c

function which approach

variable approaches

for b r a n c h e s of

z e r o as the i n d e p e n d e n t

zero.

To d e t e r m i n e the a s y m p t o t i c b e h a v i o u r of the c h a r a c t e r i s t i c frequency

loci we n e e d to o b t a i n an a p p r o x i m a t i o n

(or b r a n c h e s ) infinity.

of the loci a b o u t the p o i n t

-i

(6.2.1)

in the c h a r a c t e r i s t i c (g,s)

= ~ (g,z -I)

equation = z -q

(6.1.2)

to o b t a i n

~(g,z)

(6.2.2)

q $ n is the n u m b e r of p o l e s of g(s),

neighbourhood excluded

as k a p p r o a c h e s

F o r this p u r p o s e we w i l l put

s=z

where

s =~,

to the b r a n c h

of the v a l u e

f r o m the region)

z = 0

so that in any

(the p o i n t

z = 0 itself

the e q u a t i o n # ( g , s ) = O

is

is e q u i v a l e n t

to the e q u a t i o n ~(g,z)

= E~xygXzy

For a s t r i c t l y p r o p e r

= O system,

(6.2.3) that

is one w h e r e D is zero,

or if D is s i n g u l a r

¢oo

(6.2.4)

= 0

o t h e r w i s e C o o is n o n - z e r o which, the N e w t o n d i a g r a m , behaviour.

corresponds

as we shall see later f r o m to t h e r e b e i n g no a s y m p t o t i c

N o t e t h a t if D is n o n - s i n g u l a r

g(s)

has t h e same

n u m b e r of f i n i t e z e r o s as p o l e s a n d t h e r e f o r e we w o u l d not expect

any c l o s e d - l o o p p o l e s to a p p r o a c h i n f i n i t y .

The n e x t ~(g,z)

step is to c o n s t r u c t the N e w t o n d i a g r a m for

, from which

approximations

of the f o r m (6.2.5)

z ~cg ~ can be o b t a i n e d , real n u m b e r .

where

c is a c o m p l e x n u m b e r ,

The p r o c e d u r e

N e w t o n d i a g r a m is as f o l l o w s In a

for c o n s t r u c t i n g

and ~ a r a t i o n a l the a p p r o p r i a t e

[8].

( g , z ) - p l a n e we p l o t the p o i n t s

(x,y)

for w h i c h

106

~xy ~ 0 in equation

(6.2.3).

(l-z+2z2-25z3+29z4)g 2 +

As an example,

coincide

the smallest

g-axis point P

until this line passes A straight

the point P1 until

z-axis

is shown

The tangent

about

(the point

on figure

21)

(2,0)

the points

is then rotated

away from P1 clockwise

in figure

about

another point P2 of

is repeated

The complete

Po and PI"

the points

until the vertical

Newton diagram

for equation

22.

of the acuteangle z-axis

which the straight determines

line

the first

value of ~ , and the tangent of the acute angle which

the straight

line PIP2 makes with the verticle

the second possible of figure

value of p , etc.

z-axis

determines

For the Newton

diagram

22 we have U 1 = 1 and ~2 = ½ "

For a particular mations

clockwise

line is then drawn between

PoP1 makes with the vertical possible

axis and rotated

it passes through

The procedure

is reached.

(6.2.6)

line is then made to

through another point P1 of our net.

z-axis,

A straight

P1 and P2"

(6.2.6)

line through P 1 ' and pointing

the vertical

our net.

o

O

line is then drawn between

A horizontal towazds

=

A straight

with the horizontal

for

(z-22z2+199z3-21Oz4)g

-(33z3-594z 4) are shown in figure 21.

the points

exponent

Pt there may be several

approxi-

of the form zit-citg

~t

(6.2.7)

Note that if some root of g is implied by ~t i.e. fraction,

then it is understood

being considered.

To determine

that the principal the coefficients

Pt is a root is cit it

107

Zl

4,

3:

2

I

C

i

2

a tJ

Figure 21. Points of the Newton diagram for equation (6.2.6) z 4

3P,

0

2

a

Figure 22. Complete Newton diagram for equation (6.2.6)

108

is n e c e s s a r y equation

to s u b s t i t u t e

(6.2.3)

z = cg

corresponding

lying on the link Pt-i Pt the s u b s t i t u t i o n

~t

in the terms of

to the p o i n t s

' to e q u a t e

of our net

to zero the result of

and to solve the r e s u l t i n g

the sum of the relevant terms of e q u a t i o n

equation.

(6.2.3)

Let

have the

form x x

y~

g oy~

z

gXlzY 1 + .....

w h e r e y > ......... >YI' the n u m b e r of i n f i n i t e the terms

correspond

x~-x I

Therefore similar

and ~ is a p o s i t i v e

-

less than

Then b e c a u s e

to the same link we have _x 2 - x 1

YO-I-Yl

........

all the terms

Y2 - Yl

for the d e t e r m i n a t i o n

~t

of the e x p r e s s i o n

as a r e s u l t of the s u b s t i t u t i o n

equation

integer

zeros of the system.

Xo_l-Xl

Yc-Yl

(6.2.8)

"+ ~xlY 1

(6.2.8)

z = cg

of the c o e f f i c i e n t

Ut

become

, and h e n c e

cit we o b t a i n

an

of t h ~ a f o r m Yl ~xgy c

or d i v i d i n g

+

by

c

........ + ~XlY 1 c

Yl

= O

(6.2.9)

(c ~ O) we o b t a i n

Y~-Yl

~xay

c

+ ....

+~XlY 1 =

w h i c h has y a - y I solutions. difference

between

Consequently

enables

axis.

(6.2.10)

Note that

the o r d i n a t e s

it is equal

on to the o r d i n a t e

0

of the p o i n t s

to the p r o j e c t i o n This

is a u s e f u l

us to see at a g l a n c e how m a n y

corresponding

to the e x p o n e n t

w i t h the p o i n t w h e r e

Yo-Yl

the final

~t"

is the Pt and Pt-l"

of the link P t _ i P t fact since

coefficients

there are

A l s o the o r d i n a t e

link reaches

it

associated

the v e r t i c a l

axis

109

must give the total number of coefficients approximations

considering

all exponents.

for all the This vertical

axis point on the final link is therefore

the number of

infinite zeros of the system. Having obtained the approximations

in the form of equation

(6.2.7)

it is a simple matter of substitution

(6.1.3)

and

(6.2.1)

sit - bit k representing

to obtain the required

from equations

form

~t

(6.2.11)

approximations

loci about s =~, as k÷~.

to the characteristic

If we now apply formula

frequency (6.1.9) we

obtain the angles which the asymptotes make w i t h the positive real axis. The asymptotes,

as will be shown in sub-section

group into B u t t e r w o r t h

configurations.

6.2-1,

Each pattern has

a common intercept which has been termed the "multivariable pivot" by Kouvaritakis for calculating of the system.

and Shaked [4] who also gave a m e t h o d

the pivot given the state space description In appendix

6 it is shown how the m u l t i v a r i a b l e

pivots can be derived from the characteristic ~(g,s) 6.2-1

equation

= O. B u t t e r w o r t h Patterns

In this sub-section

it is proved that the closed-loop

poles that go off to infinity do so along asymptotes which group into several B u t t e r w o r t h is based on well e s t a b l i s h e d theory

configurations.

results

in algebraic

The proof function

[8]

Let us consider the closed-loop c h a r a c t e r i s t i c defined by equation

(6.1.5)

i.e.

polynomial

110

F (k,s) in w h i c h point

=

O

(6.2.12)

we a l l o w k to be complex.

k c the a l g e b r a i c

(6.2.12)

has

a p- fold root

in a n e i g h b o u r h o o d where

function

N of k

defined

that

at a

by e q u a t i o n of s(k)

{s.(k):i=l,2,...,p,...,n} l

{si(kc)=Sc:i=l,2,...,p}. If we a n a l y t i c a l l y

{si(k)

: i=p+l,...,n}

we return continue

Sl(k)

say,

around

around

and after

encirclements

constitute function

t . . . .

a second

o

a finite function

cyclical The

number

~i' of

Sl(k).

(6.2.13)

system

in N.

cyclical

all the functions

: i=2,3,...,p}

one of the functions

of b r a n c h e s

of the a l g e b r a i c

N of kc). s l+l(k)

Repeating system

and a n a l y t i c a l l y

the p r e v i o u s

of branches,

S~l+l(k)' .... ' s~2(k) Finally

after one

the functions

(in the n e i g h b o u r h o o d

k

we a n a l y t i c a l l y

{si(k)

S~l (k)

it around

we obtain

k c then

After

to the original

If 91

point

encirclement

say that

one e n c i r c l e m e n t

If, however,

s3(k ) say.

a cyclical

s(k)

after

one of the functions

we return

,

k c in N, then

function.

another

case we shall

Sl(k)

any one of the b r a n c h e s

the critical

we obtain

{si(k ) : i=3,4,...p},

In this

continue

to the o r i g i n a l

encirclement s2(k)

s(k)

suppose

So, and let the n branches be

c

Also

arguments

namely (6.2.14)

Sl(k), ..... Sp(k)

will be sorted

into

systems. functions

into one another

of each

according

cyclical

system

to the cyclical

pass

consecutively

law when

k goes

,

111

round the m u l t i p l e

p o i n t k c.

of b r a n c h e s

defines

function.

The n u m b e r

function

at a p o i n t

Hence

in the n e i g h b o u r h o o d of v a l u e s

in the n e i g h b o u r h o o d comprising

considered.

a single b r a n c h

the c o r r e s p o n d i n g and i d e n t i c a l Suppose

analytic

N of k

radial

the c y c l i c a l

analytic

s y s t e m being

comprises

a cyclical

the n e i g h b o u r h o o d

N is r e p l a c e d

that each n e i g h b o u r h o o d

23.

N is c i r c u l a r

point k

c

Then the n e i g h b o u r h o o d s

along the edges of the cuts so that

continue

the jth b r a n c h

(j
around

after one e n c i r c l e m e n t

can be j o i n e d

if we a n a l y t i c a l l y

k c on the jth sheet,

across

the cut into the returning

the jth branch.

The pile of n e i g h b o u r h o o d s

obviously

falls

containing

24, and each cycle

functions

defined

To d e n o t e

an a n a l y t i c

function

Sl(k) , .... , s l(k)

one or m o r e

is a d o m a i n

function

corresponding

When we c o n s i d e r

as shown

in

by a c y c l i c a l

subscript.

For e x a m p l e

to the c y c l i c a l by the symbol

the a s y m p t o t i c

to

N of k . c

determined

numerical

w i l l be d e n o t e d

sheets

(j+l)th

for one of the a n a l y t i c

in the n e i g h b o u r h o o d

system we w i l l use a Roman the a n a l y t i c

etc.

and has a

eventually

into cycles

sheet,

: i=l,2,...,n}

to its p e r i p h e r y ,

together

(j+l)th

{si(k)

the p a i r [k,si(k) ] .

in figure

b r a n c h on the

by a p i l e of

one for each of the f u n c t i o n s

as shown

figure

system,

f u n c t i o n w i l l be s i n g l e - v a l u e d

line cut from the c r i t i c a l

we move

an

N is equal to the

and let a p o i n t on the ith sheet r e p r e s e n t suppose

c

system

w i t h this branch.

n such regions,

Also

cyclical

g i v e n by this a n a l y t i c

n u m b e r of b r a n c h e s When

every

behaviour

system

si(k).

of the

112

L cut

Figure 23. Neighbourhood N of the critical point k c kc

Figure 24. Cyctes of neighbourhoods for the critical point k c characteristic branches

of the algebraic

infinity,

p

branches branches

systems.

Theorem

Suppose we have

will be arranged

into several

p

arguments cyclical

system is defined by an analytic is given by the f o l l o w i n g

[8, page 39] 4

The infinite constituting critical

infinity.

the form of which

theorem

s(k) which approach

for k =~, then by the previous

Each cyclical

function,

loci, we are concerned with the

function

as k approaches

infinite the

frequency

branches

a cyclical

point kc=~,

of the algebraic

function

system in the neighbourhood

in their totality

constitute

s(k), of its

an analytic

113

function given by a series of the form 1

-i

s I = k 9 (bO + b I k

-2

V +b2k

~ +...)

where v is the number of branches positive

(6.2.15)

in the cycle,

and I is a

integer.

Consequently

the infinite branches

particular cyclical

corresponding

system in the n e i g h b o u r h o o d

be a p p r o x i m a t e d by an analytic

to a

of k =~, can

function of the form

1 m

s I (k) ~bok~

(6.2.16)

The characteristic algebraic

function

s(k)

frequency

loci are values of the

for k going from zero to infinity

along the positive real axis. branches of the frequency

Therefore

the infinite

loci will occur in cyclical

systems which can be a p p r o x i m a t e d by equations (6.2.16).

Taking the ~ roots of k in equation

we find that the asymptotes

corresponding

themselves

of order

that is,we have ~ asymptotes

by angles of

into a B u t t e r w o r t h

as already discussed,

systems of branches

and therefore

configuration equally

spaced

.360. ° (--~-), in the complex plane.

In general, cyclical

(6.2.16)

to a cyclical

system arrange ~ ;

of the form

the asymptotes

loci will arrange themselves

there will be several

in the n e i g h b o u r h o o d of the characteristic

of k = ~, frequency

into several B u t t e r w o r t h

configurations. 6.3

Angles of d e p a r t u r e and approach To determine

the characteristic

the angles of departure frequency

and approach of

loci we need to obtain approximations

to the branches of the loci at the poles and finite zeros

114

of the

system.

departure

We w i l l

first

f r o m the o p e n - l o o p

Suppose

for the

consider

poles

the

of the

characteristic

angles

system.

equation

~(g,s ) = O we

have

s'=s-8

a pole

and also make (6.3.1)

the

=

d = s' = O,

we

and

If w e

are e x a m i n i n g

#(g,s)

so t h a t

(6.3.1) equation

construct

~

=

now

to the

reduces of

,

to o = O

the N e w t o n

diagram

of the

s =

From

substitution

approximation

case w h e r e

(excluding

(6.3.3) for the e q u a t i o n

(or a p p r o x i m a t i o n s

in the

case

of

form

e

is a c o m p l e x

equation

departing

number

(6.1.3),

g = d -I

to the b r a n c h

loci

8

(6.3.2)

(6.3.4)

where

number.

O

to the e q u a t i o n

ed ~

is o b t a i n e d ,

frequency

d - m H ( d , s ')

= O is e q u i v a l e n t

Z ~ x y d X s 'y = O

poles)

s'

a n d the

g = d -1,

=

= O an a p p r o x i m a t i o n

multiple

real

substitution

We w i l l m a k e

for e q u a t i o n

in the n e i g h b o u r h o o d

equation

H(d,s') =

Z(d,s')

at s = ~.

variable

~(d-l,s'+B)

situation

itself)

independent

pole)

becomes

~(g,s) The

(6.3.1) (or m u l t i p l e

the n e w

of

, we

the

change

therefore

(or b r a n c h e s )

from

and m a r a t i o n a l

the p o l e

of v a r i a b l e ,

arrive

at an

of the c h a r a c t e r i s t i c

B , in the

form

Hd s - B + bdk If w e n o w a p p l y given

(6.3.5) formula

(6.1.9)

the

angle

of d e p a r t u r e

8d is

as ed

=

We w i l l

argument now

{b d}

consider

(6.3.6) the

angles

of

approach

to the

finite

115

zeros

of the

equation

(6.3.1)

We w i l l that

system.

take

we h a v e

~(g,s)

a zero

characteristic

(or m u l t i p l e

reduces

the n e i g h b o u r h o o d

=

of s = 7

=

~×xy

construct

approximation

x(g,s')

to the

= O is e q u i v a l e n t x(g's')

zero)

variable

O

where

real n u m b e r . we

(excluding

to the gXs'Y= O

branches) the

zero

g = s' = O,

y itself)

af the

,×

= 0

oo

diagram

given

that

(6.3.8) for x(g,s')

an

a n d n is a r a t i o n a l

a n d the c h a n g e

of v a r i a b l e

to the b r a n c h

frequency

loci

(or

approaching

form

s -

y + b ak~a

formula

(6.1.9)

Ua w i l l

180 ° t e r m

definition

at the

Example Consider

=

(6.3.10) the

argument

always

angle

of a p p r o a c h

8 a is

angle

which zero,

{b a} ~ 180 ° (6.3.11)

be n e g a t i v e

in f o r m u l a

for the

definition

positioned 6.4

equation

as

of the

usual

number

(6.1.3)

characteristic

ea Note

in

(6.3.9)

at an a p p r o x i m a t i o n

, in the

If w e n o w a p p l y

and

equation

p is a c o m p l e x

arrive

s = y

so

form

From equation

therefore

.

(6.3.7)

oase w h e r e

the N e w t o n

of the

=

s' ~ pg~ is o b t a i n e d

at s = y

becomes

= ~(g,s'+y)

The s i t u a t i o n

If we

for the

s' = s-y as the n e w i n d e p e n d e n t

the e q u a t i o n

~(~s)

Suppose

and h e n c e

(6.3.11).

of a p p r o a c h

Also differs

is the d i r e c t i o n to see the

locus

1 an o p e n - l o o p

gain matrix

the

note by

you would arrive.

presence that

180 ° look,

this f r o m the when

116

0s

l

[3s3+s21s

s4+5s3-2s2-44s+40

8s22s+l 1

s3+79s2+44s-868

-4s3-4s2+4Os-32

w h i c h has a c h a r a c t e r i s t i c e q u a t i o n ~(g,s)

=

( s 4 + 5 s 3 - 2 s 2 - 4 4 s + 4 0 ) g2 +

(s3+l16s-432)g

-12 (s2-2s+2) (a)

T_o d e t e r m i n e Putting s = z

~(g,z)

=

= O

the a s y m p t o t i c b e h a v i o u r -i

in the c h a r a c t e r i s t i c

(l+5z-2z2-44z3+4Oz4)g 2 +

e q u a t i o n we o b t a i n

(z+l16z3-432z4)g

-12 (z2-2z3+2z 4) = O The N e w t o n d i a g r a m for ~(g,z) we o b t a i n

is s h o w n in f i g u r e 25 from w h i c h

~ = 1 , and h e n c e the a p p r o x i m a t i o n z - cg

The c o e f f i c i e n t

c is c a l c u l a t e d

in s e c t i o n

to h a v e the v a l u e s

6.2)

(using the m e t h o d -71 and ~1 .

resubstituting

for z and g = - k -I

approximations

for the c h a r a c t e r i s t i c

s -

4k

and

Two b r a n c h e s

s

-

-3k

as

described

Therefore,

, we h a v e the f o l l o w i n g

k

+

of the c h a r a c t e r i s t i c

frequency

loci

~

frequency

loci t h e r e f o r e

m o v e off to i n f i n i t y at a n g l e s of O ° and 180 ° to the p o s i t i v e real axis.

\ o

~

S

Figure 25. Newton diagram for ~z(g,z)

117

(b)

To

find

The

system

s = i, Pole

at

the

an@les

has

s = 2

of departure

four

open-loop

, s = -4+2j

, and

s = -4-~

s = 1

Putting equation

we

~l(d,s')

=

s' = s - i

and g = d

-i

characteristic

(S,4+gs,3+19S'2-29S ') + ( s ' 3 + 3 s ' 2 + l l g s ' - 3 1 5 ) d

The Newton we

in t h e

obtain

12(s'2+l)d 2 = O

-

which

poles

diagram

obtain

for

Hl(d,s')

is

shown

e I = 1 , and hence

in

figure

the

approximation

(using

the method

26

from

s' ~ e l d The

coefficient

in s e c t i o n following

e I is

6.2)

calculated

to h a v e

approximation

the

value

to t h e

the

the p o l e Pole

at

pole

s = i.

in t h e

frequency

loci

k

Therefore

the

angle

of departure

from

s = 1 is 0 °. s = 2

Putting equation

we

E2(d,s' ) =

s'

= s-2

and

g=

d -I

The N e w t o n we

in

the

characteristic

obtain (s,4+13s,3+52s,2+4Os

,) + -

which

resulting

characteristic

s - 1 + 10.86 about

-10.86,

described

diagram

obtain

for

~2 =

1

E2(d,s') , and

(s'3+6s'2+128s'-192)d

12(s'2+2s'+2)d is

hence

shown the

in

2 = O

figure

27

from

approximation

s' ~ e 2 d The

coefficient

resulting

in t h e

frequency

loci

e 2 is c a l c u l a t e d following

to have

approximation

the

value

to t h e

4.8,

characteristic

118 s-2-4.Sk about

the pole

the pole Pole

s = 2 .

Therefore

the angle

of d e p a r t u r e

from

s = 2 is 1 8 0 ° .

a t s = -4+2j Putting

s' = s+4-2j

a n d g = d -I in the c h a r a c t e r i s t i c

equation

we obtain

H3(d's')

= [s'4+(-ll+j8) s'3+(lO-j60)s'2+(88+j104)s'] + [s' 3+ (_12+96) s' 2+ (152-j 48) s ' + (-912+9 320)] d

-12 I s ' 2+ (-lO+j 4) s ' + ( 2 2 - 9 2 0 ) ] d 2 = 0 The Newton which

diagram

we o b t a i n

for

23(d,s')

~3 = i, a n d h e n c e s' ~-

The

is s h o w n

coefficient

e

in f i g u r e

28 f r o m

the approximation

3d

e 3 is c a l c u l a t e d

to h a v e

the value

912 - j 3 2 0 88 + j l O 4 resulting

in the

frequency

loci

following

approximation

s - -4+2j about

the pole

from the pole

s = -4+2j. s = -4+29 argument

Pole

(-912+j320) (88+9104)

Therefore

k

the

angle

=

110.9

of d e p a r t u r e

is -912+j32Q~ { 88+9104 j

o

at s = - 4 - 2 j By symmetry

s = -4-2j (c)

+

to the characteristic

angle

of d e p a r t u r e

is - 1 1 0 . 9 ° .

To f i n d The

the

the angles

system

has

two

s = l+j

of a p p r o a c h finite and

zeros s = l-j

from the pole

119

4i 3

2,

\ L

o

2

d

(

I

2

=d

Figure 27 Newton diagram for E2(d,s')

Figure 26. Newton diagram for ~l(d,s')

$'

$' 4'

,\ 0

I

2

O

"d

I

2

---

8

Figure 28. Newton

Figure 29. Newton

diagram for E3(d,s')

diagram for X(g,s') 3-0 2.0

2/

I

I

6'0

7"0

Figure 30. Complete characteristic frequency loci

120

Zero at s = i ~ Putting

s' = s-l-j

in the c h a r a c t e r i s t i c

e q u a t i o n we

obtain ×(g,s')

=

[ S , 4 + ( 9 + j 4 ) s , 3 + ( 1 3 + j 2 9 ) S , 2 + ( _ 5 8 + j 4 6 ) S , + ( _ 1 8 _ j 3 8 ) ] g2 +[s'3+(3+j3)s'2+(l16+j8)s'+(-318+j118)] g

12[s2÷j2s '] = The N e w t o n d i a g r a m w h i c h we o b t a i n

0

for x(g,s')

is shown

n = 1 , and h e n c e

in f i g u r e

29 f r o m

the a p p r o x i m a t i o n

s' - pg The c o e f f i c i e n t p is c a l c u l a t e d to h a v e the v a l u e -318 + jl18 j24 resulting

in the f o l l o w i n g a p p r o x i m a t i o n

to the c h a r a c t e r i s t i c

f r e q u e n c y loci s - 1 + j + a b o u t the z e r o s = l+j. to the

(318-jI18) j24

k -i

T h e r e f o r e the a n g l e of a p p r o a c h

zero s = l+j is argument

f318-jl18% _ L j24 ~ +

180 ° E

69.64 °

Zero at s = l-j By s y m m e t r y the a n g l e of a p p r o a c h

to the zero at

s = l-j is - 6 9 . 6 4 ° . To c h e c k the r e s u l t s frequency

loci are shown,

f r e q u e n c y plane, a b o u t the p o l e poles

in f i g u r e

the c o m p l e t e

characteristic

projected onto a single complex 30.

Branches

s = 1 m a k i n g the m o v e m e n t

difficult

to c o m p r e h e n d .

The

c h a r a c t e r i z e d by c o n s t a n t p h a s e c o n t o u r s of g(s)

is t h e r e f o r e

of the loci c o i n c i d e of the c l o s e d - l o o p

frequency surface

and c o n s t a n t m a g n i t u d e

shown

in f i g u r e s

31(a),

(b)

121

ChQracterJstic ×:" ~: frequency loci

(a)

I Cutjoining branch points

(b)

Figure 31.Frequency surface (a)sheetl (b)sheet2

122 2.1

f °4-05

1.95 4'0

I -3"95

] -3"9

Figure 32. Angle of departure from pole s=-4÷2j iI J 1.0

0.98

0.9~

0-94

0"9~ 0"90 0.90

Figure 33. Angle of approach to zero s=l÷j

123

to give a clearer description loci.

Small regions

s = -4+2j

in the neighbourhood

and the zero s = l+j

33 to verify the calculated 6.5

of the characteristic

Asymptotic

behaviour

In this section to determine

of the pole

are shown in figures

angles

of departure

the "Newton diagram" behaviour

loop poles of a multivariable

zero.

the optimal

The method

characteristic

of an appropriate is equivalent simplicity algebraic

function

Consider

closed-

linear regulator, criterion

is based on an association

of

loci with the branches

function.

Although

to that given by K w a k e r n a ~

of the approach

is used

of the optimal

time-invariant

frequency

algebraic

poles

approach

as the weight on the input in the performance approaches

32 and

and approach.

of optimal, closed-loop

the asymptotic

frequency

is emphasized

the procedure

[Ii] the essential in the setting of

theory.

the stabilizable

and detectable

time-invariant

linear system dx(t) dt y(t)

_

A x(t)

where

Q

(6.5.1) (6.5.2)

= and

criterion

/o~T(t)Qy(t)+puT(t) R

are positive

and the superscript vector.

Bu(t)

= C x(t)

and the performance V(~)

+

T

Ru(t) ] definite

denotes

Then it is well known

at

(6.5.3)

symmetric

the transpose (e.g.

[12])

matrices,

of a matrix

that the optimal

control action is given by u(t) where

P

= - ! P

R-IB T Px(t)

is the unique positive

or

(6.5.4) semi-definite

solution

of

124

the steady state matrix Riccati equation -PA -ATp

+ ~PBR-IBTp = cTQc P

Kwakernaak

[ii, equation

loop characteristic polynomial

(6.5.5)

(16)] has related the closed(denoted by ~c(S))

for the

optimal regulator to the open-loop characteristic polynomial (denoted by ~o(S)) as follows

~c(S)~c(-S)

where I inputs,

= ~o(S)~o(-S)

(-s) QG (s) ] det [I m + IR-IGT P (6.5.6)

is a unit matrix of order m, the number of system

m

and

G(s)

=

C(sI - A)-IB

(6.5.7)

is the open-loop transfer function or gain matrix of the system. It is obvious from

(6.5.6) that the poles of the optimal closed-

loop system, dependent on the input weighing,

are values of

s in the left half-plane which satisfy det [ Im + ~H(s) ] P

=

O

(6.5.8)

where H(s) ~ R-IGT (-s) QG (s)

(6.5.9)

Consider now the characteristic equation defining the eigenvalues of H(s), that is &(~,s)

~ d e t [ q I s -H(s)]

=

0

(6.5.10)

or by expanding the determinant ~(D,s)

= D m + al(s2)n m-I + . . . . .

+am(S2)

O

(6.5.11) where the coefficients {ai(s2) ; i=i,2, .... ,m} are rational 2 2 functions in s . The coefficients are functions in s because

125

~ (~,s)

-- det [ nI m =

-

R-IG T(-s) QG(s)]

(6.5.12)

det [ HI m - R-½GT(-s)QG(S)R -½]

and R-½GT(-s)QG(s)R -½ is para-Hermitian implying [ll, appendix A] that ~(n,s) =A(q,-s) which can only be the case 2 if the coefficients are rational functions in s • If b (s 2) is the least common denominator of the coefficients o {ai(s2)

; i=l,2,...,m}

then from

(6.5.11)

bo(S2)~(~,s)=bo(S2)~ m + bl(S2)nm-l+ ..... +bm(S 2) = O (6.5.13) where the coefficients in s 2.

{bi(s 2) : i =1,2, .... ,m} are polynomials

The function of a complex variable n(s), defined by

(6.5.13) is an algebraic function.

In general,

(6.5.11)

and (6.5.13) will be reducible into several irreducible equations over the field of rational functions in s, thereby defining a set of algebraic functions.

However,

of exposition it will be assumed that equations

for simplicity

(6.5.11) and

(6.5.13) are irreducible over the field of rational functions in s. If we compare equations

(6.5.8) and

(6.5.10) we see

that the optimal closed-loop poles can be defined as the left half-plane solutions of n (s)

=

-p

(6.5.14)

i.e. the optimal characteristic

frequency loci are the 180 °

phase contours of the algebraic function H(s) in the left half-plane.

If we now consider p

substitute for ~(s) in bo(S 2)

as a complex variable and

(6.5.13) we obtain

(_p)m + bl(S2 ) (_p)m-l+ .... +bm (s2) = 0 (6.5.15)

126

which is an algebraic equation defining the algebraic functions s(p) and p(s). the algebraic

The left half-plane branches of

function s(p), for p real and positive,

the optimal characteristic frequency loci. (6.5.15)

(Note that equation

can be obtained directly from equation

To determine characteristic

are

(6.5.8)).

the asymptotic behaviour of the optimal

frequency loci we need approximations

to the

branches of the algebraic function s(p) in the neighbourhood of s = ~, as

p approaches

zero.

These can be obtained as

the first terms in the series expansions

for the branches of

s(p) about the point s =~, as p approaches zero. purpose we put s = z

--1

.

in equation

For this

(6.5.15) and procede as in

section 6.2. 6.6

Example 2 To demonstrate the procedure the asymptotic behaviour of

the optimal closed-loop poles will be calculated for

S(s)

s + 2

6],

s + 3

1

1 (s+l) (s+2) (s+3) (s+4)

Q = I and pR = pI From this data we have H(s) ~ R-IGT (-s)QG (s) -2s 2

+

13

7s

+

15

-7s

+

151

(s2-1) (s2-2) (s2-3) (s2-4)

SO that

37

127

A(~,s)

2

=

A= det[ni 2

_

H(s)]

( -2s 2 + 50) n + (-25s 2 + 256) (s2 i) (s2 2) (s2_3) (s2_4) (s2_l) 2(s2_2)2(s2_3)2(s2_4)

O

If we now substitute

~ = -P, and multiply

least common denominator

of the coefficients,

(s2-1) 2(s2-2)2(s2-3)2(s2-4)2p2+(s2-1) +

(-25s 2

The substitution

s = z

-i

throughout by the

+

256)

=

we obtain

(s2-2) (s2-3) (s2-4) (-2s2+50) p

O.

now gives

(l-z 2) 2(l-2z2)2(l-3z~2(l-4z2)2p 2 + z6(l-z2) (I-2z2) (l-3z2)l-4z 2) (-2+5Oz2)p + z14(-25

+ 256z 2) = 0

The Newton diagram for this last equation is shown in figure 34, from which we obtain 1 1 1 ~i =~ ' ~2 = 8 ' and hence the approximationsz-elP 6 and 1 1 z=e2P~_ . To find the values of e I we substitute z = e.p± 6 i n t o I p2 _ 2z6p

=

0

i.e. the terms corresponding points on the first link

to the

of the Newton

diagram equated to zero, =

giving e I = 6/~-~ If we substitute

z

O.891 =

exp

1 e2P 8

(j2kn), k=O,1,2,3,4,5. 6

into

i.e. the terms corresponding

-25z 14 - 2z6p = 0 t

points on the second link of the Newton diagram equated to zero,

we obtain e 2 = 8/-0.08

=

to the

0.729 exp

j(~+2k~.), k=O,l,2, .... ,7 8

~o

3

LQ

~o

Q.

O

m7

T~

~D

m~

~Q O

N

c~

129

Im (s)

. . . .

-40

-20

J~

. . . . . .

J

.....

20

4 0 "Re'(S)

20

40

a)

I

-40

I

-20

R¢ (sl

(b) Figure 35. Asymptotic behaviour of the optimal characteristic frequency loci (plus right half-plane image)displayed on the Riemann surface domain of q(s) (a) sheet 1 of the surface (b)sheet 2 of the surface

130

If we now substitute back for s, and take only left halfplane solutions, we obtain the following expressions asymptotic behaviour of the optimal characteristic

for the

frequency

loci s~l.12 [ e x p -

-i ( j -~ k~ )]j p ~ ,k =

2,3,4

and s~ 1.37 [ e x p

-I j (w+2kz)] p 8 , k = 2,3,4,5. 8

Note that using Kwakernaak's notation [ii], these define Butterworth patterns of orders 3 and 4. characteristic

The optimal

frequency loci and their right half--plane

image are shown in

figure 35. Note that they have been computed

as the 180 ° phase contours of the algebraic function ~(s) and displayed on its Riemann surface domain.

References

Trans.

i]

W.R. Evans, "Graphical Analysis of Control Systems", AIEE, 67, 547-551, 1948.

2]

W.R. Evans, "Control System Synthesis by Root Locus Method", Trans. AIEE, 69, 1-4, 1950.

[3]

W.R. Evans, "Control System Dynamics", York, 1954.

[4]

B. Kouvaritakis, and U. Shaked, "Asymptotic behaviour of root-loci of multivariable systems", Int. J. Control, 23, 297-340, 1976.

[5]

D.H. Owens, root-loci",

6]

[7]

McGraw-Hill,

New

"A note o n series expansions for multivariable Int. J. Control, 26, 549-557, 1977.

U. Shaked, "The angles of departure and approach of the rootloci in linear multivariable systems", Int. J. Control, 23, 445-457, 1976. G.A. Bliss, "Algebraic Functions", (reprint of 1933 original).

Dover, New York,

1966

131

[8]

[9]

B.A.Fuchs, and V.I.Levin, "Functions of a Complex Variable", International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1961 (translation of 1951 Russian original). E.Hille, "Analytic Function Theory", Vol. 2, Ginn and Co., U.S.A., 1962.

[i0]

I.Postlethwaite, "A note on the characteristic frequency loci of multivariable linear optimal regulators", IEEE Trans. Automatic Control, 23, 757-760, 1978.

[ii]

H. Kwakernaak, "Asymptotic Root Loci of Multivariable Linear Optimal Regulators", IEEE Transo Automatic Control, 21, 378-382, 1976.

[12]

A.G.J.MacFarlane, "Dual-system methods in dynamical analysis Pt. 2 - Optimal regulators and optimal servo-mechanisms", Proc. IEE, 1458-1462, 1969.

7.

On parametric stability and future research

A feedback

system is said to be stable if all of its

closed-loop poles are in the left half-plane. of a control system is therefore parameters. parameter

Sometimes

or damage;

system the value of a

due to ageing,

in other instances

economic reasons,

dependent on its associated

in a control

is uncertain perhaps

The stability

deterioration,

it may be desirable,

to change a p a r a m e t e r value.

these cases a technique which predicts

for

In both

the relative

stability

of a system w i t h respect to a given p a r a m e t e r would be extremely useful. A dominant theme in the p r e c e d i n g

chapters has been the

association of a system with two sets of algebraic characteristic functions.

gain functions

and c h a r a c t e r i s t i c

In this Chapter c h a r a c t e r i s t i c

are introduced, root loci and

Nyquist

stability of a system, w i t h respect

frequency

parameter

and used to develop the ideas of

'parametric'

functions:

functions

'parametric'

loci from w h i c h the relative to a single parameter,

can be determined. In the final section of this chapter a few tentative proposals 7.1

and suggestions

Characteristic

for future research are made.

freqUency

and c h a r a c t e r i s t i c

parameter

functions The feedback c o n f i g u r a t i o n 36, where A(k2,k3,...,kq),

considered

B(k2,k3,...,kq),

is shown in figure C ( k 2 , k 3 , . . . k q)

and D(k2,k3,...,k q) are state-space matrices w h i c h are dependent on

(q-l)

real,

t i m e - i n v a r i a n t parameters

and k I is

133

a scalar,

time-invariant

gain parameter

common to all the loops.

-~ID(k2.....kq),l ÷

Figure 36. Feedback configuration for parameter anatysis

The closed-loop

poles for this configuration

are solutions

of det

[ si n - S(k) ] =

O

(7.1. i)

where S (k) ~A (k 2 .... ,kq) -B (k 2 .... kq)~l-llm+D(k2 is the closed-loop

....

kq) ] -Ic (k2, • • ,kq)

frequency matrix [see section 3.1].

numerical values for all the

parameters

are substituted

(7.1.i), and kj considered

into equation

a complex variable,

If

except one, kj say,

then the resulting algebraic equation

as

1:34

(which for simplicity irreducible) s(kj)

defines

and kj(s).

characteristic algebraic function

of exposition

a pair of algebraic

The algebraic

frequency

function

will be regarded

functions [ i],

function

s(kj)

function with respect

kj(s)

is called

as

is called to kj,

the characteristic

the

and the parameter

for k . (Note that the characteristic frequency 3 s(g) and the characteristic gain function g(s),

function

introduced

in chapter

3, are equivalent

to s(-kl-l)

and

-kl(S)-i respectively). The branches variation

of s (kj ), for k 3 real,

of the closed-loop

poles with respect

as such are termed parametric the parametric

clearly define

root loci.

the

to kj, and

Alternatively,

root loci can be viewed as the O ° phase

contours

of k. (s) on the frequency surface domain for k. (s). 3 ] Dual to the parametric root loci are the parametric

Nyquist

loci or characteristic

branches

of k.(s) ]

Alternatively,

parameter

as s traverses

loci which

the imaginary

the characteristic

parameter

contours

surface

which will be called

for s(kj)

for k.. 3 If, for a particular

axes.

loci can be

viewed as the ~90 ° phase domain

of s(kj)

are the

on the Riemann the parameter

surface

values

system,

for the system parameters

if any, are sensitive

introduced.

we can determine

with respect

at the set of parameter the following

we have a set of nominal

surfaces.

generalizations

to stability

which,

by looking

To help in such an assessment

of gain and phase margin

are

135

?.2

G a i n and phase margins The ~ 90 ° phase contours

of s(kj)

on the p a r a m e t e r

surface

for k. trace out the boundary between stable and unstable 3 closed-loop poles and therefore we can define parameter gain o and phase margins for k. about a stable operating point k. 3 3 which give a measure of the relative stability of the system with respect to ko. 3 Parameter gain margi n .

Parameter

gain margin is defined with o respect to a stable operating point k. as the smallest change 3 o in parameter gain about kj needed to drive the system into instability.

Let d i be the shortest distahce along the real o axis from a stable operating point k to the stability boundary 3 (characteristic p a r a m e t e r loci) on the ith sheet of the parameter surface for kj.

min{d.: i l

Then the p a r a m e t e r gain margin

i=1,2 ..... n}.

Parameter phase margin. parameter

is defined as

On each of the n sheets of the

surface

for k. imagine that an arc is drawn, 3 o centre the origin, from a stable operating point kj until it reaches the stability boundary loci).

Let

~i be the angle subtended at the origin by

the corresponding phase margin

7.3

(characteristic parameter

arc on the ith sheet.

is defined as min{~i: l

Then the p a r a m e t e r

i=1,2 .... ,n}

Example In this section an inverted p e n d u l u m p o s i t i o n i n g

(see figure

37) is considered

and its stability

with respect to one of its parameters, carriage.

system

analysed

namely the m a s s of the

136

ndutum

PlV~ll kJ

/I/I////

iage

k2

Figure 37. Inverted pendulum positioning system

This s y s t e m has also been used by K w a k e r n a a k [2],

Cannon[3],

by the f o l l o w i n g

x(t)

where

=

u(t)

and E l g e r d [ 4 ] . linearized

0

1

o

o

0

-~F

o

o

o

o

o

-~

o

~

the m o v e m e n t

equation[2]

ro]u(t ) I~ I

(7.3.1)

I |

l:I o

is a force e x e r t e d

with

x(t)+

be m o d e l l e d

can

state d i f f e r e n t i a l

on the carriage by a s m a l l motor;

M is the mass of the carriage; associated

The s y s t e m

and S i v a n

F is the f r i c t i o n

coefficient

of the carriage; and L' is given

by L'

=

J + mL 2 mL

(7.3.2)

137

w h e r e m is the m a s s of the p e n d u l u m ; the p i v o t

L is the d i s t a n c e

to the c e n t r e of g r a v i t y of the p e n d u l u m ;

is the m o m e n t of i n e r t i a of the p e n d u l u m w i t h

from

and J

r e s p e c t to

the c e n t e of g r a v i t y . The s y s t e m is s t a b i l i z a b l e

using state

f e e d b a c k of the

form u(t)

(7.3.3)

= -Kx(t)

and u s ± n g the n u m e r % c a l v a l u e s F

-

1

-

1

s

-i

M

1

kg -I

M

L'

(7.3.4) =

11.65 s

L' =

-2

O.842m

it can be f o u n d [2] t h a t

= [86.81, n.21, stabilizes

-118.4, - 3 3 4 4 ]

the l i n e a r i z e d s y s t e m p l a c i n g

poles at - 4 . 7 0 6 ~ j

(7.3.s) the c l o s e d - l o o p

1.382 and - 1 . 9 0 2 ~ j 3 . 4 2 0 .

We w i l l now look at the p a r a m e t e r s u r f a c e for M to see how v a r i a t i o n s of ikg,

affect

of the m a s s magnitude from w h i c h

in the carriag~ mass,

a b o u t an o p e r a t i n g p o i n t

the s t a b i l i t y of the system.

surface,

four s h e e t s

c h a r a c t e r i z e d by c o n s t a n t p h a s e

c o n t o u r s of s(M),

are s h o w n in f i g u r e s

the f o l l o w i n g s t a b i l i t y m a r g i n s

parameter

(mass)

parameter

(mass) p h a s e m a r g i n = 60 °

and

38-41,

are o b t a i n e d :

g a i n m a r g i n = 1 kg

The g a i n m a r g i n of ikg c o r r e s p o n d s mass to zero b e f o r e

The

to r e d u c i n g

i n s t a b i l i t y occurs.

the c a r r i a g e

The p a r a m e t e r

138

~.

:~(M)

,.,~

Figure 38. Sheet 1 of parameter (mass) surface -II)O u

!

"l'r~(l~l)l

~L - - ] 0

°

~ -~0 °

4,

-ifD°

--70 °

,z ReC~

t I0 °

"-~C

5"d~ -4

Figure 39. Sheet 2 of parameter (mass) surface

139

J..~cr43

M~

Figure 40. Sheet 3 of parameter (mass) surface ~cM) 4°¢

~.C IOaB 18o" ] -4. o

~

o

°1°

1"70-

...~c

-,4o~

Figure Z~l.Sheet 4 of parameter (mass) surface

140

surface also shows that there is a m a x i m u m mass,

for stability,

indicates 7.4

of 2.125 kg.

limit to the carriage

The phase margin of 60 °

adequate damping of the closed-loop

Future research It is thought that parameter

surfaces may prove to be

useful in the design of p a r a m e t e r dependent systems

in which a p a r t i c u l a r p a r a m e t e r

during normal operation.

range of altitudes.

controllers

suffers

For example,

aircraft engine needs to operate

for

large variations

the controller of an

satisfactorily

over a wide

A possible design scheme could be

(i)

to design real constant controllers

(ii)

to obtain an altitude interpolation",

(iii)

system.

dependent

at a number of altitudes,

controller by "matrix

and finally

to analyse the stability

of the system over the whole

w o r k i n g range using the "altitude

surface".

As indicated in this chapter the methods put forward in this work are not only applicable any single system p a r a m e t e r tests

to gain and frequency,

and frequency.

To obtain stability

in terms of more than one p a r a m e t e r variation

problem,

but one with great practical

that v a l u a b l e functions

but

is a c o m p l i c a t e d

significance.

It is felt

insight into this p r o b l e m may stem from a study of

of several

complex variables.

In recent years stability tests have been developed for two-dimensional dimensional

and m u l t i - d i m e n s i o n a l

digital

as image processing,

filters

dimensional

are used w i d e l y

and geophysics

gravity and magnetic data.

digital filters

in many fields,

for the p r o c e s s i n g

Twosuch

of seismic,

It is expected that higher

filters will also find applications;

three-dimensional

[5].

filters in holography.

which have emerged are of a N y q u i s t - t y p e

for example,

The stability

tests

and it is thought that

141

a deeper understanding may

lead to s u i t a b l e

possibly

of t h e s e d e v e l o p m e n t s

adaption

by c o n t r o l e n g i n e e r s

for u s e in the c o n t r o l

field;

in the s t u d y of s t a b i l i t y u n d e r m u l t i - p a r a m e t e r

variations. A constant

theme throughout

a s s o c i a t i o n of a m u l t i v a r i a b l e more

algebraic

appropriate algebraic with

s y s t e m w i t h one or p o s s i b l y

f u n c t i o n s e a c h of w h i c h

Riemann

function

"handles",

surface.

A Riemann

is t o p o l o g i c a l l y

surface

[6]

.

to be an i m p o r t a n t c h a r a c t e r i s t i c it m a y be r e l a t e d

of a m u l t i v a r i a b l e of s i n g l e - i n p u t , output

single-output

zero,

for an to a s p h e r e

is k n o w n as the

The g e n u s n u m b e r m a y p r o v e of a s y s t e m and it is felt that

systems.

is e f f e c t i v e l y

a set

single-

(one sheeted)

frequency

trivial

to r e d u c i n g

is, the t r a n s f o r m a t i o n

A single-input,

so that to d e c o u p l e

a multivariahle

an m - s h e e t e d

Riemann

w i t h a g e n u s n u m b e r g r e a t e r t h a n or e q u a l to zero,

to a set of

m

Consequently

the e s t a b l i s h m e n t

point

equivalent

s y s t e m to a s y s t e m w h i c h

s y s t e m w o u l d be e q u i v a l e n t surface,

surface

to " d e c o u p l i n g " ,

s y s t e m has a c o r r e s p o n d i n g

s u r f a c e of genus

is d e f i n e d on an

and the n u m b e r of h a n d l e s

g e n u s n u m b e r of the

that

this w o r k has b e e n the

trivial Riemann

singularities

topological

of r e l a t i o n s h i p s

as the g e n u s number)

line of r e s e a r c h .

interesting relationship

between branchsuch

and decoupling

In a p p e n d i x

is d e v e l o p e d w h i c h

p o i n t on a g a i n s u r f a c e c o r r e s p o n d s

shows

7

an

that a b r a n c h

to a b r a n c h p o i n t or

s t a t i o n a r y p o i n t on the c o r r e s p o n d i n g v i c e versa.

e a c h of zero genus.

(whose n a t u r e a n d l o c a t i o n d e t e r m i n e

properties

c o u l d be a f r u i t f u l

surfaces

frequency

surface

and

142 References

[i]

G.A.Bliss, "Algebraic Functions", Dover, New York,1966 (reprint of 1933 original).

[2]

H. Kwakernaak, and R. Sivan, "Linear Optimal Control Systems", Wiley, New York, 1972.

[3]

R.H. Cannon, Jr, "Dynamics of Physical Systems", McGraw-Hill, New York, 1967.

[4]

O.I.Elgerd, "Control Systems Theory", New York, 1967.

[ 53

E. I. Jury, "Inners and Stability of Dynamical Systems", Wiley, New York, 1974.

[ 6]

G. Springer, "Introduction to Riemann Surfaces", Addison-Wesley, Reading, Mass., 1957.

McGraw-Hill,

142 References

[i]

G.A.Bliss, "Algebraic Functions", Dover, New York,1966 (reprint of 1933 original).

[2]

H. Kwakernaak, and R. Sivan, "Linear Optimal Control Systems", Wiley, New York, 1972.

[3]

R.H. Cannon, Jr, "Dynamics of Physical Systems", McGraw-Hill, New York, 1967.

[4]

O.I.Elgerd, "Control Systems Theory", New York, 1967.

[ 53

E. I. Jury, "Inners and Stability of Dynamical Systems", Wiley, New York, 1974.

[ 6]

G. Springer, "Introduction to Riemann Surfaces", Addison-Wesley, Reading, Mass., 1957.

McGraw-Hill,

143

Appendix

i.

Definition

Let A(q,v) A(q,v) where

of an a l g e b r a i c

be a p o l y n o m i a l

function

in q of the form

= fo(V)q m + f l ( v ) q m - l + .... +fm(V) each c o e f f i c i e n t

{fi(v):

polynomial

in v w i t h

numbers.

T h e n an a l g e b r a i c

defined

for v a l u e s

(Al.1)

i=l,2,...,m}

coefficients

is itself

a

in the d o m a i n of c o m p l e x

function

is a f u n c t i o n

q(v)

of v in the c o m p l e x v- p l a n e by an e q u a t i o n

of the form A(q,v)

= O

The p o l y n o m i a l with

(AI.2) A(q,v)

coefficients

when considered function

can be r e w r i t t e n

which

are t h e m s e l v e s

in this w a y e q u a t i o n

which

neighbourhood

of v

expansions

It is a s s u m e d an i r r e d u c i b l e

are called o

(AI.2)

A p p e n d i x 2.

has

and in the

are r e p r e s e n t a b l e

by p o w e r

would

in

(q,v),

as a p r o d u c t then d e f i n e

or t a l g e b r a i c

In this a p p e n d i x

that A(q,v)

in

any m - s q u a r e m a t r i x The m e t h o d

(q,v).

of t p o l y n o m i a l s

t algebraic functions

method

is If

in

(q,v),

functions

of the form v(q).

to the i r r e d u c i b l e

a proposed

is

that is, that A(q,v)

or more p o l y n o m i a l s

A reduction

form.

(AI.2)

of q(v),

in the above d e f i n i t i o n

of two

of the form q(v),

canonical

an a l g e b r a i c

[i].

was e x p r e s s i b l e

reducing

in q, and

defines

equation

branches

the b r a n c h e s

polynomial

not the p r o d u c t

equation

in v

v(q).

m solutions

A(q,v)

polynomials (Ai.2)

For a fixed v a l u e of v, v O say,

series

as a p o l y n o m i a l

rational canonical

is g i v e n

to its i r r e d u c i b l e u s e d is a v a r i a t i o n

for rational on that g i v e n

form

144

P

by A y r e s

~

[2J for finding

square matrix. matrix

The r a t i o n a l

G is s i m i l a r

diagonal rather

the r a t i o n a l

blocks

outlined

canonical

to the i r r e d u c i b l e

correspond

for f i n d i n g

form e x c e p t

factors.

factors

the r a t i o n a l

canonical

X,GX,G

2

of gI-G, given

form is

definitions.

If the v e c t o r s

X,GX,G2X,...,Gt-Ix are l i n e a r l y

that its

The p r o c e d u r e

b e l o w along w i t h some n e c e s s a r y

Definitions:

form of a

form of a s q u a r e

to the i n v a r i a n t

than the i r r e d u c i b l e

b y A y r e s [2]

canonical

(A2.1)

independent

but

X,...,Gt-Ix,GtX

are not then

(A2.1)

(A2.2)

is c a l l e d a c h a i n of length

t having

X

as its leader. Procedure: (i)

For a g i v e n m - s q u a r e

let X m be the leader

for all m - v e c t o r s (ii)

over

let Xm_ 1 be the

length

(any m e m b e r

preceding members which

matrix

of a chain C m of m a x i m u m

length

F ; leader of a c h a i n Cm_ 1 of m a x i m u m

of w h i c h

is l i n e a r l y

and those of Cm)

are l i n e a r l y

G over any f i e l d F :

for all m - v e c t o r s

length

(any m e m b e r

preceding vectors

members

over

F

of w h i c h and those

which

of the v e c t o r s

of the over F

; m let Xm_ 2 be the l e a d e r of a chain Cm_ 2 of m a x i m u m

(iii)

independent

independent

is l i n e a r l y

of C

independent

of the

of C m and Cm_ I) for all m-

are l i n e a r l y

independent

of the v e c t o r s

of C m and Cm_ 1 ; and so on. E =

Then,

for

t.-I [ Xj,GXj .... G 3 ~ j + l , G X j +

...

tj+l-iXj+l ; 1 .... ,G

t -i ; X m , G X m ..... G m Xm ]

145

we have that E-IGE is the rational canonical In this approach

the chains of m a x i m u m

form of G. length are used

to pick out the invariant

factors.

Now the invariant

are made up from products

of the irreducible

characteristic

equations

which are required in the irreducible

canonical

form.

maximum

Therefore,

factors

rational

if instead of using chains of

length those of m i n i m u m

length are found,

the

t r a n s f o r m a t i o n m a t r i x E so formed is that required to give the irreducible

rational

canonical

form Q.

The problem with both of these methods indication

is that no

is given w h i c h enables one to know when a chain of

m a x i m u m or m i n i m u m

length has been obtained,

except that in

the contrary case there will appear a chain of longer or shorter length. It is interesting and m i n i m u m

to note that the chains of m a x i m u m

length form bases for the G - i n v a r i a n t

of m a x i m u m and minimum dimensions

respectively.

subspaces Therefore

the p r o b l e m of finding the chains of m i n i m u m length is equivalent minimum

to that of finding the invariant

dimension.

It also happens

dimensional

invariant

subspace

eigenvalue,

a t-dimensional

suhspaces

that just as the l-

(eigenvector)

invariant

of those of m i n i m u m dimensions)

of

picks out an

subspace

(from the set

picks out an irreducible

equation of degree t, defining t eigenvalues. Example: S(s)

To find the irreducible =

(s+3)

(s+l) 2 (s-3) (s+l)

2

-i (s+l)

2

rational

canonical

(s+2) 2 (s+l)

O

-2

0

(s+l)

2

- (s+2)

1

- - 2 2 (s+l)

s+l

form of

146

Let

X

=

( O 0 1 )t,

1 then GX = (s+l) and

X

where "t" denotes the transpose,

X

is a chain of minimum length.

Let Y = ( O 1 0

)

t

then Gy = [

s+2 (s+l) 2

-2 (s+l) 2

-(s+2) I 2 (s+l) 2

t

and G2y = [

s+2 (s+l)3

s-2 (s+]93

-(s+2) 2(s+i)3

t

1

1 - (s-~

s GY + --(s+l)3 Y

No chain of smaller length can be found and therefore Y,GY completes

the set of minimum length

chains.

The transformation

matrix E(S) is therefore given by

-[x

Y

='o

s+2

o

(s+l) O

-2

1

(s+l) 1

O

2 2

-(s+2) 2 (s+l)

2

and Q(s)

=

E-I(s)G(s)E (s)

=

s+l

O

O

0

0

s (s+l) 3

O

1

1 (s+l)

which implies that the irreducible characteristic equations are 1 s+l

(g,s) = g ~(g,s)

= g

2

1

(s+l) g

(s+l)

Note that these are also obvious from the dependence relations obtained when finding the chains of minimum length.

'r47

A~pendix

3.

The d i s c r i m i n a n t

In this a p p e n d i x

two m e t h o d s

are g i v e n for f i n d i n g

the d i s c r i m i n a n t of an e q u a t i o n of the f o r m ~(g,s) Method

= bo(S)gt + bl(s)gt-l+

=

O

1 (Barnett [ 3 ] )

The r e s u l t a n t a(g)

... +bt(s)

R[ a(g),c(g)]

of two p o l y n o m i a l s

and c ( g ) , g i v e n by n-l+ a(g)

=

a o g n + alg

c(g)

=

Cog

m

where

... + a n m-i

+ clg

+ ... + c m

e C ,is the d e t e r m i n a n t

ai,c i

aO

aI

a2

...

an

0

0

a°

aI

...

an_ 1

an m

rows

R[a(g),c(g)]

=

.oo

. , o C

o

a -i n

a

Cm_ 1

cm

co

c I ..

cI

c 2 0 . cm

n

O n

ooo

rows co The p o l y n o m i a l s

a(g)

C1

have a common factor

if and o n l y if the

R [ a(g),c(g) ]

and c o are not b o t h

..-

a n d c(g)

d e g r e e g r e a t e r t h a n zero) determinant

c2

is zero,

be d e n o t e d by a' (g).

polymonial

a(g)

-order

that a °

zero.

L e t the d e r i v a t i v e w i t h r e s p e c t a(g)

provided

(n+m)

(of

to g of the p o l y n o m i a l

T h e n the d i s c r i m i n a n t

is the d e t e r m i n a n t

of the

D g ( a o , a l , . . . , a n) d e f i n e d

148

by _

Dg(ao'''''an)

1

R

a

[ a(g)

a' (g) ]

0

The p o l y n o m i a l

a(g)

the d i s c r i m i n a n t

where

factor

a polynomial

the coefficients

g is found

in s.

{bi(s)

D

,

(s) again _

Dg(S)

called

1

b (s)

are all

of this p o l y n o m i a l

from D g ( b o , b l , . . . , b t) as d e f i n e d

g

t > O

:i=l,2,...,t}

The d i s c r i m i n a n t

b o , . . . , b t by bo(S),...,bt(s) function

if

of the type

= b o ( S ) g t + b l ( S ) g t-I +...+bt(s)

polynomials

if and only

D g ( a o , . . . , a n) is zero.

Now consider @(g,s)

has a r e p e a t e d

above

respectively.

Thus

the d i s c r i m i n a n t

in

by r e p l a c i n g there

is a

and defined

by

R[~(g,s),~'(g,s)]

O

where of

#'(g,s)

is the d e r i v a t i v e

with

respect

to g

~(g,s)

Method

2

(Sansone

and G e r r e t s e n

Consider

the p o l y n o m i a l

a(g)

dog

where

=

ai e ~

;

n

algn-i

+

[4])

+ ... + a n ,

then the d i s c r i m i n a n t

n > O

of a(g)

is given by

the e x p r e s s i o n Dg(ao'''''an ) where

=

ao2n-2p

P is a d e t e r m i n a n t P

=

given by

So

~I

•""

G1

o2

...

{~

(~

n

---

an-i gn and the elements

On-1

i

a2n_ 2

: i = 1,2,...,2n-2}

are functions

of

149

the

{a. l

coefficients

: i = O,l,...,n} =

o The

elements

o1,...,an_

1

On,...,a2n_2

can

aI

+

ao~ 1

=

0

2a 2 + al(l 1

+

ao~ 2

=

0

+

aoan_ 1 =

0

+

...

+

aoa n

=

O

ana I

+

an_lO 2

+

...

+

aoOn+ 1 =

0

a n_ i ° m + l +

•..

=

0

+

now

= bo(s)g t + bl(S)g t-I of

as

has

zero,

elements

=

polynomial

defined

above

by

respectively.

repeated

where are

this

factors

P is

determinant of

the

aoOn+ m

...

given

replacing

of

by

+ bt(s)

in g is

if a n d

P

functions

+

The

. 2t-2 DO (S) the

+

~(g,s)

the polynomial

by bo(S),...,bt(s)

is

...

an_l$1

D g ( b o , . . . , b t)

Dg(S)

from

+

discriminant

therefore

found

anO o

Consider

The

be

from

anO m

~(g,s)

and

n

(n-l)an_ 1 + an_2~ 1 ÷ and

,

found

bo,...,b t

polynomial only

~(g,s)

if

a matrix

coefficients

from

whose

{bi(s)

: i = O,l,2,...,t}

150

A p p e n d i x 4.

A m e t h o d for c o n s t r u c t i n g the R i e m a n n surface domains of the algebraic functions c o r r e s p o n d i n 9 to an o p e n - l o o p gain m a t r i x G(s)

In s u b - s e c t i o n 3.3-4 it was shown that for a c h a r a c t e r i s t i c gain f u n c t i o n of degree t the c o r r e s p o n d i n g R i e m a n n surface is made up from t sheets of the complex s-plane stitched together along cuts made b e t w e e n b r a n c h points and infinity. A l t h o u g h the cuts are in some sense a r b i t r a r y

(i.e. there is

no unique set of cuts),

it is still a p r o b l e m to choose a

set that is consistent,

and then to be able to identify in w h a t

order the sheets are connected together.

In this a p p e n d i x

a systematic m e t h o d is given for solving this problem.

The

m e t h o d is quite elegant in that the r e s u l t i n g cuts are symmetrical about the real axis and are always p a r a l l e l to the i m a g i n a r y axis except for possible cuts along the real axis. the a p p r o a c h uses the o p e n - l o o p gain m a t r i x directly,

Also, so

that there is no need to find the c h a r a c t e r i s t i c equations, and the resulting n o n - c o n n e c t e d sets of c o n n e c t e d sheets r e p r e s e n t the R i e m a n n surfaces for the irreducible c h a r a c t e r i s t i c equations. The m e t h o d is based on finding the e i g e n v a l u e s of the m a t r i x for a grid of values covering the s-plane and then sorting the e i g e n v a l u e s in a continuous form along certain lines of the grid.

If the transfer function is of order mxm,

m arrays which will e f f e c t i v e l y r e p r e s e n t the m sheets of the R i e m a n n surfaces are n e e d e d to store the eigenvalues. The p ~ o c e s s is analogous to the analytic c o n t i n u a t i o n p r o c e d u r e d e s c r i b e d in s u b - s e c t i o n 3.3-4 where i n d i v i d u a l

151

points of a circular disc are made bearers of unique functional values.

Here i n d i v i d u a l points of the s-plane sheets

(arrays) are being made the bearers of functional values. The first line along w h i c h the eigenvalues are c a l c u l a t e d and sorted is the real axis, and then the c a l c u l a t i o n and sorting is carried out along lines p a r a l l e l to the imaginary axis and e m a n a t i n g from the real axis, as shown in figure 42. The c a l c u l a t i o n is only necessary in the upper h a l f - p l a n e since the e i g e n v a l u e s in the lower h a l f - p l a n e are the complex c o n j u g a t e of those in the upper half.

C o n t i n u i n g this process

the s-plane is covered, and m arrays of e i g e n v a l u e s are o b t a i n e d which are continuous along the real axis and along lines parallel to the i m a g i n a r y axis but not n e c e s s a r i l y crossing the real axis.

I m (s)

5- plane x

×

indicates continuity between eigenvalues

Re (s)

Figure 42. Lines atong which eigenvatues ore catcutated and sorted By o b s e r v a t i o n of each array or sheet the n e c e s s a r y cuts are obvious.

If p a r a l l e l to the imaginary axis a line of

e i g e n v a l u e s is not continuous w i t h an adjacent line then these m u s t be separated by a cut starting at a b r a n c h point and e n d i n g at a branch point or infinity as shown in

152

figures

43

(a) and

(b).

Eigenvalues

of s in the upper h a l f - p l a n e in the lower h a l f - p l a n e across

corresponding

are c o m p l e x

and t h e r e f o r e

the real axis is i m p o s s i b l e

conjugate

continuity

to v a l u e s to those

of e i g e n v a l u e s

if the e i g e n v a l u e s

on the real axis are complex.

Therefore

real axis

the real axis e i g e n v a l u e s

is n e c e s s a r y w h e n e v e r

are complex.

Again

a cut a l o n g

situated

all the cuts start at a b r a n c h

and end at a b r a n c h p o i n t or infinity, X

point

as s h o w n in f i g u r e

x indicates continuity

between eigenvalues i I

the

|to00

i Cut ~--~

Cut

BrQnch points /-

< I--" Branch ~T--~-

i

t--

i

--~--

point

(a) (b) Figure 43. Cuts parallel to the imaginary axis (a) finite (b) infinite

Cut.

Im (s)

Cut

R = Real ¢i£¢nvalue

Figure 44. Cuts along the real axis

44.

153 The stitching

together of the sheets also becomes

obvious by matching

eigenvalues

sheet to those on another. complete,

in general,

on one side of a cut on one

When the matching process

there will be sets of c o n n e c t e d

is sheets

each set defining a Riemann surface. To facilitate matching

the i d e n t i f i c a t i o n

of cuts and the

of sheets it is very useful to draw the constant

phase and constant m a g n i t u d e is d e m o n s t r a t e d

contours on each sheet;

in the examples given in the main text.

Computationally

sorting the e i g e n v a l u e s

along the real

axis can be difficult because of the likelihood axis poles.

of real

This p r o b l e m is overcome if the first line of

c a l c u l a t i o n and

sorting is changed to be a line parallel

to, and a "large" distance away from, rest of the calculations along lines parallel

the real axis.

to %he imaginary

With this new approach

either along the real axis,

axis starting at this as shown in figure

the only possible as before

cuts are

(see figure 44) or b e t w e e n

complex conjugate branch points as shown in figure A Riemann surface c o n s t r u c t i o n

G(s)

=

1

5s-2

3s-18

for

2s-l]

s-8J

is shown using the original m e t h o d in figures using the m o d i f i e d m e t h o d in figures of the cuts.

46.

!

1.25(s+i) (s+2)

the arbitrariness

The

and sorting are then carried out

new line and finishing at the real axis; 45.

this

47 and 48, and

49 and 50, and illustrates

154

Ira(s} ,Y,

X

X

X

X ~K

X

z

z

S -plane X )

:K

I :_ Re

is)

Figure 45. Lines orang which eigenvaiues are calculated and sorted (modified method)

Figure 46. Cut parattet to the imaginary axis (modified method)

155

.....

Root

i

Cut

Figure 47. Sheet 1 of the Riemann surface

Cut

3_o

Figure 48. Sheet 2 of the Riemann surface

loci

156

,,*~.~

Root

loci

Cut e

Figure 49. Sheet 1 of the Riemann surface (modified method of construction)

;~;w---.~'~': R o o t loci

Cut

Figure 50. Sheet 2 of the Riemann surface (modified method of construction)

157 Appendix

5.

Extended Principle of the Argument

The required extension of the Argument Principle not seem to be readily available a suitable

in the literature,

statement of the Principle

valued analytic

function has recently

Evgrafov

[5, page 98].

chapters

4 and 5, an appropriately

Argument

is developed here.

A5.1

Therefore,

although

for a general m u l t i p l e appeared in a text by

to justify its use in extended Principle of the

Introduction The extension required is non-trivial,

problems

to be overcome.

the fact that,

in general,

source of difficulty

with two main

The first of these arises

For an example of this

consider a c h a r a c t e r i s t i c

associated with a

from

the Riemann surface of an algebraic

function will be multiply connected.

g(s)

2 × 2

gain function

open-loop gain m a t r i x G(s).

Suppose that g(s) has four branch points,

and that each

branch point is associated with a cycle of two sheets then the genus number is one.

Further

[6; 7] of the a s s o c i a t e d

Riemann surface

(s-plane)

in the way shown in figure 51.

taking two copies of the complex number sphere

number sphere),

figures

(Riemann

making cuts between the branch points,

forming the topological in the usual way

[6];

suppose that these branch points are disposed

in the frequency plane Then,

does

equivalent of the Riemann surface

[7] one obtains

52 and 53.

and having a boundary

and

a torus,

as illustrated

in

The region ~, shown shaded on this torus ~,

pair of Nyquist D-contours

corresponds

to the interiors

of the

(shown in figure 52) for the

original complex number spheres out of w h i c h the torus was constructed.

To cope with such a situation we must ensure

that the extended version of the Argument Principle holds for

158

[s)~

~-=Bronchpoint

Figure 51. Frequency ptane Cut ~ branch points _

between

-

Figure 52. Two copies of the comp{ex number sphere

Figure 53. Riemann surface shown topologically equivalent to a torus

159

a region of a Riemann surface which is n o n - s i m p l y and whose boundary ~

consists of several distinct

Jordan contours on the surface. principle

connected closed

This basic extension

is achieved via a suitable g e n e r a l i z a t i o n

of the

of the Cauchy

Residue Theorem and is covered in section A5.2. The second obstacle to be overcome suitably extended Argument

Principle

in a derivation

is more directly

associated with the branch points of an algebraic this case with their effect on the calculation via contour

function;

in its interior.

in

of residues

integrals taken round the b o u n d a r y ~

having branch points

of a

of a region

This p r o b l e m is

overcome by a change of variable and is treated in section A5.3. Finally,

in section A5.4 the results of sections A5.2 and

.3 are combined to give the required extended Principle of the Argument. A5.2

Generalized

form of Cauchy's

The basic requirement

Residue Theorem

is a g e n e r a l i z a t i o n

Residue Theorem to complex functions surface of some known algebraic of integration may be non-simply consisting

of Cauchy's

defined on the Riemann

function,

where the region

connected and have a boundary

of one or more closed Jordan contours.

generalization

may be found in Bliss [ 6 ]

is given below with slight rephrasing for the context of this

work

;

Such a

the main theorem

and change of notation

Two observations

in helping the reader u n f a m i l i a r with Riemann

may be useful

surface theory to

understand this theorem. (i)

Each n o n - s i n g u l a r place

surface of an algebraic

function

(or point)

f(s)

on the Riemann

is uniquely defined by a

160

pair of values

(f,s) which satisfy a known algebraic A(f,s)

=

0

Thus at each point on the Riemann complex function

T(f,s)

equation

surface the value of a

of the pair of complex variables

s

and f is defined. (ii)

The Riemann

an orientable positive

surface.

for an algebraic

visualize

some region ~ .

this procedure

the boundary ~

function

Thus one can u n a m b i g u o u s l y

sense in which a boundary

to "enclose"

lies

surface

~

is

define a

can be traversed so as

The simplest way in which to

is to imagine

on a t w o - d i m e n s i o n a l

someone walking round

manifold

in which

(embedded in a familiar space of three dimensions).

The positive which

sense of traversal may then be taken as that in

someone walking

forward round any p o r t i o n o f ~

always have ~ on his right-hand

side.

will

In figure 53 the boundary

is shown with a positive orientation. G e n e r a l i z e d Residue T h e o r e m Let R f(s)

be the Riemann

and ~ a portion o f ~

having a boundary

~

surface of an algebraic not n e c e s s a r i l y

consisting

function

simply connected but

of one or m o r e closed Jordan

contours not passing through any singular place o n ~ Then if a function ~(f,s) of the places o n ~ and ~ in

except possibly

.

is analytic on

for a finite number of singular points

~, we have 2~j"

~(f,s)ds

=

Z residues

~

~(f,s)

in

a

(A5.2. i)

where the boundary respect to

of

~.

~

is traversed

in the positive

sense with

161

To derive

the e x t e n d e d A r g u m e n t

apply the G e n e r a l i z e d where

f' (s) d e n o t e s

Residue

Principle

Theorem

the d e r i v a t i v e

we w i l l

to the f u n c t i o n

of f(s)

with

first

f' (s)/f(s),

respect

to s,

to give 1 /~ 2~j ~ Note that which

f' (s) ds f(s)

f(s) m u s t have no poles

are r e f e r r e d

For the p u r p o s e s also e x c l u d e calculate A5.3

of d e r i v i n g

the r e s i d u e s

Calculation

or b r a n c h

an A r g U m e n t from

zeros

represented

by a series w h i c h

clearer

step

is to

are s i n g u l a r i t i e s

as this

section

all the b r a n c h p o i n t s

of f(s)

are also poles or

(not i n c l u d i n g

with a b r a n c h point)

For a pole Pi of o r d e r t p i t h e

where

of f(s)

of a pole or a zero

are a s s o c i a t e d

f(s)

we w i l l n o w

are relevant.

those w h i c h

i.e.

Principle

~.

~.

and b r a n c h p o i n t s

In the n e i g h b o u r h o o d

f(s)

(either of

The next

in

but it turns out that only those w h i c h

represented

points

of R e s i d u e s

We w i l l c o n s i d e r

zeros of f(s)

in

on the b o u n d a r y

~.

of f' (s)/f(s)

f'(s) f(s) , a fact w h i c h w i l l b e c o m e

progresses.

f'(s) f(s)

of

to as s i n g u l a r places)

any zeros of f(s)

The poles, of

= Z residues

defines

f(s)

a single-valued

algebraic

function

can be function.

can be

by

=

=

E n=-t

a n ( S - P i )n, a_t Pi

~

~(s) t (s-Pi) Pi

~(pi ) ~ ~ and ~(s)

O

(A5.3.1)

Pi (A5.3.2)

is a n a l y t i c

in the n e i g h b o u r h o o d

162

of Pi"

This gives -t f'(s) f(s)

Pi ~pi

+

~'(s)

(A5.3.3)

~(s)

)

!

¢

and since

<s)

in the n e i g h b o u r h o o d

f' (s) has a simple pole of r e s i d u e f(s)

the f u n c t i o n

s

is a n a l y t i c

~(s)

of D. "l

-t

at Pi

Pi"

=

For a zero s. of o r d e r i be r e p r e s e n t e d by the series

the a l g e b r a i c

t

function

can

zi

co

f(s)

=

Z

Cn (s-zi)n,

n=t

O

ct

(A5.3.4)

z.

Z. 1

l

t i.e.

f(s)

where of

=

e(z i) ~ 0 and

z..

This

l

zi

(s - z i)

(A5.3.5)

e(s)

8(s)

is a n a l y t i c

in the n e i g h b o u r h o o d

gives

t f'(s) f(s)

=

zi (s - z i)

and we see that t

at

s

z. 1

=

+

f'(s) f(s)

has a s i m p l e ~ l e

(A5.3.6) of r e s i d u e

z.. 1

To d e t e r m i n e

the r e s i d u e s

necessary

to m a k e

procedure

is as follows.

a change

In the n e i g h b o u r h o o d the a l g e b r a i c form [8],

S' (s) e(s)

function

at a b r a n c h p o i n t

of v a r i a b l e [6,

page

of a finite b r a n c h

can be r e p r e s e n t e d

it is

81].

The

p o i n t b. l

by a series

of the

163 oo

f(s)

=

Z

dn(r - s/ ~ i

)n

(A5.3.7)

n___-o~

where r is the number of sheets which form a cycle at the branch point. tb Pi

If the branch point is a pole of order

and consists of a cycle of r sheets we have the series

expansion f(s)

Similarly

=

if

n~_t~id n(r s - ~ i ) n ,

the

branch

point

is

d_tbpi ~ O

a

zero

of

(A5.3.8)

order

tb

z,

we 1

have that o0

f(s)

=

Z b dn(r s - ~ i ) n n=t z. 1

, dtb z.l

~

O

(A5.3.9)

n , do

~

O

(AS. 3. iO)

otherwise 0o

f(s)

dn(r s _ ~ i )

=

n=O These expansions suitable

are multivalued

for determining

residues.

and therefore not If, however, we make

the substitution s = b. + x r

(A5.3.11)

l

we find that the series expansions

(A5.3.8,

9 and i0)

become f(x)

=

Z dnxn , d_tb ~ n=-t~i Pi

0

(A5.3.12)

f(x)

=

n~tb

O

(A5.3.13)

Zi

and

dnxn'

dtb Z,

1

164 co

f(x)

=

Z

darn, d O ~

O

(A5.3.14)

n=O

respectively, f(x)

where

~

f(b i

it is to be understood that +

The substitution

x r)

has therefore mapped the algebraic

function onto the x-plane where "single-valued"

(A5.3.15)

it can be represented by a

series expansion.

By definition in a n e i g h b o u r h o o d

(residue)bl

[6] the residue at b I of (____{s) f' analytic • f(s) of b. except possibly at b. itself is 1 1 A 1 = 249

Sf'(s) - cf(~

ds (A5.3.16)

where C is a closed positively oriented Jordan contour o n e bounding a n e i g h b o u r h o o d

analytic except possibly at b i. from

(A5.3.11)

1 27

is

If we substitute

for s

we find that

A (residue)b. = i _

f'(s) f(s)

N of b i in which

/

1 / f ' (s) 2~j I f ( s ) ds C

f' (x) dx f(x) as

ds

c X

1 f f' (x) dx 2~j , / f(x) C x where C

is a closed p o s i t i v e l y

(A5.3.17)

o~ented

Jordan contour on the

X

x-plane bounding a n e i g h b o u r h o o d f'( (s) of s----~ f residue of

of the origin.

at a branch point b.1 is therefore f' (x) at the origin. f(x)

The residue

equal to the

The residue of the function

165

f'(x) f(x) is e a s i l y o b t a i n e d f r o m its s e r i e s e x p a n s i o n . F o l l o w i n g (x) in the as b e f o r e we n o t e t h a t f' f(x----[

the same p r o c e d u r e neighbourhood (i)

of the o r i g i n has:

a simple pole with residue

-t b , if the b r a n c h Pi p o i n t is a p o l e of o r d e r t b ; or Pi a s i m p l e p o l e w i t h r e s i d u e t; , if the b r a n c h

(ii)

p o i n t is a zero of o r d e r t b l; zi otherwise

A5.4

f'(x) f(x)

is a n a l y t i c .

Extended Principle Combining

equation

the r e s u l t s of s e c t i o n A 5 . 3 w i t h that of

(A5.2.2)

2~j

of the A r g u m e n t .

we h a v e t h a t

f(s)

i

i

Z

Pi

i

Pi

~n = where

# Z

and

~ P

and zeros of f(s) in the p o s i t i v e in the R i e m a n n

-~P

are r e s p e c t i v e l y in

(A5.4.1) the total n u m b e r of p o l e s

~ e n c l o s e d by the b o u n d a r y %~ t r a v e r s e d

sense w i t h r e s p e c t to

~,

s u r f a c e of the a l g e b r a i c

that multiple poles as t h e i r o r d e r s

WZ

and w h e r e ~ lies

function

f(s).

Note

and zeros m u s t be c o u n t e d as m a n y times

indicate.

The l e f t - h a n d

side of e q u a t i o n

(4.1)

the net sum of the c l o c k w i s e e n c i r c l e m e n t s about the o r i g i n of the f-plane w h e r e

is e q u i v a l e n t of a c u r v e

set F

F is the image of ~

f(s).

D e n o t i n g this n e t sum of c l o c k w i s e e n c i r c l e m e n t s

N(F,O),

we f i n a l l y o b t a i n the r e q u i r e d e x t e n d e d P r i n c i p l e

the A r g u m e n t N(F,O)

in the f o r m =

# Z

-

~ p

to

(A5.4.2)

under by of

166

Appendix

6.

Multivariable equation

pivots

~(g,s)=O

C o n s i d e r the c h a r a c t e r i s t i c ~(g,s)

=

equation

O

and a s s u m e we h a v e characteristic

from the c h a r a c t e r i s t i c

(A6. i) f o u n d an a p p r o x i m a t i o n

frequency

loci a b o u t s =m,

for a b r a n c h of the as k ~m, of the f o r m

s - bk ~

(A6.2)

If p is the p i v o t s = p +

to this a s y m p t o t e

bk ~ 1

or k

corresponding

(A6.3)

:

(A64)

The d e p e n d e n c e equation

(6.1.5),

F(k,s) Therefore

of the c l o s e d - l o o p p o l e s on k is g i v e n by

i.e.

= O

(A6.5)

substituting

obtain

a relationship

k and

p.

for k or s in e q u a t i o n between

s and

S i n c e the o r d e r of s w i l l

the case of s u b s t i t u t i n g (A6.4)

Z(p,s) we r e q u i r e

for k.

into equation

in g e n e r a l be m u c h (A6.5), we w i l l c o n s i d e r

Substituting

for s =~,

for k from

(A6.5) we o b t a i n the e q u a t i o n (A6.6)

= 0 p

(A6.5) we

p or a r e l a t i o n s h i p b e t w e e n

g r e a t e r t h a n the o r d e r of k in e q u a t i o n

equation

then for k =~

and so w e put

s=z

-i

in e q u a t i o n

(A6.6)

to g i v e Z(p,s)

= Z ( p , z -I) = z-tn(p,z)

w h e r e t is the m a x i m u m o r d e r of s in ~(p,O)

= O

= o (A6.6).

(A6.8) The e q u a t i o n (A6.8)

167

then gives the multivariable pivot,

p.

Example Consider the open-loop gain matrix G(s)=

1

s4-s3+2s2-25s+29

~-s3-11s2-29s+92

-2Os2+35s+70

[

33s2-170s+l18

41s2-s-91

which has a characteristic equation A(g,s)=(s4-s3+2s2-25s+29)g2-(-s3+22s2-199s+210)g +(-33s+594)=O Using the method described in section 6.2 we find that as k+~ the infinite branches of the characteristic frequency loci are given by the following approximations s~k and s~+/(-33k) To find the pivot corresponding to the second-order Butterworth pattern j/(33k) we will put s=p+/(-33k) giving k=(S-P) -33

2

From the characteristic equation ~(g,s)=O we obtain

F(s,k)=~4-s3+2s2-25s+29)+(-s3+22s2-199s+210)k+(-33s+594)k2=O and substituting for k in this we obtain Z(p,s)=332(s4_s3+2s2_25s+29)_33(_s3+22s2_199s+210) (_2ps +p2)

-33s2(22s2-199s+210)+(-33s+594) (-4s3p+6s2p2-4Sp3+p4) +594s4=0 Putting s=z -1 we find ~(p,z)=332(l-z+2z2-25z3+29z 4)

-33(-l+22z-199z2+210z3) (-2p+p2z)-33(22-199z+210z 2) +(-33+594z) (-40+6zp2-4z2p3+z3p4)+594=O

188 Therefore (p, O) = 3 3 2 - 6 6 p - 3 3 x 2 2 + 3 3 × 4 p + 5 9 4 = O f r o m w h i c h we

find

p=-14.5 Appendix

7.

Association points

between branch points

on the v a i n and f r e q u e n c y s u r f a c e s

Let us s u p p o s e t h a t

f r o m an o p e n - l o o p g a i n m a t r i x

we h a v e o b t a i n e d an a l g e b r a i c e q u a t i o n gain variable

g, and the c o m p l e x

F ~ f(g,s)

G(s),

in t e r m s of the c o m p l e x

frequency variable

= O

s, g i v e n b y

(A7.1)

Let us also c o n s i d e r the derivatives

an d stationary

following equations

i n v o l v i n g the

of F:

s

and t ~F ~
--

o

CA7.3

g

The v a l u e s (A7.1 and

of s s i m u l a t a n e o u s l y

.2) are the b r a n c h p o i n t s

the v a l u e s of g s i m u l t a n e o u s l y

satisfying equation

on the f r e q u e n c y

satisfying equations

b e i n g the b r a n c h p o i n t s on the g a i n

surface; (A7.1and

.3)

surface.

If we c o n s i d e r g as a f u n c t i o n of s, the t o t a l d e r i v a t i v e of F w i t h r e s p e c t to s is dF --ds

~F

Alternatively dF

~F

+

g

dg ds

s

we can c o n s i d e r ~F

SF s

N o w from e q u a t i o n

(A7.4)

s as a f u n c t i o n of g, a n d o b t a i n

ds g

(A7.1), F is i d e n t i c a l l y

zero and

therefore

169

its total d e r i v a t i v e s (A7.4 and

m u s t also be zero,

so that

from e q u a t i o n s

.5) we have the following:

8F

DE g

dg

=

(A7.6)

ds

s

and

--

ds d-~

s

If we have a b r a n c h (A7.3)

(A7.7)

g

is s a t i s f i e d

p o i n t on the gain

and h e n c e

from e q u a t i o n

surface (A7.6)

equation we have

that (~_FF~ %~gJ s which

=

O

or

dg ds

imply that on the f r e q u e n c y

b r a n c h p o i n t or a s t a t i o n a r y surface

branch

surface (A7.7)

0

s u r f a c e we have e i t h e r

point

corresponding

a

to the gain

point.

Alternatively equation

if we have a b r a n c h

(A7.2)

is s a t i s f i e d

p o i n t on the f r e q u e n c y

and h e n c e

from equation

we have that

(~~F )s g = which

=

O

ds dg

or

o

i m p l y that on the gain surface we have e i t h e r a b r a n c h

p o i n t or a s t a t i o n a r y surface

point

corresponding

to the f r e q u e n c y

b r a n c h point.

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S.Barnett, Reinhold,

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