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ASPECTS OF NONLINEAR SYSTEM STABILITY

by

GUSTAV ' STROM CHRISTENSEN

B.Sc, University of Alberta, 1958

M.A.Sc, University of Bri t ish Columbia, 1960

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in the Department of

Elec t r ica l Engineering

We accept this thesis as conforming to the

required standard

Members of the Department

of E lec t r ica l Engineering

THE UNIVERSITY OF BRITISH COLUMBIA

September, 1966

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The Un ive r s i ty of B r i t i s h Columbia

FACULTY OF GRADUATE STUDIES

PROGRAMME OF THE

FINAL ORAL EXAMINATION

FOR THE DEGREE OF

DOCTOROF PHILOSOPHY

of

GUSTAV STROM CHRISTENSEN

B.Sc , Un ive r s i ty of A lbe r t a , 1958.M.A.Sc, Un ive r s i ty of B r i t i s h Columbia, i960

MONDAY, OCTOBER 2k AT 3;30

IN ROOM l f l .8, HECTOR MacLEOD BUILDING

COMMITTEE IN CHARGE

Chairman: L. G. James

F Noakes R. W. DonaldsonE. V. Bohn A. C. SoudackC. A. Brockley M.S. Davies .

External Examiner: A. R. Bergen

Associate Professor

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ASPECTS OF NONLINEARSYSTEM STABILITY

ABSTRACT

This thes i s t r ea t s system s t a b i l t y from threeseparate po in t s of view.

1. State Space Ana lys i s2. Complex Frequency Plane An alys i s3 Time Domain Analys i s

Asymptotic s t a b i l i t y i s considered i n s ta te space.Using s ta te space and th e gradient method an expr essioni s der ived f o r the t o t a l time de r iva t ive of the Liapunovfunc t ion . This expression i s a spec i a l case of thegeneral Zubov equation, however, i t does no t le ndi t s e l f t o an e x p l i c i t , exact s o l u t i o n except i nspec i a l cases.

Globa l asymptotic s t a b i l i t y and bounded input -bounded output s t a b i l i t y i s considered i n the complexfrequency plane. Here a method developed by Sandberghas been app l i ed to some systems the l i n e a r p a r t ofwhich has poles on the imaginary a x i s . The so lu t ion

of an example of t h i s type v i a the Sandberg methodand the Popov method shows tha t the two methods givee s s e n t i a l l y the same r e s u l t f o r the example considered.

Bounded input ~ bounded output s t a b i l i t y i sconsidered i n the time domain us ing two separatemethods. One, a method developed by B a r r e t t us ingVol te r ra se r ies has been extended t o cover cases

w i t h a non l inea r i ty of 2nd and kth degree. Two, amethod depending on the con t rac t ion mapping p r inc ip l ei s developed and app l i ed to severa l types of systems.I t i s shown tha t t h i s method generates the V ol te r ra

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GRADUATE STUDIES

F i e l d of Study; E l e c t r i c a l Engineering

E l e c t r i c a l Power SystemsNetwork TheoryServomechanisms

Analog ComputersNumerical An a lys i sHeat TransferElectromagnetic Theory

Design of E l e c t r i c a l Machine ryNuclear PhysicsNonlinear SystemsIntegra l EquationsD i g i t a l Computers

F, NoakesA. D Moore

E, V. BohnW* Die t icker

C FroeseWm. Wolfe

G W. Walker

J. SzablyaG. M, G r i f f i t h s

A. C* SoudackE. Macskasy

E. V. Bohn

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GUSTAV STROM CHRISTENSEN. ASPECTS OF NONLINEAR SYSTEM STABILITY.

S u p e r v i s o r A. Co Soudack.

ABSTRACT

This thesis treats system stability from three separate points

of view.

1. State Space Analysis

2. Complex Frequency Plane An al ys is

3. Time Domain An al ys is

Asymptotic s tabi l i ty is considered in state space. Using

state space and the gradient method an expression is derived

fo r the to ta l time der iva tiv e of the Liapunov fun ct ion . This expression

is a special case of the genera l Zubov equa tion , however, i t does not

lend i t s e l f to an ex p l i c i t , exact so lu t ion except in special cases.

Global a symptotic s t a b i l i t y and bounded inpu t - bounded output

s t a b i l i t y is conside red in the complex frequency plan e. Here a method

developed by Sandberg has been applied to some systems the linear part

of which has poles on the imaginary axi s . The so lu ti on of an example of

th is type v i a the Sandberg method and the Popov method shows that the

two methods give essentially the same re su lt fo r the example cons idere d.

Bounded input - bounded output s t a b i l i t y i s consi dered i n the

time domain using two separate methods. One, a method developed by

Barrett using Volterra series has been extended to cover cases with a

nonl inear i ty of 2nd and 4th degree . Two, a method depending on the

co ntr ac ti on mapping p ri n c i pl e is developed and ap pl ie d to s eve ral types

of systems. It i s shown that this method generates the Volterra series

found by Barret t ' s method, and thus we can actually determine a region

where the s ol ut io n of a given d i ff e re nt i a l equation can be represented

in the form of a Vo lt er ra se ri es .

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TABLE OF CONTENTS

Page

L i s t o f I l l u s t r a t i o n s v

Acknowledgement v i i

1. INTRODUCTION .1

2. STABILITY ANALYSIS USING STATE SPACE 5

2.1 In t roduc t ion 5

2.2 D e f i n i t i o n s and S t a b i l i t y Theorems 8

2.3 The S t a b i l i t y o f Linea r,Autonomous Systems 11

2.4 S t a b i l i t y Domains f o r Nonlinea rSystems by Zubov's Method 12

2.5 S t a b i l i t y Domains f o r Nonlinea rSystems by Gradient Method 13

2.6 Discuss ion o f Resul ts 22

3. STABILITY ANALYSIS USING THE COMPLEXFREQUENCY PLANE 2 3


3.2 A Frequency S t a b i l i t y C r i t e r ionf o r Nonl inear Systems 25

3.3 Systems with Poles on theImaginary Ax i s 35

3.4 The S t a b i l i t y o f Some

Nonl inear Systems 39

3.5 Discuss ion o f Resul ts 554. STABILITY ANALYSIS USING THE TIME DOMAIN 57


4.2 S t a b i l i t y v i a Vo l t e r r a S e r i es 59

4.3 The S t a b i l i t y o f Two Speci f icNonl inear Systems 65

4.4 The S t a b i l i t y o f a System witha N o n l i n e a r i t y o f Second Degree ?5

i i i

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Page

4.5 The S t a b i l i t y o f a System w i t h aNonl inea r i ty o f Fourth Degree 80

4.6 Comparison to Desoer's Method 87

4.7 The Contract ion Mapping P r i n c i p l e 88

4.8 S t a b i l i t y v i a Contract ion Mapping 92

4.9 The S t a b i l i t y o f some NonlinearSystems 96

4.10 The S t a b i l i t y o f some NonlinearSystems w i t h a Simple Pole a t th eO r i g i n 107

4.11 The Va l i d i t y o f Vo l t e r r a SeriesRepresentation 117

4.12 Discuss ion o f Resul ts 122

5. CONCLUSIONS 124

5.1 Summary 124

5.2 Recommendations f o r Future Work 125

APPENDIX A .' 127

REFERENCES 134

i v

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LIST OF ILLUSTRATIONS

Figure Page

2.1 Phase Plane Tra jec tory 6

2.2 In tegra t ion o f the Function S 18

3.1 Linear Feedback System wi th ConstantParameters 23

3.2 Nyquist P l o t fo r F Q ( s ) = s(1+ Q.04s) 2 5

3.3 C r i t i c a l C i r c le (a > 0) 29

3.4 C r i t i c a l C i r c le (a < 0) 31

3.5 Bounds on 0 and K 33



3.8 Block Diagram Representation o f (3.47) 39

2 - 13.9 K(jco) Locus f o r + x 1 - 2x1 x 2 Q e '=o) 4 2

3.10 K ( Locus f o r (x + x + 0 (x,t) = 0)e=l 443.11 K( ju ) Locus f o r Equation (3.74). 48

3.12 S t a b i l i t y Sector f o r Equation (3.75) 49t

3.13 W (ju)) Locus f o r Equation (3.80) 53

3.14 S t a b i l i t y Sector f o r Equation (3.80) 54

4.1 Block Diagram Representation o f (4.1) 57

4.2, Graph of Equation (4.18) 63

4.3 Boundary Values o f X 66

4.4 | h ( t ) | versus Time 67

4.5 S t a b i l i t y Sector f o r Equation (4.27) 70

4.6 S t a b i l i t y Sector f o r Equation (4.49). 74

4.7 Graph of Equation (4.76) 774.8 Boundary Values o f X (Equation (4.76)) 79

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Figure Page

4.9 80

4.10 83

4.11 86

4.12 87

4.13 92

4.14 93

A.l 128

A.2 130

A. 3 131

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ACKNOWLEDGEMENT

I wish t o express my g ra t i tude t o my supervisor

Dr. A. C. Soudack and t o the head of t h i s department

Dr. F. Noakes f o r encouragement and guidance during

the course o f t h i s study. Further, s incere thanks are

given t o Drs. H. P. Zeiger, E. V. Bohn and R. W.Donaldson

fo r t ak ing time t o l e v e l cons t ruc t ive c r i t i c i s m a t th i s

projec t . Also I wish to acknowledge i n t e r e s t i n g d i s

cussions he ld wi th my f e l l o w graduate students concerning

the topic treated in t h i s t hes i s and subjec ts re l a te d

thereto. I n p a r t i c u l a r I wish t o mention Mr. J .

Sutherland in t h i s connection.

Acknowledgement i s g ra t e f u l ly g iven t o the

Nat iona l Research Council f or pro vidi ng assistance fo r

the session 1963-1964 and to the Uni ver s i t y o f B r i t i s h

Columbia f o r awarding U.B.C. graduate fe l lowships for

the sessions 1964-1965 and 1965-1966.

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1. INTRODUCTION

The s t a b i l i t y of an undriven ph ys ic al system i s determined by

i t s behaviour when subjec ted to e xter nal per turb atio ns which di sp la ce

the system from i t s or ig i n al rest po si ti on . In th is thesis a system w i l l

be sai d to be stabl e i f i t remains close to it s o ri g i n al rest pos i t ion

for a l l times afte r the distu rbance has ceased, and the system w i l l be

said to be asymptotica lly stable i f i t , in time, returns to it s or ig in al

rest pos it io n aft er the disturbance has ceased.

In many cases i t happens that a system exhibits this behaviour

only within a l imi ted region around the original res t point, and this

region is then called the region of attraction or the stability region

of the giv en system. The exact l oc at io n of the boundary of th is reg ion

is i n most cases very d i f f i c u l t to determine . I f no such boundary exists

the system is ca ll ed glo bal ly or absolutely stab le. Also a system may

have more than one stable position, and may indeed come to rest at a

point other than the or i g i n al one i f the applied disturbance is large

enough. However, i n general at te nt io n i s focussed on one res t point .

Further, i f the motion of a system does not ex h ib it any of the

cha rac ter ist ics jus t mentioned, i t is sa id to be unstable.

In consi deri ng the st a b i l i t y of dri ven systems we gen erall y usea d if f er en t concept of s t a b i l i t y and speak of bounded input - bounded

output s t a b i l i t y . That i s , we attempt to determine the cl as s of f i n i t e

inputs which produce a f i n i t e output.6*

There are other de fi ni ti on s of s t ab i l i t y ; however, these

w i l l not be discussed or employed here.

* References placed above the l i n e of text re fer to the bib liog rap hy.

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2

The modern concept o f s t a b i l i t y o f phys ica l systems was f i r s t

in t roduced by Lagrange l a t e i n the eig hteen th century. He showed that

i n order f o r a mechanical system to be s table i t s po t en t i a l energy mustbe a minimum at the s ingu la r point . However, t h i s appl ies on ly to

conservat ive systems which form a very r e s t r i c t e d c l a s s o f systems i n

that the forces a c t ing must be der ivable from a s ca l a r po t en t i a l function.

The ne xt major con t r ibu t ion to the theory o f s t a b i l i t y o f

phys ica l systems was made by A.H. Liapunov i n 1892. H i s docto ral

d i s s e r t a t i o n on t h i s subject i s ava i l ab le i n book form under th e t i t l e

"Probleme General de l a S t a b i l i t e du Mouvement" ed i ted by Princeton

Unive r s i ty Press. The general contents o f t h i s d i s s e r t a t i o n are

a v a i l a b l e i n the Engl ish l i terature- '- '^ '^. Liapunov con side red a c lass

of systems o f very general natu re, and developed two d i s t i n c t methods

f o r i nves t iga t ion o f system s t a b i l i t y . In the l i t e r a t u r e these are

genera l ly denoted Liapunov's " f i r s t method" and "second method".

The " f i r s t method" cons is ts o f a s t a b i l i t y ana lys i s from

approximate so lu t ions of the system equati ons of the pertu rbed response,

obtained by means of a success ive approximati on procedure. That i s ,

t h e " f i r s t method" a c tua l ly comprises a l l procedures i n which the

e x p l i c i t form of the so lu t ions i s used when represented by i n f i n i t e

series 0 .

The "second method" which i s a l so o f t en termed the "direct

method" gives information about system s t a b i l i t y d i r e c t l y witho ut

knowledge o f the de t a i l ed motion of the system. Here spec i a l functions,

genera l ly c a l l e d "Liapunov Funct ions" , are formed and u t i l i z e d to

i nves t iga te the s t a b i l i t y behaviour o f the system i n ques t ion. I t

fol lows that Lagrange's concept o f system s t a b i l i t y belongs to the

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3

"second method".

As developed by Liapunov, th e "second method" cons t i tu tes

s u f f i c i e n t condi t ions f o r

system s t a b i l i t y ; th e

inverse problem, tha t i s ,the exis tence o f a Liapunov funct ion fo r a system exh ib i t i ng stable

motion was not i nves t iga ted by Liapunov. However, t h i s problem has been

success fu l ly so lved and found to be t rue f o r most cases through the

e f f o r t s o f severa l Russian s c i e n t i s t s dur ing the past twent y-five y ears.

Probably th e most s i g n i f i c a n t developments r e su l t i ng from these

i nves t iga t ions were made by L u r e4

and Zubov5

. Lure developed a construc

t i o n procedure f o r Liapunov funct ions r e l a t i n g t o c losed loop cont ro l

systems, and Zubov devised a cons t ruct ive pro of which shows Liapunov's

condi t ions a re both nece ssary and s u f f i c i e n t to ensure system s t a b i l i t y.

While Liapunov's "second method" u t i l i z e s s ta te space ana lys is

and thus leans heav i ly on the d i f f e r e n t i a l equ ati on approach t o

i nves t iga te s t a b i l i t y there ar e other methods ava i l ab le f o r t h i s purpose.

The more w e l l known o f these a re the "Ny quis t Frequency Locus", "Bode

Diagrams", and the "Root Locus Method". These s t a b i l i t y c r i t e r i a 7 were

developed on t h i s cont inent during the past t h i r t y years and u t i l i z e the

complex frequency plan e. However, these methods app ly onl y to

e s s e n t i a l l y l i n e a r systems, although some attempts have been made to

extend these methods to non l inea r systems as w e l l 7 . Unfo rtun atel y these

extensions to nonl inear systems are qu i t e d i f f i c u l t to use.

In 1961 V.M. Popov developed a frequency s t a b i l i t y c r i t e r i o n

f o r nonl inear systems 4 which aroused widespread i n t e r e s t , and gave

impetus t o i n t ens ive inves t iga t ions regarding the gene ra l i ty and

l i m i t a t i o n s o f th e method. These i nves t iga t ions are s t i l l continuing.

However, the Popov c r i t e r i o n has only been proven to cons t i tu t e a

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4

s u f f i c i e n t condi t ion f o r s t a b i l i t y ^ . On the other hand, Lure's

const ruct ion procedure has s ince been shown t o be c l o s e l y connected to

the Popov method. In f a c t , i t has been shown t h e o r e t i c a l l y tha t the

Popov cond i t ion (see Appendix A) i s necessary and s u f f i c i e n t f o r the

exis tence o f a Liapunov funct ion of the type found by Lure's method^.

There are o f course many more workers be side s those al re ad y

mentioned who have cont r ibuted to the f i e l d of s t a b i l i t y theory, and

probably th e most complete b ib l iog raph ies r e l a t i n g to the subject are

given i n references 1,4, and 8.

In th i s t hes i s both s ta te space, frequency plane and time

domain methods are u t i l i z e d , and t h e i r respect ive meri ts w i l l become

evident as var ious methods ar e developed, disc uss ed, and employed to

inves t iga te the s t a b i l i t y of several systems.

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7

G e n e r a l l y such a set of equat ions w i l l be represented by

x = Ax

= (x) (2.7)

where t h e no n - subsc r i p t ed v a r i a b l e s x, x and f denote n -d imens iona l

column v e c t o rs , and A denotes an square n by n ma tr ix w i t h const ant

o r v a r i a b l e e lements . Further , i f t ime t does no t occur e x p l i c i t l y i n

eq ua t i on (2.7) i t i s c a l l e d autonomous, wherea s i f t ime t does occur

e x p l i c i t l y equ at ion (2.7) i s c a l l e d nonautonomous.

o r tho gona l c o o r d i n a t e s , XT, x 2 , x n , i n s t a t e space, and that we

can form equations of the type

e t c . The s o l u t i o n of these equat ions w i l l the n gi ve a phase t r a j e c t o r y

( p r o v i d e d a set of i n i t i a l co nd i t io ns a re g iven) i n the space repr ese n ted

by the n co or d in a t es . However, g e n e r a l l y the se eq ua tio ns cannot be

s o l v e d e x p l i c i t l y .

Th e d i s t ance between two po in t s i n s t a t e space can be defined

as the e u c l i d i a n norm

I t w i l l be no te d from eq ua t io n (2.5) t h a t we now have n

|jxj| = ( x x 2 + x 2 2 + ... + x^)h

(2.8)

where x denotes t he tra ns pos e o f the column v e c t o r x.

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8

In the phase plane, points where

dx 2 = f 2 ( * i , * 2 ) _ 0

3xY f i ( X ! , x 2 ) 0

are called singular points. It should be noted that i f we desire to

inves t iga te the s t a b i l i t y of a singular point which is not located at

the origin we simply translate the coor dinate s, such that the singular

point i n question i s located at the o r ig in .

2.2 Def in i t ions and St ab i l i ty Theorems'^

In mathematical terms the system (2.7) is said to be stable

with respect to the solut ion x = 0 if , given a small positive number e,

there always exists another positive number u , such that any solut ion of

equation (2.7) which i n i t i a l l y s a t i s f i e s |x(t=0)| i u a l so sa t i s f i e s for

a l l t ^ 0 the inequali ty |x(t):| < e - I t w i l l be noted that th i s assures

system s t a b i l i t y i f , b y choosing sufficiently small i n i t i a l conditions,

we can guarantee the solut ion w i l l remain smaller i n magnitude than any

predetermined positive number. I f i n addition

lim x(t) =0t -

the system i s said to be asymptotically s table with respect to the

solut ion x = 0.A function i s ca l led def in i te i n a domain D containing the

o r ig in i f i t has values of only one s ign and vanishes only at

x ^ = x 2 = . . . = x n = 0. In the l i t e r a t u r e , Liapunov functions are

generally denoted by V = V ( X p x n ) . Such a function i s cal led

defini te (posi t ive or negative) i n a ce r t a in domain D : |xjj< H

(H > 0 i s a constant) i f the s ign of V is invariant , and V vanishes only

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9

for X T = x 2 = ... = ^ =0

Example

V = x T x

i s a posi t ive def ini t e function.

On the other hand, V i s cal led semidefini te i f V = 0 fo r

values other than x n = x 2 = . . . = x n = o.

Example

V = x x 2 + (x 2 + x 3 ) 2

i s a semidefinite function since i t vanishes at the point

JCXX = 0, x 2 = -x 3 ) .

The total time derivative of V = V (x i, . . . , x n ) i s found in the

standard manner

v a t L 6 H d T 4 - , 11 =1 1 =1

f .

by use of equation (2.7).

Liapunov's Theorem on S t ab i l i t y

Given the di ff er en ti al system (2.7) with the sing ular point

x = 0 located i n a domain D, then i f we can f i n d a pos i t ive def in i te

function V with a to ta l time deriv ative V which sa ti sf ie s 0 the

system (2.7) i s stable i n D with respect to the so lu t ion x = 0. For

proof see reference 3.

Example

*1 = X 2

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10

x.2 = -x- 1

Let

V = 1/2 ( x x 2 + x 2 2)

then

V = x n x n + x 0 x

= 0

Liapunov's Theorem on Asymptotic St ab il it y

Given the differential system (2.7) with the singular point

x = 0 located i n a domain D, then i f we can f i nd a posi t ive de fini te

function V which has a total time derivative which i s negative def ini te

i n D, then the system (2.7) is asymptotically stable with respect to

the solution x = 0. For proof see reference 3. (Note: V may be

semidefinite i f the points V = 0 do not form a trajectory.)

Example

= -x.1 - x.2

Let

V - + x

then

+ X 2 X 2

Clear ly V< 0 and V> 0 also ensures asymptotic s t a b i l i t y .

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2.3 The Stabil i ty of L inear, Autonomous Systems

Since we i n thi s section consider only lin ea r systems i t

follows that the

domain of

s t a b i l i t y wiHl

either include a l l

points i n

state space or none at a l l . That i s , the region of asymptotic s ta bi li t y

for such systems i s never f i n i t e .

Consider the li ne a r system

x = Ax (2.9)

where the elements of the matrix A , a j , are constants. For such a

system Liapunov used the following procedure to inves t iga te s t ab i l i t y.

Let

V = - ||x|| 2 (2.10)

and

V = x T Bx (2.11)

then from equation (2.11)

V = x T Bx + x T Bx

since B i s a constant matrix ; further", su bs ti tu ti ng from equation (2.9)

V = xTABx + xTBAx (2.12)

We then equate the ri gh t hand side s of equations (2.10) and

(2.12) and obtain

ATB + BA = -I (2.13)

where I is the identi ty matrix. If the eigenvalues of the matrix A a l l

have negative real parts we w i l l on solving for the elements of B,

b ^ j , i n equations (2.13) f ind that B i s a symmetric, po sit ive de fi ni te

matrix. Pos itiv e definiteness i s characterized by the fact that a l l

the principal minors of B are p o s i t i v e . It therefore follows that

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12

Liapunov's Theorem on Asymptotic S ta bi li ty stated on page 10 is s a t i s f i ed

12

fo r the system (2.9) . Gibson gives a p ra c t ica l appl ica t ion of this

procedure.

2.4 S t a b i l i t y Domains f or Nonlinear Systems by Zubov's Method

As mentioned in the introduction,Zubov^ developed a

con stru ctiv e method which shows Liapunov's c r i t e r i a to be both necessary

and suff icient for a given system to be asymptotically s tabl e. Hisq

method was invest igated i n de t a i l by Margolis , and the sa l ien t features

of Zubov's method are the fol lowi ng. Consider the systemx =f(x) (2.14)

and l e t D be an open domain i n state space while the closure of D i s

denoted D. Assume D contains the o r i g i n and that x = 0 is a singular

point of the system (2.14). Then necessary and su ff ic ien t condit ions

for D to be the exact domain of at traction of the equil ibrium of

(2.14) are the existence of two functions v(x) and 0(x) with the

following propert ies ,

1. v(x) i s defined and continuous in D.

2. 0(x) i s defined and continuous i n the whole state space.

3. 0(x) i s pos i t iv e def in i te for a l l x.

4. v(x) i s pos i t ive d ef in i te fo r x e D, x f 0.

5. 0 < v(x) < 1

6. I f y e D - D then lim v(x) = +1;x -> y

also lim v(x) = +1 provided that the l a t t e r l i m i t process can be x* co

car r ied cut for x e. D.

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13n

7. dv = y 5v_ f = . (J ) C x) ( i . v ( x ) ) ( 2.15)at

In cases when equation (2.15) can not be solved explicitly

(which i t i n general can not) i t may be so lved by assuming a series

solution of the form

V =

j=2 k=l

d j k V _ I P r i * l (2.16)

for the two dimensional case. Similar series solutions may be assumed

for higher dimensions j however, i t i s evid ent that the complexity w i l l

then increase r a p i d l y 9 , 1 3 The advantage gained in using the form

(2.16) i s j t h a t recurrence relat ions a r ise for the coef f icien ts d ^ .

Generally the approximation to the required stability boundary improves

as the number of terms included i n v is i ncr ease d, however, i t is

usual ly found that the convergence i s s lo w^ ' l^ . j n theory though, as

n oo v > +1 as required by condition 6 stated above.

2.5 S t a b i l i t y Domains fo r Nonlinear Systems by Grad ient Method

From the previous s ec tio n i t is ev ident that a dir ect sol utio n

of Zubov's equation is in most cases d i f f i c u l t . Further, numerical

methods are di f f i c u l t to apply fo r an ar bi tr ar y 0(x) i n (2.15). For

this reason a somewhat different approach w i l l be developed here which

r e s u l t s i n a par tic ul ar choice for (f)(x) i n (2.15) The approach taken

i s t h a t we seek to obtain the time variation of the Liapunov function

V ( x 1 , . . , x n ) , by using the gradient m et ho d 1 0 . Thus we ob ta in V i n a

form which r e su l t s i n grad V being di re ct ed along the system

t ra j ectorie s.

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Consider the n-dimensional system

x = f( x) (2.17)

which has a stable singular point at

f(0) = 0 (2.18)

Then l e t us postulate the existence of a continuous, po sit ive d ef in it e

funct ion V(x) such that

V(0) = 0 (2.19)

I t then fo l lows-^ that i n order V has a form which results i n the

steepest descent of V along any trajectory we require

x^ = - K~ (K = Gain factor) (2.20)

In the l i t e r a t u r e ^ K i s often required to be a cons tant; however, here

we le t

SV_ = - S ( x i , . . . , x n ) X i

Further, using (2.21) we can writ e

cW /5x k x k

(2.21)

5v75% = ; (2.22)

but

dx k ^ \

^

*m (2.23)

Hence i t fol low s that grad V is directed along the t ra jector ies of

(2,17) as requir ed.

We can now f i n d an expression fo r the to ta l time deri vati ve of

V, Thus, since

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15

we f ind using (2.21) i n (2.24)

V = - S ( x 2 2 + . . . + x ^ ) (2.25)

S x i 2

or

n . n

j = 1 J = 1 (2.26)

wh ic h i s a pa r t i a l d i f f e r en t i a l equa tion in V.

We can wri te (2.21) as

s _ !_ bV

X i & x i (2.27)

then using (2.27) i n (2.26) we find

n nbV 1 dVe>x,

x j =_ i ^ i z _j = i 3 x 2

i " F l " j (2.28)

or

n . nY x . dV V ^ d x i ^ " 5xT Z _ x 'j = l J 1 j = i 1

(2.29)

which i s also a pa r t i a l d i f f e r en t i a l equat ion i n V . Here i may have

any one valu e from one to n .

We can now compare equation (2.25) to Zubov's equation

(2.14), and i t w i l l be observed that i f we l e t

V = - l n ( l - v) (2.30)

then equation (2.25) becomes

= (S k. 2 )d - v) (2.31)

j-1

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Therefore i n order that (2.15) and (2.31) are equivalent we require

n2,

j=l

0 = (S ^ x / ) (2.32)

which requires S to be a pos i t ive d ef in i te funct ion of the state

variables (x^, > . From equ ation (2.21) i t follows

(grad V ) T = (Sx x , S x 2 , S ^ ) (2.33)

and evidently we can require

cu r l grad V = 0

which implies that the mat rix formed by

d xj

is symmetrical.

This means that i n the two-dimensional case we can write

d* 2 d x l C 2 ' 3 4 )

For the two-dimensional case we ther efo re have the following

equations to solve

av a/ 2 ' ?-vat = " S ( x l + x 7 2 )

(2.35)

X l &q 2 &q - S ^ - bx 2 ) (2.36)

On the boundary of the stability domain D we must have

V = 0 and therefore S = 0. This occurs because the s t ab il it y boundary

of the system (2.17) must necessarily be a trajectory of the system.

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Therefore i t appears that we only need to solve equation (2.36) for S

14through the use of the ch ara ct eri sti c equations

dx dx2 1 dS

x i x 2 &2

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and integrate the ch ar ac te ri st ic equations obtained from (2.37)

along a cha ract eris t ic curve. I t w i l l be noted that we know only

one point on the solution surface of equation (2.36), and we can

therefore only trace out one char acte ris t ic curve.

Characteristic Curve

S t a b i l i t y Boundary S = 0

Fig . 2.2 Integration of the Function S

There i s , however, an inherent d i f f i c u l t y i n th i s approach

which i s best i l l u s t r a t e d by an example. Consider the system

X l " " h +

V X 2 (2.41)

x = - x2 2

then equations (2.39) and (2.40) become

dx 2\HxjJ s

-x^ + 2xi z X 2 i _ Xi

x 2 x 2(2.42)

dx 2 dxi

= ~ X2 i ^ 2V 2 " x l- x 1 + 2 x x ^ 2

(2.43)

i n the small , Solut ion of these two equations i n the small gives

respectively

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19

+ x :2 ) S = c x c, = constant.

= c 2 c-, = constant

Therefore using analog or digital methods to solve equation (2.36)

simply r e su l t s i n the variables x^ and x 2 remaining c lose to their

i n i t i a l values so that the nonlinear terms i n x- and x 2 remain

ins ign i f i can t numerically. The i n i t i a l values were perturbed i n

various ways i n an attempt to overcome th is d if f i c u l ty , but to noa v a i l . The ref ore th is approach was abandoned.

becomes evident that S i s simply an integrat ing factor fo r the

equation

We know that such a function always exists fo r the two-dimensional

cases, and provided certain conditions are met such a function also

exists for higher dimensional c a se s 1 5 - However, even i n the cases when

S can be shown to exist we have no guarantee that S i s posi t ive

de f in i t e .

For example, fo r the system (2.41) we may assume a series

so lu t ion of equation (2.36) of the form

If we take a clo ser look at equations (2.35) and (2.36) i t

dx 2

3x7 = ( x i . * 2 )

n j+1 j-k+1 k-1

k=l

x1 (2.44)

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20

however, when we then attempt to determine the coeff ic ients

i t soon becomes evident that the form (2.44) does not sa ti sf y equation

(2.36).Consider again equation (2.33) and make the following

assumption

OV

^ x 2 = S x 2 = F 2^ x 2^

But i n general , for the two dimensional case

dV = $L dx, + dx 0d* i 1 d* 2 2

= F 1 ( x 1 ) d x 1 + F 2 ( x 2 ) d x 2

thereforeV = G ^ ) + G 2 (x 2 ) ( 2 < 4 6 )

which means we can not have cr oss- produ cts i n V . This is of course

a severe l imitation; however, consider the following example,X l = - x l + 2 x i 2 x 2 (2.47)

then from (2.45)

x 2 * " x 2

s = F l ( x l ) _ F 2 ^

which means

x l x 2

cur l grad V = 0

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The to ta l time deri vati ve of S then becomes

21

dS = & i . + Idt 3x7 1 5x7

2

n 2 S x 1= - 2x n L

1 x (2.48)

therefore the character is t ic equations are

dx.. dx1 2 dS

1 x l

the solution of

i s

dx^ dx^

X l = x 2

x 1 = ^ I

X2 + C

c = constant (2.50)

Then using the value fo r x from equation (2.50) and subs t i tu t ing i n

to equation (2.49) we f ind

S = S -cO 2 rx L c

2

S Q = i n i t i a l value of S

o r subs t i tu t ing fo r c from equation (2.50) we obtain

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22

It should be noted that S = 0 at = 1 i s the exact

9 13stability boundary for system (2.47) '

However, unfortunately this approach cannot be used i n

general since:

1. the state equations can i n general not be so lved e xp l i c i t l y.

2. the Liapunov function V i s generally =)= G^(x^) +

2.6 Discussion of Results

From the foregoing i t i s evident that the equation developed

in sect ion 2.5 for the Liapunov function V, vi a the method of steepest

descent i s a spec ia l case of Zubov's general eq uat ion . However,

unfortunately we can not solv e equation (2.25) f or the general case.

The reason for th i s i s that a stra ight forward sol utio n using the

method of characteristics i s usua l ly as d i f f i c u l t as so lv ing the

nonlinear equation i t s e l f . Further, we al so impose an addit ion al

r e s t r i c t i o n on V by requi r ing i t to have a form which results i n the

steepest descent of V along any trajectory.

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23

3. STABILITY ANALYSIS USING THE COMPLEX FREQUENCY PLANE

3.1 Introduction

s t a b i l i t y c r it er io n for li ne ar systems. The methods used i n thi s

chapter to investigate the stability of nonlinear systems are

somewhat similar to that developed by Nyquist; however, while his

cri ter ion provides both necessary and sufficient conditions for :

s t a b i l i t y t o exis t the methods used here w i l l only provide

s u f f i c i e n t c ond itio ns f or a given non line ar system to be st ab le .

Because o f the relatively close correspondence between the

l ine ar and nonlinear cr i t er ia Nyquist 's cr i t er io n w i l l be described

b r i e f l y. Consider the li nea r feedback system shown i n F i g . 3.1.

F i g . 3.1 Linear Feedback System wi th Constant Parameters

Since the system is linear with constant parameters i t i s governed

by a d i f f e r e n t ia l equation of the type (2.4) when the co ef fi ci en ts

glf g n are constants. Therefo re F 0 (s ) represents the Laplace

transform o f that type, only in this case we have a d r iv ing force

V][(t). Then using standard techniques we f ind

It was mentioned earlier that Nyquist developed a

V 2 (s)

v ^ I i L v

R(s) F Q (s)

V2

( s ) = 1 + F0

(s)

(3.1)

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2 4

Since we require the system to be stable (i n the sense

that i t must have finite output for every f i n i te input) the

expression F(s) = 1 + F 0 (s) can have no zeroes with posi t ive real

par t s . This can be shown to be equivalent to the requirement that

as we p lo t the frequency locus F(jio) = 1 + F Q (jaj) for

- o o < (jj < oo i n the F(ju ) plane then for oi increasing

N = Z-P

where

N = number of encirclements of the o r i g i n in the clockwise d ir ec ti on .

P = number of poles of 1 + F 0 (s ) (or of F 0 ( s ) ) with pos i t iv e re al

parts .

Z = number of zeroes of 1 + F 0 ( s ) with pos i t iv e real part s .

But since we require Z = 0 we obtain

N = - P

Further, since P f o r 1 + F Q and P f or F Q are equal, and

N for 1 + F 0 with respect to the o r i g i n i s equal to N fo r F 0 with

respect to the point (-1,0) we do i n general consid er N and P fo r

the open loop tra ns fe r fu nct ion F Q . The Nyquist locus i s usual ly

plo t ted as shown in Fig. 3.2 where the function

F fs) = 4 l J s ( l + 0.04s)

was chosen arbitrarily for i l lu s t ra t i ve purposes.

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27

a) 0 n ( O , t ) = 0 t E ( n - l , . . . . N)

b ) 0n (x,t) (n = 1, . . . , N ) i s a measurable function(3.7)

of t whenever x(t) is measurable.

Let a and $ denote real numbers with a ^ 6, then we shall

say 0 e 8 0 ( a , 6 ) i f and only i f

a < 0 n ( x , t ) < 3 (n = 1, N) (3.8)

,-1

for t e Jb,-) and a l l x f 0

Let M denote an ar bi tr ar y matrix and M* and M" A the

complex conjugate and the inverse respecti vely of M. Further, let

A (M) denote the lar ges t eigenvalue of (M*M) and l e t 1 denote the

i den t i ty matrix of order N. We s h a l l say k is an element of the

set 6 ( a , 3 ) i f and only i f k e Kj^ and, with

K(s) = J e _ s t k(t) dt

0

and

det I N + l ( a + 6) K(s) f 0 for . 0

(3.9)

(3.10)

7 (3 - a) sup XL \

- O O < (jj < 0 0

J

N +

|( a + 3) K(ja>)-1

K(ja)) < 1 (3.11)

I t should be noted carefully that the condit ion k e

means that the s t ab i l i t y c r i t e r i on does not hold f o r cases where

K(s) has poles on the imaginary axis.

Conditions (3.10) and (3.11) can be shown to be s a t i s f i e d

i f a ^ 0 and j K(ju)) + K(jto)*j i s nonnegative definite f o r a l l a ^ -The preceeding two pages are extracted from reference 19.

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28

In this thesis we sh al l only consider cases where

equation (3.2) has only one component; hence N = 1. In that case

(3.10) and (3.11) reduce to the simple form

1 + i ( a + 6 ) K(s) / 0 for (f> 0 (3.12)Li

j ( 8 - a ) sup- OO < < oo l4 (e +cOK(juO l +|(e +o)K(-ja)) _

< 1

(3.13)

Condition (3.12) is noticed to be s a t i s f i e d i f K(ju>)

does not enc i r c l e the point (- ^ + > 0) i n the K(ju>) plane.

The second condition (3.13) can be invest igated i n the following

way. First take the square of both sides of the inequal i ty (3.13)

then

l ( g - a ) 2 K

1 + (g+a)Re(K) + ^(g+a) 2 | K |< 1 (3.14)

and

or

-ct6 | K l 2 < 1 + (8+a)ReK

- i _ < (I + 1) ReK + I K | 2 i f "6 > 0

(3.15)

(3.16)

or

i f a6 > 0 (3.17)

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We w i l l now consider the inequality (3.15) for some

separate cases

1. a = 0

From (3.15) i t follows

ReK >6

2. a > 0

Consider Fig. 3.3 below

(3.18)

ReK

Fig . 3.3 C r i t i c a l C i r c l e (a > 0)

The closed contour shown i s a c i r c l e C, of centre P ( -d , 0) and

radius r Q , where

(3.19)d = - \ (i D

r - 1 r 1 \r 0 - 2 ( + g)

Then using the law of cosines

rQ

2 = |K ' | 2

+ d2

- 2Kd cos(180 - 0) (3.21)

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then using the law of cosines

r Q 2 = | K ' | 2 + d 2 _ 2dK* cos 0' (3.29)

which upon substitution from equations (3.27) and (3.28) reduces

to

- h = IK'I 2 ( | 4 ) R e K ' ( 3 - 3 0 )

Now consider a c i r c l e of radius r c < r 0 , and l e t a point

on the frequency locus K, form a triangle with d and r c .

[ K | 2 + d 2 . 2dK cos 0

+ d 2 . 2dReK cos 0 (3.31)

which upon substitution for r Q and d becomes

-h > l K l 2 + + F ) R e K C 3 - 3 2 )

and i t - w i l l be noted (3.32) is i d e n t i c a l to (3.17) which must be

s a t i s f i e d fo r a < 0. Therefore the locus of K must l i e en t i re ly

inside the c r i t i c a l c i r c l e C, i n this case.

It should be observed that condit ion (3.12) i s met i f

any one of the three conditions just examined i s s a t i s f i e d .

The foregoing proofs relating to (3.13) are to the

au thor 's knowledge new; however, we do of course only prove an

es tabl i shed re la t ion i n this case.

I t has been pointed out by Profess or A. R. Bergen,

Universi ty of C a l i f o r n i a , that these proofs can also be obtained

See Fi g. 3.4. Then

or

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33

by using the trigonometry used in der iving "M - c i r c l e s" .

Then to rec ap itu lat e, the sa li en t features of this

s t a b i l i t y cr it e ri o n, as used here, are the following:

Given a free, nonlinear system (that i s f a system without a driving

force) w ith one no nl in ea ri ty , where the tra ns fer fun cti on K(s) of

the li n e a r pa rt of the system has (simple) poles only in the l e f t

hand plane, the c ri te r i on provides su ff ic ie nt condit ions for the

system to be globally asymptotically stable, provided the

nonl inear i ty 0, s a t i s f i e s

0 (x, t)

where 8 iL a are r ea l numbers.

I t w i l l be observed, that here we do not view this

c r i t e r i o n as a bounded input - bounded output re la ti on sh ip .

Geometrically the cond it io ns enumerated above mean the

following:

Case 1. a = 0

Fig . 3 . 5 Bounds on 0 and K

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Case 2. a > 0

Fig . 3.7 Bounds on 0 and K

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36

then combining equations (3.34) and (3.36) we obtain

L* ( X) + 0 '( x, t) = 0 (3.37)

which can now be t reated by using the s t ab i l i t y cr i te r ion descr ibedin section 3.2. That i s , we p lo t K(jco) = rr^r^y* and from th is

plot we gr ap hi ca ll y determine a and 8. Then

5 0 J x 1 t O _ K 6 ( 3 > 3 8 )a = x

or

0 (x ,t ) - exa _< 2- _ 0 small)

a

therefore

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4 2

ImK

(3.59)

I f we l e t vi = 0 we must remove the eq ual it y signs from

(3.59) to s a t i s f y (3.13) . In that case (3.59) becomes

-1 < -2XjX2Qe < co

(3.60)

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or

( i - U) ) + j u

44

(3.64)

Hence we can determine the K(jw) locus shown i n F i g . 3.10 below

Fig . 3.10 K j o j ) Locus for (x + x + 0 ( x , t ) =0 ) e = 1Then using equation (3.39) we require

x = + 1 _< J &p ^ 3 + 1

but from F i g . 3.10

a = -3 = 0

1 - v

hence

0 (x , t) iA

(3.65)

where p > 0 i s smal l .

Hence we obtain quite a conservat ive bound on the

nonl inear i ty i n th is example when we assume e = 1.

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45

However, we can also use the following procedure. Using

e d i r e c t l y i n P(p) without spe cif yin g i t s value we have

K O ) = ^(e - w ) + ju>

(e - OJ ) loiTl 7 ' 2 2 2

(e - ID ) +oj ( e - w ) + u) ( 3 . 6 6 )

then c lear ly, as e becomes large the locus shown i n F i g . 3 . 1 0 w i l l

shrink to a po in t at the o r i g i n i n the K(joi) pl an e.

Hence, given a spe c i f ic va lue of e i n ( 3 . 6 6 ) i t i s clear

we can always pi ck a and g such that the frequency locus K( joj)

l i es en t i re ly ins ide the c i r c l e whose extremities on the ReK axis

are given by - \ and - . Thenp ot

- a + c ^ P C x , t ) e + ( 3 - 6 7 - )

Further as E + i t fol lows.from ( 3 . 6 6 ) that we can l e t

a and 8 -> i n such a way that the K( j o j ) locus l ies with in the

c r i t i c a l c i r c l e .

Therefore

( 3 . 6 8 )

0(x,t)

where y ^ > 0 i s small and y > 0 i s a r b i t r a r i l y l a rg e, and

y = - a + e

y 2 = B + e

It follows that i n the l i m i t as e -*

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46

may l i e anywhere i n the f i r s t and t hi rd quadrant.

Example 3 (See reference 4 page 86)

Consider the system

x l = " c x l + x 2 " ^ x l ^

x 2 " x l + x 3 (3.69)

x 3 " ' c x l + b { 3 ( x i ^ '

where c > 0, b > 0 are two constant parameters. We can reduce

(3.69) to a single equation through elimination of x 2 and x^.

Then we obtain.

X l + C X 1 + X l + C X 1 + ' b ( 3 ( x l ^ = 0 (3.70)

This equation is of the type (3.40) since from (3.70)

L l ( s J (s 2 + l)(s + c)

which has two conjugate poles on the imaginary axis

s = + j

Therefore we can rewr ite equation (3.70) i n the form given by

(3.41) s and we obta in

x + (c + e) x 1 + x x + (c - be)x 1 + L 2 0 l ( x ] ) = 0 (3.71)

where

L 7(p) = P 2 - b (3.72)

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Fig . 3.11 K(jco) Locus for Equation (3.74)

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c r i t i c a l c i r c l e s and to vary e i n order to determine i f a

s t a b i l i t y sector could be found containing 0 (x^) en ti re ly . From

F i g . 3.11 i t is readily seen that the locat ion of the s t a b i l i t y

sector i n the (0 ,x^) plane depends on which c r i t i c a l c i r c le i s

chosen. Further, i n reference 4 (see pages 8 and 9) i t i s

indicated that varying e w i l l r e su l t i n a ro ta t ion of the

s t a b i l i t y sector i n the (0 ,x^) pl an e. On the other hand, i t is

clear that the s t a b i l i t y sector can not include the x^ axis i n this

case since this would r e su l t i n system (3.69) not beingasymptot ically st ab le . However, we sha l l now investigate (3.69)

by use of the Popov method i n order to show that the two

s t a b i l i t y criteria give approximately the same r es u l t .

Example 3 by Popov's Method (See Appendix A)

1C X , + X

2 " 0 C x l )

X2

X , + X3

(3.79)

x,3 cx^ + b0 (x^)

which can be rewrit ten as

(p 3 + cp + p + c)x, + (p 2 - b )0 ( x j = 0 (3.80)

Then

(3.81)

o r

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51

when

(1 - co )( c + j u ) (3.83)

It w i l l be noted that we have a pa r t i cu la r case i n that

W(p) has poles on the imaginary axis

P a 1 3 -To inves tigat e equation (3.80) for s t a b i l i t y - i n - t h e -

l im i t we must determine

d Q + j e Q = l im (p - j) W (p)P+J

2 b

Now, le t

1 ( i + j c ) (3.84)1 + c

c = 2

b = 1

as before; then

, 1 ( 1 + bs nd o = 2 ( ~ $ > 0

and the condit ion fo r s t a b i l i t y - i n - t he - l i m i t i s s a t i s f i e d . See

Appendix A.

Using equation (3.83) we can separate W ( j c o ) into i t s

rea l and imaginary pa rt s, and

Re

w . -

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then

I m W = Cu 2 + B O O(1 - a) )(4 + u )

X = ReW

Y = u ImW

Y Y _ (to 2 + 1) (2 + qto 2)A q Y j T

(o> - 1)(4 + to )

(g>2 +

D(2)

52

4 + u, 2 C 3 - 8 5 ^

for q= -2

Hence equation (3.85) states that

X - qY ^ 0

therefore

X - qY + ^ > 0 (3.86)

the "Popov Condition" i s s a t i s f i e d f or ^ = 2 except at the point

a) = 0 0 (see F i g , 3.13), however, as mentioned i n reference 4 this

is permissible since a s li g h t change i n q w i l l s a t i s f y (3.86) for

0 < to < oo. i t therefore follows system (3.79) is asymptotically

stable when the nonl inear i ty 0 (x^) is contained i n the sector shown

i n Fig 3.14.

We can now compare the res ults obtained by the

Sandberg method and the Popov method. For th is purpose we compute

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54

F i g . 3.14 St a bi li ty Sector for Equation (3.80)

(u > 0 small)

the angular magnitude of each sector i n F i g . 3.12 and 3.14. Hence

6g = tan" 1 1.0 - tan" 1 0 .50 = 1 8 . 9

e = ta n" 1 0.50 = 26 .6 P

and it is seen that

e > e cP s

when e = 1. This is to be expected since a rotation of any sector

away from the x^-axis causes this sector to decrease in angular

magnitude^ The value s of 6 and 0 found above il lu st ra te s thiss p

fact .

i

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55

No attempt is made here to determine the optimum value

of e or the optimum c r i t i c a l c i r c l e , sinc e the primary purpose

here is merely to show how we can apply Sandberg's method t o

some systems whose l i n e a r par t has two conjugate poles on the

imaginary axis.

3=5 Di scus sion of Results

I t w i l l be noted that we have only treated systems

for which Sandberg's vector equation has only one? component i n

th is chapter . However, i t is evident that a large class of

systems belongs to th is group. Fur the r, sinc e th is chapter i s

pr imar i ly concerned with undriven systems we have used the

Sandberg cr it er io n to investi gate such systems for asymptotic

s t a b i l i t y . That i s , we have not used the cr it er io n as an

input - ou tput s tab i l i ty c r i te r ion .

From the so lu ti on of examples 2 and 3 i t is cl ea r that

i t is pos si bl e to apply Sandberg's c r i te r i o n to some systems

with poles on the imaginary axis.

Two separate cases are considered.

1. One simple po le at the o r i g i n

2. Two simp le , pu re ly complex, conjugate p o le s.

Fu rt he r, i t i s shown that this development applies to driven

systems as we l l , p rov ided the system inputs are square in teg ra bl e.

AlsOj, Example 1 shows the ap pl ic at io n of Sandberg's

cr it er io n to an equation i n which the non lin ea rit y includes time

e x p l i c i t l y. It i s evident from the de sc ri pt io n give n of the

cr it er io n and the so lu ti on of thi s example that Sandberg allows

/

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56

the non linea rity to have a certa in e xp li c it , func tion al

dependence on time.

Example 3 il lu st ra te s the sol ut ion of a system with a

nonlinearity which does not depend explicitly on time. This

example has been solved by both the Popov and the

Sandberg c r i t e r i o n . From the resul ts obtained i t i s evident that

both methods in this case give approximately the same answer. I t

w i l l be recalled that the Popov criterion! does not apply to systems

having no nli nea rit ies containing time ex p li c it l y .

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57

4. STABILITY ANALYSIS USING THE TIME DOMAIN

4.1 Introduction

In this chapter we shall consider some specific, non

l inear systems with arb itr ary inp uts , and inves tiga te these driven

systems fo r bounded inpu t - bounded output s t a b i l i t y . In order to

define our terms consider the following system:

LCXj) + 0 ( x 1 ) = y(t ) ( 4 > 1 )

where L is a li ne ar , tim e-in varia nt, di ff er en ti al operator, e is aconstant, 0(x^) is a nonl inea r fun cti on of x^, and y(t ) is a fo rc in g

funct ion. System (4.1) may be represented i n bloc k diagram form as

shown i n F i g . 4.1 below, (p = ^ )

yCt) +,

e 0 (x x)

o1 x-, (t)

L(p)

0 0

Fig . 4.1 Block Diagram Representation of (4.1)

It w i l l be observed that Fig. 4.1 is identical to Fig.

3 . 8 , however, whi le we i n chapter 3 employed the complex frequency

plane fo r st a b i l i t y anal ysis we s ha ll in thi s chapter use the time

domain ins te ad. Fur ther , i t should be noted that while the previous

two chapters dea lt mainly wit h undriven systems, we s h a l l here consider

driven systems, as already mentioned above.

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59

4 . 2 S ta bi l i ty v ia Vol ter r a Series23

Barrett developed an in ter es tin g it er at iv e method fo r

determining the input - output stability of the system

L(x) + ex 3 = y(t)

which is of the form (4.1). His procedure w i l l be described in detail

i n the following paragraphs.

Consider equa tion (4.3) where L(p) is a l i n e a r , time-

inva rian t , d if fe re nt ia l operator with s imple zeroes only i n the l e f t

hand pla ne, and y(t ) is an ar bi tr ar y dr iving function, and let us assume

the i n i t i a l conditions

x(0) = 0

y(O) = 0 (4.4)

We can then form the no nlin ear in te g ra l equation3 ,

x(t) + e f h ( t - x ) x 3 ( i )dT = f h ( t - T ) y ( x ) dT on-* ca-'

(4.5)

where h(t ) is the impulse response corre spond ing to the li near tr an sf er

function T \ \ , It should be noted that we can use the limits (-,)L(pJ

sin ce x = y = 0 f o r t< 0 and h( t - T) = 0 for T > t. Equa tion (4.5)

can clearly be considered from the point of view that we endeavour to3

determine the output x ( t ) , by con vol ving the input y( t) - ex , wi th

the impulse response of the system h(t). This in te rpre ta t ion agrees

with F ig . 4.1. ^

We s h a l l now solv e equation (4.5) by i te r a ti o n . For

th is purpose l e t us assume as a f i r s t approximationoo

x(t) = I h(t T )y(t) d i (4.6)

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and substitute this into equation (4.5). Then we obtain

60

CO

x(t) = h( t - T)y (x)dT - E h ( t - x) h (x - T 1 ) y ( T 1 ) d t 1

(4.7)

which upon expansion becomes the second approximation to the true

solut ion of equation (4 .5) . The la st term i n (4.7) can be written i n

the following wayoo

3 h (t - x) h(t - T 1 ) y ( T 1 ) d x 1 dx

h(t - x) h(x - x . 1 )y(x 1 )dx 1 |h(x - x 2 )y (x 2 )dx 2

h(x - x 3 )y (x 3 )d x 3 dx

Let us now defin e

h 3 ( t - x 1 } t - x 2 , t - x 3 ) =

h(t - x)h(x - x ^ h f x - x 2 ) h(x - x 3 ) dx

then (4 08) becomes

e h ( t - x) h(x - T 1)y(T 1 )dr 1 dx =

(4.8)

(4.9)

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61

h 3 ( t - x 1 , t - T 2 , t - T 3 ) y ( T 1 ) y ( T 2 ) y ( T 3 ) d T 1 d x 2 d T 3

(4.10)

and subs t i tu t ing (4.10) into (4.7) we obtain

- co -oo -oo

00 00 00

x(t) = h( t - x ) y ( x ) d x - e / I h 3 y ( T 1 ) y ( T 2 ) y ( T 3 ) d T 1 d T 2 d T 3

(4.11)

If we desire higher order approximations we must then

substitute equation (4.11) in to (4.5). We w i l l then obtain the so lu t io n

of equation (4.3) i n the form of a Vo l t e r r a series as fol lows.

x(t ) = |h (t - x )y (x)dx - e j j j h ^ t x ^ y ( x ^ y f x ^ d x ^ x ^

+ + (4.12)

At this point we sh al l digress to consider the v a l i d i t y

of this series represe ntation. That i s , when does the output x ( t ) ,

obtained i n th is manner represent the so lu t ion of the o r ig ina l

d i f f e r e n t i a l equation (4.3).

When we conside r li ne ar forc ed equations we are always

ensured of the existence of a unique steady state solution, however, i n

general nonlinear different ial equations do not exhib i t this

charac te r i s t i c . Here sev eral steady state solut ions may ob ta in , however ,

some o f these may not exis t due to i n s t a b i l i t y.

Vol t e r r a introduced the fun ctio nal representation used22

in (4.12) and estab lishe d the concept of an ana ly t ic funct ional . He

defined a funct ional F [y( t, a)] to be ana ly t ic i f F[y( t ,a)] . is an analyt ic

function of the parameter a, when y ( t , a ) is an analy t ic funct ion of t

and the parameter a.

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63

H = / |h(t)| dt

0

and consequently

sup |x(t)| < x

(4.16)

If we consider the algeb raic equation

.3

(4.17)

X - |e| HX J = HY (4.18)

and attempt to solve i t by i t e r a t i o n , i t becomes clear that 0 (Y) i n

(4.14) is a series solut ion of (4.18). Equation (4.18) may be graphed

as shown i n F i g . 4 .2 below.

In order to determine the values X 2 and Y 2 (see Fig.

4.2) we f i n d , usin g equation (4.18),

Y

D(X~, Y )

\ 0\ 1 / X

C ( - X 2 , -Y 2 )

F i g . 4 .2 Graph of Equation (4.18)

dYdX

= 0

1 - |e| 3HX

or

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64

written

X 9 = j (4.19)2 (3 | e |H)*

Y 2 = l e l ^ S H ) 5 / 2 C4.20)

For the range |Y|

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65

L(x) + ex 3 = y( t ) (4.26)

where L(p) is a l i ne ar , t ime-invariant di ff er en ti al operator with

(simple) zeroes only in the le f t hand plane and x = y = 0 for t ^ Othen for sup |y(t)| sup |y(t) | versus

X > sup |x(t)|. This graph clearly shows that we have a unique output

x ( t ) , for a given input y ( t ) , only along the branch COD. When we in clu de

points outside this branch we do not have a unique output for a given

input i n that to each value of y(t ) there may correspond three values

of x ( t ) .

However, the problems of ex is te nce , uniqueness and time-

invariancy w i l l be treated i n d e t a i l i n sect ion 4.11 and w i l l not be

considered further here.

4.3 The S t a b i l i t y of Two Specific Nonlinear Systems

Consider F i g . 4.3 shown below. I t w i l l be observed that

F i g . 4.3 i s i d e n t i c a l to Fi g . 4.2, however, the purpose here is to

indicate where the maximum values of X (X^, - X ) are located for

|Y|

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66

Y

\ F ( - X b , Y 2 ) D (X 2 ,Y 2 )

l \I \1 \1 \i \i \

0 1 X

C ( - X 2 , - Y 2 ) Wb , - Y 2 J

Fig . 4.3 Boundary Values of X

In order to i l l u s t r a t e the u t i l i t y of the theory presented

i n sec t ion 4.2 we w i l l solve two examples.

Example 1

x l + x l + x l " x l 3 = y ( t ) ( 4 , 2 7 )

where

x(0) = y(0) = 0

Comparison of (4.27) to equation (4.3) shows that with

t = + 1

L(p) = p 2 + p + 1

then we f ind

(4.28)

(4.29)

= h(t) (note X = j)

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68

but

( 2 > 1 ) Fat

e s i n bt dt = *7 ( a s i n D t - bcosbt)a +b

2k;

(2k+l)

(2k)

bea ( 2 k )

a 2 + b

F aiT/b

7 (1 + e ) (4.34)

and

(2k+2)|

ate s i n bt dt =

"2 (1 + e b J

(2k+l),a + b'

(4.35)

hence, i f we su bs ti tu te equations (4.34) and (4.35) in to (4.33) we

obtain

t 2ak?-i O / K \ a ( 2 k + l )

h(t) I dt - - ^ ( 1 + e ^ / b ) ( 2 e + e b ) ( 4 > 3 f i )

'0 k=0

1 11

l k F. if

a*

k=0

I t w i l l be r e c a l l e d the p a rt i a l sum o f a geometric series i s

1 - r n + 1

S n - V T V - (4.38)

and from equa tion (4.37)

air/b

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69

but from (4.30) a < 0, therefore r

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70

We can now determine from equations (4. 18) , ( 4.41), and

(4.42). Hence

3Xb - 1.39X b = (1.39)(234)

Solution of th is equation by t r i a l and e r ror yields

X b = .983

which can be compared to X b for (4.27) when i t i s undriven 13

x b - 1.0

(4.43)

(4.44)

(4.45)

It i s evident that here we can al so speak of a s t a b i l i t y sector,

although the l a t t e r i s f i n i t e i n this case. See F i g . 4.5 below

0(x) = -x-5

\

" X b

Fig . 4.5 S t a b i l i t y Sector for Equation (4.27)

Example 2 (Rayl eigh' s Equation i n reversed time)3

x, = x 9 + u (- x,) (4.46)

x 2 " X l

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71

We can rewrite (4.46) i n the following way

x

3

Tx i = x 2 " I +

w - at 3

or

x

3

x 1 + y x 1 + x 1 - yp - j - = 0 (4.47)

which i s Rayleigh's equation. This we can i n turn write as

2 x 3

P + y p P + 1 ( x p - y - f - 0 (4.48)

and i f we now apply a dr iving force y( t ) , we obtain

X l 3

L(x x) - p - f = yCt) (4.49)

Comparing (4.49) to (4.3) we f ind

L(p) = P 2 y P * 1 (4.50)

' " + %

We ag ain assume the i n i t i a l conditions

x(0) = y(0) = 0

Then

X _ 1 ( n T r ) = ^ ~ T T s i n [(1 - X 2 ) t + 6] (4.51)U s J (1 - \ T

= h(t) (note X = j)

where

e - t an " 1 ( 1 " ^e tan ( 4 > 5 2 )

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If we le t2 J.

b = (1 - x z ) :

then

and

a = -X

e a t

h(t) = Sg- [sinCbt + 8 ) ]

H = j Z r | s i n ( b t +e)| dt

0

To evaluate this int eg ra l le t

bz = bt + 8

then

dz = dt

and equation (4.55) becomes

8

H = jL e ^ ( e a z |sin bz|dz - J e a z |sinbz|dz)

0 0

However, from equations (4. 31) , (4. 32) , and (4.40) we f ind

e f -af- a

F1 ^ D / az i . , i , e 1 + eb~ / e l s i n b z l d z = T T ? aT

a + b 1 - e F

Therefore we need only evaluate the in tegra le/b

-ae1 . TI = e u / e a z I si n bz| dz

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73

then

To deteremine I explicitly assume

v = 0.7

= t a n _ y J

Therefore

= 1.93 radians

= 2.06 radians

and

B"< TT

It therefore follows that we can remove the absolute va lue si gn i n the

in tegra l (4.58) and determine I d i r e c t l y. Then

- a e

or

e a z s i n bz dz = ^ ^ j (a s i n bz - b cos bz)

a +b

1 e

ae1>

0

eE"

0 (4.59)

I =a +b (4.60)

We can now substitute (4.57) and (4.60) into (4.56), whence

o 9 ^

H = e C e - 1)

1 - eair (4.61)

Evaluating this expression numerically there results

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74

e = .308

= 1.83

Then using equation (4.20) we obtain

Y - 2 ...Y 2 " 37T

(4.62)

( 3 ) C 5 ' 4 9 )

= .322 (4.63)

We can now determine from equation (4.18) by using (4.62) and (4.63)

Thus

X b - (0.7) ( i ^ 5 - ) , X b 3 = (1.83) (.322)

Solut ion of (4.92) by t r i a l and e r ror yields

X b = 1.77

(4.64)

(4.65)

In this case the f in i te s tab i l i ty sector has the appearance

shown in Fig. 416

X.

- x b

^ ^ ^ ^ ^

X

Fig . 4.6 S t a b i l i t y Sector for Equation (4.49)

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75

4.4 The S t a b i l i t y o f a System wi th a Non l inea r i ty o f Second Degree

In t h i s sec t ion Barre t t ' s development (see s ec t ion 4.2) w i l l

be extended t o cover systems of the type

L(x) + e x 2 = y ( t ) (4.66)

where L(p) i s a l i n e a r , t ime- invar iant , d i f f e r e n t i a l operator with

(simple) zeroes only i n the l e f t hand plane, e i s a constant, and y ( t )

i s an a rb i t r ary dr i v in g funct ion . Inspect ion o f F i g . 4.1 shows that

(4.66) can be represented i n block diagram form as w e l l .Let us assume the i n i t i a l conditions

x(0) = y(0) = 0 (4.67)

We can then form t he fol l owin g nonlinear in te gr al equation from (4.66)

x ( t ) = / h ( t - T ) y ( i ) d i - e / h ( t - T ) X 2 ( T ) d x (4.68)

I t i s c le ar tha t (4.68) can be solved by i t e r a t i o n as was

equation (4.5). In t h i s case we obtain the Vol ter ra ser ies

OO 00 00

x( t ) = / h ( t - T ) y ( i ) d T -e / / h 2 y ( T 1 ) y ( T 2 ) d T 1 d x 2

OO *>00 00

+ ... (4.69)

where

h 2 ( t - i 1 P t - T 2 ) = J h ( t - T ) h ( T- T 1 ) h ( t - T 2 ) d T (4.70)-00

We can then take the absolute value o f both s ides o f (4.69) whence

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76

sup |x(t)| < sup |y(t)| / |h(t)|dt + |e| sup |y(t) | 2 ( J |h(t) |dt) 3

0 0

(4.71)

We then observe that each term i n (4.71) i s dominated by the

corresponding term i n the power series

X = 0 (Y) = HY + |e| H 3 Y2 + . . . (4.72)

where

sup |y(t)| < Y (4.73)

H = J |h(t)| dt (4.74)

0

and thus

sup |x(t)| < X (4.75)

It w i l l now be shown that (4.72) is a series so lu t ion of the

algebraic equation

X - |e| HX 2 = HY (4.76)

Let

X = HY (1st Approximation)

then

X = HY + | E | HV (2nd Approximation)

and

X = HY +|e |H 3Y2 +2|e| 2 HV + | E | 3 H 7 Y 4 + . . . (4.77)

and i t i s seen that (4.72) is indeed a series so lu t ion of (4.76).

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77

Consider the graph of equation (4.76) shown i n F i g . 4.7

below. I t w i l l be noted that the points A and B are

F ig . 4.7 Graph of Equation (4.76)

easily determined to correspond to

1 1A ( X 2 2 lei H Y2 =

H) (4.78)

B = B(X = 1eTH

, Y = 0 )

We must now inquire into the region of convergence of the

series (4.72). For th is purpose we s h a l l employ the r a t io t es t . Then

th |n term(n-1) st term

, n - l

^ 2

H 2n-1 Y n jn

H 2n-3 Y n-1 jn-1

< 1 (4.79)

where I _^ and I are the integer co effic ient s of the (n- l )s t

and the nth term res pec tive ly. Let us now assume

1Y

2 4 lei H (4.80)

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(see F i g . 4.7) then (4.107) becomes

78

nth term| 1 I

n(n-l)st term| = J 1 (4.81)

We can now investigate the respective values of I n and I n _ ^

I f , i n ( 4 . 7 7 ) w e disregard the factors i n |e| and H ( i t w i l l be noted

that these factors cancel i n (4.79) when (4.80) i s substitut ed into

(4.79)) then the series (4.77) can be wri tten,

X* - ajY + a 2 Y2

+ a 3 Y3

+ a 4 Y4

+ a g Y5

+ ...

But

a l = 1

a 2 = (a x )

a 3 = 2 a l a 2

a 4 = ( a 2 )2

+ 2 a i a 3

a 5 - 2 a x a 4 + 2 a 2 a 3

etc.

and i f we determine the f i r s t few coe ff i c i en t s i t soon becomes evident

that

n x* as n + n

that i s

l |x n - x*|| - 0

n -+ oo

Then re fe rr in g to De fi ni ti on s 1 and 2, a complete

normed vector space is called a Banach space. Thus the space

of a l l r ea l, continuous functions of the in te rv al -

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||x(t) I = sup |x(t) | (4.109)

- oo < t < 0 0

Let S be a l inear operator i n a Banach space E which

maps points of E i n to i t s e l f . Fu r the r, l e t the impulse response

of 6 be h( t) , then i f the input to Q is Y^(t) the output Y Q (t)

i s found from

Y Q (t) = h(t - T ) Y . ( t ) d T

I f we now take abso lute values on both sides we find

0 0

||Y 0II < ||Y.|| J |h(t)| dt

0

=< ||V i| h i

Therefore the noim of the l inear operator g, i s

0 0

II31| = J |h(t)| dt (4.109a)

0

since fo r physical systems we can assume h( t) = 0 fo r t =< 0.

Further, i f0 0

Hell = j |h(t)| d t < o

i f f o l l o w s 2 4 that 6 i s a bounded operator and i s therefore

24

continuous . This means $ maps points of E in to E i n a

cont inuous manne r .

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92

Then to r ecap i tu la t e . I f the mapping, li ne ar or

nonlinear,

A(x) = x (4.115a)

s a t i s f i e s the contraction condition (4.110) then we are

guaranteed that a solut ion of (4.115a) exis ts . Further, i f

i n addition the fixed point condition (4.112) is s a t i s f i ed

then we obtain a region where this solut ion i s unique. Hence

we obtain the so lu t ion o f (4.115a) i n the form of a convergent

series

x* = x + (x, - x ) + . . . + (x - x , ) + . . .o v l o J K n n - l '

4.8 S t a b i l i t y v i a Contraction Mapping

Consider the system

L(x) = y( t ) (4.116)

where L is a l inea r, t ime- invar ian t d i f fe r ent ia l operator with

(simple) zeroes only in the left hand plane and y( t) i s an

arbitrary, bounded dr iving funct ion. We can represent (4.116) i n

block diagram form as shown i n F i g . 4.13 below/ Cle arl y we have

L' ( X) + x = y( t) (4.117)

y + L (p)x

L ' C P )

X

r L ' C P )X

Fig . 4.13 System (4.116)

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93

when i t follows from (4.116) and (4.117) that

L(p) = L'( P) + 1 (4.118)

which means F i g . 4.13 represents a s table linear system with

input y(t) and output x ( t ) .

Now consider the fol low ing nonl inear system

L(x x) + e0 (x 1) = y(t ) (4.119)

where L is a l inear, t ime-invarian t, d i f fe r ent i a l operator with

(simple) zeroes only i n the left hand plane, e i s a constant,

0 (x 1 ) i s a nonlinear functi on of x^, and y(t) is an arbi trary,

bounded dr iv in g fu nc ti on . System (4.119) can be represented i n

block diagram form as shown i n F i g . 4.14 below where N = x^+ 0 (x^).

Further, i t follows that, as in F i g . 4.13, we have

L(p) = L (p) + 1 (4.120)

s L ( p ) x 1 1 x 1 = X + y ) " L'(P)

x x + 0 ( x 1 )

N

F i g . 4.14 System (4.119)

We can now compare (4.116) and (4.119), and i t w i l l be

observed that (4.119) is i den t i ca l to (4.116) i f e = 0 .

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Therefore we s h a l l i n essence consider system (4.119) as a

perturbed version of system (4.116) and assume L and y are

i den t i ca l i n both systems, whereas

y (t) A x x ( t ) - x( t) (4.121)

as shown in Fi g. 4.14. This point of view i s c lose ly akin to

17

that taken by Desoer .

Now consider (4.119) and l e t us wri te that equation i n

the form

x x = gy - eg 0 (x x ) (4.122)

Here g is a linear operator with impulse response h(t), and from

(4.109a) the norm of g isoo

II 3II = j |h(t)| dt (4.123)0

Further, i n reference 29 the following theorem i s

proven. A necessary and suf f ic ien t condit ion that a bounded

input y(t) , to a l i ne ar , time-i nvarian t system gives ri se to a

bounded output x(t) is that

oo

Hell = J |h(t)| dt < ( 4 _ 1 2 4 )0It w i l l be observed that th is constitutes part of Barrett 's

theorem.

It follows that (4.116) can be wr i t ten as

x = gy .(4.125)

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95

Therefore substituting (4.121) and (4.125) into (4.122) we

obtain

By + y = By - eB 0

(By + y)or

y = -cB 0 (By + y) (4.126)

In what follows we s h a l l , fo r the sake o f s impl i c i ty,

use the following notation

H I - u

II e|| = H (4.127)

l|x|| -X

llyll -Y

I t w i l l be observed that (4.126) is equivalent to a

nonlinear mapping. Therefore usi ng the terminology of section

4.7 we haveA(y ,y) A -eB 0 (By +y) (4.128)

In order that (4.128) is a contraction mapping, condition (4.110)

must be sa t i sf ie d. Therefore we require

||A(y 2,y) - A O i ^ y ) ! =

|| - eB 0 (By + y 2 ) + eB 0 (3y + y j) || ,

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Consider the case when the nonlineari ty 0 i s such that

0 _< 0 (gy + y) _< k(6y + y)

where k > 0 is a constant. Then (4.129) may be written

|e| Hk ||y 2 .< K ||y 2 - y j

whence the contractio n condi tion (4.110) reads

e l H k ^ K (4.129a)

where 0 < K < 1.

We must now determine a sphere (4.111) within which

the f ixe d point con dition (4.112) is sa t i sf ie d. Therefore we

f ind from (4.128)

" l = IIAC v 0,y|| =< |e| H- ||0 .(6y)||

< | e | H 2kY

Hence, the fixe d point condition (4.112) becomes

|e| H 2kY < (1 - K)a

or

o ^ |e|H 2kY A L ' i 1 - k (4.129b)

In this case the contraction condition (4.129a) is

c l e a r l y d i f f i c u l t to sat isf y unless e or H i s small (note

0

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LCxp + e a ^ * = y ( t )n=2

For this system (4.128) becomes

kA(y ,y) A - e$ a n ^ y + ^

n=2

97

(4.130)

(4.131)

and (4.129) becomes

k

|e| H || Y, a n (6y + y 2 )n

- Y a

n ( * Y +

^ /n =2

But i n general

(By + y 2 ) m - (By + y n )

k

In=2

< K | y 2 - y x |

(4.132)

1' /

m-1

(By + y 2 ) - (By + y ^ ] (By + y 2 ) m " 1 " 1 ( 6 y + M l } 1

i=0

We can therefore write (4.132) as

k

I e | H | | y 2 - y 1 | | Y n | a n | (HY + U ) n _ 1 ,< Kn=2

where we have put

m-1|| Y ( ^ + v 2 ) m " 1 " i ( 8 y || =< m(HY + U)

i=0

(4.133)

i n 2 " w x l(4.134)

m-1(4.135)

Relation (4.135) is j u s t i f i e d on the basis that we have

llvjl < IM

< y

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98

where y i s given by (4.121). From (4.134) we can now determine

the condition f o r (4.131) to be a con tr act ion mapping. This

condition i s seen to bek

i n _ 1 - v (4.136)E| H ^ n a n (HY + U) K * < K

n=2

where 0 < K < 1. Or equ ivalently

k

| e | H ^ n a n (HY + U ) 1 1 " 1 < 1

n=2

(4.137)

We must now determine a region within that specif ied

by (4.137) where the fixed point condition (4.112) is s a t i s f i ed .

To do this we observe that from (4.131) we obtain

A(y 0 , y ) = y 1 (y 0 - 0)

or

n(HY) n (4.138)

n=2

Therefore combining (4.136) and (4.138) the fix ed point cond ition

reads

k

n(HY) n < U( l - K) (4.139)

n=2

where we have put a i n (4.112) equal to U.

Refer to equation (4.121). This equation may be

written

X 1 _< X + U (4.140)

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or using (4.125) we f ind

Xx ^ HY + U

Therefore (4.137) is equivalent to

k

(4.141)

nX ^ ' 1 < 1

TTTTn=2

(4.142)

Further, i f we choose U and K such that (4.137) and

(4.139) are s a t i s f i e d fo r the largest possible value of Y which

we denote by Y^ then

and from (4.141)

Y < Y,

X l = ^ 1 + U

(4.143)

(4.144)

Thus we have shown that i f (4.142) and (4.143) are

sat isf ied then equat ion (4.131) has a unique solut ion. This i n

essence means that the so lu t ion of (4.131) obtained by i t e r a t i on

y = y o + (a 1 - y Q ) + . . . + ( j i n - y ^ ) + . . .

i s a convergent se ri es , and s in ce .

(4.145)

v n = A k n - l ' r t

we have

y n + l " y n < K n l n - l

Therefore, i n the region specif ied by (4.143) and (4.144) the

system (4.130) has a unique output x -^ t ) , fo r every inpu t y(t)

Hence we have determined a region where (4.130) exhibi ts

bounded input - bounded output s t a b i l i t y .

We can now s ta te the following theorem:

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1Q0

Theorem 4. Given the non line ar system

k

L(Xl) + e Y, =yCt) (4

'1

n=2

Where L(p) is a l i nea r, t ime-invari ant , d if fe re nt ia l operator with

(simple) zeroes only in the left hand plane, and = y = 0 for

t =

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101

H = 1.39

For (4.149) the nonlinear mapping (4.131) becomes

A(j i ,y) = - E g (By + y ) 3 (4.151)

the contraction condition (4.148) becomes

3 | E | H (HY + U ) 2 4 K (4.152)

and the f ixed point condit ion (4.149) is

| E | H (HY) 3 < U ( l - K) (4.153)

The contract ion condit ion (4.152) can be wr i t t en as

follows (on removing the equ alit y sign)

or

x

l , 1.41 |e| H (4.180)

where 0 < K < 1.

We must now ensure that there exists a sphere (4.111)

within which the fixed point condition (4.112) is s a t i s f i e d .

From (4.176) we f ind

||A(y 0,y)|| < |e| ||3|| ||sin6y|| (y Q = 0) (4.181)

Therefore the fixed point condition (4.112) becomes.

|e| H ^ (1 - K)a (4.182)

or

|e| H

a = 1 - K (4.183)

I t i s evident that (4.183) can be s a t i s f i e d for any

0

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107

Theorem 5. Given the non linea r system

LCx.^ + e s i n x = y( t) (4.184)

where L(p) is a l ine ar, t ime -invariant , di ff er en ti al operator

with (simple) zeroes only in the left hand plane, and x = y = 0

for t ^ 0 then for sup |y(t) | < we have sup |x^(t) | <

provided (2** |e| H) < 1.

This theorem states that the system (4.184) exhibits

bounded input - bounded output stability provided (2^ |e| H) < 1.

It follows that i f instead of e s i n x^ we had e cos x^

i n (4.175) a sim ila r res ult would obtain. Further, i t i s evident

that an equation having a nonlineari ty consist ing of a combination

of trigonometric functions can be treated by a similar procedure.

However, i t is not considered necessary to investigate further

examples of thi s type.

4.10 The S t a b i l i t y of Some Nonlinear Systems with a Simple Pole

at the Or ig in .

Consider the following system

L ' ( X I ) + e 0 , ( x 1 ) = y( t ) (4.185)i

where E i s a constant, 0 (x^) is a nonlinear functi on of X p

y(t) i s an arbi trary dr iving function and L (p) i s a l inear,

t ime-invari ant , di ff er en ti al operator such thatL' ( P ) = pF(p) (4.186)

Here F(p) has (simple) zeroes only in the left hand plane.

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Clear ly (4.185) can not be treated by the method

developed i n sec t ion 4.8 because (4.124) i s not s a t i s f i e d .

However, here we can employ the following procedure. Let us

rewrite (4.185) as follows

i / ( X j ) + ax x + e 0 , ( x 1 ) - ax x = y (t ) (4.187)

It w i l l be observed we have simply added and subtracted the

term ax^, (a > 0) from (4.185). Therefore we can writ e

L (x 1 ) = L ' CX ^ + ax (4.188)

0 (x x ) = 0 , ( x 1 ) - a X l (4.189)

It w i l l be noticed that L(p) has zeroes only i n the left hand

plane provided we choose a such that i t does not cause L(p) to

have pu re ly complex conjugate zer oes . Hence we can wr i te (4.187)

as follows

L ( X l ) + 0 (x x ) = y( t ) (4.190)

which is of the same form as (4.119) and can therefore be treated

in a similar manner.

In th is case the nonlinear mapping (4.128) becomes

A(ji ,y) = - ef3 0 (By + w) + aB (By + M ) (4.191)

where B now i s a linear operator derived from L(p), and the

corresponding impulse response h ( t ) , thus depends on a.

The contract ion condit ion (4.129) becomes, i n this

case,

|E | H|| 0 (By + v 2 ) - 0(6y + MX) II + a| | B(By + y 2 ) -

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eCey + y x )

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