Lectures on nonlinear systems

Lectures on nonlinear systems

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Barrett, J. F. (1976). Lectures on nonlinear systems. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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roep Meten en Regelen

cfd. der Elektrotechniek echnische Hogeschool

indhoven

LECTURES ON NONLINEAR SYSTEMS

J.F. Barrett

CONTENTS

1. Preliminary remarks on power series

1.1 Power series in one variable

1.2 Power series in many variables 1., Power series between veotors 1.4 Analytic functionals

2. Volterra series representation of nonlinear systems 2.1 Discrete-time systems

2.2 Continuous-time systems

2.' The practical importance of majorant series

,.

The solution of forced nonlinear differential equations

'.1

Introductory example

'.2

The forced pendulum problem

,.,

Discussion of a general type of equation

'.4

On reversion of power series

3.5

An

equation occurring in the theory of oscillations

3.6

Formulae for reversion

4. Systems of analytio differential equations in state variable form 4.1 Introduotory' remarks

4.2 Transient stability of autonomous systems 4.3 Forced motion of non-autonomous systems

5.

General analyti? systems

5.1

Definition of a class of bounded analytic

5.2

On the use of delta funotions: inversion

5.3

Algebraic notation

5.4

Homogeneous and multilinear operators

5.5

Cascading in algebraic notation

5.6

Inversion in algebraic notation

5.7

Feedback: functionals Page 1

,

5 "7 11

14

17

19

2'

27

'3

36 38 39

43

48

51

55

61

63

66

69

74

6. Multidimensional system transforms

6.1 Multidimensional system transforms in continuous time 77

6.2 Cascaded systems 84

6.3

System transforms of inverse 87

6.4 Calculation of output from input using Laplace transform 91

6.5

Calculation of response to sinusoidal input 101

6.6 Multidimensional system transforms in discrete time 109

7.

Response of nonlinear systems to random disturbances

7.1

Determination of output statistics using Volterra series 7.2 The use of orthogonal polynomials for instantaneous

non--linear operations

7.3

The use of Hermit. functional expansions to discuss the passage of Gaussian noise through a time-dependent nonlinear system

7.4

Hermite functional expansions relative to white noise

Appendix:

Bibliography on Volterra series,! Hermite functional expansions and related subjects

1932 - 1974

111 114

122

1. PRBLI)\iINARY !Bf'!ARKS O~l PO\ih""R SERES

1.1 Power series in one variable:

A power series in one real variable has the form

Such.a series, when convergent, defines y as an analytic function of x:

By f1acLaurin' s theorem, the coefficients of the power series are given by

Convergence is conveniently discussed by the method of majorant series. We say that the series

is a majorant series to the previous series if

-. . _.- . . . ··e.tc

\

If

'xl

i.X, every term of the original series is less or equal to the corresponding

te~ of the majorant series in absolute value. Consequently, convergence of the

Further, i f I x

I

~ X, it follows from the iriequali ties above that also \ y \ :S. Y. For

I

\j I

I

Clo -+- <=I., x:. +

<

I

Qo J

+ \

CI. I

I l

~ I

+ \

a. 'l \

t::e \

'2 +- ..

<

A (:)

.Ir AI

X

-t A'l)<.':t

-+ ..

-::

Y ..

The conv.~rgence of the majorant series is easier to discuss than the conv~rgence of the orig~nal series. For the original series may have negative teros whereas when

X ~ 0, the majorant series has only non-negative terms and must ther~foreei thar

converge to a finite posi ti ve limit or to + DO. Since, if it is convergent for

some positive velue of X then it must also be convergent for smaller positive values

of

X,

there are just two possibilities:

Case 1 e.g. '(

=

1 + )(.

+

\ X'1+

2~

)!. .,

.

::

e

The rnajorant series converges for all X> O.

The original series converges for all x.

. '2

(\

X

)

-,

Case 2 e.g. '( :: \

-+

x

+

l~)

+ ..

XI X,

~e _{majorant series converges for O~X'SXl'}

The original series converges for

lx'

~Xl"

The radius of convergence of the original series is ~ _Xl

These two PQssibili ties may be illustrated by constructing the graph of Y

=

F(X)

where F(X) is the sum of the majorant aeries. F(X) is clearly positive and

increasing when X ~O· and either (case 1) remains finite for all positive v3lues of

y

x

Case 1 Case 2

1.2 Po.er series in ~~y variables:

The power series expansion of y as a function of n variables x

1,x2, ••• ,xn takes the form (0) Q,. 'Vl

+ \"

L

(I) 0.. t :X:t

+

""'I T\..

V~

LL

tl~1 i2.=' , " ,

Where the coefficients a. . are usually assumed symmetrical in the indices

1.

1].2 •• i

1,12, •• This makes the expansion uniquely deterr!une the coefficients.

Assuming convergence, the series ~~11 define an analytic function

and by MacLaurin's theorem, the coefficients are given.by

to)

f

\x:::o

Q..

-en

_af

\"'=0

0..

-

-C>:x:.

_\ ll)

,

_a'lf

\ '<

=

0 ~

-2\,

~:c;"dXj

.

.

..

..

..

, .. te:. •

To

:1iscuflS convergence, we introduce a :najorant series

Y

=

A lO)

+

A (I) X + A t7.) X'2..

+- .

where it is assumed that

A 'tOl

>

I

'0. (0) \ -n (1)

L

~ t.. 1) A

_/

Q.. I.

-.

. I

-=,

"'1\ _'l1

t'2. )

"

~

l'2..\

A

~

a..

,

/

_/

l \ 1..1.

-l.,':1

t l'l .:: I

. .

.

, ,

.

If' \ x.\~X, i

=

1, ••• ,n, the sum of the terms of any degree in the original series

1

·is less or equal to the term of the same degree in the majorant series, both being positive or zero. e.g.

'I'l 'l1

<.

~I

\'

_r

_Q

_i.i

2

\,~

i.

IlJc:

&2'

/-,

_~

.

• L,::I 12. :: t n 'Yl

\

-<:.

>

IQ~Ii.2.1 Xl.

/

-

-.

l..=.

l';4 :: , .(

A

_t2,

X'-Consequently \convergence of the mnjorant series implies convergence of the original series when 'xil~:{ and we further deduce that lY\~ Y.

1.3 Power series between vectors:

In a similar way it is possible to discuss'the multiple power series arising from functional relations between two vector quantities

::

1

~I)

_.

-which will have the form.

::1,

::

SI

\.:t,) .... , . x : " , )

1

-

,-

.-

_{. " "} \ " " , .

~

i'.1 ... =.

5"",

lx,,--

):)c"") , or

The corresponding power series relations have the form

-no 'Vt. '"r'\. (Q)

L

(I) .~

I

ti)

11\t

-

...

0. k..

+

_{ct k ; l} .::t.

₊

_{0.. ,"}

L

x. ::t.

_{+ ..}

l _{11:.)\112- I, I}

2

-I. ': , _{l\ :: \}

.

_{i '2. ::,}

Relations of~,tMs type may also be discussed by the method of majorant series, for

it

is easily verified that the series

'( =

A

to\

+

A (1) X

+

A ~'2.) X '2.

+ . _,

.'

is a majorant series for each power series ( i.e. for k

=

1,2, •• ,m) provided that

A (0)

~

I

Cl

<..~

I

A

l2)~

--(6)

i.e. provided that

A

to) ">

\

~ len \ ~ (n

>

"

lC1..

k :t \

A

-

""" 0..)(

L-k:. ':\) "'\ "" I ': I l'l.) Y'\ \1\.

A

~ r.... 00..)<

'"

'"

.

\

_{Q.,,- ,.}

l

lot ::\)'" ')~

- -

/ . / . : ' \ \ '2. I I -::: I . I 2.:: \ \

..

.

.

, e.. 1-1:

If \ x:\6. X , i = 1,2, ••• , n , then convergence of the majorant series implies

J. .

convergence of each of original s.erles for k

=

l,2, ••• ,m and further we deduce that

,y

j\ ~ Y . , j = 1,2, •••

,m.

It is convenient to introduc~ the following notation giving norms of the various

quantities:

tl

'X.

\I

\ :1:;. • \ \

,

~::I).",V\

\

,

.

Then we have that

~alytic functionals~

Consider again the analytic function

to) C\.

+

+

2.2:

a... •• (a> .::t:. x. I, 12. II 1'4

+ ' ...

i 1:1

'2:'

and think of x

=

1

Xl'" • ,x

n \ as giving a discrete rep.cesentation of a functi. on as in the following diagram.

l1T

_{,Q(. .}

11

Keeping the interval fixed, let n ~Oo so that the sampling interval becomes

smaller and smaller and tends to zero. Then in the limiting case, we might expect that the summation in (1.2) hecomes replaced by integration, the vector x having become a function: X· t. ~') ... ,vt. _-7 :lC Lt.)

«. S

t

<

_(!.> I. )

,

-'VI.,

(3

L

~

S

ctt .

.

_{t. ::., 0<.}

The above expansion might thus be expected to take the form:

'J

:: ~ l:x:)

~

(0)

~

(.I) CL + 0.. (t) x. tt) d...t

+

~

(8)

S'ince x is now a. function, f(x) is a function of a function or II functional ".

A functional b?ving an expansion of the form (1.3) is said to be an analytic function with Volterra series expansion, after V.Volterra (1860-1940) who first investigated this type of expansion.

It is possible to discuss convergence of Volterra series by the method of majorant series provided that the following boundedness condition is fulfilled:

(?>

5 \

a-

(n It)

I

~

<

00

~ ,

.

e.tc.,

Then

is said to be a majorant series for the Volterra series if

(3

.

5

I

a..

(I)

~

t ) \ clt'

oi.

(3

f3

S~

If ., x (t) \

~

X for I)(. <. t <.

~

, then each term of t he Vol terra. series is less or equal

to the term of the same degree in the ma.jorant series. e.g.

(3

(3

0 (3

l

s

_C

_Q.. l~)

_l

_{t \ ) t2,) .::c.lt.).:t: tt'2.,) c:U I d..t::;t}

\

_<.

j

_{} \ "U'\t"t,)\d-t,J.t2}

«.

eX. ol..

<

A

(2)

X

'2.. .

>

Convergence of the majorant series consequently implies convergence of the Volterra series when

1

x(

t)I~X

for

Ol~

t

~ ~

and further we deduce that

I

y

I

~

y •

Corresponding to relations of the type

Y'I 1') VI (0)

I ()

_"

_I

lZ)

CL""

+

0.... k .

:x:..

1-

a..

;);:. t X. t

+ ,.

~ I. I.

L.

Ie ~I

t'l.

\ I I

"2-i.

=-1 i J :: I i 2,

=,

which were considered above, there will be Volterra series of the form

,~ tt;\ -( 0) 0. (t)

+

~

S

o.(I'tt;

t.)x.lt,)d.t,-+

+

~

(.S t

S

~

Q. 2.)

tt

~

t,. t'1 )xlt,). :clt'2.)c:U-,.J..J:-2., ~ ~

in which the values y(t) of ,one function are expressed as analytic functionals

of the values x(t) of another function either 'over the same or different t-interval. There ia thus' defined an anlytical nonlinear relation y =: rex) where x ani y both

represent functions. To use the method of majorant series it is then necessary to assume that the following boundedness conditions are satisfied

~

II

Q.... (I)

II

-

"rr\ Q.)C

5

I

Q. Cn

l

t':

t-, ) \

c:U;,

<-

₀₁₀

t-

o(,. 0.. l'l)

lJ

~ ~

II

:: MCl..X

Sf

I

Q. ('l)

It

.

_{t-,.t'l)t cU:\d.t~} )

<

_QO

t

ot. 0( - . . . e..t-c.

(10)

A majorant series then has the for.n

where , l2.\

A

~

I

a... (s) \

\I

a... (t)

l\

, , ,

Convergence of the majorant series then implies convergence of the Volterra series when \x(t)\ 5X for all t and further 'we deduce that \y(t)\SY . for all t.

It is thus seen

that

that the theory of analytical functionals possessing Volterra series expansions is strictly analogous to the theory of analytical functions of several variables. There is however one important difference which makes the theory of analytical functionals very much more complic~ted than that of analytical functions: not all analytical functionals possess an expansion of the Volterra series type. For example, if

f; ( )

and

tf ( )

are analytic functions of 2 and 3 variables respectively, then the functionals

{!>

S

+

l

t ,

~

(t:\,

~

(tl) dr

0(.

should certainly be called analytical and they possess a power series type

or

expansion although this expansion is not the same as the Volterra series type given above.

(11)

2. VOLT3R.!1A SEES i{!::PRESENTATION OF NOiiLnmittt SYSTEI~

2.1 Discrete-time systems

Consider a general nonlinear syste~ with single input x and single output y.

NON l \ N E A f<.

':J ....

--->~I ~---7,

$'( S TE. M.

Denoting the value of the input at instant k by ~, the input over the whole time range - ()O( k < + 00 is given by the infim te vector

1 ...

Similarly, the output is given by the infinite vector

,

.,

The system output Yk at instant k must be a function of the inputs ~'~-1'~-2""

at and before the instant

k.

Consequently we can write

which equations define a functional relation

between the input and output vectors. we shall have

Supposing the functional relation analytic

0'0 Qo ()o to)

,

ll)

>

-

l'l)

"

'dk. -=

"-k +-

_L

CL ~<." • J:.

+

L

a.. J::. :lI:.

+ ...

, I.,. l. _{k~t.i.'2. I. I L'l.} \ \ t :::-00

_L,

=-00

_{t~: -().:;)}

(12)

The coefficient arrays will be uniquely determined if it is assumed, as is usually

done, that they are symmetrical in the suffices i

li2"".

The physical realizability condition: since the output at any time ca~ot depend on

future values of the input (so that Yk is not a function of x

k+1,xk+2"" ) it ~~ll follow that t ,)

a.

k .. i. 0

if

...

,.

k l'1l

''2 .,.

k e l k " _~"'\I.'l. 0

'f

etf-he~ L I .,. k ov

It is tHerefore also possible to write

k

>

replacing the upper limit in each summation by k. It is,ho~ever, usually more

convenient to make the summation to infinity bearing in mind the su~~ation includes

zero terms.

To measure the total contribution of the coefficients of any degree, the following

norma are defined on the coefficient arrays:

n

a. tOl \\

,f

a..

cn

H

Il

0.. l'2., 1/

.

,

.

-oo<..k<.oo "

.

...

..

\"

"

L L

.

.

.

In order to define a majorant series for. the. Volterra expansion, it must be assumed that the norms of all coefficient arrays are finite:

It is then said that

'( ::. A

to)

+

A (.1) X

+

A

(7..) X 1.

+ ...

is a majorant series for the Volterra series if

If \ x

k\ ~ X for all k, convergence of the majorant series will imply convergence of the Vol terra series and further that 'Ykl~Y for all k. Denoting by F(X) the sum of the majorant series as a function of X we thus have

where input- and output norms are defined as

Il~\\

- 0 0

<

k <. ~ -OO<'K<~

The time-invariant case: a system is said to be time-inv£riant if the input-output relation does not change with time. The coefficients in the Volterra expansion must then only depend on time ~ifferences and so the expansion takes the form

00 ₀₀ 0'I:l.

h

lo)

'"

(n

_"

_"

h

<"2)

'dk

=

+

1, k .

::x:..

₊

::c.

~

+

~

..

L

- I. ~

/

I

k-~l _{k ~ ~"2. \.}

,

_~'2.

.

-

-t

=-00

_{",;. -00 \'l ': -00} 00 ₀₀ 00"

<2-)

~ tOl

+

~ (n ~

'"

I

h

"(' .::c k _-('

+

h..,-

or ~k.~v _{X k} v +-~

..

L

L

. , . • - 1. , 'f :. _00

r, ;:

..(b ""2. ':: _OQ

The physical realizability condition now becomes

='0

if

=0

, ,

.

't..tc.

and the norms of the coefficient arrays b~come

Q.o

Il

h

<n \\ ::: .

Yn 0-)(--Coo

<

k<. 00

H

h~)

\\:::

hot Q.)(

_"""'<'

k<.oo 2.2 Continuous-time systems V'.: -00 \'

~

(1.) , k -i I l<.-i 1-~l;-~ ~.:t:::-fA) OQOQ

- L

L \

h;:~~

\

The formulae of the previous section extend to continuous time with little modifica tion •.

The input and output are taken to be Input:

,~ \. x.

l

~) 1 - OQ <. i: <. ()o)

Output:

I;!

-A general nonlinear system is characterised by a functional

givi~g output y( t) at any time t in'terms of present aM pa$~,1nputs x(

t')

t' ~

t.

]

The functional relation may also be\regarded as defining a linear transformation

\ ~ ('

between total input ~ and total outp~t

Z.

A system is said to be analytic if the input-output functional relation can be expanded in a functional power series. AS will be seen later, this functional power series can take various forms. The most important form however, is the Volterra series

to)

r

,t·>

a.. It )

+

j

0...

l

t , t \ ) ::c. t t, ) eLt \

',.

00 00

+ .

S

~

a.t'L)

l~",

t •• t'l.) ;;lett.) ;'t:.\.t.,) .:::lrloU

Z

+'"

-0<:1 -000

(0) (1)(

where the functions a (), a ) , . • .€called. the "kernels" of the expansion) are regular integrable functions. ,The kernels completely characterize the nonlinear system. They are usually assumed symmetrical in the indices i

1,i2, ••• in order to make the expansion unique.

The physical realizability condition is

In .

0..

It; \:; \) ::

0

...

,

Because of these conditions, the upper limits of the integrals in the Volterra expansion may also be taken to be t.

The norms of the kernels are defined as

II

0-to) \\

-

'\'Y\ 0. X

I

a.. to) It:;) \

- 0:1 <. t: <. 00

..

QQ 0)

S· \

C!) ,

.

It

Q.. ₁₁ :: ' 'YI-'\CLX

_a..

l

_{t ., t;}

1)\

ct.t

~ -00 <.. t-<. 00

.

-OQ \ (2.) ()o 0."

It

\I

'h'\ CL::t:..

Sf

t'2.} Cl.

_l

Cl.

{t-;

t,.t.'I..)\oU:-,ol.t':t

-00 <. t<. 00 -0.:. -~

,

.

,

..

,

To use the method of majorantseries, i t i~ assumed that the norms of all the kernels are finite. Conve~gence of the Volterra series then follows from that of the majorant series as before and if Y

=

F(X) is the msjorant series relation we shall have that if lx(t)\sX for all t, then ly(t)ls.y = F(X). This result may also be expressed in terms of input and output norms defined as

--<\;<'00

I· Time-invariant systems: for tirne-:invariant systems, the kernels depend only on time differences and the Volterra expansion takes the form

lilt \

- 0 0

h

(0)

+

- 00 oa eo

+

S

f

k

L'2.)

lc-

t ' ; " t - t2)X

U

:;,):lClt2

Jd.J:.

1

cU-:2-+

-00-00 0.0 ~

+

S

f

h

l'2.)

l'tl\

't1.):c

l

t -1:\);;:C

It:-l:~)d

1:, d."C2."i' . -Oc) - CX)

The physical realizability condition becomes

h.

(1)Ce.) ':: 0

if

1:.

>

0

h(z)

ttl

~'l..\

::: 0

• ~ • '" ... , e..1::" c

and the norms of the kernels become

~ H",CDn::: _'WI.

_0."

S

_I'"

_{It -}

_{t ,)\ d..t \} .... 00<.t:<..4;X> _~

U

h'-?)

H:::

Yv\ 0. 'X - OQ <. t <.~ "" ... " ... ~.

2.3 The practical importance of majorant series f.i.ajorant series can be used

(a) to discuss system stability

(b) to find the maximum error arising from truncation of a Volterra series. Let us consider each of these in turn.

(a) Suppose a sys~em has a majorant series

"( =

A to) -+ A

<.-'

X

+

A\,2,\ )(2.+...

=

F,K) . .

Then if the input is bounded by \x(t)l$X for all t, then the output Will be bounded bYly(t)I~Y = F(X). So we have the following picture:

-+X

0

1

A

.

~

.

:::r:r~

-;x.

-,

.

The system is thus stable under the influence of canstantly acting disturbances. Stability'in this sense is called "bounded-input, bounded-output stability"

(sometimes abbreviated as BIBO stability) The relation between input and output bounds is conveniently shown by the graph of Y

=

FCX) As has already been remarked, the graph curves upwards and sometimes becomes infinite after a certain value X

=

Xl' In this case, the system will be stable when the input bound is less than Xl but may go unstable when the input bound is greater than ~.

y

I

'" I

. -,:

1~

x

\, Ii ... ;\-

ot-

c.e .. +Q., .... s: ta.b \ li l.)

(18)

(ll) Fo!' .the pur!'ose of calculation, only the first few terms of a Volterra series

are normally used. It is therefore important to be able to estimate the error

made by neglecting higher order terms. This can be done by using the majorant series since the neglected terms will have absolute value less or equal to the corresponding terms of the majorant series.

Suppose, for example, that a system has input-output relation

wi th majorant series

.'( =

A (n X

+

A t2l

x:

'2. + ...

In linearising the system, i.e. making a one-term approximation, the maximum error will be, when \x(t) \~Xf

.-

....

The maximum error is consequently equal to the vertical distance between the graph Y = F(X)

and the tangent to this. curve at the origin. The graph shows clearly when the system departs significantly from linearity.

y

(19)

THE SOLUTION OF FORCED NONLINEAR DIFFERe.:NTIAL EQUATIONS BY VOLTERRA S'8RIES

1 Introductory example

t us consider the equation

\()(>o)

the infinite time interval - CN<t <+.)0 • Here x(t) is the forcing functin~

l:?'l.t) and yet) the response (outout). ',ilien

i

is zero, the system is ~inear with lution

. ..)0

,~.

h.

l

t - C \ ) ,x: It I ) cU: \

- 00

.s lineal' solution may be regarded as a first approxima.tion to the solution when

.S small bU1rtonzaro

'f€'rential equation as

<

lj \.:t: l T- .)1.. ':j \. t 1

M10 get further approximations in this case, write the

l solve again, regarding the nonlinear term on the right hand side as a forcing function.

~ ~'

w

+

-

'/'U

Ls is a nonlinear integral equation which is 'equivalent, on the infinite interval, the original differential equ,!'!tion. I t is very suitable for finding the solution successive approximation. On substituting the linear auproxirnation for y2 on the ?;ht han::! side we get a second approximation which w"ill be correct to terms in ~ :

(20) c>C)

j

Vt

l

t -

t.)

x l<. )

cIJ::.

"I-- 0\;;1 In putting I ~:2.)

It

y ' " , ) n '.. \.,) \.. - t l. ::::: It--t.I )

L

rs

h\t'-t,)h~t'-t.)

.-' " ' " - \)Q

),,-It,)

x \..t2) C\.t\GU'2,

\~I

) h It- t ' ) h lti- \;,) h

I.e ..

t2.) dJ::.'

-. vv I

e- ,-" \.

I:' - C . )

[I

.:::x...

e -

~ It - t ,) [ I

o

t2., - 'A. \. \: - t: ',2.)

1

t.

(21)

The process of successive approximation may now be continued, the solution just

found being substituted in the y2 term on the right hand side. On retaining terms

to order l2 the first three terms of the Volterra series will result. The Volterra series is thus generated term by term.

To find

a

majorant series for the Volterra aeries solution, we set up what is called

•

a comparison equation for the nonlinear integral equation. In the present case, the comparison equation is

y=

H)(+ \E\H,(2..

which is written down by observing the form of the nonlinear integral equation.

Because the comparison equation is similar in form to this equation, it may be solved

by successive approximation in exactly the same way i.e. for ~= 0 we have the

linear approximation

which, by substitution in the right hand side, gives the next approximation

and'soon, thus generating an infinite series solution

It is clear that if

I

x( t)\~X and

H

~

t:

I

h l"<)

I

J.

'<.

-each term generated will be greater or equal in absolute value to the corresponding

term of the Volterra series. The procedure consequently gives a majorant series.

Now the series thus found is nothing else than

one of the solutions for

Y

of the comparison

equation. It is in fact,

J,_4~£'H'2.X

2.\t\H

which corresponds to the branch OA of the graph in the diagrlL'll.

The majorant series will consequently be convergent when

4la\

H'2. X

S. \

ie. when

\ ~

0

<.

X

<

--

41£ \

H 2.

The corresponding bound on Y is

j \

O

< y < .

-

2 I.E' H

<

-

•

2.

IE l

When X exceeds the above bound, the majorant series will be divergent and no conclusion

can be drawn regarding the convergence or otherwise of the Volterra series.

The existence of a Volterra series solution in only a limited region round the value

x

=

0, y = 0 may be understood by considering the stebility properties of the

undist--urbed differential equation

Supposing that O()O, the equation has two points of equilibrium: a stable equilibrium . at y :i:

°

and an unstable equilibrium at y

=

rx./s..

The trajectories of the differential

equation converge and diverge respectively from these two equilibria as shown in the diagram below for the case ~ ') O.

If a small disturbance acts on the system when it is in the neighbourhood of the

equilibrium point ~

=

0, it ~~ll continue to remain in the neighbourhood. If a large

disturbance acts, then it is clear that the system could be transferred to a region of

instability and y would then become unbounded. The limits found above for the Volterra

series solution, restrict the motion to the region PQRS.

3.2. The forced pendulum problem

.

Consider the motion of a damped pendulum under the action of a force applied at right angles to the string. Denote by ~ the p~gle of the string to the vertical and by f(t) the force applied at time t. With a suitable choice of time unit, the equation of

L ,

motion is'

which we consider in the form

,

•

(D'2.+

₁

Z;D

t-t)Blt) -

~3lt)

+ eS(t\ - ... ::

Slt)

:3! S~

grouping the linear terms together on the left

hand

side.

On

taking the nonlinear terms to

the right hand side, the equation becomes:

where

3

as

e - _

+ ...

31

s

~

On inverting the linear differential opera tor on the infinite time interval - ().:;)( t ( + C\.:)

the equation is converted to the nonlinear integral equation

00

S

h\.t-t,)5-t

t

_,)cLt

_l

00

+

S

h,lt-t')S<.9lt,l)d.t-,

-co -co

where h( ) is the impulse response function corresponding to the linear operator:

t ) 0 ) }'-

=

~

\.

-c;

'?<.o

The equation is now in a form suitable for solution by successive approximation. If the force remains small at all times, then the pendulum will perform sm~ll motions in the neighbourhood of the stable equilibrium at (;

=

0 and so G will remain small. The nonlinear terms represented by g( )on the right hand side will then be at least of the third order. They are neglected for the first approximation.so giving the linear solution:

~his first approximation is now substituted in the g( ) term on the right hand side Lnd for the next approximation the cubic term is retained giving

e

\.t) _

00'

j

hlt:-

t'l)flt\)d.t:, _00

+

00 O r . : ) .

S

h

~I:.

t')

l )

\-t

It'.

t.

H

It. )

cU:c₁

Y

d.J::'

-~ -~ .

6

co

S

h\.t-b\)

5-lt,)

elk,

- 00

+-

5)

~

h

l3

\c_t"t_t.,.

tot,)

flt.)Slt2)~lt})cLt.d.t2d.t3

_ ~ - 0 0 - 00 rhere 00

h

\1»

l

_{t - t I \ t - t '2, 1::: - t 30)}

=

6"

l

S

_h

_{l \: -}

_{t I )}

_h

_l

e -

_{t t )}

_{~ l}

_{t/ - t 2.) "" It I - t 3 ) cU:.. 1} --00

~e next approximation will be obtained by resubstituting this expression in the

.onlinear term g( ) and retaining terms to the fifth order and 50 on. The higher

,rder terms may also be obtained more systematically using the method described in ection 3.4 below.

'0 discuss the convergence of the Volterra aeries solution the comparison equation is

et up which has the form

rhere

+ -. .. -

-

s~v-.40

- G

a solving the comparison equation by successive approximation in the same way as he nonlinear integral equation was solved. we get

irst approximation:

E>

=. \-U: econd approximation:

e::.

HF

+

.!-. \-\4

F

3> 6

•

nd so on. It is easy to see that each of the approximations obtained in this way

ill be majorant for the corresponding approximations to the Volterra series solution

f \ r(

t}\

~

F and

- ~'C •

e

!S'\'\tu."C[

d..t:

~.

co

r\"

rhe infinite series solution of the comparison equation

0:::::

qill then give a majorant series for the Volterra series solution.

)n plotting the graph of the comparison equation a

~rve of the form shown results. There will be a

rartieal tangent when dFjdQ = 0. i.e. when

cos \,

G

\ +

,bleh will always have just one solution for ~.

3ince the majorant series was found by assuming Gand F small, it will represent

it least a part of the curve OA leading from the origin O. It cannot represent

the curve beyond A since the curve turns back and becomes a two valued function

)f F. For values of F corresponding to points to the right of A, the majorant

series will certainly diverge. There is consequently a critical value of F, F

I,

such that the majorant series is only convergent if F ~ FI' • The value of FI

is actually the value corresponding to A and so the construction of the graph

~f the comparison function shows immediately the range of convergen~e of the

najorant series (see below).

rhe existence of a Volterra series solution only for If(t)!<F

I can be understood

~n physical grounds. If the maximum value of the force is sufficiently small, the

pendulum can only perform motions with a limited deviation from the equilibrium at However if the permitted maximum value of the force is allowed to increase, then it is clear that there will be a critical value beyond which the character of

the motion changes and the pendulum performs complete revolutions. 9(t) may then

increase indefinitely and so mathematically, the response is unbounded.

For a given bound F on the force, it is clear physically that the maximum amplitude of swing will occur when

•

~nd using this equation, the maximum bound corresponding to a given maximum force can

be computed numerically for any value ~. The diagram below shows the bounds for

e

----

---,

.. -Curve for comparison / 1/ /

\

;,. /, / / , I Exact relation I between input-I and output-,bounds. ~---~---~----~~---~----~>

F

/ .. 0

c-s

3.3

Discussion of a general type of equation

In this section we will discuss the general type of equation corresponding to the

two examples introduced in 3.1 and 3.2. The equation is

where L(D) is a linear differential operator of stable type i.e it satisfies: L(8) has all its zeros in He s (0

and g( ) consists of nonlinear terms only:

,~

Cj ~ I.:!) - 'x '2. 'i

~T--The equation is first written in the form

h lD) '; ~ t:}

and then, by inversion of the operator L(D) on the infinite interval, transformed to the nonlinear integral equation

~J \ t ) (

J

:x, '1 I ... v \ '" t I) x l t ,) c!l: I

,l<.'

..'l.;)

S

I-, l t; . t \ )

<J

l

'::I I,J:; \ ) ,

cl~.

.- ->u where

,. - I

,.J...

L l~) )

From this, the Volterra'series solution may be generated term by term by the successive approximation method or, more systematically, by the formulae given in the next section. The comparison equation for the nonlinear integral equation is defined as

where

(28)

The method of successive apDroximation applied to the comparison equation then gives a series

which under the condition

00

H

~

S

I

h l-cl , J 7.

-OQ

is a m~jorant series for the Volterra series solution.

A region of convergence of both the method of successive approximation and of

the Volterra series,olution can then be determined from the graph of the comparison

equation. The graph of the comparison equation curves upward from the origin and

has a vertical tangent when dX/dY 0 which can

always be shown to have just one positive solution since the equation is

and the right hand side increases steadily from

o

to +.~ as Y increases over the same range. Denoting this value by Yl and the corresponding VAlue of X by Xl the following facts may be proved:

o

(a) on any interval OB' lying within OB i the method of successive

approximation is convergent both for the comparison equation and the nonlinenr integral equation.

(b) over the same range the majorant series is convergent and so is the Volterra series solution.

For a proof of these statements see Barrett (1974)

C29}

1. The same method can be used to study time-varying systems of the same type or to study the transient behaviour of such systems. For example, let us consider the

transient behaviour for t~ t of a'system with operator

o

90 that we are solving the equation

with initial conditions

:1 Lto) ,

S

It

0) J • ' .

Following the previous procedure, we divide by L(D), i.e. apply the inverse operator,

and get

S It)

+

o

which equation is interpreted as

~ It)

+

¢>

tt)

wher~ h( ) is the impulse response function corresponding to the operator L(D) and pet) is a solution of the linearized form of the differential equation i.e.

L

t.O)

4-

It) ::: 0

Now since

we find, by setting t = t in the integral equation and its first n-l derivatives,

o

that

f \:" 1 ..J.. It:;) • I t )

1. (

,

u ("'-l) lLO) __ ..J ("'-<) ll-o\ ~ \. 0 =. '+' \: Q ,!::S 0

=

"f-' \l:o' • " . ";.} \. 't' \. ,

so that ¢(t) has the same initial conditions as yet) and is consequently uniquely

determined.

The integral equation is now written

t

~ ~lt-t\)Bly(tl))d.t\

to

and solved by the method of successive approximation. Notice that, for the

transient solution, no stability condition is required.

2. The nonlinear integral equation has a feedback form

+

_.,. y H(o) .... ~ r' r /

-

'\

-"

sl

) where

In this form it would be solved on an analogue computer. Since g( ) is nonlinear

and of higher order than the first, feedback will not be important for small values

of y and the system is effectively linear with operator

R(D).

At larger values of

• Further examples of occurrence of this type of differential equation are a) a circuit with a nonlinear element represented by the diagram

f i

I

~l)

I

1

E

1

0

-l

.

\,i':z. l

_1----.

1

~

__ J

Here the voltage drops across the linear and nonlinear parts are respectively,

)

and so the equation for the circuit is

E

Z

l

D ) "-

+

:3

l-t ')

Here the linear contribution of the nonlinearity may be thought of as included in

z{ )

so, for example, if we have an L,R circuit with a nonlinear resistance

then we may put

Z

\D) _

_R,

₊

_LD

_.

)

(b) a DC motor has an equation

(c) an FM detector (phase-locked loop)

A signal cos «(.)t + e) is received and it is required to detect;{j-. This is done by locking on to a locally generated signal cos (wt +6J by using a feedback system. A signal propertional to sin (e - e) _o is generated by correlation of the rece! ved signal with the quadrature component sin (<.;oJ t + ~.:) of the locally generated. signal. The phasee is then driven towards

9

according to the equation

o

.

eo'::

K

s \."

l

e -

e.

0)

(f the deviation E) -

0

₀ is small, this equation approximates to

md so

eo

follows

e

with a first order lag. I f the deviation is large, nonlinearity

~auses loss of performance and if

e

changes too quickly,

e

will not be able to keep

c

lp with it. That this is so may be seen by writing the equation as

where e

=

e -

eo

is the error. If the maximum value of

e

remains small, e will

~main in the neighbourhood of the stable equilibrium at e

=

O. When the maximum ralue of

e

is allowed to increase, the bounds of e increase but the system still

~mains bounded-input, bounded-output stable. Finally when

e

may exceed a

~ritical value, e may leave the neighbourhood of the equilibrium e

=

0 completely md move near another eqUilibrium e. g. at e =TI This is what is called "cycle-~kipping". It is analogous to the behaviour observed with the forced pendulum.

:he system is usually considered in feedback form as either of

+

e ;-

t~

k

~()---7

D

+

k

e.

1.4 On Reversion of Power Series

~i ven x as a power series in y :

;he problem of series reversion is to find y as a power series in

x,

i.e. to ,etermine the coefficients ~ •. ~;:.) ,_, in the expansion

~e systematic way to do this is to write the first equation in the form

md then substitute the formulae for the powers of y :

~

(61

x. -+

{.).,.

x. 2- _-+

r :,

v:3 .1. 4--2 " '2. '2.

+

2.

,2>1

(3'

l x: '!> y l::> I .:x: -r

...

';)

(31

3 x: '3

_-+

l1 ::::. "t

':I

:::.

.

.

e tee

3y equating coeffiqients of x we get the formulae

'6.:;. =

etc.

The method of power series reversion may be used for the solution of forced

nonlinear differential equations. e.g. the equation of the forced pendulum

may be written

and we see that the problem of expressing

e

in terms of the forcing function f(t) is one of ' series reversion where the coefficient of

e

is an operator.

More generally, the equation

has the form

and so we have to revert a power series where method, we write the equation in the form

=

.l: It)

0( = L(D).

I Following the above

which, since the operation 1/ L(D) means convolution with the impulse response : function

h( )

corresponding to L(D), can also be written in the form

This is just the nonlinear integral equation previously considered. Now, since

Iwe know that the solution has the form of a Volterra series, we may put

I

1j (I: l

.'2.(

Y

t..)

- ( 3 5 )

fow by substitution and comparison of kernels we get a series of equations

h.lt--t)

I V\. , In

h

l

t- -

t I) I:: - \: 01. ) (.) (1)

::Xl

~

i-t

U:. -

tt) h

tt'-

t.} h

It(-

t:z.}

d..~1

,

-~

.;,0

- 0(..3

Sh

It-

t / ) h(.') It(-t.) hQ\(I- t--2.) \.,<J'(e_

t"3,.d

- 0\;;)

from which the kernels may successively be determined. This procedure does not

give the symmetical form of the kernels. If necessar,y, the kernels must be made symmetrical at each step.

rhe reversion formula thus give a systematical way of calculating higher order

~ernels. It is easy to write down the formulae for doing this using only the

expressions for algebraic reversion of ordinar,y power series. Tables of such

3.5

An

Equation Occurring in the Theory of Oscillations

As a fUrther example of the reversion method we consider the equation

where gl(y) and g2(y) are analytic functions. This type of equation is

of common occurrence in the theory of oscillations. It is usually possible

to assume that at small amplitudes,

The small amplitude equation is consequently

~

ltl

+

c,

y

Lt'

+

d

I :y (4:;l

-=

~ (I:)

We shall assume that this linear approximation is stable so that the roots of

lie in the region Re s

<

0 of the complex s-plane. The condition for this is that

Ct :> 0

Now consider nonlinear behaviour. Put

All the coefficients are now operators. Using the reversion method, we write the equation as where

~

l

t:)

'*

x.

It

~

-

S

_~1.

h .'

t;) '2 \ \

*

':J It\

+

~

...

[-t

L

- I et-c,

l

S1

+

CI S

+

d.., \ ) C'l. S it- d? ~ ) S'2.+ C

\iS+d, )

From this follows the Volterra series for yet)

Convergence may be discussed using the comparison equation

y

\

3.6 Formulae for Reversion

The following formulae are given for reference. Let

Then.

~I

.-

+

\ l

CI.\

~'2.

::: -

~

I / do, )

0(

'2.

~1'1.

~?J

::.

- (I

1()l\'\""~~I'3+- 2o('2~'~'l.1

~

+ ::: -

II

I

0( I ,

l

c( 4-

~

It\-

+

3

0( 3

~

\'2.

~

'1

+

0( 2.

(~;

+

2

PiP

'3 ) }

~5 ~

- (I

IrA,' \

o(

SPl

5

+

4-0(4f>?(i1...·+

30<3

(~I~;+ (d1'l~3)

+-

0( 3 (

~}

+

'3

(3

1~ ~

....

+

b

(31

~:4 ~

3 ')

4. SYSTEMS OF ANALYTIC DIFFERENTIAL EQUATIONS IN STATE VARIABLE FORM

4.1 Introduotory Remarks

The equations to be considered are of the following types:

(8) autonomous systems

i.. ::1,2., ...

,,,,-where y is the state variable:

and fi(y), i

=

1,2, ••. ,n are analytic functions of the variables Yl""Yn

(b) non-autonomous systems

where Y is the state variables as before and x is the input variable:

representing a control or disturbance (or both) applied to the system. The

funotions f.(y,x) i

=

1,2, ••• ,n are assumed to be analytic functions of

l.

the variables Yl""'Yn; xl •• ··,xm•

toIost of the results can be extended to equations of the types

I

! it~\ =S\.\.~lt)t)

'"

(40)

The behaviour of systems of the autonomous type (a) can be described by the fixed pattern of trajectories in the n-dimensional state space of points (Y1'Y2""'Yn) Of great importance are the fixed pOints of equilibrium that are determined as solutions y = Y of the equations

t =: I) 2, ... ,

"-Near to a point of equilibrium, the motion is conveniently studied by

transform-ation to a local variable 'h

,=

(

1

T

:ry'.l1'1.)···.'t)" by

The point of equilibrium then becomes the point

">J ::

0

For small deviations from the point of equilibrium we have

~

a.. ".', "h J'

+

l ~ I "'O"'t\n~af' t-evVY\$ where 0. •• 1)

Referred to .~ the equations of motion take the quasi-linear form:

J - I

or in matrix form

where A is the matrix with i j element a

The instability or stability of the point of equilibrium

"?

== 0 is determined by

the nature of the eigenvalues of the matrix A. If these eigenvalues have

negative real part then the equilibrium will be asymptotically stable in the sense of Lyapunov i.e. every trajectory starting sufficiently near to',= 0

will tend asymptotically to ~= O.

In the case of non-autonomous systems, the behaviour of the system cannot be similarly described by a fixed pattern of trajectories in the state space. In

particular, there are not fixed points of eq~librium. In practice however,

it is possible to described the behaviour of the system in terms of moving equilibria because we are usually interested in the behaviour of the system

when the input disturbance x varies near to some fixed level

x.

For any

fixed level there will correspond points of equilibrium y

=

y

determined by

the equations

t=.I,2) ... ,V\...

Then for motion near to this equilibrium we shall have

r.

where i) and

I

are small. We can wri te

where

6,.

'J

(

~

_{i )}

d:x..

and then the equations of motion have the quasi-linear form "11 _'Y>1 •

I

y

~ ~ \t)

-

_a.q1')j\t)

+

L

b

ij

S

l

It)

+

J =t _J

= \

which may be put into matrix form as

r

.", (t)

= A

'?

(t)

+

B;

(t)

+

hQ ....

t.

",.ecv.r t.e'lr"f'If'\IS where A and B are the matrices with elements a .. and b .. in the i j th place.

l. J l.J

The stability properties of the system under sufficiently small disturbance are again dependent on the matrix A (which now however depends on the working level

x

and the corresponding equilibrium y) If the matrix A has eigenvalues with negative real parts then i t may be shown (see below) that if

S

remains within certain bounds, then"7 will also remain within certain bounds. If these bounds s.re exceeded, then the system may become unstable or move off to the neighbou.r--hood of some other equilibrium corresponding to the input level

x.

Again if the input level"i changes, then the pattern of corresponding equilibria

y

will change, points of equilibria will move their position and stable equilibria will becomes unstable and vice versa.

4.2 Transient Stability of Autonomous Systems

We consider the motion of a system •

":::It

\t) =.

f

~

"

'j

~t)

)

_{{ =}

_l,

_{'2.. "_. ""}

for t ~ 0 in the neighbourhood of a point of equilibrium which is taken to be

at y

=

O. This equilibrium will be assumed to be stable.

To illustrate the ideas, consider first the one-dimensional system

..

~ I t ) - f~~lH) t '> c_

in the neighbourhood of a point of stable equilibrium at y

=

O.

This problem

is a special case of the problem considered in section

3.

Expanding fey) about the point y

=

0, we may write the equation as

where, by the assumption of stability,

DI.'"

O.

to

This equation may now be converted

~ 2 3 1

r

-1::- -"" ll:: - t') \"

e

a z

'i l

t I ) + Q..3 ~

l

t )

+ ...

1

cL::'

o

Now using either the method of successive approximation or the method of reversion we may-solve this equation to give

giving an explici~ expression for yet) when tt-O in terms of powers of y and

o

To discuss convergence we take the comparison equation

where

The series

A

'2

Y

=:

l

+ \-\

~2

+---

F lZ) .

derived from this comparison equation will be a majorant series for the series solution for y(t) found above provided that

The majorant series will be convergent when Z <'Zl where Z1 is the value of Z for which there is a vertical

tangent in the graph of Y vs.

Z.

This value of Z is

also characterized by the condition dZ/dY

=

0 i.e.

from the comparison equation. by

o

Y.

This equation determines the value Y

l• The corresponding value Zl is found

from the comparison equation.

Note that on the range O~ Z <'Zl of convergence of the majorant series. we have that

HG'lY)

<. i

which is also the condition of convergence of the iterative method of solving the comparison eQuation.

Using the majorant series we see that the series solution for yet) found above is convergent for all t>... 0 provided that

and further that

This last inequality shows there is asymptotic stability of all solutions

whose initial condition satisfies the previous inequality. This fact is

also clear from the expression for yet) in terms of decaying exponentials.

The corresponding results for the multidimensional case are essentially the same: they are only complicated by notational difficulties and by the fact that the graph of Y vs. Z cannot be drawn since now Y and Z are vecto~B.

The equations of motion are written, in the neighbourhood of an equilibrium at y = 0,

"

~ t (t) T

L

J ::. I 1'\ 'V\. <X ij

'Jj

\t)

L~

• _{k ::: \} J :: I Y\ "'V\ 'l'\

"'\'

~

-t-

L,~6

j::1 k:'-\ 3::'1

l'2. )

\j . (t) Q. 'j k.. \.t) i

'd

k. j

where the matrix A with i j th element 'X .. will be assumed to have eigenvalues

1J

rhe equation is next converted to 'VI.

t

L

h~Slt-)':J5~O)

j

= \

+

I )

h'jlt-t')g/~(\'))dXl

, 0 J

= \

where h

ij ("t) is the i j element of the matrix

h

~L) ._ ) exp

-AT:.

[ 0

1:: <.. 0

The series solution for y.(t) i

=

1,2, ••• ,n may be generated from the integral

).

equation. The explicit series solution will be fairly complicated but in many

cases this detailed solution will not be required. If only a numerical solution

is needed, this may most easily be found by iterative solution from the integral

equation. And if a definition of stability region is required, this can be

i ound from the comparison equation which is

-

Z·

\

L

H,jCTj(Y)

J

=

1 where

r

OOl

0 hlj ~ '( ) \

ch:.

\r1 v1

G

i ~ "1' )

"

'"

l2 )

L

L

A

i ,; I...

1"

j

Y

k;. .J :: I 1<.-::1

+. "\ "\ \:'"

(3)

L

L- /

A

Y.

'i'

'f

- t;jk.~ j k. S J :: i k = j s:.:,

>

1 , '"

The comparison equation can be solved for the y, as J. ~ "V'I ...

y.

_-=

_Z·

₊

L

H 'l

\

'"

"

A

\'2) l \.. ~ ~ j " k s, J -= I k.::: \ S :: \

and this can be u.sed.:a.s a majorant series provided that

Yl

\ L.

h.

j

\t)

';/j

(0) \

~

Z.

l

J =: (

7

Z

)+ ...

- k , S

Since the gra.phical construction for the range of convergence of the majorant series is no longer possible, an alternative approach is used based on the identification of the region of convergence of the majorant series with the region for which the iterative method of solving the comparison equation is valid.

to be

The condition for convergence of the iterative method can be shown

where

A

[

]

denotes the maximum eigenvalue of the matrix enclosed wi thin

max

the square brackets. The matrices H and J have i j th elements

LHJ

,,)

[:.r1

'J

Note that all elements of the matrices H and J (and therefore of their product HJ)

are non-negative. By the theorem of Frobenius - Perron, this implies that the

eigenvalues of HJ will have non-negative real parts.

Although the definition of the region of convergence of the majorant series is not so clear here as in the one-dimensional case, it should be possible to

compute this region in any particular Case and so find a region of asymptotic

stability arouud the point of equilibrium.

The method given here may be regarded as an extension of Lyapunov's first method of discussing stability which is based on explicit solution of the differential equations by series and a proof of the convergence of these series.

4.3 Forced Motion of Non-autonomous Systems

Now consider steady motion of a non-autonomous system

•

':!

i. \ t ) ==

:t

t \

'd

(t) , x. (t) ) ~ = 1,2., ... ,Y\.

Suppose that x

=

0 means that there is no disturbance and that in this case

the system has a stable equilibrium at y

=

O. By expanding the functions

f

i( ) about x

=

0 ; y

=

0, we may put the equations in the form

~

'1-

~

0< i j

~

.I\. t ) -

~

i (

'Y

~

t) , x (t) )

j :: I

where

¢' (

y, x) consists of linear and non-linear terms in x and nonlinear

t terms in y so that ~

'V

\(0)

=

a.

~

;"j )C. J V2o)

a.

JC. ,. l:;:-l;jk \ k. J :: \

+

L

\'

(02)

+.

/

C \ , . ,

'j."'j \

."---..1 I. • J k J

k..)

.J = \ k:. \

-+ , ...

'He...

I t is assumed that the matrt'X A with i j element 0<. ij has eigenvalues

A

wi th

Re

A

">0.

where h

ij ("t.) is the i j th~element of the matrix h("'C) defined as

Z<o

Using the integral equation, the y.(t) may be developed into multidimensional

l.

Volterra series in terms of the inputs xl(t), ••• , xm(t). The comparison

equations are

Y.

where Oo:b

\

\hq\"t)\Jt.

0 ~ _{V'" V'V\}

L\

A

\\0)

X.

₊

i. : 'j J

(

'-~

_V

\'2.0'

X.

'X\<,.

-~\ .---\ I

A

l J :: \ VI" V'\

)\

t-

V

A

l4.) / L_I {~\

}k

, J :: \ k :: \ T ' - . _ttc

A

(\0) _~

I

QtIO~

A

_{l, '2 () \} ~'I~

-<}

l ~ j K j :: \

)<

i

I'K

+->

a.. (20) \ -' k=,

"

V\

"

\ '

G

6

- I k :: \ \ _{',I k} \0 2.1

A

y,

_Y

i. ~j k. l ~

Solving the comparison equations for the Y. as multiple power series in Xl, ••• ,X _{, 1 m} we find majorant functions for the Volterra series solutions: