Theory of nonlinear systems

Theory of Nonlinear Systems?_ by Y. H. KU1

Moore School of Electrical PA 19104, U.S.A.

presented

a paper

Moscow. Professor the synthesis Laguerre

University

of Pennsylvania,

Philadelphia,

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the

ABSTRACT:

author

Engineering,

“Theory

Norbert

and analysis

coejicients

of Nonlinear

Control”

at the First

Wiener, who attended this Congress, of nonlinear

systems

IFAC

Congress

at

drew attention to his work on

in terms of Hermitian

polynomials

in the

ofthe past of the input.

Wiener’s original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre

network gives ul, u2,.

gives V(a)‘s. Applying

. . , us, and then to a Hermite polynomial generator

the same input to the actual nonlinear system gives output c(t). Putting c(t)

and V(a)‘s through a product averaging denotes time average

device, we get c(t)V(a) = AJ(2n)S”,

and A,‘s can be considered

system. A desired output z(t) The Volterrafunctional

as characteristic

method suggested

by Wiener in 1942 has been greatly developedfrom

The method involves a multi-dimensional

dimensional

The associated

A. A. Wolf(J. Franklin byforming

as input. Volterra functions From

an

set offunctionals

kernels

of nonlinear

and Wiener

of the

linear

= Au(r).

This extension Measurement

Parallel to the Volterra transforms

Wiener extended

can be correlated

system

to the

and form

nonlinear

developed

is given by &,(a)

forms

the characteristic

system,

the

input-output

of system impulse response

= Ah,(o),

the basis of an optimum

h,

while the autocorrelation method for

nonlinear

is

system

of kernels can be made through proper circuitry. series and the Wiener series, another series based on Taylor-Cauchy

since

1959

are given for

comparison.

The

method can be applied in the analysis of simultaneous

nonlinear

Volterra

transform

functional

the Volterrafunctionals

using Gaussian white noise

Ip,,. Using the white noise as input, where its power density spectrum is a

constant, say, A, the crosscorrelation identijication.

kernels

integral with multi-

are given by Y. H. Ku and

known as G-functionals,

&, can be shown to be equal to the convolution

with the autocorrelation &(z)

convolution

transforms

systems.

extension

crosscorrelation

multi-dimensional

Inst., Vol. 281, pp. 9-26,1966).

an orthogonal

of the nonlinear

may replace c(t) to get a new set of A,‘s.

1955 to the present. kernels.

where the upper bar

coeficients

method

and the Taylor-Cauchy

Taylor-Cauchy

transform

systems. It is noted that the method

give identical jinal

results.

A selected Bibliography theory

but also

identijcation economic

to show

to problems

is appended

not only to include other aspects of nonlinear

the wide application in biology,

ecology,

of nonlinear

physiology,

system

cybernetics,

characterization control

theory,

system and socio-

systems, etc.

t This paper is dedicated to the Centennial Celebration of Electrical Engineering at Massachusetts Institute of Technology, September 1982. An abridged version was presented at the 25th Midwest Symposium on Circuits and Systems, held at Michigan Technological University, Houghton, Michigan, 30 and 31 August, 1982. $ Emeritus Professor of Electrical and Systems Engineering.

!f? The Franklin

Institute

00164032383~010001-26$03,00/O

1

YH.Ku

I. Introduction

to Wiener’s Theory

This paper gives a general review of the Theory of Nonlinear Systems over the last twenty years. In 1960, the author presented a paper: “Theory of Nonlinear Control” (1,2) at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work (3) on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input. According to Wiener, the most general probe for the investigation of nonlinear systems is Gaussian white noise with a flat power density spectrum, because there is finite probability that this noise will, at some time, approximate any given time function arbitrarily closely over any finite time interval. Now Gaussian noise, with a flat power density spectrum, can be approximated by the output of a shot-noise generator. Thus, one can characterize a nonlinear system by a set of coefficients A, and these coefficients can be obtained from a knowledge of the response of the nonlinear system to shot-noise excitation. Accepting the shot-noise effect as the standard probe for nonlinear systems, the input r(t) is to be characterized by a particular set of orthogonal functions, namely, the Laguerrefinctions. As shown in Fig. 1, a Laguerre network can be used to obtain the Laguerre coefficients ul, u2,. . ,us,. . . When the input is shot-noise, the output (the Laguerre coefficients of the past of the shot-noise input) has the following properties : (1) they are Gaussianly distributed, (2) they are statistically independent, (3) they all have the same variance. The proofs of the second and third properties are given by Bose in Ref. (4). Since the output c(t) of an actual nonlinear system depends on the input r(t), which can be characterized by the Laguerre coefficients, the output c(t) can be expressed as a function of these coefficients. Thus c(t) = F[u,,

z42,.

.

.)

u,, . . .].

(1)

The Hermitefunctions are chosen as the expansion for the function F of the Laguerre coefficients, because they form a complete orthogonal set over the interval - co to co, and are particularly adapted to a Gaussian distribution. In terms of normalized

Ul

Shot Noise Generator

r(t)

Laguerre fJ2

-

I ‘-----, I INonlinear w , System

l

V(a)

Hermite Polynomial Generator

v(B) V(y)

.

i 1c(r) -

Product Averaging Device

_

coV(cl)=-

A, (2a)s’2

L-_---l

FIG. 1. Block diagram

2

of the circuitry

for the characterization

of a nonlinear Journal

system.

of the Franklin Institute Pergamon Press Ltd.

Theory of Nonlinear Hermite

polynomials,

the output

Systems

c(t) is given by

c(t) = s_cai=l lim 1 j=l 1 . . . ,zl

ai,j,...,h’[ni(u,)nj(u,)...nh(u,)l

eF’*

(2)

where ai,j,...,h denotes a constant coefficient A,, ni(U1)nj(U2). . . nh(u,) denotes a particular term V(N), and u2 = U: + U; + . . + u,* . In the practical case, we can use a finite number of Laguerre coefficients and Hermite functions. Then (2) can be written as c(t) = 1 A, V(a) e-“‘*. a In Ref. (3), page 96, Wiener

said :

“I shall be able to get my coefficients in terms of averages. These averages will be averages involving a (alpha). What I can get easily enough in a circuit is a time average. The time average is, in general, not the same as the a averages, but under the ergodic hypothesis, time averages will almost always be c( averages. Therefore, if we have ergodicity, this will work. Furthermore, the ergodic hypothesis is satisfied by the Brownian motion.. . . In other words, by taking a particular shoteffect input and obtaining time averages, we can obtain a averages that will be what we need for obtaining coefficients.” Thus, using the output of a shot-noise generator as a standard choosing the least weighted mean-square error criterion, according Bose method (3,4), we get the final result,

test input, and to the Wiener-

A, = (27~)“” c(t)V(a), where the upper bar denotes the time average. The above expression provides the basis for the experimental determination of the characterizing coefficient A,. Referring to Fig. 1, if the nonlinear system (shown in dotted lines) is actually given and connected to the input r(t), then c(t) represents the actual output. As shown, both c(t) and V(a) are, fed into the product averaging device, so that we get the time average of c(t)V(a) which is related to the characterizing coefficient A, by (4). If, instead of a given nonlinear system, there is given a desired output z(t), Fig. 1 shows the circuitry for characterizing and optimizing a desired nonlinear system to be designed by sustituting z(t) for c(t). Figure 2 shows the circuitry for the synthesis of a nonlinear system, based on (3). With ul, u2,. . . , u, fed to the exponential generator, we get e -G/2 = e -(U;+U$+...+U:)/z

(5)

Applying gain A, to V(a) and taking the sum give the summation of &l/(a), which is to be multiplied by e-““* to get the output c(t). We have assumed that at our disposal is an ensemble member of the input time function r(t) and the corresponding member of the desired output z(t) or c(t). By recording or making direct use of a portion of the given input time function, we get Vol. 315. No. I, pp l-26. January Prmted m Great Britam

1983

3

YH.KU Gain

r(t)

-

FIG.2. Block diagram

-c Exponentiol Function Generator

of the circuitry

e

for the synthesis

of a nonlinear

system.

the ensemble member of r(t). The ensemble member of c(t) can be determined in different ways. For prediction problems, c(t) is obtained from r(t) by a time shift. For filter problems involving the separation of signal from noise at the receiver, we can record a portion of the desired output z(t) at the transmitter, and the corresponding portion of r(t) at the receiver. For the radar type of problem, c(t) can be generated corresponding to signals r(t) received from known typical targets. For further discussion concerning Figs. 1 and 2, see Ref. (5). II. The Volterra Functionals and Kernels In 1942, Norbert Wiener gave a method for finding the response of a nonlinear device to noise (6). Recent developments of the Volterra functionals (7) are given in Refs. (8)-(39). A study by Ku and Wolf (27) appeared in this Journal in 1966, by Ku and Su (30) in 1967, and by Ku and Lin (35) in 1971 and 1973. Comparison of the Volterra functional method with the Taylor-Cauchy transform method (40, 41) is given in this Journal in Ref. (44) in 1982. See also Appendices in Refs. (30) and (35). Given a nonlinear system described by the differential equation + 4(x, x’, . . .) = r(t),

2(&(t)

(6)

where Z(D) is a linear differential operator, with D = d/dt, and 4 is a nonlinear function of x and its derivatives (x’, x”, . . .) subject to the condition : jc$[v(t)] -$fu(t)]l According xi(t)

to the Volterra

< A4 e-“‘Iv(t)-u(t)1

functional

method,

for all t 2 0.

the solution

is given by the sum of

:

x(t)

=

f

xi(t),

i = 1,2 ,...,

n,n+l,...

(7)

i=l

where, for i = n, m x,(t) =

4

s -cc

...

m M~l,~ 2,. . . ,z,)r(t-~~)r(t--TJ.. s -cc

. r(-z,)

dz, dz,. . .dz,,

(8)

Journal of the Frankhn Inataute Pergamon Press Ltd.

Theory of Nonlinear ’

I I I y(t) I I I

Linear

z(t)

Squarer

FIG. 3. A second-order nonlinear

where h, denotes an n-dimensional n-dimensional convolution integral.

Systems

system.

kernel. Equation (8) can be considered For n = 1, we get the linear response

as an

m x1(t) =

s -a,

hl(rl)r(r-rl)

dri,

(9)

where h,(z,) denotes the impulse response of the linear portion of the overall system. For r(t) = u(t), the unit step, xl(t) is the step response. For n = 2, we get the quadratic or second-order response m x&) =

‘xl

s -co s -m

&(r,, M-r&o-J

drr dr,,

(10)

where h,(z,, z2) denotes the second-order Volterra kernel, which may be considered as the two-dimensional impulse response. Figure 3 shows a nonlinear system with a square following a linear network H. The second-order Volterra kernel is shown in Fig. 4. These figures are taken from Ref. (42) pp. 40 and 41. Note that in Fig. 3, the input is x(t) and the output is y(t), while z(t) is the input to the squarer. In this example, the second-order kernel is symmetric, that is, h,(z,, z2) = h,(r,, TV). In other cases, the second-order kernel may not be symmetric. Symmetrization can be made according to Section 3.3 of Ref. (42), pp. 4143. Example 1 Given the nonlinear

system with random

input

x’ +x2 = r(t) = random where

(r(t) : - co < t < m} denotes

FIG. 4. The second-order Vol. 315, No. I, pp. l-26, January Printed III Great Britain

1983

a sample

input,

function

(11) from a strictly

kernel for the system given in Fig. 3.

stationary

YH.Kl.4 source in which the moments denoted by the symbol

of all orders are bounded.

W)>f

=

for average with respect to a random According to the Volterra functional is given by

s

_mo.fI(f)

Let the ensemble-average

be

(12)

df

processf(t), where p(f) denotes its probability. method, the linear response or first-order term

(13) The second-order Mt)>, The third-order

or quadratic

response

= (-j-e&)

di)g

or cubic response

is given by

= -l:(r2>,r2

is given by f 1 = 2 o (r”)q” > 0

The fourth-order

or quartic

(14)

dz = -
response

2

dr = (r’),,,

ts.

(15)

s

is given by

(16) The fifth-order

or quintic

response

is given by

(17)

Thus we get (x(t)),

= (r)rt-(r2)rg

+

+
(18)

For r(t) = unit step and x(0) = 0, we get simply x(t) = tanh t. Journal

6

of the Franklin Institute Pergamon Press Ltd.

Theory of Nonlinear Example 2 Given the nonlinear

system with random

x” + 3x’ +2x + a’

input

= r(t) = random

where {r(t): - co < t < co} denotes a sample function source in which the moments of all orders are bounded, The linear response is given by

Xl@)=

f s

r(z)h(t - z) d7 =

0

t s

input

(19)

from a strictly stationary and ,Uis a constant.

h(z)r(t -7)

dz,

(20)

0

where h(t) denotes the impulse response of the linear portion the numerical example, the impulse response is given by

of the overall system. In

h(t) = e-*-ePzt. Taking

the ensemble-average

Systems

(21)

of (20), we get

(22) where f Xl”@) =

u(z)h(t-z) s

1 dz = 2-ePr+Z

1

emzl,

(23)

0

in which u(t) = unit step. Hence xl,(t) denotes the step response of the overall system. The quadratic response is given by CM>,

of the linear portion

= (r2(t)>,x2&)

(24)

where xZu(t) = -p

’ h(t-z) s0

Note that x’Jt) is equal to the impulse Taking the derivative of (25) gives

[x;,(z)]’

response

[

dz

h(t) given in (21).

x;,(t) = ,u t eC’+2(1-r)e-2’-3e~3’f~ The cubic response

1.

(26)

is given by
Vol. 315, No. I, pp. l-26, January Prmted in Great Britam

em4’

=

,~3,(0T

(27)

1983

7

YH.Ku where x3,(t) = - 2~

’ h(t - z)x;,(z)x;,(z) s0

11

-St_

_

+3Se

1

dr

1 .

e-6t

30

(28)

Similarly, x4,,(t)

- p

=

’ h(t

-7)

{ [x;,(z)12

+

~x;,(z)x~,,(z)} dz.

(29)

s0 Taking

the sum of (x,(t)),

for i = 1,2,3,. . . gives

, = Q)A. Example 3 Consider the nonlinear

+


+


+


+

.+ *

(30)

system (Refs. (22) and (27)),

x’+sin

1 -x3+ 3!

x = x/+x-

1 -x5+ 5!

... = r(t),

(31)

where r(t) is a forcing fucntion which, if deterministic, is bounded or, if stochastic, has bounded moments of all orders. For the linear portion of the overall system, the impulse response is h(t) = eP’. The linear response is given by (32) With x,(t) known,

t s

we get

x3(t)

With xl(t) and x3(t) known,

=

f

x:(t--)e-'

(33)

dz.

. 0

we get

f xf(t-z)x,(t-,)e-’

x,(t) = ;

dz-

$

s0

.

*x:(t-r)e-’ s0

dr.

(34)

Following Ref. (27), Example 5, we shall develop the third- and fifth-order kernels in multi-dimensional transforms. Note that the convolution in the real domain corresponds to multiplication in the complex domain. The linear transfer function corresponding to e-’ is L,(s) = l/(s + 1). From (33), the convolution integral corresponds to the transform

H1(S1)H1(S2)H1(S3)

8

=

(&)

(&)

(35)

(&).

Journal

of the Franklin Institute Pergamon Press Ltd.

Theory of Nonlinear According

to Theorem H&i,

7, Ref. (27) we get the third-order

:!(s~+s~:s3+1)(~)(~)(~) (36)

from (34), we get the fifth-order

H,(s,, sz, . . >4

=

(

s,+s,+...+s,+l

For further discussion

kernel

1

x

III. Measurement

1

(&)(&)-&Ji(&I)].(37) [gm,s,,s,)

on Associated

Transform

Pairs, see Refs. (lo), (22), (27).

of Volterra Kernels by Cross-correlation

In linear system theory the system input-output input x(t) with the output yl(t) is defined by 4Xy,(a) = x(t - a)yl(t)

= lim -

From (9) by changing

T

x(t -o)yl(t)

4Xy1(~) of the

dt.

s -T

xl(t) to yl(t) and r(t) to x(t), we get Y&) =

u” h,(z)x(t - z) dz. s -co

(39) into (38) and inverting

the order of integration

&y,(o) = lrn h,(r) dr [ ft -02 =

sm -a)

where the autocorrelation &,(r)

crosscorrelation

1

~-rnzT

Substituting

kernel

s2>ss) = ; L,(s, + s2 + s3)Hl(sl)Hl(s*)H,(s,) =-

Similarly,

Systems

h,W&

&

(39) gives

x(t - o)x(t - z) dt

1; T

- 7) dz,

1 (40)

is defined by = x(t)x(t +z) = Ji:

&

x(t)x(t + z) dt. s

(41)

T

From (40) it is seen that the input-output crosscorrelation of a linear time-invariant (LTI) system is equal to the convolution of the system impulse response with the autocorrelation of the input. A time function x(t) is called white noise if its power density spectrum is a constant so that @‘,,(jw) = A. Vol. 315, No. 1, pp. l-26, January 1983 Pnnted ID Great Bntain

(42) 9

The autocorrelation

function

of x(t) then is &X(T) = Au(z).

Substituting

(43) into (40) gives &,,(c$

=

;a h,(z)Au(o-7)

dz = Ah,(o).

s From (38) and (44) we get

Figure 5 shows the arrangement for the measurement of h,(o) by crosscorrelation. The input is x(t), and the output through the linear network H, is yl(t). The averager gives the time average of the product of the output yl(t) and the delayed input x(t - a), which may be designated by D, [x(t)]. From (lo), by changing x2(t) to y2(t) and r(t) to x(t), we get

W)

=

m m h,(z,,z,)x(t-z,)~(t--~) s -m s -a,

dz, dz,.

(46)

The second-order Volterra kernel h, is a function of two variables. Thus, to extend the measurement technique for first-order kernels, we average the product of the response yz(t) with a two-dimensional delay of the white Gaussian input x(t), as shown in Fig. 6. This is second-order crosscorrelation. As shown, yz(t) is the output through H,. The product of x(t - ~JJ and x(t - cr2) gives a delay functional Dz[x(t)]. After multiplying D,[x(t)] by the output y2(t), we take the time average, which can be expressed as YzW(t

-

~l)X@- a21=

sI cc

m

-m

pa,

h,(z,,z,)x(t

-zl)x(t-~z,)x(t-oa,)x(t-~o,)

dz, ds,. (47)

Using the result for the average = Au(z), we get

of the product

yz(t)x(t - o,)x(t - CT2)= 2A%,(C,,

n

FIG. 5. Measurement 10

of Gaussian

variables,

CJ,)+ [ A ’ s_‘- h,(z, z) d+(o,

with 4X,(~)

- az).

(48)

y.(t)

of the first-order Volterra kernel by crosscorrelation. Journal

of the Franklin Institute Pergamon Press Ltd.

Theory of Nonlinear

FIG. 6. Measurement

of the second-order

Volterra

Systems

kernel by crosscorrelation.

The second term in the average is an impulse. So it is zero for o1 # CT~.Hence, we get UC,,

cZ) = &

y2(t)x(t - a,)x(t - c2)

for o1 z oz.

(49)

Note that by crosscorrelation the second-order Volterra kernel h,(z,,z,) can be determined everywhere in the z1-z2 plane, except along the 45” line z1 = z2. Similarly, we can extend the measurement technique to find the third-order Volterra kernel by using the product of y3(t) and the three-dimensional delay D3[x(t)] of the Gaussian input x(t) and taking the time average. The result is given by h,(a,,

Finally,

~72903)

=

&

Y3(t)X(t-01)X(t

we get the general

-a,)x(t

-03)

for

cl

Z

02

+

03

#

DI.

(50)

result

.,‘Ay,(t)x(t

hn(~l,o2,...,~n) = n

- al). . . x(t - 0”)

for no two b’s equal.

(51)

Figures 5 and 6 were given in Ref. (42), as Figs. 11.1-l and 11.1-2. The historical development of the measurement techniques is given in Refs. (13), (14), (23H26).

IV. The Wiener G-Functionals and Kernels Define a Volterra

functional

of order n by

H,Cx(Ol. For n = 0, we have H,[x(t)] = h,, where h, is a constant. H,[x(t)] is homogeneous, since a change in the amplitude of the input results in a change of the output amplitude but not of the output waveform. If this condition is not satisfied, then the functional is nonhomogeneous. Wiener called the orthogonal functionals from the Volterra functionals Gfunctionals, since they are orthogonal when the input is a white Gaussian function. Because the convergence of an orthogonal series is a convergence in the mean, the class of nonlinear systems that can be described by the Wiener G-functionals is much larger than the class that can be described by a Volterra series. Vol. 315, No 1. pp I 26, January Printed m Great Britam

1983

11

YH.Ku We now develop the Wiener G-functionals functionals

as a set of nonhomogeneous

Volterra

; x(t)1 g,Ck,, k, - l(n)9. . . 2k O(n) for which

; .+)I = 0 for m < n H,C.$~)lg,Ck,, k,- lcnj,.. . , k Ocnj when ~Jz)

x(t) is a white Gaussian time function with the autocorrelation = Au(r). The set of functionals so derived are G,Ck,

; 441

=

function

; x(t)1

Wiener G-functionals, and k, are Wiener kernels. (Volterra h,.) The zeroth and first-order Wiener G-functionals are GoCko

(52)

kernels are denoted

by

ko

(53) G,Ck,

;.+)I =

m k,(~M-~,) s -00

dz,.

Note that G,[k, ; x(t)] is simply the response to the input x(t) of a LTI (linear) system with the impulse response k,(t). The second-order Wiener G-functional is the nonhomogeneous Volterra functional g&,

k,(,,, ko(2) ix(t)1= KzCx(t)l+K,(,,Cx(t)l+Ko(,)Cx(t)l co

03

=

~~~~~~~~~~~~~~~~~~~~~~~~~~dz, s -50s -m

(54) with the following

Substituting

properties,

as specified by (52) :

HoCxWlgzCk,, kuwko(zj ; x(t)] = 0

(55)

ff,t-Wlszh

(56)

k,p,,ko~,,;xMl = 0.

(54) into (55) gives, for 4=,(r) = Au(r), m

cc

0 = h, s -cc s -m

s

k2(Z1,Z2)~(t--Z1)~(t-Z2)

dz, dz,

cc

+ho

= h,A

12

_

a, k&d

x(t -71) dz, + hokop,

m k&,2 71) dz, +h,kcw s -m

(57)

Journal of the Franklin lnstsute Pergamon Press Ltd.

Theory of Nonlinear

Systems

Note that the second integral is zero since it involves the average x(t - z) which is zero. The condition given in (55) is satisfied for any constant h, only if k O(2)

=

cm

--A

k,@,,~,)

dT,.

(58)

s -cc

Substituting

(54) into (56) gives, for 4x,(z) = Au(z), cc

o=

h,(a)x(t-a)

da

s -m CC _m h,(o)x(t-co)

+ s

m _ m h,(+(t--)

+ [s

m 3o k2(~1,ZZ)x(t-Z1)X(t-Z2) s pm s -02 do

m kr(zj(rAx(r-7,) s -m

da

dr, dr,

dr,

1 kow.

(59)

The value of the first term in (59) is zero, since it involves the average of the product of three zero-mean Gaussian variables. Similarly, the value of the last term is zero since it involves the average x(t - a). The second term can be rewritten as m

Cc m Mrr)kr&)

co

o=

hl(o)kl~2~(~1)x(t - o)x(t - zl) da dz, = A s -Cc s -cc

dr,.

s (60)

The solution of (60) is k,&~,) = 0. The functional g2 which satisfies (55) and (56) is Cc G,Ck,

; x(t)1=

m k2(~1,52)x(t-~1)x(t-z2)

dr, dz,-A

s -02 s -cc

m Ur,,r,) s -m

dz,. (61)

Note that the derived Wiener kernel koc2) is determined uniquely in terms of the leading kernel k, and the power level A of the white Gaussian input. Similarly, the third-order Wiener G-functional is given by G3[k3;x(t)]

=

m m m k 3(zl,z2,~3)x(t-~l)x(t-~2)x(t-~3) s pm s -a, s -a, cc + k&M-71) dr, s -cc

dr, dr, dr,

(62)

in which (63) The fourth-order

Wiener

G-functional

is given by

G,Ck,; x(t)1= KsCx(t)l+ KzcqCx(t)l + KocqCx(t)l Vol 315. No. 1, pp. I-26, January Printed in Great Britain

(64)

1983

13

YH.Ku in which

The fifth-order

Wiener

G-functional

is given by

G,Ck,; 401 = K,Cx(t)l+ Kw)Cx(t)l+K~(s)Cx@)l

(67)

in which

(68) (69)

For a fifth-order system, the relation kernels is given below :

between

the Wiener kernels and the Volterra

h5 = k,,

h, = k4

h, = k, + kw

h, = k2 + kqq

h, = k, + kw, + kw,

ho = ko + koc,, + kow

See Ref. (42), Table

V. Determination

(70)

12.5-1, p. 260.

of Wiener Kernels by Cross-correlation

Let the output of a nonlinear system N to input x(t) be y(t), as shown in Fig. 7. In terms of the Wiener G-functionals, we get y(r) = f

‘3%;

fl=O

x(t)l.

(71)

From (52), the time average of Ho[x(t)] = ho multiplied by G,[k,;x(t)] n 2 1. Set ho = 1. We get, for x(t) = white Gaussian input, G,[k,;x(t)]

n

FIG. 7. Measurement

14

= 0

is zero for

for n 2 1.

(72)

y(t)

of the first-order

Wiener kernel by crosscorrelation. Journal of the Franklin Instmte Pergamon Press Ltd.

Theory of Nonlinear Then the time-average

Systems

of y(t) is

y(t) = :

x(d = GJk, ; xW1= k,.

&I%,;

(73)

n-0

The zero-order Wiener kernel k, is just the average value of the system output y(t) for the white Gaussian input x(t). Comparing Fig. 7 with Fig. 5, the network H, in Fig. 5 is replaced by the actual nonlinear network N. The input x(t) is the white Gaussian noise in both figures. In Fig. 7, the output of the actual nonlinear system is y(t), while in Fig. 5, the output of linear network H, is yl(t). An adjustable delay gives Dl[x(t)] = x(t-co,) in both figures. Multiplying y(t) by Dl[x(t)] and taking the time average gives y(tP,

CxWl= f Gd-k; x(Wl Cx(Ql.

(74)

II=0

For n = 0, we have G,[k,;x(t)]D,[x(t)]

= k,x(t-a,)

= 0.

(75)

For n = 1, we have

=s m

k,(~Jx@-~&@--a,)dz,

-co

=

Thus we obtain

the first-order

A

O” k,(t,)u(z, s -m

Wiener

-al)

dr, = Ak,(a,).

kernel of the nonlinear

(76)

system :

k,(o,) = ;Y(WC-WI.

(77)

In practice, the kernel is determined at some finite number of values of ol. The graph of k,(a,) then is obtained by connecting these points by a smooth curve. Since Dl[x(t)] is a functional of the first order, the Wiener functionals for G, for n > lare orthogonal to Dl[x(t)], in accordance with (52). For the measurement of second-order Wiener kernels, we refer to Fig. 8. The input

FIG. 8. Measurement Vol. 315, No. I, pp. l-26, January Pnnted in Great Britam

of the second-order

Wiener kernel by crosscorrelation.

1983

15

EH.KU x(t) is Gaussian white noise. The actual nonlinear system is shown as N. The output of the actual nonlinear system is y(t) as in Fig. 7. Using the two-dimensional delay DJx(t)] = x(t-a,)x(t-0J, we get ~(r)&Cx(r)l

= f G&i n=O

x(t)l&Cx(r)l.

(78)

For n = 0, we get G,[k,;x(t)]D,[x(t)]

= k,x(t-o,)x(t-Da,)

= k,Au(o, -a*).

(79)

The average for n = 1 is = r m k,(z,)x(t-~z,)x(t-cro,)x(t-~o,)

G,Ck,; x(t)l&Cx(t)l

dr, = 0.

The average is zero since the average of the product of an odd number Gaussian variables is zero. The average for n = 2 is

G,Ck,;xWMWl

(80)

of zero-mean

= 1 m 1 O”k2(71, zZ)X(~-Z~)X(~-Z~)X(~-~~)X(~-~~) dr, dz, J-mJ-m

-A

m k,(z,, 21) dz,x(t-o,)x(t--a,) s -co

= 2A2k,(o,, c2).

(81)

Since D2[x(t)] is a functional of the second-order, the Wiener G-functionals G, for n > 2 are orthogonal to D2[x(t)] in accordance with (52). The final result gives 1 k,(o,, a2) = ~y(r)D,Cx(t)l 2A2 Similarly,

the third-order

Wiener

for cl Z c2.

(82)

kernel is given by

1 k&l,

~~2,631

=

-3!A3

~(t)&Cx(t)l

(83)

for g1 + o2 # e3 f ol.

More generally, k,(ao

~2,.

. . ,a,,)

=

&p

Y(@%lCx(t)l.

Measurement of the Wiener kernels can be made in accordance with (77), (82), (83) and (84). Note that in these equations, y(t) represents the actual output of a nonlinear system subjected to a probe by the Gaussian white noise. In case we have a desired output z(t) instead of the actual output y(t), a new set of Wiener kernels can be measured by crosscorrelation. VI. The Taylor-Cauchy

Transforms

Parallel to the Volterra series and the Wiener series, another series based on the Taylor-Cauchy Transforms may be mentioned for comparison. In Ref. (40) TaylorJournal

16

of the Franklm

Inmtute Ltd

Pergamon Press

Theory of Nonlinear

Systems

Cauchy Transforms were first presented at the IRE National Convention at New York, March 1959. Two companion papers appeared in the IRE Proceedings, May 1960. (See Refs. (40) and (43).) Given the nonlinear system represented by (6), which becomes (85) below by the Taylor-Cauchy Transform :

Z-,CWMt)l +-EL&x,x’, . . .)I = FXr(t)l where z,

the direct transform, rJF(n)]

(85)

is defined by =

1 27rJ

adi, s c;In+i

n = 0,1,2 ,...

where jl is a complex time-variable which replaces the real time-variable t in this transform method, and F(1) is any of the quantities inside the brackets in which t is replaced by ,J and x(t) is replaced by W(i,), C denotes a closed contour in the A-plane enclosing the singularities of F(1), and n is a discrete variable taking on values 0, 1, properties. The highest 2 ,... The new transform has additivity and commutivity derivative term is assumed to be analytic for a class of nonlinear systems and can be expressed by W”‘(n) = f w,;l n=O

(87)

where Wck)(,?)denotes the kth (highest) derivative. Note that if the highest derivative is analytic, integrating with respect to 2 for a number of times will ensure a suitable solution of IV(L) and hence x(t). Applying the Taylor-Cauchy Transform to both members of (87) gives ~[w(k’(n)] The inverse transform

= w,,

n = 0, 1,2,. . .

(88)

is then given by 02

rc:- ‘[W”] =

c

w,P = Wk)@).

It=0 Extension of this Transform to the case of random inputs was given at the NEC, Chicago, in Oct. 1959. (See Ref. (41)) This transform method can be applied to a system of simultaneous nonlinear differential equations, as shown in Example 4, Ref. (44). (Also see Ref. (45)) The series given by (87), characterized by w,‘s, may be termed Taylor-Cauchy series, or symbolically Fc series. Example 4 Given the nonlinear

system with random x1+x2

input

= r(t) = random

input

where (r(t): - 00 < t < a) denotes a sample function source in which the moments of all orders are bounded. Vol. 315, No. I. pp. I-26, January Prmted m Great Britam

1983

(90) from a strictly stationary Changing x(t) to W(A) and

17

Y

H. Ku

r(t) to G(A) gives w,(A) + [W(A)]’ = G(A).

(91)

Let the Taylor-Cauchy series be given by (87) for the first derivative Integrating with respect to A gives

in this example.

W(A) = z %_lR”. n=l n The square of W(A) can be expressed

as a double

(92) summation (93)

In the deterministic

case, taking

the transform

“-1

wn+k=l c

of (91) gives, for r(t) = u(t),

Wk-~W,-k-l

k(n-k)

=

48

(94)

where 6, = 1 for n = 0 and 6, = 0 for n > 0. In the case of random input, we introduce the symbol ( ) for ensemble-average as given in (12). Taking the ensemble-average of (94) gives

+“f!
(w > n

w

k(n - k)

k=l

where
the ensemble-average (we>,


= (r),,

= - ; <%%)W

= (r),

In the deterministic solution is simply

(95)

of the input r(t). Solving recursively

gives

(w,>u’ = -(w&’
in w,‘s gives the series for W’(A). Integrating

(~(t))~ = (r),t-(rz)rS

6,:

$ (w;>,,

and changing

+
.. .

(96)

WQ) to x(t)

...

(97)

case, where the input is a unit step, (ri),. is replaced by 1, and the

x(t)=t--~3+i25t5-~t7+-..=tanh

t.

It is of interest to note that the Taylor-Cauchy transform method and the Volterra functional method give identical final results. (See Refs. (30), (35) (44))

Acknowledgements

The author wishes to acknowledge the encouragement

of Emeritus University Professor

J. G. Brainerd, Dean Joseph Bordogna of SEAS, and Dr Sohrab Rabii, Chairman Department of Electrical Engineering and Science, University of Pennsylvania.

18

of the

Journal of the Franklin Institute Pergamon Press Ltd.

Theory

of Nonlinear

Systems

References (1) Y. H. Ku, “Theory of nonlinear control”, Proc. First Int. Congress oflFAC on Automatic Control, Moscow, U.S.S.R., 1960. J. Franklin Inst., Vol. 271, pp. 108-144, 1961. (2) Y. H. Ku, “Analysis and Control of Nonlinear Systems”, The Ronald Press, New York, 1958. (3) N. Wiener, “Nonlinear Problems in Random Theory”, MIT Press, Cambridge, MA; John Wiley, New York, 1958. (4) A. G. Bose, “A theory of nonlinear systems”, MIT RLE Tech. Report No. 309,1956. (5) Y. H. Ku, “On nonlinear networks with random inputs”, Trans. IRE, Vol. CT-7, pp. 479490, 1960. (6) N. Wiener, “Response of a nonlinear device to noise”, MIT Radiation Lab. Report No. 129, April 1942. (Also published as U.S. Department of Commerce Publications PB-58087.) (7) V. Volterra, “Theory of Functionals and of Integral and Integro-ditferential Equations”, Dover, New York, 1959. (8) A. H. Nuttall, “Theory and application of the separable class of random processes”, MIT RLE Tech. Report No. 343, 1958. (9) M. B. Brilliant, “Theory of the analysis of nonlinear systems”, MIT RLE Tech. Report No. 345, 1958. (10) D. A. George, “Continuous nonlinear systems”, MIT RLE Tech. Report No. 355,1959. (11) D. A. Chesler, “Nonlinear systems with Gaussian inputs”, MIT RLE Tech. Report No. 366,196O. (12) G. D. Zames, “Nonlinear operators for system analysis”, MIT RLE Tech. Report No. 370, 1960. (13) Y. W. Lee and M. Schetzen, “Measurement of the kernels of a nonlinear system by crosscorrelation”, MIT RLE Quarterly Report No. 60, pp. 118-130, 1961. (14) M. Schetzen, “Measurement of the kernels of a nonlinear system by correlation with Gaussian non-white noise”, MIT RLE Quarterly Report No. 63, pp. 113-l 17, 1961. (15) D. A. Chesler, “Optimum multiple-input nonlinear systems with Gaussian inputs”, Trans. IRE, Vol. IT-8, pp. 237-245,1962. (16) H. L. Van Trees, “Synthesis of Optimum Nonlinear Control Systems”, MIT Press, Cambridge, MA, 1962. (17) M. Schetzen, “Some problems in nonlinear theory”, MIT RLE Tech. Report No. 390, 1962. (18) R. H. Flake, “Volterra series representation of time-varying nonlinear systems”, Proc. 2nd Int. Congress of IFAC on Automatic Control, Basel, Switzerland, Vol. 2, pp. 91-99, 1963; Trans. AIEE, Vol. 81, part 2, pp. 33&335, 1963. (19) G. Zames, “Functional analysis applied to nonlinear feedback systems”, Trans. IEEE, Vol. CT-lo, pp. 392404, 1963. (20) J. F. Barrett, “The use of functionals in the analysis of nonlinear physical systems”, J. Electron. Control, Vol. 15, pp. 567-615, 1963. (21) J. F. Barrett, “Hermite functional expansion and the calculation of output autocorrelation and spectrum for any time-invariant system with noise input”, J. Electron. Control, Vol. 16, pp. 107-113, 1964. (22) H. L. Van Trees, “Functional techniques for the analysis of the nonlinear behavior of phase-locked loops”, Proc. IEEE, Vol. 52, pp. 894911, 1964. (23) M. Schetzen, “Measurement of the kernels of a nonlinear system of finite order”, Int. J. Control, Vol. 1, pp. 251-263, 1965. (24) M. Schetzen, “Synthesis of a class of nonlinear systems”, Inr. J. Control, Vol. 1, pp. 401414, 1965. Vol 315, No. 1, pp. l-26, January Printed in Great Bntain

1983

19

(25) Y. W. Lee and M. Schetzen, “Measurement of the Wiener kernels of a nonlinear system by cross-correlation”, Int. J. Control, Vol. 2, pp. 237-254, 1965. (26) Y. W. Lee and M. Schetzen, “Some aspects of the Wiener theory of nonlinear systems”, Proc. NEC, Vol. 21, pp. 759-764, 1965. (27) Y. H. Ku and A. A. Wolf, “Volterra-Wiener functionals for the analysis of nonlinear systems”, J. Franklin Inst., Vol. 281, pp. 9-26, 1966. (28) A. M. Bush, “Some techniques for the synthesis of nonlinear systems”, MIT RLE Tech. Report No. 441, 1966. (29) R. B. Parente, “Functional analysis of systems characterized by nonlinear differential equations”, MIT RLE Tech. Report No. 444, 1966. (30) Y. H. Ku and C. C. Su, “Volterra functional analysis of nonlinear varying-parameter systems”, J. Franklin Inst., Vol. 284, pp. 344-365, 1967. (31) Y. H. Ku, “Volterra functional analysis of nonlinear systems with deterministic and stochastic inputs”, Proc. 4th Int. Conf: on Nonlinear Oscillations, Prague, Czechoslovakia, 1968. (32) R. B. Parente, “Nonlinear differential equations and analytic system theory”, SIAM J. appl. Math., Vol. 18, pp. 41-66, 1970. (33) M. Schetzen, “Power-series equivalence of some functional series with applications”, Trans. IEEE, Vol. CT-17, pp. 305-313, 1970. (34) E. Bedrosian and S. 0. Rice, “The output properties of Volterra systems driven by harmonic and Gaussian inputs”, Proc. IEEE, Vol. 59, pp. 1688-1708, 1971. (35) Y. H. Ku and T. S. Lin, “Analysis of nonlinear systems with stochastic input and stochastic parameters”, J. Franklin Inst., Vol. 292, pp. 313-331, 1971; Vol. 295, pp. 4233430,1973. (36) M. Schetzen, “A theory of nonlinear system identification”, Int. J. Control, Vol. 20, pp. 577-592,1974. (37) M. Schetzen, “Theory of pth-order inverses of nonlinear systems”, Trans. IEEE, Vol. CAS-23, pp. 2855291, 1976. (38) E. G. Gilbert, “Volterra series and response of nonlinear differential systems: a new Proc. Conf: Information Science & Systems, Johns Hopkins Univ., approach”, Baltimore, MD, 1976. (39) E. G. Gilbert, “Functional expansions for the response ofnonlinear differential systems”, Trans. IEEE, Vol. AC-22, pp. 909-921, 1977. (40) Y. H. Ku, A. A. Wolf and J. H. Dietz, “Taylor-Cauchy transforms for a class of nonlinear systems”, IRE National Convention Record, Part 2-Circuit Theory, pp. 49-61, March 1959; Proc. IRE, Vol. 48, pp. 911-922, May 1960. “Taylor-Cauchy transforms for analysis of varying-parameter systems”, Proc. IRE, Vol. 49, pp. 10967, June 1961. (41) Y. H. Ku and A. A. Wolf, “Transform-Ensemble method for analysis of linear and nonlinear systems with random inputs”, Proc. NEC, Vol. 15, pp. 441-455, 1959. (42) M. Schetzen, “The Volterra and Wiener Theories of Nonlinear Systems”, John Wiley, New York, 1980. (43) Y. H. Ku and A. A. Wolf, “Laurent-Cauchy transforms for analysis of linear systems described by differential-difference and sum equations”, Proc. IRE, Vol. 48, pp. 923931,196O. (44) Y. H. Ku, “On the analysis of nonlinear stochastic systems”, Presented at 9th Int. Conf. on Nonlinear Oscillations, Kiev, U.S.S.R., 30 Aug.-5 Sept. 1981. J. Franklin Inst., Vol. 313, pp. 233-244, 1982. (45) Y. H. Ku, “Heat transfer problems solved by the method of nonlinear mechanics”, Int. J. Nonlinear Mech., Vol. 1, pp. l-16, 1966.

20

Journal of the Franklin Institute Pergamon Press Ltd.

Theory

of Nonlinear

Systems

Additional Bibliography 1. R. H. Cameron and W. T. Martin, “The orthogonal development of nonlinear functionals in series of Fourier-Hermite functional?‘, Ann. Math., Vol. 48, pp. 385-392, 1947. 2. N. Wiener, “Cybernetics”, John Wiley, New York, 1948. 3. N. Wiener, “Extrapolation, Interpolation and Smoothing of Stationary Time Series”, John Wiley, 1949. 4. Y. W. Lee, “Application of statistical methods to communication problems”, MIT RLE Tech. Report No. 181,195O. 5. Y. W. Lee, T. P. Cheatham and J. B. Wiesner, “Application of correlation analysis to the detection of periodic signal in noise”, Proc. IRE, Vol. 38, pp. 116551171, 1950. 6. L. A. Zadeh, “A contribution to the theory of nonlinear systems”, J. Franklin Inst., Vol. 255, pp. 387468,1953. 7. Y. H. Ku, “Nonlinear analysis ofelectromechanical problems”, J. Franklin Inst., Vol. 255, pp. 9-31,1953. 8. Y. H. Ku, “A method for solving third and higher order nonlinear differential equations”, J. Franklin Inst., Vol. 256, pp. 229-244, 1953. 9. Y. H. Ku, “Analysis of multi-loop nonlinear systems”, Trans. IRE, Vol. CT-l, No. 4, pp. 6-12,1954. 10. Y. H. Ku, “Analysis of nonlinear coupled circuits”, Trans. AIEE, Vol. 73, part 1, pp. 626 631, 1954 ; Vol. 74, part 1, pp. 4399443, 1955. 11. Y. H. Ku, “Analysis of nonlinear systems with more than one degree of freedom by means of space trajectories”, J. Franklin Inst., Vol. 259, pp. 115-131, 1955. 12. J. F. Barrett and D. G. Lampard, “An expansion for some second-order probability distributions and its application to noise problems”, Trans. IRE, Vol. IT-l, pp. 10-15, 1955. 13. Y. H. Ku, “Boundary layer problems solved by the method of nonlinear mechanics”, Proc. 9th Int. Congress of Applied Mechanics, Brussels, Belgium, Vol. 4, pp. 132-144, 1956. 14. J. F. Barrett, “The use of functionals in the analysis of nonlinear physical problems”, Statistical Advisory Unit, Ministry of Supply, Great Britain, Report No. l/57, 1957. 15. A. Bose, “Nonlinear systems-characterization and optimization”, Trans. IRE, Vol. CT-6, Special Supplement, 1959. 16. Y. W. Lee, “Statistical Theory of Communications”, John Wiley, New York, 1960. 17. A. M. Letov, “Stability in Nonlinear Control Systems”, Princeton Univ. Press, Princeton, N. J. (English translation), 1961. 18. J. LaSalle and S. Lefschetz, “Stability by Liapunov’s Direct Method with Applications”, Academic Press, New York, 196 1. 19. Y. H. Ku, “On nonlinear oscillations in electromechanical systems”, Proc. IUTAM Int. Symposium on Nonlinear Vibrations, Kiev, U.S.S.R., Vol. 3, pp. 180-199, 1961; J. Franklin Inst., Vol. 272, pp. 253-274, 1961. 20. V. M. Popov, “Absolute stability of nonlinear systems of automatic control”, Automation and Remote Control, Vol. 22, pp. 857-875, 1961. 21. R. L. Kalman, “Liapunov functions for the problem of Lur’e in automatic control,” Proc. Nut. Acud. Sci., Vol. 49, pp. 201-205, 1963. 22. Y. H. Ku and N. N. Puri, “On Liapunov functions of high order nonlinear systems”, J. Franklin Inst., Vol. 276, pp. 349-364, 1963. 23. M. A. Aizerman and F. R. Gantmacher, “Absolute Stability of Regulator Systems”, Holden-Day, San Francisco, CA (English translation), 1964. 24. Y. H. Ku, Ralph Mekel and C. C. Su, “Stability and design of nonlinear control systems via Liapunov’s criterion”, IEEE Znt. Convention Record, Vol. 12, pp. 154170, 1964. Vol. 315, No. 1, pp l-26, January Printed in Great Britain

1983

21

25. Y. H. Ku, “Liapunov function of a fourth-order nonlinear system”, Trans. IEEE, Vol. AC-9, pp. 276278, 1964. 26. Y. H. Ku, “On stability of some fourth-order nonlinear systems with forcing functions”, Int. Colloquium on Forced Vibrations in Nonlinear Systems, Marseilles, France, Sept. 1964. 27. Y. H. Ku, “Stability and boundedness considerations in some nonlinear systems”, Proc. NEC, Vol. 21, pp. 787-792, 1965. 28. S. Lefschetz, “Stability of Nonlinear Control Systems”, Academic Press, New York, 1965. 29. Y. H. Ku and H. T. Chieh, “Extension of Popov’s theorems for stability of nonlinear control systems”, J. Franklin Inst., Vol. 279, pp. 401416, 1965. 30. B. N. Naumov and Y. Z. Tsypkin, “Frequency criterion for absolute process stability in nonlinear automatic control systems”, Automat. Remote Control, Vol. 25, No. 6, pp. 765-778,1965. 31. C. A. Desoer, “A generalization of the Popov criterion”, Trans. IEEE, Vol. AC-lo, pp. 182-185, 1965. 32. I. W. Sandberg, “On generalizations and extensions of the Popov criterion”, Trans. IEEE, Vol. CT-13, pp. 117-118, 1966. 33. R. W. Brockett, “The status of stability theory for detefministic systems”, IEEE International Convention Record, Vol. 14, pp. 125-142, 1966. 34. Y. H. Ku and H. T. Chieh, “New theorems on absolute stability of non-autonomous nonlinear control systems”, IEEE Int. Convention Record, Vol. 14, pp. 260-271, 1966. 35. Y. H. Ku and H. T. Chieh, “Stability of control systems with multiple nonlinearities and multiple inputs”, J. Franklin Inst., Vol. 282, pp. 357-365, 1966. 36. R. E. Kalman, “Pattern recognition properties of multilinear machines”, IFAC Symposium Technical and Biological Problems of Control, Yerevan, Armenian SSR, U.S.S.R., Sept. 1968. 37. Y. H. Ku, “On nonlinear control system analysis”, Fourth All-Union ConJ: on Automatic Control, Tbilisi, Georgia SSR, U.S.S.R., 30 Sept.-5 Oct. 1968. (See Ku, Collected Scient$c Papers, pp. 101 l-1057, 1971.) 38. A. Sandberg and L. Stark, “Wiener G-functional analysis as an approach to nonlinear characteristics of human pupil light reflex”, Brain Res., Vol. 11, pp. 194211, 1968. 39. G. S. Christensen, “On the convergence of Volterra series”, Trans. IEEE, Vol. AC-13, pp. 736737,196s. 40. C. D. Gorman and J. Zaborszky, “Functional calculus in the theory of nonlinear systems with stochastic signals”, Trans. IEEE, Vol. IT-14, pp. 528-531, 1968. 41. Y. H. Ku, “On the application of diakoptics to nonlinear systems”, J. Franklin Inst., Vol. 286, pp. 634-642, 1968. 42. R. E. Maurer and S. Narayanan, “Noise loading analysis of a third-order nonlinear system with memory”, Trans. IEEE, Vol. COM-16, pp. 701-712, 1968. 43. L. Stark, “The pupillary control system : its nonlinear adaptive and stochastic engineering *design characteristics”, Automatica, Vol. 5, pp. 655-676, 1969. 44. P. A. V. Hall, “Generalization of Wiener’s theory of nonlinear systems for process identification”, Trans. IEEE, Vol. AC-14, pp. 312-313, 1969. 45. G. Marchesini and G. Picci, “On the functional identification of nonlinear systems from input-output data records”, Trans. IEEE, Vol. AC-14, pp. 757-759, 1969. 46. Y. H. Ku, “Stochastic stability of nonlinear control systems”, Trans. IEEE, Vol. AC-14, pp. 599-601,1969. 47. Y. H. Ku, “Stochastic stability of nonlinear oscillating systems”, Proc. 5th Int. Con& on Nonlinear Oscillations, Kiev, U.S.S.R., Vol. 2, pp. 233-254, 1970. J. Franklin Inst., Vol. 288, pp. 305-317, 1969.

22

Journal oftheFranklin Pergamon

Institute Press Ltd.

Theory of Nonlinear

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94. M. Rudko and D. D. Weiner, “Volterra systems with random inputs: a formalized approach”, Trans. IEEE, Vol. COM-26, pp. 217-226, 1978. 95. I. W. Sandberg, “On the mathematical foundations of compartmental analysis in biology, medicine and ecology”, Trans. IEEE, Vol. CAS-25, pp. 273-279, 1978. 96. G. Palm and T. Poggio, ‘Stochastic identification methods for nonlinear systems: an extension of the Wiener theory”, SIAM J. appl. Math., Vol. 34, pp. 524-534, 1978. 97. S. A. Billings and S. Y. Fakhouri, “Identification of a class of nonlinear systems using correlation analysis”, Proc. IEE, Vol. 125, pp. 691-697, 1978. “Random vs. pseudorandom test signals in nonlinear system 98. V. Z. Marmarelis, identification”, Proc. IEE, Vol. 125, pp. 4255428, 1978. 99. R. B. Swerdlow, “Analysis of intermodulation noise in frequency converters by Volterra series”, Trans. IEEE, Vol. MTT-26, pp. 305-313, 1978. “Analysis of nonlinearly loaded 100. T. K. Sarkar, D. D. Weiner and R. F. Harrington, multiport antenna structures over an imperfect ground plane using the Volterra-series method”, Trans. IEEE, Vol. EMC-20, pp. 278-287,1978. 101. B. J. Leon and D. J. Schafer, “Volterra series and Picard iteration for nonlinear circuits and systems”, Trans. IEEE, Vol. CAS-25, pp. 7899793, 1978. 102. C. Lesiak and A. J. Krener, “The existence and uniqueness of Volterra series for nonlinear systems”, Trans. IEEE, Vol. AC-23, pp. 1090-1095, 1978. 103. H. A. Barker and R. W. Davy, “Measurement of second-order Volterra kernels using pseudorandom ternary signals”, Int. J. Control, Vol. 27, pp. 277-291, 1978. 104. S. Yasui, “Stochastic functional Fourier series, Volterra series and nonlinear system analysis”, Trans. IEEE, Vol. AC-24, pp. 230-242, 1979. 105. G. A. Parker and E. L. Moore, “The identification of single-valued separable nonlinear systems based on a modified Volterra series approach”, Proc. 5th IFAC Symposium Identijcation &System Parameter Estimation, Darmstadt, Sept. 1979. 106. S. Klein and S. Yasui, “Nonlinear system analysis with non-Gaussian white stimuli: general basis functionals and kernels”, Trans. IEEE, Vol. IT-25, pp. 4955500, 1979. 107. M. Rudko and D. D. Weiner, “Optimum nonlinear Wiener filters”, J. Frank/in Inst., Vol. 308, pp. 57-65, 1979. 108. P. J. Lawrence, “A Volterra series approach to the solution of time varying linear differential equations”, Int. J. Control, Vol. 30, pp. 581-586, 1979. 109. V. Z. Marmarelis, “Error analysis and optimal estimation procedures in identification of nonlinear Volterra systems”, Automatica, Vol. 15, pp. 161-174, 1979. 110. G. E. Mitzel, S. J. Clancy and W. J. Rugh, “On transfer function representation for homogeneous nonlinear systems”, Trans. IEEE, Vol. AC-24, pp. 242-249, 1979. 111. J. F. Barrett, “A Bibliography on Volterra Series, Hermite Functional Expansion and related subjects”, Dept. of Electrical Engineering, Eindhoven Univ. of Technology, Eindhoven, The Netherlands, 1980. 112. M. Maqusi, “Identification of Volterra kernels of a class of nonlinear systems by Walsh function techniques”, J. Franklin Inst., Vol. 310, pp. 65-75, 1980. 113. J. A. Sharp and C. J. Stewart, “Parameter identification in nonlinear models of socioeconomic systems”, Trans. IEEE, Vol. SCM-lo, pp. 652-655, 1980. 114. H. J. Kushner, “Diffusion approximations to output processes of nonlinear systems with wide-band inputs and applications”, Trans. IEEE, Vol. IT-26, pp. 715-725, 1980. 115. H. J. Kushner and Y. Bar-Ness, “Analysis of nonlinear stochastic systems with wide band inputs”, Trans. IEEE, Vol. AC-25, pp. 1072-78, 1980. 116. M. Lambiris, “On the adjoint equation of stochastic linear systems with small correlation times”, J. Franklin Inst., Vol. 310, pp. 239-245, 1980. Vol. 315, No. I, pp. I-26, Printed in Great Brmin

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ZH.Ku 117. W. J. Rugh, “Nonlinear System Theory-The Volterra/Wiener Approach”, Johns Hopkins Univ. Press, Baltimore, MD, 1981. 118. D. P. Atherton, “Stability of Nonlinear Systems”, Wiley-Interscience, New York, 1981. 119. L. 0. Chua and Y-S. Tang, “Nonlinear oscillation via Volterra series”, Trans. IEEE, Vol. CAS-29, pp. 15&168, 1982. 120. S. Yasui, “Wiener-like Fourier kernels for nonlinear system identification and synthesis (nonanalytic cascade, bilinear, and feedback cases)“, Trans. IEEE, Vol. AC-27, pp. 677-685,1982.

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