systems which possessg- andh- representations

CIRCUITS SYSTEMS SIGN A L PROCESS VOL. 8. NO. 2, 1989

S Y S T E M S W H I C H POSSESS g- AND h-REPRESENTATIONS*

David Ball 1 and Irwin W. Sandberg 2

Abstract. Examples are given concerning the range of applicability of recent representation results that provide a means of studying the input-output properties of nonlinear systems in terms of the familiar impulse-response concept, and which extend the concept of integral transformation to nonlinear maps. We show that such representations, which we call "g-'" and "h-representations," exist for important classes of systems governed by nonlinear integral equations. In particular, it is proved that a large class of maps that have Volterra series representations also have these representations.

1. Introduction

This p a p e r conta ins two examples concern ing the range o f app l i cab i l i t y o f results in [1]. In [1] i n p u t - o u t p u t r ep resen ta t ion results are given for non l inea r systems whose inputs and ou tputs are rea l -va lued func t ions on the whole real l ine R, the half - l ine [0, co), or R m. F o r example , cons ide r the case o f systems with inputs and ou tputs def ined on R. Cond i t i ons are given unde r which a no t -necessa r i ly -causa l m a p H def ined on a set S o f essent ia l ly b o u n d e d funct ions s: R ~ R has a r ep resen ta t ion o f the form

s e S . (1)

We call (1) a " g - r e p r e s e n t a t i o n . " Cond i t i ons are also given u n d e r which the m a p H in (1) has the " h - r e p r e s e n t a t i o n "

( H s ) ( t ) = l ~ h ( t , ' r , P , s ) s ( ' r ) d ' r , t c R , s~S , (2)

* Received October 4, 1988; revised December 27. 1988. 1 Applied Research Laboratories, University of Texas at Austin, Austin, Texas 78713-8029,

USA. 2 Department of Electrical and Computer Engineering, University of Texas at Austin,

Austin, Texas 78712, USA.

146 BALL AND SANDBERG

where h in (2) may often be interpreted in terms of impulse responses of linearized systems. These results considerably extend related previous results [2] which concern causal maps defined on functions on the half-line.

The examples in this paper show the existence of g- and h-representations for important types of systems governed by nonlinear integral equations. In particular, we show that these representations exist for large classes of Volterra series input-output operators.

We begin in Section 2 with a summary of the results needed from [1]. Section 2.1 contains preliminaries and notation. We present g- and h- representation results in Sections 2.2 and 2.3. (We repeat here only those results needed for our examples; see [1] for a complete list of results, with proofs.)

Section 3 contains our examples of the range of applicability of the results in Section 2. Section 3.1 concerns systems which can be described in terms of a certain general integral operator model. We show that these systems have g-representations. In Section 3.2 we consider maps which are defined in terms of Volterra series. We prove that a large class of such maps possess g- and h-representations, and we present explicit formulas for obtaining g in (1) and h in (2).

2. Our representation theorems

2.1. Preliminaries

This section contains definitions and notation, in which R and R " denote the set of real numbers and the set of real m-vectors, respectively.

Let ~ ( R ) (~1(0, co), ~qPl(Rm)) denote the set of functions f : R ~ R (f : [0, co )~R, f : R ' ~ R ) which are Lebesgue integrable with respect to Lebesgue measure on R([0, co), Rm). The integral of fcS~(R) ( fc ~1(0, co), f c 5~ over a Lebesgue measurable set A in R([0, co), R m) is written as Saf('r) d z (SAf( r l , . . . , " r m ) d ' r 1 �9 �9 �9 drm in the R m case) or, if f is a function of several variables and, say, f(a,.)~ ~I(R) we write SAf(a,r) dr to denote the integral of f(a,.) over A. Given f c~ l (R ) ( f c ~1(0, co)), we write the integral of f over R([0, co)) as S_~o~f(r)dr (Iof(r) dr), and given an interval [a, b] in R([0, co)), we write the integral o f f over [a, b] as I~f(r) dr. We define ~~176 (~~ co)) to be the set of Lebesgue measurable functions f : R - R (f : [0, oo) -. R) which are bounded except on a set of Lebesgue measure zero.

Given t->0, the sets ~1(0, t) and ~ ( 0 , t) are defined similarly to ~1(0, co) and ~~ co). The set of functions f : [0, t] m ~ R which are Lebes- gue integrable with respect to Lebesgue measure on [0, t] m is denoted by ~1([0, t]m). Given f : [0, co)-~ R, we say that f is an element of ~1(0, t) ( ~ ( 0 , t)) if f" Xt0.t] is in ~ ( 0 , co) (5r176 co)), where X~o.o denotes the characteristic function of [0, t].

SYSTEMS WHICH POSSESS g- AND h-REPRESENTATIONS 147

The standard ~1 and ~ norms are denoted by I[" II and II" Lo, respectively. Whenever we say that a property holds a.e. (almost everywhere) or that a function or set is measurable, we are referring to Lebesgue measure.

Let M(AT/) denote the set of functions f : R --> R (f : [0, ~ ) --> R). Given ~" in R([0, oo)), we define the truncation operator P~: M -> M (P,: ~r --> h4) by

(P~f)(t) = ~f(t) , t < _ % [ 0 , t > r,

for each f in M(MI), and given ~r->O, we define the map Q~,~: M ~ M by

I f ( t ) , t<--~ ", (Q~,~ f ) ( t l=t f ( r ) , z < t--< z+cr, f e M ,

~0, t > .r + ~r.

We may think of Q~.~ as a "truncate plus hold" operator, for given f in M, Q~.~f=P~f+f(r)xcn~+~ 1. Given r in R, we define the delay operator T~: M--> M by

( T ~ f ) ( t ) = f ( t - z ) , t ~ R , f ~ M .

Let S(S) denote any set of functions in M(hT/) satisfying S c ~ ( R ) ( g c ~ ( 0 , oo)) which is closed under the truncation operator P~ for each z in R([0, co)). Define W as the set of piecewise-constant functions of bounded support in M.

Given cr > 0, define w~ ~ W by

w~(t)=~l/~, t~(0,~], [0, otherwise.

As or ~ 0, w~ approaches the unit impulse at the origin (in the usual intuitive sense). Let 0 denote the zero function.

We say that a map H: S--> Ar is causal if P , H = P , HP, for each a in [o, ~).

Finally, whenever we say that a limit exists, we mean that it exists as a real number.

2.2. g-Representations

Theorems 1 and 2 of this section provide representations (which we refer to as "g-representat ions") for maps describing systems whose inputs and outputs are real-valued functions defined on R or [0, oo), respectively.

Theorem 1, which deals with maps that need not be causal, is used in our example in Section 3.2. The following hypothesis is referred to in Theorem 1:


A.1. The limit l imn-~o~((HPns)( t ) - (HP-ns) ( t ) ) exists and equals (Hs) ( t ) .

Theorem 1. Let H: S ~ M, and let t ~ R and s ~ S. Assume that A. 1 is satisfied and that there is a function w ~ M such that ws ~ ~LPX ( R ) and

I( HPbs)( t) -- ( HPas)( t)l <-- I[W( Pb -- Pa)s [[ (3)

for all a and b in R. Then

(a) the limit

g(t, r, s):= lim i ( (HP,+~s)( t ) - (HP , s ) ( t ) ) o-~0 or

exists for almost all r in R, and (b) the function g ( t , . , s) belongs to ~ I ( R ) and we have

(Hs)(t)=f~oog(t,r,s)dr. Notes. 1. Theorem 1 provides a representat ion for (Hs ) ( t ) for arbitrary t and s; it therefore yields a representation for H. Specifically, it is clear f rom the theorem that if there is a function w: S x R x R-~ R such that, with w = w(s, t, �9 ), the hypotheses o f Theorem 1 are satisfied for each t c R and s c S, then

(Hs ) ( t ) -= f~-o~ g(t, "r, s) d'r, t c R , s e S ,

where g ( t , . , s) is defined as in (a) for each t c R and s c S. It is of course sufficient that there exists w: R- '~ R such that, with w = w(t, �9 ), (3) and the other condit ions are met for each s for each t ~ R. Such a w exists in many important cases (e.g., in the cases addressed in this paper).

2. The condit ions under which Theorem 1 holds are essentially continuity condit ions on H. Hypothesis A.1 asserts that (Hs ) ( t ) is the limit of a sequence o f differences o f responses to certain truncations o f s. Condi t ion (3) requires that the difference in responses at t to truncations o f s at points a and b is bounde d by the "mass" of s on the interval [a, b] multiplied by some weighting funct ion w. A careful study of the p roo f o f Theorem 1 in [1] reveals that these condit ions are quite natural for the existence of such a representat ion for H.

3. Observations similar to those above apply to Theorem 2 below. Further comments on Theorem 1 appear in [1].

SYSTEMS W H I C H POSSESS g- AND h -REPRESENTATIONS 149

We refer to the following corollary to Theorem 1 when we address "h- represen ta t ions" in Section 2.3 (for a discussion of the interpretat ion of Corol lary 1, see [1]).

Corollary 1. Let H: S-~ M, and let t c R and s c S. Assume that the conditions o f Theorem 1 are met, and assume further that s is continuous almost everywhere in R and that, for almost all ~" in R, Q~.~s ~ S for tr > 0 sufficiently small. Finally, assume that there is a constant K, such that, for almost all T in R and tr > 0 sufficiently small,

]( HQ~,~s )( t ) - ( HP~+~,s )( t )l <- K,I] ( Q~.~ - P~+~)s H.

Then

1 lim -- ( ( HQ,.~S)( t ) - ( HP~s )( t ) ) o ' ~ 0 0 "

exists and equals g( t, T, s) for almost all "r in R.

Theorem 2, which deals with causal maps of functions defined on the half-line [0, co), is appl ied to our example in Section 3.1.

Theorem 2. Let H: S--> if4 be causal, and let t c [0, co) and s ~ S. Assume that there is a function w ~ if4 such that ws c 5~(0, t) and

I( HPbs)( t) -- ( HPos)( t)l <- ]1W(Pb -- Po)s I]

for all a and b in [0, t]. Then

(a) the limit

(b)

g( t, ~', s ) := lim 1 ( (HP,+~s)( t ) - ( H P , s ) ( t ) ) o-~0 O"

exists fo r almost all .r in [0, t], and the function g ( t , . , s) belongs to ~Lfl(0, t), and we have

fo ( H s ) ( t ) = (HPos ) ( t )+ g(t, r, s) dr. (4)

2.3. h-Representations

In this section we present Theorem 3, which gives condit ions under which the integral representat ion obtained in Theorem 1 takes the form (2).


Let H: S ~ M , and let t ~ R and s ~ S . Assume that the conditions of Theorem 1 and Corollary 1 are met. We refer to the following hypothesis in Theorem 3 (recall that Q~.~s = P,s + s(r)x(~,~+~l):

A.2. There is a map LH : R • S x {r. XA : r E R, A a bounded interval in R} ~ R such that

(a) for almost all ~- in R

1 -- ( ( H ( P~s + s('c) X(,,,+~l) )( t ) - ( HP, s)( t ) - Ln( t, P~, s( ~') X(~,~+~l) ) ~ 0 or

as or --> 0, and (b) LH satisfies the homogeneity condition that, for/3 ~ R,

Ln( t, P~s, /3X(,.,+~j) = f lLn( t, P~s, X<,,,+~1).

We now present Theorem 3.

Theorem 3. Let H : S ~ M, and let t ~ R a n d s ~ S. Assume the conditions of Theorem 1 and Corollary 1 are met, and assume that A.2 holds. Then we have

(Hs) ( t ) = (t, ,r, P,s)s(~) d,r,

where h( t , . , Pc.)s) is defined by

h( t, .c, p ~ ) = {~m~_~o Ln( t, P~s, TTw~) i f the limit exists, (5)

otherwise

for each -r in R.

The function LH (t, P~s, �9 ) often has a natural interpretation in terms of the concept of linearization. Indeed, if the map Ln(t , P~s, �9 ), assumed in A.2(b) to satisfy a certain homogeneity condition, is in fact linear, then Ln(t , P~s, T~w~) has the interpretation that it is a linearization of the func- tional H ( . ) ( t ) at the point P~s for the increment T~w~. Of course, if we drop the supposition that LH (t, P~s, �9 ) is linear, a corresponding interpretation holds in terms of what might be called the concept of"homogeniza t ion ." In light of (5), h(t, r, P~s) has the interpretation that it is the response at time t to an impulse at time r of the linearized or "homogenized" system described by LI4 ( ' , P~, �9 )., We thus have a natural extension o f the concepts of impulse response and integral representation for nonlinear maps satisfying the conditions of Theorem 3.


3. Examples

3.1. Representation of a system governed by integral operators

In this section we consider systems governed by the model

y = Nx,

x = As + Cy, (6)

w = Ds + By,

in which s is the input, w is the output, A, B, C, and D are linear operators, and N is nonlinear. Models of this kind have been used in [3], [4], and elsewhere. Here we assume that A, B, C, and D are maps from ~?~(0, ~ ) to ~ ( 0 , ~ ) defined by

( A v ) ( t ) = a ( t - ~ ' ) v ( r ) d%

(Bv)( t ) = b ( t - ~ ' ) v ( z ) dT,

fo' (Cv)( t )= c( t -~-)v( , ) d~-,

I0' (Dv)( t ) = d ( t - r ) v ( r ) dr

for each v e ~ ( 0 , oo) and t_> 0, where a, b, c, and d are elements of ~ ( 0 , oo)• ~ ( 0 , oo) and Ilcll~> 0, i.e., C is nontrivial. We assume that N is a memoryless map from ~ ( 0 , oo) to ~ ( 0 , oo) defined by ( N v ) ( t ) = 1V(v(t)), v e ~ ( 0 , oo), t >-O, where N maps complex numbers to complex numbers and real numbers to real numbers. We assume that ~/- is Lipschitz with constant K and N(0) = 0. We also assume that N is locally invertible about O, and that cb, the local inverse of N, satisfies

(i) 4~ is continuously Fr~chet ditterentiable throughout some open neighborhood of O,

(ii) (dd~(O) - C) is an invertible map of ~ ( 0 , oo) onto ~~176 oo).

These last assumptions on N guarantee the existence of a 8 > 0 such that, for each s in ~ ( 0 , oo) satisfying Ilslloo < 8, there are "locally" unique y, x, and w in ~ ( 0 , ee) satisfying (6) (see the proof of Theorem 6 of [3]).

We will show that input-output maps for systems in this class meet the conditions of Theorem 2 and thus possess g-representations. As we will see, this implies that a large class of system maps which have Volterra series representations also have g-representations.


Returning to the system above, and defining S = {s ~ ~~ oo): II sll~ < ~), we may define maps H and H on S by H s = y and H s = w for s ~ S, where y and w are the " loca l ly" unique funct ions in ~~ 0o) satisfying (6). Then

I~Is = N ( A s + CI~Is ),

H s = B H s + D s

for s e S, H s is the system output cor responding to the input s, and we may show H to be causal [3]. Given a and b in [0, oo) with a <- b, t -> O, and s c S,

I( HPbs) ( t ) -- ( UPas) ( t )l <_ IB( I~IPbs -- I~IPas)( t )] + ID( Pb - P~)s( t )l

<--I~ [ b ( t - z ) l . [(I2lPbs)(~ ") --(HP~s)(~')] d~"

+ Ld(t-~-)~(~-tl d~- a

-~ I]bll~ I~ I(~Pbs)( ,)- (APos)O-)l d~-

+ lid I1~ II(Pb - Po)s II. Now, using the fact that N is Lipschitz, given t ~ [0, 0o) and s c S,

I( I~IPbs)( t) - ( I~IPas)( t)l

= I N ( A P b s ( t ) + C (HPbs) ( t ) ) - N ( A P a s ( t ) + C(I2IP~s)(t))]

<- K I A ( Pb - Pals( t )] + K] C ( I~IPbs -- I~IP~s)( t)l

Io Io <-K l a ( t - ' r ) s ( - Q I d r + K I c ( t - ~ ' ) l " ](I~IPbs)(T)--(I2IP~s)(~')I d~"

Io <_K]]a l l~] l (Pb-P~)s l l+ K[]c[[~ ](I~-IPbs)('O-(I~IPas)(.)l d.r,

so, using the Be l lman -Gronwa l l inequality,

[( HPbs l ( t) - ( AP~s) ( t)] <_ K II a IIo~ [l(Pb - Po)sl[ e '~l~ll~*

and

I( HPbs)( t) -- ( UP~s)( t)[ <-- lid IIo~ II(P~ - Po)s II

Io + IIbll~gllallooll(Pb - P~)sll e Kl~ll~" d~"

-~ { IIdllo~ Ilalloollbll~ (eKiicll~, 1)lll(P~- Polsll.

Then defining w( t, �9 ) c 1~I by

w(t,~')=lld[lo~+llal~llbl[| ~'~ [0,~) , [Iclloo

SYSTEMS W H I C H POSSESS g- A N D h - R E P R E S E N T A T I O N S 153

we have [(HPbs)(t) - (HPas)(t)[-< [] w(t,. )(Pb - Pa)s(" )[[ for each t ~ [0, oo) and each s e 5~. Clearly, w(t, �9 )s(. ) c ~](0, t), so the condit ions of T h e o r e m 2 are met for each t->0. Now (APos)(t)=(DPos)(t)=O, t c [ 0 , oo), so (HPos)(t) = 0 and we have

;o' ( N s ) ( t ) = g ( t , .~, s ) d~, t e [0 , oo) , s e ~,

where, for t - 0 and s c S,

g(t, r, s) = lim 1 ((HP,+~s)(t) - (HP, s)(t)) o '40 O"

for a lmost all z in [0, t]. We thus have a gTrepresentat ion for H. Now suppose that ~r also satisfies

the condit ions that (dN/dz ) ( z ) exists locally (in the complex plane) abou t 0 and (dN/dz ) (O)r 0, and that

-fo ~ ( 0 ) c(r) e-ZT d r r l Re(z)_>0. for

Under these condit ions [3], there a r e ~1 > 0 and b o u n d e d integrable kernels hi, h2 , . . . , such that the restriction of g to $1 = {s e ~ ( 0 , oc): [Is [[~ < ~il} is given by

(gs)( t ) = ~ h g ( t - ~ ' l , . . . , t - - T k ) S ( T 1 ) " ' " S(Tk) d ( 7 " l , . . . , Tk) =1 O,t] k

Io' = g(t, r, s) dz

for t ~ 0 and s ~ $1. Therefore , there exist g - representa t ions for an impor tan t class of Volterra series operators .

3.2. Representation of Volterra series operators

In Section 3.1 we showed that an impor tan t class of system maps which have Volterra series representat ions also have g-representa t ions . In this section we consider maps which are defined directly as Volterra series operators . It is shown that each map in a large class of such opera tors , which includes all finite Volterra series opera tors with integrable kernels defined on a set of essentially bounded inputs, have g- and (under addi t ional condit ions) h-representa t ions , and the funct ions g and h are given as explicit functions of the inputs and kernels of the Volterra series.

Cons ider a m a p H : S ~ M defined by

(Hs)( t )= ~ [ pk(t, z , , . . . , ~'k)S(Zl) " " " S(~'k) d(~ ' l , . . . , r~), k = l ,] R i"

t c R , soS ,


where S = {s ~ ~ ( R ) : [[ s [[o~ < 6} for some 6 > 0 and Pk (t,") E ~ ( R k) with pk(t, r ~ , . . . , rk) symmetric in the r~'s for each k and t. Assume that the pk'S satisfy the summability condition

c o

6k[Ipk( t , ' ) l l<~, t ~ R , k=l

where II pk(t," )U denotes the ~ ( R k) norm of pk(t , . ). Of course, this condition is met if H is a finite Volterra series operator, i.e., if Pk -- 0 for k > K for some K.

It should be noted at this point that, while we study Volterra series maps on the whole line (i.e., maps H: S - M) in this section in order to provide an example of the application of Theorem 1, it becomes clear that the development can be modified routinely to obtain g- and h-representations for corresponding half-line (H: S - M) or R " cases.

The representations for H are obtained by first finding g- and h: representations for each Hk : S ~ M defined by

(Hks)(t) -= i rk pk(t, Z l , . . . , rk)S('rO " " " S(~'k) d ( r l , . . . , rk),

t ~ R , s ~ S , (7)

and then summing the integrals. Thus consider Hk ; given t ~ R and s e S, A.1 clearly holds by the Dominated Convergence Theorem.

Given a, b e R with a -< b, t ~ R, and s ~ S,

( HkPbS )( t ) -- ( HkPaS )( t )

= [ pk(t , Z l , . . . , "rk)(Pbs(T1)" " " Pbs(zk) d R k

- Pas(rO" �9 �9 P,,S(Zk)) d(7" , , . . . , "rk)

= f pk(t, '7"1 . . . . , T k ) S ( ' r l ) " ' " S ( T k ) d ( z l , . . . , Zk). J( (--c~. b ]k --(--oo .a ] k )

We use the following proposition, which is proved in the Appendix.

Proposition 1. Let f ~ ~l (Rn) , and assume f ( x ~ , . . . , xn) is symmetric in the x~'s. Then given a and b in R with a <- b,

f( f ( x , , . . . , xn) d ( x 1 . . . . . Xn) (--o~,b]n--(-oo,a] ")

= ~ f ( X l , . . . , X n ) d(Xl . . . . ,Xn), j ~ O j --oo.a]Jx(a, b] n - j

where (-0% a]~215 (a, b]" = (a, b]" and, as usual,

(;) j ! ( n - j ) ! "


Using Propos i t ion 1 and the symmet ry o f pk(t , ' ) , we have

[( n k e b s )( t ) -- ( n k e a s )( t )[

~ ~ j=o j J( . . . . ]~x(a,b] k-'pk(t'

j=0 \ j : . . . . l'• bl*-: Ipk(t, r , , . . . , Tk)S(7.1)''' S(Tk) I d ( r l , . . . , 7.k)

<- I p k ( t , 7 . 1 , - - , , T k - 1 , r ) s ( r l ) " " " S ( r k - 1 ) [ da I . j=0 -cX3 a]Jx(a,b] k-J-1

�9 d (7 . , , . . . , rk -0} Is(r)l d r

fo { ~ k - 1 ~ ]Pk( t, r l , . . . , 7 . k - l , 7.)1 j=O --OO,a]J• b] k J-1

�9 d ( r , , . . . , rk_,)} Is(r)l dr

<-- kak-1 ~ Ipk(t, 7.1,... , 7.k-,, r)l j = 0 j --oo,a]]x(a,b] k- l - I

�9 d(7., . . . . , r~_,)} Is(r)l dr

<- k6 k - ' Ipk(t , r l , . . . , rk-1, 7.)[ d ( 7 . 1 , . - . , rk-1 IS(r)I d7., k--I a

so defining Wk(t," ) by

Wk(t, r) = k6 k-' [ Ipk(t, 7 . , , . . . , rk-,, 7") I d ( r l , �9 �9 �9 7.k-,), 7. ~ R, d Rk-1

we have

L (HkPbS)(t)-(HkP~s)(t) <- Iwk(t, 7.)s(r)l dr.= 11 wk(t, ")(Pb-P,)s(" )[I

for each t e R and s e S. Since Wk(t, ") ~ ~ ( R ) for each t e R (by F u b i n r s theorem) , we have Wk(t, " )S(" ) ~ 5C~(R). Thus the cond i t ions o f T h e o r e m 1 are met, so

(HkS)( t )=f~ogk(t , 7.,s) dr, t ~ R , soS ,

where gk(t, " ,S) , t ~R , s~S , is defined by

gk(t, 7., S) = lira I ((HkP~+~s)(t) - (HkP~S)(t)) o'~0 or

for a lmos t all 7. in R.

1 5 6 B A L L A N D S A N D B E R G

F o r t ~ R and s c S, gk(t, ", S) is given by (using the symmet ry ofpk( t , ") and Propos i t ion 1)

gk(t, 7, S ) = lim 1 ((HkP~+o`s)(t)- (HkP~S)(t)) o ' ~ 0 o r

= l i r a . . . . 1 [ pk(t, z l , . . . , Zk)S(ZO S(Zk) o`~0 or .l((_oo,~.+o`]k (_oo,~.]k)

�9 d ( z l , . . . , Zk)

= ~ l im 1 Zk)S(ZO' '" S(Zk) j=0 j ~-,o o" . . . . lJ• k jpk( t, Zl, . . . ,

�9 d ( 7 1 , . . . , rk) for a lmos t all z in R. G iven f ~ ~f l (R) ,

;5 l im 1 f ( h ) dh = f ( r ) o'-~0 Or

for a lmost all z in R [5, p. 141], and

l * ,r+o"

lira f ( h ) dh = O, r c R , o- --~ 0 "r

so using Fub in i ' s theorem,

gk(t, z, s) =- ~ tim pk(t, T I , . . . , Zk-1, T) j = 0 j (_oO,~.] j • (T,T+Cr ]k--j 1

�9 S ( Z , ) " �9 �9 S ( Z k _ , ) S ( Z ) d(z , , . . . , Zk_,)

�9 d ( z , . . . . , z k - 1 ) ,

gk(t, z, s) = ks(z) [ pk(t, ZX , . . . , Zk--1, Z) d (_oo. . ] k-1

�9 S ( Z l ) " " " S ( Z k - , ) d ( ' r l , �9 �9 �9 , Z k - 1 )

for a lmost all z in R. N o w cons ide r the app l i ca t i on o f Theorem 3 to Hk. We must first show

that the cond i t ions of Coro l l a ry 1 are met. Let s ~ S be a.e. con t inuous on R; clearly, Q,,~s c S for a lmos t all r e R and each Or> 0. Given t ~ R, o -> 0, and z ~ R such that Q,,o`s c S, by Propos i t ion 1 and the symmet ry o f pk(t, �9 ),

I( HkQ~,,~s )( t ) - ( HkP~+r )( t )]

= f pk(t, zi . . . . , Zk)(Q~,o`S(zl)''" Q ~ s ( z k ) dR k

- P,+~S(Zl) �9 �9 �9 P~+o`s(zk)) d(Zl . . . . , rk)l

m

I

S Y S T E M S W H I C H P O S S E S S g - A N D h - R E P R E S E N T A T I O N S 1 5 7

= j=o j . . . . ]'• pk(t , r l , . . . , r k ) ( s ( r , ) �9 �9 �9 s ( r j ) s k - J ( r )

- s ( r , ) . " s(rk)) d ( ~ , , . . . , rk)

<- ~ Ipk(t , T I : , ' " �9 :, T k ) S ( T 1 ) " " " S ( ~ j ) ( s k - J ( T ) j = 0 j ec~,r]lx(r,r+o-] h J

- - S ( 5 + 1 ) " �9 " S ( T k ) ) I d ( T , , . . . , ' r k ) .

Given a, b ~ , . . . , b, c R,

n n--1 ( a - b ]a n- i - lh " " " b , , a - b l " " b , = ( a - b , ) a ' - l + • i = l , - i , ~,-i+1

so again using the symmetry of pk( t , " ) and Proposi t ion 1,

I( nko~,~s )( t ) - ( nkP,.-(.s )( t )l

<- F, Ipk(t, ~ l , . . . , r k ) s ( r O " " s ( r j ) l " I s ( , ) - s(rk)[ 3=o j -~,,]'• ~ ,

s k - j - l ( 7 ) + }~ k - j - i - I �9 s ( r ) s ( r k - i ) ' ' ' s ( r k - , ) d ( r , . . . . , r k ) 1=1

<-- 3~ ( k - j ) a k - ' Ipk(t , r l , . . . , rk)] 3=o j -OO,r]lX(r,r+(r] k-I

�9 Is(~)-S(Tk)[ d ( ~ , , . . . , rk)

<_

I.j=O j _aO,r]aXlr, r+cr]~, j-1

�9 d(T,,..., T~_,)} Is(T)-~(T01 d~

<<- ka k-' I p k ( t , r , , . . . , r k ) l d ( r , , . . . , r k - , Is(r)--s(rk)ldrk r R k 1

<-f[+~ Iwk(t, MI Is(T)-*(T01 d w

Then if Wk(t, " ) c S ~ ~ 1 7 6 for each t c R , we have

I(HkO,,~s)(t)-(HkP,+~s)(t)l<-Ilwk(t, .)l[~ll(O,.~-P,+=)sll, t e R ,

and the condi t ions of Corol lary 1 are met for each t e R and each a.e.

con t inuous s 6 S. Thus suppose that Wk(t, " ) C ~ ~ t ~ R . We seek a map L/4~ which

satisfies A.2 for each t c R and a.e. con t inuous s c S. Given t and r in R


and an a.e. con t inuous s e S, we might expect a " l inear iza t ion" of the map Ilk in (7) about P,s to be defined by

Lnk(t, P , s , f ) = f pk(t, r , , . . . , rk)f(r,)(P,s)(r2)" ' ' (P,s)(zk) d R ~

�9 d ( q , . . . , rk)

+ f pk(t, % , . . . , zk)(P,s)(rl)f(z2)(P~s)(r3)" "" (P,s)(rk) d R k

�9 d ( r l , . . . , rk)

~-" " "'JV f p k ( [ , T I , . . . , rk)(P~s)(r,)" " " (P,s)(rk_,)f(rk) d R k

�9 d ( r , , . . . , rk)

= k f pk(t, r , , . . . , rk)(P~s)(rO" "" (P: ) ( rk - , ) f ( rk ) d R k

�9 d ( r l , . . . , rk)

for each f c W. The map LHk(t, P : , �9 ) clearly satisfies A.2(b), and

l i r a - - LHk(t, P,s, s(r)x~,:+~l) = l~m k pk(t, % , . . . ,rk) o'~0 O" _c%rlk-t x(r,r+o.]

�9 s ( r l ) �9 �9 �9 s(rk_,)s(r) d ( r , , . . . , rk)

= k ( pk(t, % , . . . , rk-1, r) 3~ _oo.7]k 1

�9 S ( ' F 1 ) �9 �9 �9 S ( ' F k _ I ) S ( T ) d ( ' r l , . . . , " l ' k _ l )

=gk(t, r,s)

for almost all r in R, showing that A.2 is satisfied and that the condi t ions of Theorem 3 are met for each t e R and a.e. con t inuous s E S. We have for

t e R

hk( t, r, P : ) = lim LHk ( t, P : , T~w~) triO

f pk(t, r , , . . . . , rk-,, r ) s ( r , ) ' ' ' S(rk-,) d(rl . . . . , k T k ) O( - - 0 0 7 ] k ~ l

for almost all r in R, and

(Hks)(t) = f~oo hk(t, r, P~s)s(r) dr, t c R,

for each a.e. cont inuous s E S.

S Y S T E M S W H I C H P O S S E S S g - A N D h - R E P R E S E N T A T I O N S 159

We now return to the question of representing H. We have

( H s ) ( t ) = (Hks)(t) = E gk( t , r , s ) dr, t e R , s e S . k = l k = l .

I f the gk'S satisfy oo

E I g k ( t , ' , s ) l c ~ l ( R ) , t ~ R , s ~ S , k = l

then foo

(Hs)( t ) = J-o~ g(t, % s) d~', t ~ R, s ~ S,

o o

where g=~k=l gk, by the Dominated Convergence Theorem. Using the Monotone Convergence Theorem, we see that it is sufficient that

I lgk(t , . ,s) l l<oo , t ~ g , s ~ S . k = l

Now, for each K

k = l k = l -oo,7] k-1

�9 s(r,) �9 �9 �9 S(rk) d ( T 1 , . . . , ~'k-1) drk

K

<_ y. k[Isllk[Ipk(t, ")[[ k = l

co

<- E kllsll~llpk(t,')ll <~176 k = l

for each t ~ R and s E S by our summability condition and the fact that IIs[l| ~. Thus

IIgk(t,.,s)ll<oo, t~g, s~S, k = l

and we have the desired representation with g=~k~176 gk. An argument similar to that above yields

(Hs) ( t )=f f h(t,r,P,s)s(r)dr, t e R ,

- ~ 9 hk, for each a.e. continuous s ~ S if Wk(t, " ) ~ ~ for where h - k=l e a c h t ~ R for each k.

The development above yields g- and h-representations for H if H is a finite Volterra series operator. Now suppose H : S--> M is given by

= ~ I pk(t, ' / '1 , . - . , Tk)S(T1) ' ' ' S(Tk) d ( ' t l , ' . . , ( I2Is)( t) "rk ), k = l J [0 , t ] k

t~ [0, oo), s~g,

1 6 0 B A L L A N D S A N D B E R G

where S = {s c L#~176 co): ]]Slloo < 8} for some ~ > 0 and pk( t , " ) ~ Lfl([0, t] k) for each t --- 0 and each k. Since pk(t , " ) ~ 5E1([0, t]k), by Fubini's theorem

�9 , . . � 9 oo OC) ~to.tlkpk(t, r l , . . r k ) s ( r l ) �9 S(rk) d ( r l , . . Zk) is invariant on oY ( 0 , ) under any permutation of the order of integration for each t >-0, so H is unchanged if we replace each Pk with the kernel ilk defined by

1 ilk(t, r l , . . . , rk) = ~.v. E pk( t, r+(1), . . . , re,(k)),

where the sum is taken over all permutations 4~ of the sequence { 1 , . . . , k} (see p. 28 of [6])�9 Since ilk(t, r ~ , . . . , rk) is symmetric in the r~'s for each k and each t c [0, oo), a development similar to that described above yields

S/ (/~s)(t) = g ( t , r , s ) dr, t>-O, s a S ,

where g(t, -, s), t ~ [0, co), s c S, is given by

g(t, ~, s) = ~(r) ~ k [ ilk(t, r, . . . . , rk-, , r) k = l . ] [ 0 , r ] k-z

�9 s ( r , ) �9 �9 �9 s ( r k _ l ) d ( r , , . . . , r k - , )

for almost all r in [0, t], and we have a g-representation for /4 . Also, using an approach similar to that employed above, we may "linearize" each term in the expansion for H about P , s for each z ~ [0, t] to obtain

(I2Is)(t) = h(t , r, P , s ) s ( r ) dr, t>_O,

for each a.e. continuous s e S, where h ( t , . , P(.)s), t e [0, oo), s e S, is given by

h(t , r, P , s ) = ~ k [ ilk(t, r l , . . . , Tk-1, r) k = l . ] [0 ,1] k - I

�9 S ( r l ) " " " S ( r k - 1 ) d ( r l , . . . , r k - 1 )

for almost all r in [0, t], if each ilk satisfies

t o , n k _ , l i l k ( t , r l , . . . , r k - 1 , " ) [ d ( r l , �9 �9 �9 , r k _ l ) E ~ c ~ ~ 1 7 6 t )

for each t - 0. We thus have an h-representation f o r / 4 in this case. This is of particular interest because it shows that the large class of

"approximately finite" memory systems which can be approximated arbitrarily well by finite Volterra series of the form of H (see [4]) can also be approximated using g- and (possibly) h-representations. This makes it clear that another important class of systems possesses integral representations of the type presented in this paper, in this case in the sense that arbitrarily good approximations can be obtained.

SYSTEMS W H I C H POSSESS g- A N D h - R E P R E S E N T A T I O N S 161

Appendix. Proof of Proposition 1

For m = 1, we have

f ( ( - , b ] - ( - .a]) dx l = f = ta,b] j=0 \ J / .l (-c~.a]S• -s

Now suppose the proposition holds for m = n - 1 ; then using Fubini's theorem and the symmetry of f ,

f( f ( x l . . . . . x,,) d ( x , , . . . , Xn) (-co b ] " - (-oO,a ]")

=(ff "" ffoo-ff "" f_~)f(Xl,...,x~176 =(L ... IL-M... f; I: ... ;I)

�9 f ( x , , . . . , x , ) dx , . �9 �9 dx,

=fLf<< . . . . ],,-l) " [ ( x l ' ' ' ' ' x n ) d ( x l . . . . . x,,-1) dx,,

+[ - f ( x x , . . . , x , ) d ( x i , . . . , x , ) --oz),a]"-l•

" f ( X l , . . . , Xn) d ( X l , . . . , Xn-1) dx,,

+ [ f ( x l , . . . , x,,) d ( X l , . . . , x , , ) d~ --oo,a ] "-1 •

�9 f ( x l , . . . , x , ) d ( x , , . . . , x , )

(f~ + ~"-2{(n-1 '] ( + n-1)}ft / \ j . . . . ]J~<(a.b] n-] a,b]n j = l j - 1

+f(n-l~ }f ) + 1 f ( X l , . . . , Xn) d ( x l , . . . , Xn) [ \ n - 2 ] a( . . . . ]n-lx(a,b]/

I< ; ) = + ~ + n a,b]" j = l -oo,a]S• "-~' (--oo a] n lxta, b ]

" f ( X l , �9 �9 �9 , Xn) d ( x l , � 9 x , , )

= 2 ~=o j . . . . ]Xx(a.b] " - ' f ( x l ' ' ' ' ' Xn) d ( X l , . . . , Xn),


where we have used the identity

n

[7, p. 20]. The proposition follows by induction. []

References

[1] D. Ball and I. W. Sandberg, g- and h-Representations for Nonlinear Maps, Journal of Mathematical Analysis and Applications (to appear).

[2] I. W. Sandberg, Linear Maps and Impulse Responses, IEEE Transactions on Circuits and Systems, 35, 201-6, 1988.

[3] I. W. Sandberg, Expansions for Nonlinear Systems, Bell System Technical Journal, 61, 159-99, 1982.

[4] I. W. Sandberg, Nonlinear Input-Output Maps and Approximate Representations, AT&T Technical Journal, 64, 1967-83, 1985.

[5] W. Rudin, Real and Complex Analysis (3rd edition), McGraw-Hill, New York, 1987. [6] N. Wiener, Nonlinear Problems in Random Theory, M.I.T, Press, Cambridge, MA, 1958. [7] W. Trench, Advanced Calculus, Harper & Row, New York, 1978.

systems which possessg- andh- representations

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