The formal Laplace Borel transform of Fliess operators and the composition product

OPERATORS AND THE COMPOSITION PRODUCT

Received 28 July 2005; Revised 14 April 2006; Accepted 25 April 2006

The formal Laplace-Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace-Borel transform is shown to provide an iso-morphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition prod-uct. Finally, the formal Laplace-Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the op-erational calculus of Fliess for computing the output response of an analytic nonlinear system.

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1. Introduction

Letu:R→Rbe a function which is real analytic at a pointt0∈R. Its Taylor series

ex-pansion is denoted by

u(t)=∞

n=0

cu(n)

t−t0

n

n! . (1.1)

Within its radius of convergence,uis completely characterized by the sequence of coeffi -cients{cu(n)}∞n=0. From this, one can construct a formal power series representation ofu

by introducing an abstract symbol setX0= {x0}, namely,

cu=

∞

n=0

cu(n)xn0. (1.2)

The formal Laplace transform in this setting is the mapping

ᏸf:u−→cu. (1.3)

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International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 34217, Pages1–14

Its inverse is the formal Borel transform. Whenuis entire, its one-sided integral Laplace transform is often well defined. Settingt0=0, it can be written in the form

ᏸ[u](s) := _∞

0 u(t)e

−st_dt=∞

n=0

cu(n)

_∞

0

tn

n!e−stdt

=s−1 ∞

n=0

cu(n)

s−1n,

(1.4)

using the Laplace transformation pair

tn_⇐⇒ᏸ _n!s−1n+1_{, n≥0. (1.5)}

Clearly, the two transformsᏸf andᏸare related via

ᏸ[u](s)=x0ᏸf[u]|x0→s−1. (1.6)

An alternative notion of the formal Laplace-Borel transform appears in [23] as a mapping between two formal power series.

The formal Laplace-Borel transform was first used by Fliess et al. for nonlinear sys-tems analysis in [6–8,12,19]. A related approach was later given by Minh in [22]. In each instance, an algebraic representation of an analytic function was used to produce a type of operational calculus for computing the output responses of a nonlinear system given various analytic input functions. In [9–11] Fliess also applied the formal Laplace-Borel transform via a tensor product to examine some classical problems regarding transfer matrix representations of linear time-varying systems and generalized state space systems. Later in [3], Delaleau and Rudolph characterized the properness of linear time-varying systems using this approach. In [18], Hautus used the formal Laplace-Borel transform to characterize the invertibility properties of so-called smooth linear systems. More recently, in [2] it was applied to obtain conditions for well-posedness of the input-output mapping for several classes of boundary control systems. The formal Laplace-Borel transform ap-plied in the analysis of linear systems, as with the analysis of input and output functions, mostly involves formal power series over a single-letter alphabet, and thus is commuta-tive in nature. What is absent in the nonlinear systems literature is an explicit notion of computing the formal Laplace-Borel transform of an input-output operator, an idea that is familiar and useful in linear systems theory. In this setting, noncommutative formal power series arise as a natural generalization.

The main purpose of this paper is to define the formal Laplace-Borel transform of a class of analytic input-output operators using noncommutative formal power series, and then to show how this notion can provide an alternative interpretation of the op-erational calculus of Fliess when combined with the composition product of two formal power series [4,5, 13–16,20]. The specific class of operators considered is known as Fliess operators. LetX= {x0,x1,...,xm}denote an arbitrary alphabet of noncommutative

letters, andX∗the set of all words overX, including the empty word∅. For eachcin the set of all formal power seriesR_{X, one can formally associate a corresponding}

For a measurable functionu: [a,b]→Rm_{, defineu}

p=max{uip: 1≤i≤m}, where

uipis the usualLp-norm for a measurable real-valued function,ui, defined on [a,b]

whenp∈[1,∞), andui∞=sup_t∈[a,b]{|ui(t)|: 1≤i≤m}. For any fixedp∈[1,∞], let

Lm

p[a,b] denote the set of all measurable functions defined on [a,b] having a finite · p

norm andBm

p(R)[a,b] := {u∈Lmp[a,b] :up≤R}. Witht0,T∈Rfixed, andT >0,

de-fine inductively for eachη=xiη∈X∗the mappingEη:Lm1[t0,t0+T]→Ꮿ[t0,t0+T] by

Eη[u]t,t0:=

_t

t0

ui(τ)Eη[u]τ,t0dτ, (1.7)

whereE∅≡1 and u0≡1. The input-output operator corresponding to cis the Fliess

operator

Fc[u](t) :=

η∈X∗

(c,η)Eη[u]t,t0. (1.8)

All Volterra operators with analytic kernels, for example, are Fliess operators. When there exist real numbersK,M >0 such that

_{(c,η):=max(c,η)j: 0≤j≤}

≤KM|η||η|!, (1.9)

the formal power seriesc is said to be locally convergent. (Here|η| denotes the num-ber of symbols in η∈X∗.) The subset of all locally convergent series is denoted by

R

LCX. When c∈RLCX, it is known that Fc constitutes a well-defined

opera-tor fromBm

p(R)[t0,t0+T] intoB_q(S)[t0,t0+T] for sufficiently smallR,S,T >0, where the

numbersp,q∈[1,∞] are conjugate exponents, that is, 1/ p+ 1/q=1 with (1,∞) being a conjugate pair by convention [17]. Therefore, the specific operator class of interest in this paper is the set of Fliess operatorsᏲ:= {Fc:c∈RLCX}. When=m, this set forms a

semigroup under operator composition, as does the setRm

LCXunder the composition

product [4,20]. It will be shown here that the formal Laplace-Borel transform provides an isomorphism between these two semigroups.

The paper is organized as follows. InSection 2, the notion of a formal Laplace-Borel transform of a Fliess operator is defined. Then its basic properties are developed and a set of examples is given. InSection 3, the composition product is introduced and its relation-ship to the formal Laplace-Borel transform is described. Using the concepts developed in Section 3, the formal Laplace-Borel transform is shown to provide a compact interpreta-tion of the operainterpreta-tional calculus of Fliess inSection 4. Examples are given to illustrate the computation of the output response of an analytic system.Section 5concludes the paper with a brief summary of the main results.

2. The formal Laplace-Borel transform of a Fliess operator Consider a causal linear integral operator

y(t)= _t

t0

where the kernel function,h, is analytic att=0. The operator is completely characterized by the Laplace transform of the kernel function, namely,H(s) :=ᏸ[h](s)= k>0ch(k)s−k,

when it exists. Therefore, the usual definition (1.3) of the formal Laplace-Borel transform over a single symbol applies directly to this case. In the more general context of Fliess operators, a generalization is required. The following preliminaries are needed to ensure the new definition is well-posed.

Theorem 2.1 [24, Corollary 2.2.4]. Supposec,d∈R

LCX. IfFc=FdonBm∞(1)[t0,t0+

T] for some finiteT >0, thenc=d.

Given the nested nature of the set of spacesLp[t0,t0+T] forp=1, 2,...,∞, the

follow-ing result is immediate.

Corollary 2.2. Supposec,d∈R

LCX. IfFc=Fd onBmp(R)[t0,t0+T] for some p∈

{1, 2,...,∞}and real numbersR,T >0, thenc=d.

In light of this uniqueness property, there exists a one-to-one mapping between the set of well-defined Fliess operators,Ᏺ, and the set of locally convergent formal power series,

R

LCX. This guarantees that the following definition is well-posed.

Definition 2.3. LetX= {x0,x1,...,xm}. The formal Laplace transform is defined as

ᏸf :Ᏺ−→RLC

X

:Fc−→c. (2.2)

The corresponding inverse transform, the formal Borel transform, is

Ꮾf :RLC

X−→Ᏺ

:c−→Fc.

(2.3)

Note that whenm=0, this definition is consistent with that given in (1.3) in the sense that a fixed functionucan be represented as the constant operatorFcu, that is,u=Fcu[v]

for allv∈Bm

p(R)[t0,t0+T] andᏸf[u]=cu=ᏸf[Fcu].

It is next shown that many of the familiar properties of the integral Laplace transform also have counterparts in the present context. To facilitate the analysis, two concepts are needed.

Definition 2.4. For anyxi∈X, the left-shift operator,x−i1:RX →RX, is defined

as

x−1 i (c)=

η∈X∗

(c,η)x−1

i (η), (2.4)

where

x−i1(η)=

⎧ ⎪ ⎨ ⎪ ⎩

η:η=xiη, η∈X∗,

Definition 2.5. A Dirac series,δi, is a generalized series with the defining property that

Fδi[u]=ui(t) for any 1≤i≤m.

The main result concerning elementary properties of the formal Laplace-Borel trans-form is stated below.

Theorem 2.6. Given any c,d∈R

LCXand scalarsα,β∈R, the following identities

hold.

(1) Linearity:

ᏸf

αFc+βFd

=αᏸf

Fc

+βᏸf

Fd , Ꮾf

αc+βd=αᏮf[c] +βᏮf[d].

(2.6)

(2) Integration:

ᏸf

In_F

c

=xn

0c, Ꮾf xn 0c =In_F

c,

(2.7)

whereIn_{(·) is thenth order integration operator so that}

InFc[u](t)=

_t

0

_τ1

0 ···

_τn−1

0 Fc[u]

τndτn···dτ2dτ1. (2.8)

(3) Differentiation:

ᏸf

DFc

=x−1

0 (c) + m

i=1

δi

x−1

i (c)

,

Ꮾf

x−1

0 (c) + m

i=1

δi

x−1

i (c)

=DFc.

(2.9)

Ifxn0is a left factor ofc, that is,c=x0ncfor somec∈RLCX, then

ᏸf

Dn_F

c

=x−n

0 (c),

Ꮾf

x−n

0 (c)

=Dn_F

c,

(2.10)

whereDn_{(·) is thenth-order differentiation operator so thatD}n_F

c[u](t)=dnFc[u](t)/

dtn_{, anddenotes the shuffle product (see, e.g., [21]).}

(4) Multiplication:

ᏸf

Fc·Fd

=ᏸf

Fc ᏸf Fd ,

Ꮾf[cd]=Ꮾf[c]·Ꮾf[d].

(2.11)

a Fliess operator is

d

dtFc[u](t)=Fx−01(c)[u](t) + m

i=1

ui(t)Fx−1

i (c)[u](t). (2.12)

Applying the formal Laplace transform to this equality gives the first pair of equations in part (3). Now ifx0is a left factor ofc, thenF_x−1

i (c)[u](t)=0 fori=1, 2,...,m. In this case,dFc[u](t)/dt=Fx−1

0 (c)[u](t). Proceeding inductively, the second pair of equations

follows.

The following definition is utilized in the examples which follow.

Definition 2.7 [1]. Let c∈RX be proper (i.e., (c,∅)=0). Then the star operator applied tocis defined as

c:=

n≥0

cn_:=(1−c)−1_{, (2.13)}

wherecn_{denotes the catenation power.}

Observe that whencis not proper, it is always possible to writec=(c,∅)(1−c). In which case, there exists ac−1_{∈RXsuch that under the catenation productcc}−1₌₁

andc−1_{c=1. Specifically,}

c−1₌ 1

(c,∅)(1−c)−1= 1

(c,∅)(c). (2.14)

Example 2.8. LetX= {x0,x1,x2}andFc[u](t)=exp(

_t

0u1(t)+u2(t)dt). Observe thatFc[u]

can be expanded as

Fc[u](t)=

n≥0

1 n!

_t

0u1(t) +u2(t)dt

n

=

n≥0

_t

0

u1

τ1

+u2

τ1

τ1

0

u1

τ2

+u2

τ2

× ··· _τn−1

0

u1τn+u2τndτn···dτ2dτ1.

(2.15)

Therefore,

ᏸf

Fc

=

n≥0

x1+x2

n₌

x1+x2

. (2.16)

This result can be viewed as an operator generalization of the integral transform pair

et_⇐⇒ᏸ _(1−s)−1_{. (2.17)}

Table 2.1. Some formal Laplace-Borel transform pairs.

Fc ᏸf

Fc

Fc:u−→1 1

Fc:u−→tn n!xn0

Fc:u−→

_n−1

i=0

i n−1

i! aiti

eat _1−ax

0

−n

Fc:u−→ 1

n!

t

t0 k

j=1 uij(τ)dτ

n

xi1+xi2+···+xik n

Fc:u−→

n≥0 an n! t t0 k

j=1 uij(τ)dτ

n

n≥0 an

xi1+xi2+···+xik n

Fc:u−→exp

t

t0 k

j=1 uij(τ)dτ

xi1+xi2+···+xik

Fc:u−→

t

t0 k

j=1

uij(τ)dτexp t

t0 k

j=1 uij(τ)dτ

xi1+xi2+···+xik

1−xi1+xi2+···+xik 2

Fc:u−→cos

_t

t0 k

j=1 uij(τ)dτ

1

1 +xi1+xi2+···+xik 2

Fc:u−→sin

t

t0 k

j=1 uij(τ)dτ

xi1+xi2+···+xik

1 +xi1+xi2+···+xik 2

Example 2.9. LetX= {x0,x1,...,xm}. SupposeFchas the generating seriesc= η∈X∗η, andFξis given for some fixed wordξ∈X∗. Then

ᏸf

Fc·Fξ

=ᏸf

Fc ᏸf Fξ =cξ = ν∈X∗

ν ξ

ν, (2.18)

where (_ξν) denotes the number of subwords ofνwhich are equal toξ(see [21, page 127]).

3. The formal Laplace-Borel transform and the composition product

Definition 3.1. For anyc∈R_Xandd∈Rm_{X, the composition product is defined}

as

c◦d=

η∈X∗

(c,η)η◦d, (3.1)

where

η◦d= ⎧ ⎪ ⎨ ⎪ ⎩

η:|η|xi=0, ∀i=0,

xn+1 0

di(η◦d):η=x0nxiη, n≥0, i=0.

(3.2)

(Here|η|xi denotes the number of symbols inηequivalent toxianddi:ξ→(d,ξ)i, the ith component of (d,ξ).)

Consequently, if

η=xnk

0 xikx0nk−1xik−1···x n1

0 xi1x n0

0 , (3.3)

whereij=0 forj=1,...,k, it follows that

η◦d=xnk+1

0

dikxn0k−1+1

dik−1···x n1+1

0

di1x

n0

0

···. (3.4)

Alternatively, for anyη∈X∗one can uniquely define a set of right factors{η0,η1,...,ηk}

ofηby the iteration

ηj+1=xnj

+1

0 xij+1ηj, η0=x n0

0 , ij+1=0, (3.5)

so that η=ηk with k= |η| − |η|x0. In which case, η◦d=ηk◦d, where ηj+1◦d=

xnj+1+1

0 [dij+1(ηj◦d)] andη0◦d=x n0

0 . It was shown in [15,16] that the composition

product of two series is always well defined since the family of series{η◦d:η∈X∗}is locally finite for any fixedd∈Rm_X.

It is easily verified that for any seriesc,d,e∈Rm_{X, the composition product is left}

distributive over addition. That is,

(c+d)◦e=c◦e+d◦e, (3.6)

but in generalc◦(d+e)=c◦d+c◦e. An exception is the class of series called linear series. A seriescis linear if

supp(c)⊆η∈X∗:η=xn1

0 xixn00,i∈ {1, 2,...,m},n1,n0≥0

. (3.7)

The composition product is associative, that is, (c◦d)◦e=c◦(d◦e); hence (Rm_X,◦)

forms a semigroup [4,20]. In [16], it is shown that the composition product of two locally convergent formal power series is always locally convergent; therefore the setRm

LCXis

closed under composition, and (Rm

LCX,◦) also forms a semigroup. The central

prop-erty of the composition product is its relationship to the composition of two Fliess oper-ators. Namely, for anyc∈R

(Fc, Fd) FcÆFd=Fc Æd

(c, d) cÆd Æ

Æ

Ꮾf ᏸf ᏸf Ꮾf

Figure 3.1. The isomorphism between the semigroups (Ᏺ,◦) and (Rm

LCX,◦).

Fliess operators is always another Fliess operator with generating seriesc◦d, that is,

Fc◦Fd=Fc◦d. (3.8)

Therefore,Ᏺforms a semigroup under operator composition when=m. The following theorem shows that composition is preserved in a natural sense under the formal Laplace-Borel transform.

Theorem 3.2. LetX= {x0,x1,...,xm}. For anyc∈RLCXandd∈RmLCX,

ᏸfFc◦Fd=ᏸfFc◦ᏸfFd,

Ꮾf(c◦d)=Ꮾf(c)◦Ꮾf(d).

(3.9)

Proof. The proof follows directly from the definitions. For any well-definedFcandFd,

ᏸf

Fc◦Fd

=ᏸf

Fc◦d

=c◦d=ᏸf

Fc

◦ᏸf

Fd

. (3.10)

Similarly, for any locally convergent formal power seriescandd,

Ꮾf(c◦d)=Fc◦d=Fc◦Fd=Ꮾf(c)◦Ꮾf(d). (3.11) FromTheorem 3.2, when=m, the formal Laplace-Borel transform provides an iso-morphism between the two semigroups (Ᏺ,◦) and (Rm

LCX,◦), as shown inFigure 3.1.

Example 3.3. LetX= {x0,x1,x2},Fc[u](t)=cos(

_t

0u1(t) +u2(t)dt), andd=(d1d2)T∈ R2

LCX. Define

Fe[u](t)=

Fc◦Fd

[u](t)=cos t

0Fd1[u](t) +Fd2[u](t)dt

. (3.12)

The formal Laplace transform ofFeis then

ᏸf

Fe

=c◦d=

i≥0

(−1)i_x 1+x2

2i_◦

d

=−x1+x22◦d= 1

1 +x1+x22

◦d.

Example 3.4. LetX= {x0,x1,...,xm}andc∈RmLCX. It is easily verified by induction

that forn≥1,

xn i ◦c=

1 n!

x0ci

n

, i=1, 2,...,m, (3.14)

where (·)n_{denotes the shuffle power.}

Applying the formal Borel transform to both sides of this identity gives

Ꮾf

xn

i ◦c

=Ꮾf

₁ n!

x0ci

n = 1 n! Ꮾf x0ci

_n

= 1 n!

_t

0Fci[u](τ)dτ

n

.

(3.15)

Example 3.5. First consider the linear ordinary differential equation

dn_y(t)

dtn + n−1

i=0

aid i_y(t)

dti = n−1

i=0

bid i_u(t)

dti (3.16)

with initial conditionsy(i)_(0)=0,u(i)_{(0)=0, and wherea}

i,bi∈Rfori=0, 1,...,n−1.

Integrate both sides of the equationntimes and assume there exists ac∈RLCXsuch

thaty=Fc[u]. Applying the formal Laplace transform to both sides of the equation gives

δ+

n−1

i=0

aixn0−1−ix1

◦c=n−

1

i=0

bixn0−1−ix1,

1 +

n−1

i=0

aixn0−i

c=n

−1

i=0

bixn0−1−ix1.

(3.17)

Therefore,

c=

1 +

n−1

i=0

aixn0−i

−1_n−1

i=0

bix0n−1−ix1. (3.18)

Rephrased in the language of the integral Laplace transform, this is equivalent to

Y(s)=

1 +

n−1

i=0

ai_sn1_−i

−1_n−1

i=0

bi_sn1_−i

U(s)

=

sn₊n− 1

i=0

aisi

−1_n−1

i=0

bisi

U(s).

(3.19)

Now consider the nonlinear differential equation

dn_y(t)

dtn + n−1

i=0

aid i_y(t)

dti + k

j=2

pju(t)yj(t)= n−1

i=0

bid i_u(t)

withy(i)_(0)=0,u(i)_{(0)=0, and wherea}

i,bi,pj∈Rfori=0, 1,...,n−1 and j=2,...,k.

Again integrate both sides of the equationntimes and assume y=Fc[u]. Applying the

formal Laplace transform gives

1 +

n−1

i=0

aixn0−i

c+

k

j=2

pjxn0−1x1

cj₌n− 1

i=0

bixn0−1−ix1. (3.21)

As in [12], a recursive procedure can be applied to solve the algebraic equation iteratively so that

c=c1+c2+··· (3.22)

with

c1=

1 +

n−1

i=0

aix0n−i

−1_n−1

i=0

bixn0−1−ix1 (3.23)

and forn≥2

cn=

1 +

n−1

i=0

aixn0−i

−1

xn−1 0 x1

k

j=2

pj

ν1≥1,...,νj≥1 ν1+ν2+···+νj=n

cν1cν2···cνj. (3.24)

4. Operational calculus for the output response of Fliess operators

In [7,12,19], an operational calculus was developed to compute the output response of an analytic nonlinear system represented by a Fliess operator. In this section, the formal Laplace-Borel transform pair in conjunction with the composition product are employed to produce a clear and compact interpretation of this methodology.

Theorem 4.1. LetFc∈Ᏺand letube an analytic function with formal Laplace transform

cu∈RmLCX0. Then the output function y=Fc[u] is analytic and has formal Laplace

transformcy=c◦cu∈RLCX0.

Proof. The analyticity of y follows from [24, Lemma 2.3.8]. Let cy denote its formal

Laplace transform. Local convergence is preserved under the composition product; there-fore the formal power seriesc◦cuis locally convergent, and, for any admissible input

v∈Bm

p(R)[t0,t0+T],

y=Fcy[v]=Fc

Fcu[v]

=Fc◦cu[v]. (4.1)

Applying the formal Laplace transform to both sides of the equation givescy=c◦cu.

The following examples further illustrate the applications of this result for computing the output response of an analytic input-output system.

Example 4.2. Consider the linear time-invariant systemy(t)=0th(t−τ)u(τ)dτ, where

u z 1 y

1 z

Figure 4.1. A simple Wiener system.

Ifu(t)= k≥0(cu,xk0)tk/k!, then it follows thaty(t)= n≥0(cy,x0n)tn/n!, where

cy=c◦cu

=

k≥0

c,x0kx1

xk0x1◦cu

=

k≥0

c,x0kx1

xk0+1cu.

(4.2)

Therefore,

cy,x0n

=n−

1

k=0

c,xk

0x1

cu,x0n−1−k

, n≥1, (4.3)

which is the conventional convolution sum.

Example 4.3. Consider the simple Wiener system shown inFigure 4.1wherez(0)=0. The mappingu→ycan be written as

y(t)= ∞

n=0

Ex1[u](t)

n₌∞

n=0

Fxn

1 [u](t)=

∞

n=0

n!Exn

1[u](t). (4.4)

Therefore y=Fc[u] with c= n≥0n!x1n. When u(t)=tk/k!, for example, the formal

Laplace transform ofuiscu=x0k. FromTheorem 4.1and (3.14) it follows that

cy=

∞

n=0

n!xn1◦xk0=

∞

n=0

x0k+1

_n =

∞

n=0

(k+ 1)n!

(k+ 1)!nx

(k+1)n

0 . (4.5)

Consequently, the output response is

y(t)= ∞

n=0

(k+ 1)n!

(k+ 1)!n

t(k+1)n

(k+ 1)n!= ∞

n=0

t(k+1)n

(k+ 1)!n=

1

1−tk+1_{/(k+ 1)!}. (4.6)

5. Conclusion

References

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Yaqin Li: Department of Electrical and Computer Engineering, University of Memphis, Memphis, TN 38152, USA

E-mail address:[email protected]

W. Steven Gray: Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, VA 23529, USA