Generating Series for Nonlinear Cascade and Feedback Systems

Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002

ThA11-3

Generating Series for Nonlinear Cascade and Feedback Systems W. Steven Gray Yaqin Li Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia 23529, U.S.A. [email protected] [email protected] Abstract

condition on the coefficients of c insures that there exist positive real numbers R and T0 such that for all piecewise continuous u with kuk∞ ≤ R and T ≤ T0 , the series defining Fc converges uniformly and absolutely on [t0 , t0 + T ]. Under such conditions the power series c is said to be locally convergent. More recently, it was shown in [8] that the growth condition (1) also implies that Fc constitutes a well defined operator from Bpm (R)[t0 , t0 + T ] into Bq` (S)[t0 , t0 + T ] for sufficiently small R, S, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents, i.e., 1/p + 1/q = 1 with (1, ∞) being a conjugate pair by convention.

Given two analytic nonlinear input-output systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection and feedback connection. In each case, the corresponding generating series is produced, when one exists, and conditions for convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series is generalized to the multivariable setting, and its set of known properties is expanded. In addition, the notion of a feedback product for formal power series is introduced and characterized.

In many applications input-output systems are interconnected in a variety of ways. Given two Fliess operators Fc and Fd , where c, d ∈ R`  I  are locally convergent, Figure 1 shows four elementary interconnections. In the case of the cascade and feedback connections it is assumed that ` = m. The general goal of this paper is to describe in a unified manner the generating series for each of the four interconnections shown in Figure 1 and conditions under which it is locally convergent. The parallel connection is the trivial case, and the product connection was analyzed in [12]. They are included for completeness and some of the analysis is applicable to the other two interconnections. It was shown in [3] for the SISO case (i.e., ` = m = 1) that there always exists a series c ◦ d such that y = Fc [Fd [u]] = Fc◦d [u], but a multivariable analysis of this composition product is apparently not available in the literature, nor are any results about local convergence. So in Section 2 the composition product is first investigated independent of the interconnection problem. This material is a continuation of that which appeared in [7]. In Section 3, the three nonrecursive connections: the parallel, product and cascade connections are analyzed primarily by applying the results of Section 2. In Section 4 the feedback connection is considered in the case where the system inputs are being generated by an exosystem which is itself a Fliess operator. This leads to an implicit characterization of the feedback product, c@d, for formal power series.

1 Introduction Let I = {0, 1, . . . , m} denote an alphabet and I ∗ the set of all words over I. A formal power series in I is any mapping of the form I ∗ 7→ R` , and the set of all such mappings will be denoted by R`  I . For each c ∈ R`  I , one can formally associate a corresponding m-input, `-output operator Fc in the following manner. Let p ≥ 1 and a < b be given. For a measurable function u : [a, b] → Rm , define kukp = max{kui kp : 1 ≤ i ≤ m}, where kui kp is the usual Lp norm for a measurable real-valued function, ui , defined on [a, b]. Let Lm p [a, b] denote the set of all measurable functions defined on [a, b] having a finite k · kp norm and Bpm (R)[a, b] := {u ∈ Lm p [a, b] : kukp ≤ R}. With t0 , T ∈ R fixed and T > 0, define inductively for each η ∈ I ∗ the mapping Eη : Lm 1 [t0 , t0 + T ] → C[t0 , t0 + T ] by Eφ = 1, and Z t Eik ik−1 ···i1 [u](t, t0 ) = uik (τ )Eik−1 ···i1 [u](τ, t0 ) dτ, t0

where u0 (t) ≡ 1. The input-output operator corresponding to c is then X Fc [u](t) = (c, η) Eη [u](t, t0 ), η∈I ∗

which is referred to as a Fliess operator. In the classical literature where these operators first appeared [4, 5, 6, 9, 10, 11], it is normally assumed that there exists real numbers K > 0 and M ≥ 1 such that |(c, η)| ≤ KM |η| |η|!,

∀ η ∈ I ∗,

2 The Composition Product

(1)

The composition product of two series c, d ∈ Rm  X  over an alphabet X = {x0 , x1 , . . . , xm } is defined recursively in terms of the shuffle product. For

where |z| = max{|z1 | , |z2 | , . . . , |z` |} when z ∈ R` , and |η| denotes the number of symbols in η. This growth

0-7803-7516-5/02/$17.00 ©2002 IEEE

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(a) Parallel connection.

(b) Product connection.

(c) Cascade connection. (d) Feedback connection. Figure 1: Elementary system interconnections. any η ∈ X ∗ let  η : |η|xi = 0, ∀i 6= 0 η◦d = 0 k 0 xk+1 [d tt (η ◦ d)] : η = x0 xi η , k ≥ 0, i 6= 0, i 0

and supp(c) := {η ∈ X ∗ : (c, η) 6= 0} denotes the support of c. Hence, for any ξ ∈ X ∗ Id (ξ)

where |η|xi denotes the number of symbols in η equivalent to xi and di : ξ 7→ (d, ξ)i , the i-th component of (d, ξ). Consequently, if n

η = xn0 k xik x0 k−1 xik−1 · · · xn0 1 xi1 xn0 0 ,

|η|−|η|x0

=

(2)

n

x0 k−1

+1

[dik−1

tt

· · · x0n1 +1 [di1

tt

xn0 0 ] · · ·]].

supp(c) ⊆ {η ∈ X ∗ : η = xn0 1 xi xn0 0 , i ∈ {1, 2, . . . , m}, n1 , n0 ≥ 0}. The set R  X  forms a metric space under the ultrametric

nj + ord(dij )

j=1

dist

|η|−|η|x0

= |η| +

X

For c, d ∈ Rm  X  the composition product is defined as X c◦d= (c, η) η ◦ d.

but in general c ◦ (d + e) 6= c ◦ d + c ◦ e. A special exception are linear series. A series c ∈ R`  X  is called linear if

Proof: Given an arbitrary η ∈ X ∗ expressed in the form (2), it follows directly that k X

ord(dij ),

j=1

where the order of c is defined as  inf{|η| : η ∈ supp(c)} ord(c) = ∞

ord(dij ) ≤ |ξ|}.

It is easily verified for any series c, d, e ∈ Rm  X  that (c + d) ◦ e = c ◦ e + d ◦ e,

Theorem 2.1 Given a fixed d ∈ Rm  X , the family of series {η ◦ d : η ∈ X ∗ } is locally finite, and therefore summable.

= n0 + k +

X

η∈X ∗

The next theorem insures that the composition product of two series is well defined.

ord(η ◦ d)

{η ∈ X : |η| +

Clearly this latter set is finite, and thus Id (ξ) is finite for all ξ ∈ X ∗ . This fact implies summability [1].

η◦d= tt

∗

j=1

where ij 6= 0 for j = 1, . . . , k, then it follows that

x0nk +1 [dik

:= {η ∈ X ∗ : (η ◦ d, ξ) 6= 0} ⊂ {η ∈ X ∗ : ord(η ◦ d) ≤ |ξ|}

: :

R  X  × R  X  7→ R+ ∪ {0} (c, d) 7→ σ ord(c−d) ,

where σ ∈ (0, 1) is arbitrary. The following theorem states that the composition product on R  X  × R  X  is at least continuous in its first argument. The result extends naturally to vector-valued series in a componentwise fashion.

: c 6= 0 : c = 0,

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Theorem 2.2 [7] Let {ci }i≥1 be a sequence in R  X  with limi→∞ ci = c. Then limi→∞ (ci ◦ d) = c ◦ d.

=

The metric space (R  X , dist) is known to be complete [1]. Given a fixed c ∈ R  X , consider the mapping R  X  7→ R  X  : d 7→ c ◦ d.

2.

X

|ν| X

X

i=0

ξ∈X i |ν|−i ¯ ξ∈X

¯ ν) ≤ (ξ tt ξ,

ξ∈X i |ν|−i ¯ ξ∈X

¯ ν) ¯ tt ξ, (c, ξ)(d, ξ)(ξ

  |ν| , 0 ≤ i ≤ |ν|, i

where X i := {ξ ∈ X ∗ : |ξ| = i}. Theorem 2.3 [7] For any c ∈ R  X  with X = {x0 , x1 }, the mapping d 7→ c ◦ d is a contraction on R  X .

Now for any η ∈ X ∗ with X = {x0 , x1 } one can uniquely define a set of integers nj ≥ 0 and a set of subwords ηj ∈ X ∗ , j ≥ 0, by the following iteration

It is shown in [3] by counter example that the composition product is not a rational operation. That is, the composition of two rational series does not in general produce a rational series. But in [2], it is shown in the SISO case that a special class of rational series produce rational series when the composition product is applied. The following definition is the multivariable extention of the defining property of this class, and the corresponding rationality proof is not significantly different.

n

ηj+1 = x0 j+1 x1 ηj , η0 = xn0 0 ,

(3)

so that η = ηk where k = |η|x1 (cf. equation (2)). Using the set of subwords {η0 , η1 , . . . , ηk } define a family of corresponding functions:

Definition 2.1 A series c ∈ R  X  is limited relative to xi if there exists an integer Ni ≥ 0 such that sup |η|xi ≤ Ni .

Sη0 (n)

=

Sη1 (n)

=

Sηj (n)

=

1 , n≥0 |η0 |! 1 Sη (n), n ≥ |η1 | (n)n1 +1 0 n−|ηj |

1 (n)nj +1

X

Sηj−1 (n − (nj + 1) − i),

i=0

n ≥ |ηj |, 2 ≤ j ≤ k,

η∈supp(c)

where (n)i = n(n − 1) · · · (n − i + 1) denotes the falling factorial. Using these functions the following theorem provides a bound for |(ηj ◦ d, ν)| using only the coefficient bounds for d, the triangle inequality, and the inequality in Lemma 2.5. It therefore produces a least upperbound on the growth of the coefficients of ηj ◦ d in the general case.

If c is limited relative to xi for every i = 1, 2, . . . , m then P c is input-limited.` In such cases, let Nc := A series c ∈ R  X  is input-limited if i Ni . each component series, cj , is input-limited for j = 1, 2, . . . , `. In this case, Nc := maxj Ncj . The property of local convergence is next considered in this setting.

Theorem 2.6 For X = {x0 , x1 } let d ∈ R  X  be locally convergent with growth constants Kd and Md . For any η ∈ X ∗ with subwords {η0 , η1 , . . . , ηk } as defined in equation (3) it follows that

Theorem 2.4 [7] Suppose c, d ∈ Rm  X  are locally convergent series with growth constants Kc , Mc and Kd , Md , respectively. If c is input-limited with Nc ≥ 1 then c ◦ d is locally convergent with

−|ηj |

|(ηj ◦ d, ν)| ≤ Kdj Md

|ν|

Md |ν|! Sηj (|ν|), ∀ν ∈ X ∗ ,

where j = 0, 1, . . . , k. (Since (ηj ◦ d, ν) = 0 for |ν| < |ηj |, the value of Sηj (|ν|) when |ν| < |ηj | can assigned to be any nonnegative value.)

|(c ◦ d, ν)| < Kc KdNc (Nc + 1) (M (Nc + 1))|ν| |ν|!, for all ν ∈ X ∗ and where M = max{Mc , Md }. It was demonstrated by example in [7] that inputlimited is only sufficient for local convergence. But removing this assumption introduces significant complexity into the analysis. This issue is considered further here. The main tools employed are the basic properties of shuffle product given below.

Proof: The proof is by induction. The case j = 0 is trivial. When j = 1 observe that employing the leftshift operator [12], denoted by (·)−n , gives |(η1 ◦ d, ν)| = (x0n1 +1 (d tt xn0 0 ), ν) −(n1 +1) n0 (ν)) = (d tt x0 , x0 {z } | θ X (d, ξ)(ξ tt xn0 0 , θ) = ξ∈X |θ|−n0

Theorem 2.5 [12] For arbitrary c, d ∈ R  X  and any ν ∈ X ∗ : X ¯ ν) ¯ tt ξ, 1. (c tt d, ν) = (c, ξ)(d, ξ)(ξ ∗ ¯ ξ,ξ∈X

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X

≤

|ξ|

(Kd Md |ξ|!) (ξ tt xn0 0 , θ)

≤

ξ∈X |θ|−n0

  |θ| |θ|−n0 (|θ| − n0 )! ≤ Kd Md n0 −|η1 |

= Kd Md

(Kdj M −|ηj | M |ν| |ν|! Sηj (|ν|)) ≤ Kc (Kd M )|ν| |ν|! ψ(|ν|).

|ν|

Md Sη1 (|ν|) |ν|!.

In order for c ◦ d to be locally convergent in general, it is necessary that ψ(n) ≤ βαn for some α, β ∈ R. At present this is an open problem. It appears that some form of combinatoric analysis is required. When c and d are both linear series, it can be shown directly that |(c ◦ d, ν)| < Kc Kd M |ν| |ν|!, ∀ν ∈ X ∗ . 3 The Nonrecursive Connections

¯ = 0 for |ξ| ¯ < |ηj | it follows that Since (ηj ◦ d, ξ) |(ηj+1 ◦ d, ν)|

≤

X

X

i=0

ξ∈X i |θ|−i ¯ ξ∈X

|ξ| (Kd Md |ξ|!)

In this section the generating series are produced for the three nonrecursive interconnections and their local convergence is characterized.

·

  ¯ −|η | |ξ| ¯ θ) ¯ Sη (|ξ|) ¯ (ξ tt ξ, Kdj Md j Md |ξ|! j −|ηj+1 |

≤ Kdj+1 Md |θ|−|ηj |

X i=0

Theorem 3.1 Suppose c, d ∈ R`  I  are locally convergent power series. Then each nonrecursive interconnected input-output system shown in Figure 1 (a)(c) has a Fliess operator representation generated by a locally convergent series as indicated:

|ν|

Md ·

  |θ| i! (|θ| − i)! Sηj (|θ| − i) i −|ηj+1 |

= Kdj+1 Md

|ν|

Md |ν|!

1 · (|ν|)nj+1 +1

1. Fc + Fd = Fc+d 2. Fc · Fd = Fc tt d

|θ|−|ηj |

X

Sηj (|ν| − (nj+1 + 1) − i)

3. Fc ◦ Fd = Fc◦d , where ` = m, and c is inputlimited.

i=0 −|ηj+1 |

= Kdj+1 Md

(Kc M |ηj | |ηj |!) ·

i,j=0 ηj ∈X i

Now assume that the result holds up to some fixed j ≥ 1. Then in a similar fashion −(nj +1) |(ηj+1 ◦ d, ν)| = (d tt (ηj ◦ d), x0 (ν)) {z } | θ |θ| X X ¯ ν) . ¯ tt ξ, = (d, ξ)(ηj ◦ d, ξ)(ξ i=0 ξ∈X i |θ|−i ¯ ξ∈X |θ|−|ηj |

|ν| X X

|ν|

Md |ν|! Sηj+1 (|ν|). Proof:

Hence, the claim is true for all j ≥ 0.

1. Observe that The main application of this result is below. Fc [u](t) + Fd [u](t)

[(c, η) + (d, η)] Eη [u](t, t0 )

= Fc+d [u](t). Since c and d are locally convergent, define M = max{Mc , Md }. Then for any η ∈ I ∗ it follows that

|(c ◦ d, ν)| ≤ Kc (Kd M )|ν| |ν|! ψ(|ν|), ∀ν ∈ X ∗ , where

|(c + d, η)| ψ(n) :=

X η∈I ∗

Theorem 2.7 For X = {x0 , x1 } suppose c, d ∈ R  X  are locally convergent series with growth constants Kc , Mc and Kd , Md , respectively. If Kd ≥ 1 then

n X X

=

= |(c, η) + (d, η)| ≤ (Kc + Kd )M |η| |η|!

|ηj |! Sηj (n), n ≥ 0

or c + d is locally convergent.

i,j=0 ηj ∈X i

and M = max{Mc , Md }.

2. In light of the componentwise definition of the shuffle product, it can be assumed here without loss of generality that ` = 1. Thus,

Proof: Since η ∈ Id (ν) only if |η| ≤ |ν|, it follows that |ν| X X |(c ◦ d, ν)| = (c, ηj )(ηj ◦ d, ν) i,j=0 ηj ∈X i

Fc [u](t)Fd [u](t) X X = (c, η)Eη [u](t, t0 ) (c, ξ)Eξ [u](t, t0 ) η∈I ∗

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ξ∈I ∗

X

=

(c, η)(d, ξ) Eη [u](t, t0 )Eξ [u](t, t0 )

=

η,ξ∈I ∗

X

=

X

(c ◦ d, ν)Eν (t, t0 )

ν∈I ∗

= Fc◦d [u](t).

(c, η)(d, ξ) Eη tt ξ [u](t, t0 )

η,ξ∈I ∗

Local convergence of c ◦ d under the stated conditions follows from Theorem 2.4.

= Fc tt d [u](t). Applying Lemma 2.2.7 from [12] and using the fact that n + 1 ≤ 2n , n ≥ 0, gives

c being input-limited is only a sufficient condition for the composition product to produce a locally convergent series. If in Theorem 2.4 it is instead assumed that both c and d have finite Lie rank, then the mappings Fc and Fd each have a finite dimensional analytic state space realization, and therefore so does the mapping Fc ◦ Fd . The classical literature then provides that the generating series c ◦ d must be locally convergent [11]. An example of this situation is given below.

|(c tt d, η)| ≤ Kc Kd (2M )|η| |η|!. Thus, c tt d is locally convergent. 3. For any η ∈ I ∗ and d ∈ Rm  I  the corresponding Fliess operators are Fη [u](t)

= Eη [u](t, t0 ) X = (d, ξ)Eξ [u](t, t0 ).

Fd [u](t)

Example 3.1 Consider the state space system

ξ∈I ∗

z˙ y

Therefore, (Fη ◦ Fd [u])(t) = Eη [Fd [u]] (t, t0 ).

It is easily verified that y = Fc [u] where the only nonzero coefficients are (c, 1 · · 1}) = k!, k ≥ 0. So c | ·{z

If |η| = |η|x0 then (Fη ◦ Fd [u])(t)

= z 2 u, z(0) = 1 = z.

= Eη [u](t, t0 ) = Fη [u](t) = Fη◦d [u](t).

k

is not input-limited. But the mapping Fc◦c clearly has the analytic state space realization,

If, on the other hand, η = 0 · · 0} iη 0 with i 6= 0 then | ·{z k times

(Fη ◦ Fd [u])(t) = E 0···0 iη0 [Fd [u]](t, t0 ) |{z} k times Z t Z τ2 = ··· Fdi [u](τ )Eη0 [Fd [u]](τ, t0 ) · t t | 0 {z 0 }

z˙1

= z12 u, z1 (0) = 1

z˙2 y

= z22 z1 , z2 (0) = 1 = z2 ,

and therefore the generating series c ◦ c must be locally convergent. 4 The Feedback Connection

k+1 times

=

Z |

t t0

··· {z

τ2

Fdi tt (η0 ◦d) [u](τ, t0 ) dτ1 . . . dτk+1 t0

k+1 times

= F

0···0

|{z}

The general goal of this section is to determine when there exists a y which satisfies the feedback equation corresponding to Figure 1(d). In particular, when does there exist a generating series e so that y = Fe [u] and

dτ1 . . . dτk+1

Z

}

[di

Fe [u] = Fc [u + Fd◦e [u]]. tt

(η 0 ◦d)] [u](t)

The feedback product is then defined by c@d = e. To make the analysis simpler, it is assumed through out that the inputs u are supplied by an exosystem which is itself a Fliess operator as shown in Figure 2. That

k+1 times

= Fη◦d [u](t). Thus, (Fc ◦ Fd [u])(t) X = (c, η)Eη [Fd [u]](t, t0 ) η∈I ∗

=

X

(c, η)Fη◦d [u](t)

η∈I ∗

=

X η∈I ∗

=

X ν∈I ∗

(c, η)  

"

X ν∈I ∗

X η∈I ∗

(η ◦ d, ν)Eν [u](t, t0 )

# Figure 2: A feedback configuration with a Fliess operator exosystem providing the inputs.



(c, η)(η ◦ d, ν) Eν [u](t, t0 )

is, u = Fb (v) for some locally convergent series b. In

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Example 4.1 Consider a generalized series δ with the defining property that δ is the identity element for the composition product, i.e., c ◦ δ = δ ◦ c = c for any c ∈ R  X . Then (mapping the symbols xi 7→ i) Fδ [u] = u for any u, and a unity feedback system has the generating series c@δ. Setting b = 0 in Figure 2 (or effectively setting u ≡ 0), a self-excited feedback loop is described by Fe˜[v], where e˜ = c ◦ e˜ and e˜ = e ◦ 0. In this case e˜ = limk→∞ c◦k and c@d|d=δ = e = e˜ = limk→∞ c◦k . From Theorem 4.1, c@δ is well defined in general. Analogous to the situation with the composition product, if c has finite Lie rank then c@δ will always be locally convergent. For example, when P c = 1 + x1 it is easy verified that c@δ = k≥0 xk0 so 2 that Fc@δ [0](t) = et for Pt ≥ 0. Whenkc = 1 + 2x1 + 2x1 it follows that c@δ = k≥0 (k + 1)! x0 and Fc@δ [0](t) = 1/(1 − t)2 for 0 ≤ t < 1.

this setting, a sufficient condition is given under which a unique solution y of the feedback equation is known to exist, and y is characterized as the output of a new Fliess operator when a certain series factorization property is available. This leads to an implicit characterization of the feedback product c@d. Theorem 4.1 Let b, c, d ∈ R  I  with I = {0, 1}. Then: 1. The mapping S

: R  I  7→ R  I  : e˜i → 7 e˜i+1 = c ◦ (b + d ◦ e˜i )

has a unique fixed point e˜. 2. If b, c, d and e˜ are locally convergent then the feedback equation has the unique solution y = Fe˜[v] for any admissible v.

References [1] J. Berstel and C. Reutenauer, Les S´eries Rationnelles et Leurs Langages, Springer-Verlag, Paris (1984). [2] A. Ferfera, Combinatoire du Mono¨ide Libre Appliqu´ee ` a la Composition et aux Variations de Certaines Fonctionnelles Issues de la Th´eorie des Syst`emes, Th`ese de 3`eme Cycle, Universit´e de Bordeaux I (1979). [3] , Combinatoire du mono¨ide libre et composition de certains syst`emes non lin´eaires, Analyse des Syst`ems, Soc. Math. France, 75–76 (1980) 87–93. [4] M. Fliess, Fonctionnelles causales non lin´eaires et ind´etermin´ees non commutatives, Bull. Soc. Math. France 109 (1981) 3–40. [5] , R´ealisation locale des syst`emes non lin´eaires, alg`ebres de Lie filtr´ees transitives et s´eries g´en´eratrices non commutatives, Invent. Math. 71 (1983) 521–537. [6] M. Fliess, M. Lamnabhi, and F. LamnabhiLagarrigue, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits Syst. 30 (1983) 554–570. [7] W. S. Gray and Y. Li, Interconnected systems of Fliess operators, Proc. 15th International Symposium on Mathematical Theory of Networks and Systems, South Bend, Indiana, U.S.A., 2002. [8] W. S. Gray and Y. Wang, Fliess operators on Lp spaces: convergence and continuity, Systems & Control Letters 46 (2002) 67-74. [9] B. Jakubczyk, Existence and uniqueness of realizations of nonlinear systems, SIAM J. Contr. Optimiz. 18 (1980) 445-471. [10] , Local realization of nonlinear causal operators, SIAM J. Contr. Optimiz. 24 (1986) 230-242. [11] H. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems, Math. System Theory 12 (1977) 263-284. [12] Y. Wang, Algebraic Differential Equations and Nonlinear Control Systems, Doctoral Dissertation, Rutgers University (1990).

3. If e˜ = e ◦ b for some locally convergent series e then c@d = e. Proof: 1. The mapping S is a contraction since via Theorem 2.3: dist(S(˜ ei ), S(˜ ej ))

< dist(b + d ◦ e˜i , b + d ◦ e˜j ) = dist(d ◦ e˜i , d ◦ e˜j ) < dist(˜ ei , e˜j ).

Therefore, the mapping S has a unique fixed point e˜, that is, e˜ = c ◦ (b + d ◦ e˜). 2. From the stated assumptions concerning b, c, d and e˜ it follows that Fe˜[v]

= Fc◦(b+d◦˜e) [v] = Fc [Fb [v] + Fd [Fe˜[v]]]

for any admissible v. Therefore the feedback equation has the unique solution y = Fe˜[v]. 3. Since e is locally convergent y = Fe˜[v] = Fe [Fb [v]] = Fe [u], thus c@d = e. This result suggests several open problems. Are there conditions on b, c, and d alone which will insure that e˜ above is locally convergent ? When does there exist a factorization of the form e˜ = e ◦ b, where e is locally convergent ? Can the theorem be generalized to the case where the inputs are simply from an Lp space and not filtered through a Fliess operator a priori ?

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