A Unified Approach to Generating Series for Nonlinear ... - IEEE Xplore
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
FrC01.4
A Unified Approach to Generating Series for Nonlinear Cascade Systems W. Steven Gray
Abstract— The goal of the paper is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input-output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of Fliess operators or memoryless maps provided that the composite system is well defined and resides in one of these two classes.
I. I NTRODUCTION In many applications, input-output systems are interconnected to form more complex systems. Describing the nature of the composite system and providing some explicit parametrization for it are generally nontrivial problems when the subsystems are nonlinear. In this paper, cascade connections involving two classes of analytic nonlinear inputoutput systems are considered. The first class of systems are those that can be represented by memoryless maps of the form f : Rm → Rℓ , which are real analytic on some domain of interest. Thus, each f can be written in terms of a Taylor series whose coefficients are associated with a formal power series c over a commutative alphabet. Such a map is denoted by fc , and c will be called its generating series. The second class of systems under consideration are those that can be represented as Fliess operators [6]–[8]. This class includes any input-output system that can be written as a Volterra operator having analytic kernels (including the linear, time-invariant case). Such an operator Fd can be expressed in terms of a generating series d, which is a formal power series over a noncommutative alphabet. In general, Fliess operators are not memoryless except for the trivial case where their generating series corresponds to a constant polynomial d = y0 ∈ Rℓ , in which case Fd [u] = y0 for every input u. The specific goal of the paper is to describe in a unified fashion the generating series for the cascade connection of any two systems from one or both of these system classes as shown in Fig. 1. It will be shown that a single general definition of a composition product involving formal power series can be formulated to produce the generating series for any possible cascade connection, as long as the composite system is well defined and resides in one of these two classes. Certain elements of this problem are well understood by other means. For example, the composition of analytic maps in one variable is introduced in most classic books on power series, e.g., [17]. More advanced versions of the problem are addressed in [2], [9]. The interconnection of memoryless maps (not necessarily analytic) with linear time-invariant dynamical systems, so called Wiener and Hammerstein systems, are analyzed in [13], [14], [19], [20]. The author is affiliated with the Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, USA.
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
u
fd
v
fc
y
(a) cascade of memoryless maps
u
Fd
v
fc
y
Fc
y
(b) Wiener-Fliess system
u
fd
v
(c) Hammerstein-Fliess system
u
Fd
v
Fc
y
(d) cascade of Fliess operators Fig. 1.
Cascade systems under consideration
The composition of Fliess operators was considered in [3], [4], [10], [18]. The primary interest here, however, is in showing that all of these interconnections have a common algebraic/combinatoric source. From this vantage point it is easier to see how certain cascade connections fit easily into existing theoretical frameworks, while others are more problematic. For example, in the case of a Wiener system, replacing the linear dynamical system with an arbitrary Fliess operator to produce a Wiener-Fliess system does not require any additional mathematical machinery. It can be viewed as a generalization of existing results for the case when fc is a polynomial [6], [21], [22]. On the other hand, a Hammerstein-Fliess system, as shown in Fig. 1(c), can easily fall outside the domain of the theory presented here without significant restrictions. The paper is organized as follows. In the first section some preliminaries are presented to properly frame the problems under consideration. In Section III the general notion of a composition product is introduced along with some specific examples and conditions under which it is well defined. In the subsequent section, the concept is applied to the
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FrC01.4 four cascade connections shown in Fig. 1. The final section summarizes the main contributions of the paper. II. P RELIMINARIES A finite nonempty set of noncommuting symbols X = {x0 , x1 , . . . , xm } is called an alphabet. Each element of X is called a letter, and any finite string of letters from X, η = xi1 · · · xik , is called a word over X. The length of η, |η|, is the number of letters in η, while |η|xi is the number of times the letter xi appears in η. The set of all words with length k will be denoted by X k . The set of all words including the empty word, ∅, will be denoted by X ∗ . It forms a monoid under catenation. A language is any subset of X ∗ . Any mapping c : X ∗ → Rℓ is called a formal power series. The value of c at η ∈ X ∗ is written asP(c, η). Typically, c is represented as the formal sum c = η∈X ∗ (c, η)η. The coefficient (c, ∅) is referred to as the constant term, and c is called proper when this coefficient is zero. The support of c is the language supp(c) = {η ∈ X ∗ : (c, η) 6= 0}. The order of c is defined as min{|η| : η ∈ supp(c)} ord(c) = ∞
: :
c 6= 0 c = 0.
η∈X
where X
η=ξν
(c, ξ)(d, ν)
t0
where xi ∈ X, η¯ ∈ X ∗ , and u0 = 1. The input-output operator corresponding to c is the Fliess operator X (c, η) Eη [u](t, t0 ). Fc [u](t) = η∈X ∗
The collection of all formal power series over X is denoted by Rℓ hhXii. It forms an associative R-algebra under the (Cauchy) catenation product and a commutative R-algebra under the shuffle product, designated by ⊔⊔ . Rℓ [[X]] denotes the set of all formal power series when the letters of X commute. In this situation it is more natural to impose the ordering x0 < x1 < · · · < xm on X and then assume that the support of c is only on ordered words, specifically, words of the form η = xi1 · · · xik , where ij ≤ ij+1 for all j = 1, 2, . . . , k−1. For c, d ∈ R [[X]] an alternative definition of the catenation product is preferred, namely, X η (cd, η) , cd = η! ∗ (cd, η) =
|(c, η˜)| ≤ KM |˜η| |˜ η |!, where |z| := max{|zi | : 1 ≤ i ≤ ℓ} ˜ ∗. when z ∈ Rℓ , for fixed constants K, M > 0 and all η˜ ∈ X Such a power series is called locally convergent, and the ˜ If instead set of all such series is denoted by RℓLC [[X]]. c ∈ Rℓ hhXii, one can formally associate a causal m-input, ℓ-output operator, Fc , in the following manner. Let p ≥ 1 and t0 < t1 be given. For a measurable function u : [t0 , t1 ] → 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 [t0 , t1 ]. Let Lm p [t0 , t1 ] denote the set of all measurable functions defined on [t0 , t1 ] having a finite k · kp norm and Bpm (R)[t0 , t1 ] := {u ∈ Lm p [t0 , t1 ] : kukp ≤ R}. Define recursively for each η ∈ X ∗ the map Eη : Lm 1 [t0 , t1 ] → C[t0 , t1 ] by setting E∅ [u] = 1 and letting Z t ui (τ )Eη¯[u](τ, t0 ) dτ, Exi η¯[u](t, t0 ) =
η! ξ!ν!
and η! := |η|x0 ! |η|x1 ! · · · |η|xm !. For a single letter alphabet, this yields what is commonly called binomial convolution i X i (c, xj )(d, xi−j ). (cd, xi ) = j j=0 So the catenation product in the general commutative case will be referred to as the multinomial catenation product. Let X = {x0 , x1 , . . . , xm } be a noncommutative alphabet ˜ = {˜ and X x1 , x ˜2 . . . , x ˜m } a second alphabet, distinct from ˜ one can associate X and commutative. For each c ∈ Rℓ [[X]] a formal function X z η˜ (c, η˜) , fc (z1 , z2 , . . . , zm ) = η˜! ∗ ˜ η˜∈X
˜ik . If fc is analytic ˜i1 · · · x where z η˜ := zi1 zi2 · · · zik if η˜ = x then its coefficients must satisfy a Cauchy growth condition
If |(c, η)| ≤ KM |η| |η|! for all η ∈ X ∗ , then Fc constitutes a well defined operator from Bpm (R)[t0 , t0 + T ] into Bqℓ (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 [11]. The set of all locally convergent series in this setting is denoted by RℓLC hhXii. III. C OMPOSITION P RODUCTS In this section a class of formal power series products called composition products is considered. Each specific product introduced comes from essentially the same con˜ = struction process. Let X = {x0 , x1 , . . . , xm } and X {˜ x0 , x ˜1 , . . . , x ˜m ˜ } be two arbitrary alphabets and consider the ˜ ˜ formal power series c ∈ Rℓ hhXii and d ∈ Rℓ hhXii. Assume that RhhXii is endowed with an associative R-algebra having product 2 and multiplicative identity element 1. Definition 1: A composition product of c and d is any binary operation ˜ ˜ Rℓ hhXii × Rℓ hhXii
(c, d)
˜
→ Rℓ hhXii X (c, η˜) η˜ ◦ d, 7→ c ◦ d = ˜∗ η˜∈X
where η˜ ◦ d is the unique extension of ˜∗
x ˜i ◦ d = ρi (d), i = 0, 1, . . . , m ˜
to X given by (˜ xik x ˜ik−1 · · · x ˜i1 ) ◦ d = ρik (d)2ρik−1 (d)2 · · · 2ρi1 (d) with ρi : Rℓ hhXii → RhhXii such that ρi (∅) = 1, i = 0, 1, . . . , m ˜ and ∅ ◦ d = 1. This type of construction arises in many forms when systems are interconnected to produce new systems. The particular form of the composition product depends precisely on the nature of the systems being connected. Whenever possible, the same notation will be used for any given composition product, and the specific definition will be evident from the context. It is easily verified that any composition product
8003
FrC01.4 is in general R-linear in its left argument but not its right argument. It should also be mentioned that a composition product can be viewed in terms of transductions [5], [16]. ˜ X)-transduction is any formal power series t ˜ ∈ An (X, XX ˜ [RhhXii]hhXii. For a given d ∈ Rℓ hhXii one can define the ˜ X)-transduction (X, X X (˜ η ◦ d) η˜. (tX X˜ , η˜) η˜ = tX X˜ = ˜∗ η˜∈X
˜∗ η˜∈X
X
X
˜ = X, Example 3: Suppose X = {x0 , x1 , x2 , . . . , xm }, X ℓ m c ∈ R hhXii and d ∈ R hhXii. Define their composition as in (1)-(2) with d0 = 1. This type of composition product describes a Hammerstein-FliessPsystem, Fc ◦ fd = Fc◦d , m provided that fd (z1 , . . . , zm ) = j=0 (d, xj )zj with z0 = 1. Otherwise, as explained in the next section, Fc ◦ fd has no Fliess operator representation. Example 4: Using the same setup as in the previous example, define the family of mappings
In which case, c◦d
=
˜∗ η˜∈X
(c, η˜) η˜ ◦ d =
= (c, tX X˜ ).
Dxi
(c, η˜)(tX X˜ , η˜)
˜∗ η˜∈X
Before pursuing any technical issues, such as conditions under which a composition product is well defined, some specific examples are introduced. ˜ = Example 1: Suppose X = {x1 , x2 , . . . , xm }, X ℓ˜ ˜ m ˜ {˜ x1 , x ˜2 , . . . , x ˜m ˜ }, c ∈ R hhXii and d ∈ R hhXii. Let di denote the i-th component series of d, that is, (di , ξ) = (d, ξ)i for every ξ ∈ X ∗ . Consider the composition product defined by letting ρi (d) = di , i = 1, . . . , m, ˜ and (˜ xi1 x ˜i2 · · · x ˜ik ) ◦ d = di1 di2 · · · dik =: dη˜ {z } |
(1)
(xik xik−1 · · · xi1 ) ◦ d = Dxik Dxik−1 · · · Dxi1 (1) = Dη (1). {z } | η
In which case,
c◦d=
with d∅ = 1. Therefore,
X
(c, η˜) dη˜.
(2)
˜∗ η˜∈X
˜ commute, then the composition product If the letters in X is written in the exponential form X dη˜ (c, η˜) . c◦d= η˜! ∗ ˜ η˜∈X
If, in addition, the letters in the alphabet X commute, then the power dη˜ is taken to be the multinomial catenation power having coefficients X η! , η ∈ X ∗. (dη˜, η) = (di1 , η1 ) · · · (dik , ηk ) η ! · 1 · · ηk ! η ···η =η 1
k
It is shown in the next section that this type of composition product describes function composition, fc ◦ fd = fc◦d . ˜ = Example 2: Let X = {x0 , x1 , x2 , . . . , xm }, X ˜ and d ∈ Rn hhXii. If ρi (d) = {˜ x1 , x ˜2 , . . . , x ˜n }, c ∈ Rℓ [[X]] di , i = 1, . . . , n, and ˜ik ) ◦ d = di1 ˜i2 · · · x (˜ xi1 x | {z }
⊔⊔
di2
⊔⊔
···
⊔⊔
dik =: d ⊔⊔ η˜
η˜
with d ⊔⊔ ∅ = 1 then
c◦d=
X
(c, η) Dη (1).
η∈X ∗
This type of composition product describes the composition of two Fliess operators, Fc ◦ Fd = Fc◦d . A variation of this product, where instead Dxi
: RhhXii → RhhXii : e 7→ xi e + x0 (di ⊔⊔ e)
for i = 1, 2, . . . , m (the i = 0 case is unchanged), can be used to describe the feedback connection of two Fliess operators [10], [12]. The well definedness of a composition product is considered next. The following two theorems will cover all the examples described above. Theorem 1: Consider a composition product defined by (ρ, 2), where 1) ord(ρi (d)) > ord(d), i = 0, 1, . . . , m, ˜ d ∈ Rℓ hhXii ′ ′ 2) ord(d2d ) = ord(d) + ord(d ), d, d′ ∈ RhhXii. ˜ ˜ and d ∈ Rℓ hhXii the composition c◦d For any c ∈ Rℓ hhXii ˜
is a well defined series in Rℓ hhXii. Proof: It suffices to show that the family of formal power series {˜ η ◦ d}η˜∈X˜ ∗ is locally finite, and hence, summable [1]. For a fixed d ∈ Rℓ hhXii defined the integers ri = ord(ρi (d))−ord(d) > 0, i = 0, 1, . . . , m, ˜ and r = mini ri > ˜ ∗: 0. Then given any nonempty word η˜ ∈ X ord(˜ η ◦ d)
d ⊔⊔ η˜ . (c, η˜) η˜! ∗
X
˜ η˜∈X
RhhXii → RhhXii e 7→ x0 (di ⊔⊔ e),
i = 0, 1, . . . , m with d0 = 1. Assume D∅ is the identity map on RhhXii. Such maps can be composed in the obvious way so that Dxi xj := Dxi Dxj provides an R-algebra which is isomorphic to a subalgebra of the usual R-algebra on RhhXii under the catenation product. In this setting, consider a formal power series composition product defined by letting ρi (d) = Dxi , i = 0, 1, . . . , m, and
η˜
c◦d=
: :
=
˜i1 ◦d) ˜ik−1 · · · x ord(˜ xik x | {z } η˜
=
This type of composition product describes a Wiener-Fliess system, fc ◦ Fd = Fc◦d .
8004
=
ord(ρik (d)2ρik−1 (d)2 · · · 2ρi1 (d))
k X j=1
ord(ρij (d))
FrC01.4
=
k X
ord(d) + rij
j=1
≥
|˜ η | (ord(d) + r).
Since ord(d) + r > 0, ord(˜ η ◦ d) increases at least proportionally as the length of η˜ increases. So for a fixed ξ ∈ X ∗ the set ˜ ∗ : (˜ Id (ξ) := {˜ η∈X η ◦ d, ξ) 6= 0} must be finite since (˜ η ◦ d, ξ) = 0 when the length of η˜ is such that
A. Cascades of Memoryless Maps In the context of Example 1, consider the following theorem. Theorem 3: Suppose U and V are two neighborhoods of ˜ ˜ ˜ the origin. Let fc : U ⊂ Rm → Rℓ and fd : V ⊂ Rm → Rm be real analytic functions with fd (V ) ⊂ U and having Taylor series about z˜ = 0 and z = 0: X z˜η˜ (c, η˜) fc (˜ z1 , z˜2 , . . . , z˜m ˜) = η˜! ∗ ˜ η˜∈X
fd (z1 , z2 , . . . , zm ) =
X
(d, η)
η∈X ∗
|˜ η | (ord(d) + r) > |ξ| . In which case, the family of series in question is locally finite. Theorem 2: Consider a composition product defined by (ρ, 2), where 1) ord(ρi (d)) ≥ ord(d), i = 0, 1, . . . , m, ˜ d ∈ Rℓ hhXii ′ ′ 2) ord(d2d ) = ord(d) + ord(d ), d, d′ ∈ RhhXii. ˜ ˜ and proper d ∈ Rℓ hhXii the For any c ∈ Rℓ hhXii
zη , η!
˜ ˜ ˜ and proper d ∈ Rm respectively, with c ∈ RℓLC [[X]] LC [[X]]. Then fc ◦ fd = fc◦d . Proof: The composite function fc ◦fd corresponds to setting z˜i = fd,i (z), i = 1, 2, . . . , m. ˜ Since d is proper, fd (0) = 0. Thus, by direct substitution
fc ◦ fd (z) =
˜
composition c ◦ d is a well defined series in Rℓ hhXii. Proof: Following the same logic as in the previous theorem, one can conclude here that
=
ord(˜ η ◦ d) ≥ |˜ η | ord(d).
=
Since d is proper, ord(d) > 0. In which case, Id (ξ) is finite since (˜ η ◦ d, ξ) = 0 when the length of η˜ is such that |˜ η | ord(d) > |ξ| . This proves the theorem.
=
Example 5: Suppose ρi (d) = di as in Examples 1-3. Clearly, ord(ρi (d)) = ord(di ) ≥ ord(d), so in these cases, Theorem 2 applies. Therefore, the respective composition products are well defined if d is proper. Example 6: Consider the composition product defined in Example 4. Observe that for any e ∈ RhhXii
=
ord(Dxi (e)) = ord(di ) + ord(e) + 1. So it makes sense to define ord(Dxi ) = ord(di )+1. It then follows that
=
ord(ρi (d)) = ord(Dxi ) = ord(di ) + 1 > ord(d) =
for i = 0, 1, . . . , m. In which case, Theorem 1 applies and c ◦ d is well defined everywhere on Rℓ hhXii × Rm hhXii.
z˜ (c, η˜) η˜! ˜∗ η˜∈X
X
η∈X ∗
IV. N ONLINEAR C ASCADE S YSTEMS
z˜=fd (z)
fd,i1 (z) · · · fd,ik (z) (c, η˜) η˜! ˜∗ η ˜∈X X 1 X z η1 (c, η˜) (di1 , η1 ) ··· η˜! η1 ! ˜∗ η1 ∈X ∗ η ˜∈X ηk X z (dik , ηk ) ηk ! ∗ ηk ∈X X X 1 (di1 , η1 ) · · · (dik , ηk ) · (c, η˜) η˜! ˜∗ η1 ,...,ηk ∈X ∗ η ˜∈X z η1 · · · z ηk η1 ! · · · ηk ! " X X 1 X (c, η˜) (di1 , η1 ) · · · (dik , ηk ) · η˜! ∗ η ···η =η ∗ ˜ η∈X 1 k η ˜∈X η z η! η1 ! · · · ηk ! η! X zη 1 X η˜ (d , η) (c, η˜) η˜! η! ˜∗ η∈X ∗ η ˜∈X X X 1 zη (c, η˜) (dη˜, η) η˜! η! ∗ ∗ X
η∈X
=
η˜
X
˜ η˜∈X
(c ◦ d, η)
zη η!
= fc◦d (z).
In this section the generating series for the four cascade connections shown in Figure 1 are computed directly and then related to the composition products in Examples 1-4. Local convergence of the resulting series is also considered.
In which case, fc ◦ fd = fc◦d as claimed. It is well known that the composition of two analytic functions is again analytic, therefore c ◦ d is a locally
8005
FrC01.4 convergent series. In the case where fc and fd are merely formal functions, c ◦ d is the generating series for the formal function fc ◦ fd . B. Wiener-Fliess Systems In the context of Example 2, consider the following theorem. Theorem 4: Given a Fliess operator Fd , d ∈ Rn hhXii and a formal function fc : Rn → Rℓ with generating series ˜ at z = (d, ∅), that is, c ∈ Rℓ [[X]] fc (z) =
X
(c, η˜)
˜∗ η˜∈X
(z − (d, ∅))η˜ , η˜!
X
(c, η˜)
˜∗ η˜∈X
˜ η˜∈X
Finally, using Stirling’s approximation, the asymptotic approximation 1 2ℓ ∼ √ 4ℓ , ℓ ≥ 1 ℓ−1 πℓ holds, and, in particular, 2ℓ ≤ 4ℓ , ℓ ≥ 1. ℓ−1
the cascade connection fc ◦ Fd has the generating series c◦d=
In addition, the following identities hold: ℓ X ℓ+k−1 2ℓ = , ℓ≥1 k−1 ℓ−1 k=1 X k! = nk , k ≥ 0. η ˜ ! k
(d − (d, ∅)) ⊔⊔ η˜ . η˜!
That is, fc ◦ Fd = Fc◦d . Proof: The proof follows from basic properties of the shuffle product. Define the proper series d˜ := d−(d, ∅), and observe that X (z − (d, ∅))η˜ (c, η˜) fc ◦ Fd [u] = η˜! z=Fd [u] ∗
Now the proof. The case where ν = ∅ is trivial. For any nonempty word ν ∈ X ∗ , the properness of d˜ and the assumption that nKd Mc ≥ 1 gives |(c ◦ d, ν)| X ˜ ⊔⊔ η˜, ν) ( d (c, η˜) = η˜! η˜∈X˜ ∗
=
X
(c, η˜)
˜∗ η˜∈X
=
≤
(Fd [u] − F(d,∅) [u])η˜ η˜!
X (c, η˜) (Fd˜[u])η˜ η ˜ ! ∗
˜ η˜∈X
=
X (c, η˜) F ˜ [u]Fd˜i [u] · · · Fd˜i [u] 1 k−1 η˜! dik ∗
X (c, η˜) F˜ η˜! dik ∗
˜ η˜∈X
=
⊔⊔
|ν|
Kc Mck k!
≤ Kc (4nKd Mc Md )|ν| |ν|!.
˜ η˜∈X
=
|ν| X X
Kdk Md (|ν| + k − 1)! η˜! (k − 1)! ˜k k=1 η˜∈X |ν| X |ν| + k − 1 |ν| Md |ν|! = Kc (nKd Mc )k k−1 k=1 2 |ν| (nKd Mc Md )|ν| |ν|! ≤ Kc |ν| − 1
˜ η˜∈X
This completes the proof.
d˜ik−1 ··· ⊔⊔ d˜i1 [u]
Example 7: Consider an analytic state space system
X (c, η˜) F ˜ ⊔⊔ η˜ [u] η˜! d ∗
z˙
= g0 (z) +
˜ η˜∈X
m X
gi (z) ui , z(t0 ) = z0
(3)
i=1
= Fc◦d [u].
y
= h(z).
(4)
Using Theorem 2, the properness of d˜ ensures that c ◦ d is a well defined series in Rℓ hhXii.
It is well known that (3) has the solution z = Fcz [u], where
Convergence properties are considered in the next theorem. ˜ and d ∈ Rn hhXii Theorem 5: Suppose c ∈ RℓLC [[X]] LC with growth constants Kc , Mc > 0 and Kd , Md > 0, respectively. Then c ◦ d ∈ RℓLC hhXii. Specifically, if nKd Mc ≥ 1 (which is no loss of generality) then
[6], [15]. Here the iterated Lie derivatives with respect to the vector fields gi are written as
|(c ◦ d, ν)| ≤ Kc (4nKd Mc Md )|ν| |ν|!, ∀ν ∈ X ∗ .
Proof: The proof uses several facts. First, a direct extension of a growth condition for shuffle products appearing in [21] to shuffle powers is ⊔⊔ k |ν| (|ν| + k − 1)! (d , ν) ≤ Kdk Md , ∀ν ∈ X ∗ , k ≥ 1. (k − 1)!
(cz , η) = Lgη I(z0 ), ∀η ∈ X ∗
Lgη = Lgi1 · · · Lgik , η = xik · · · xi1 ∈ X ∗ with L∅ := I being the identity map, i.e., I(z) = z. The corresponding Wiener-Fliess system is then y = h(z) = h(Fcz [u]) = Fch ◦cz [u], where ch denotes the generating series for h. From Fliess’s fundamental formula (ch ◦ cz , η)
8006
= =
Lgη (h ◦ I)(z0 ) Lgη h(z0 )
FrC01.4 =:
(c, η)
⊔⊔
for all η ∈ X ∗ . Therefore, y = Fc [u], that is, the state space system (g, h, z0 ) realizes the input-output system Fc . C. Hammerstein-Fliess Systems First an example is presented to justify the restriction introduce on the function fd in Example 3. Example 8: A correlator corresponds to a Hammerstein system where Fc = Ex1 and fd (u) = u2 . By direct substitution Z t u2 (τ ) dτ Fc ◦ fd [u](t) = 0 Z t Z τ2 = δ(τ2 − τ1 )u(τ2 )u(τ1 ) dτ1 dτ2 . 0
0
When viewed as a second order Volterra operator, it is clear that this system does not have a Fliess operator realization since the kernel function w2 (τ2 , τ1 ) = δ(τ2 − τ1 ) is not analytic.
Now consider the following theorem in the context of Example 3. Theorem 6: Given a Fliess operator Fc , c ∈ Rm hhXii, m and a function fd : Rm → RP with generating series d ∈ m R hhXii of the form d = xj ∈X (d, xj )xj , the cascade connection Fc ◦ fd has the generating series X (c, η)dη , c◦d= η∈X ∗
where d0 = 1. That is, Fc ◦ fd = Fc◦d . Proof: With u0 = 1 observe that for any xi ∈ X Z t X (di , xj ) uj (τ ) dτ Exi [fd (u)](t, t0 ) = t0 x ∈X j
X
=
(di , xj ) Exj [u](t, t0 )
xj ∈X
= Fdxi [u](t). Inductively, it is easy to show that Eη [fd (u)] = Fdη [u] for any η ∈ X ∗ . Therefore, X Fc ◦ fd [u] = (c, η) Eη [fd (u)] η∈X ∗
=
X
(c, η) Fdη [u]
η∈X ∗
= Fc◦d [u]. The properness of d ensures that c ◦ d is well defined. Since d is a polynomial, it is straightforward to show that if c is locally convergent then so is c ◦ d. D. Cascades of Fliess Operators In the context of Example 4, the composition product of a word η ∈ X ∗ , written generically in the form n
η = xn0 k xik x0 k−1 xik−1 · · · xn0 1 xi1 xn0 0 ,
where ij 6= 0 for j = 1, . . . , k, and a series d ∈ Rm hhXii was defined in [3], [4] as η◦d
= xn0 k +1 [dik
n
⊔⊔
x0 k−1
+1
[dik−1
⊔⊔
···
x0n1 +1 [di1
⊔⊔
xn0 0 ] · · · ]].
It is easy to verify that this composition is identical to Dη (1), and thus, this original notion of series composition is equivalent to the one defined here for Fliess operator composition. The following theorem is included for completeness. Theorem 7: [3], [10] For any c ∈ Rℓ hhXii and d ∈ m R hhXii, the identity Fc ◦Fd = Fc◦d is satisfied. In addition, if c and d are locally convergent then c ◦ d is also locally convergent. V. C ONCLUSIONS A unified treatment of generating series for cascade connections of analytic input-output systems was presented. In particular, it was shown that a single general definition of a composition product can be formulated so as to describe the generating series for any possible cascade connection, as long the composite system resides in one of the two systems classes of interest: memoryless systems and inputoutput systems representable as Fliess operators. R EFERENCES [1] J. Berstel and C. Reutenauer, Rational Series and Their Languages, Springer-Verlag, Berlin, 1988. [2] J. Chaumat and A.-M. Chollet, On composite formal power series, Trans. AMS 353 (2001) 1691-1703. [3] A. Ferfera, Combinatoire du Mono¨ıde Libre Appliqu´ee a` la Composition et aux Variations de Certaines Fonctionnelles Issues de la Th´eorie des Syst`emes, Doctoral Dissertation, University of Bordeaux I, 1979. [4] , Combinatoire du mono¨ıde libre et composition de certains syst`emes non lin´eaires, Ast´erisque 75-76 (1980) 87-93. [5] M. Fliess, Transductions de s´eries formelles, Discrete Math. 10 (1974) 57-74. [6] , Fonctionnelles causales non lin´eaires et ind´etermin´ees non commutatives, Bull. Soc. Math. France 109 (1981) 3-40. , R´ealisation locale des syst`emes non lin´eaires, alg`ebres de [7] Lie filtr´ees transitives et s´eries g´en´eratrices non commutatives, Invent. Math. 71 (1983) 521-537. [8] M. Fliess, M. Lamnabhi and F. Lamnabhi-Lagarrigue, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits Systems 30 (1983) 554-570. [9] X.-X. Gan and D. Knox, On composition of formal power series, Int. J. Math. Math. Sci 30 (2002) 761-770. [10] W. S. Gray and Y. Li, Generating series for interconnected analytic nonlinear systems, SIAM J. Control Optim. 44 (2005) 646-672. [11] W. S. Gray and Y. Wang, Fliess operators on Lp spaces: convergence and continuity, Systems Control Lett. 46 (2002) 67-74. , Formal Fliess operators with applications to feedback intercon[12] nections, Proc. 18th Inter. Symp. Mathematical Theory of Networks and Systems, Blacksburg, Virginia, 2008. [13] W. Greblicki, Continuous-time Hammerstein system identification, IEEE Trans. Automat. Contr. 45 (2000) 1232-1236. [14] , Continuous-time Hammerstein system identification from sampled data, IEEE Trans. Automat. Contr. 51 (2006) 1195-1200. [15] A. Isidori, Nonlinear Control Systems, 3rd Ed., Springer-Verlag, London, 1995. [16] G. Jacob, Sur un th´eor`eme de Shamir, Inform. and Control 27 (1975) 218-261. [17] K. Knopp, Infinite Sequences and Series, Dover Publications, Inc., New York, 1956. [18] Y. Li and W. S. Gray, The formal Laplace-Borel transform of Fliess operators and the composition product, Int. J. Math. Math. Sci. 2006 (2006) Article ID 34217. [19] W. J. Rugh, Nonlinear System Theory, The Volterra/Wiener Approach, The Johns Hopkins University Press, Baltimore, Maryland, 1981. [20] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, John Wiley, New York, 1980. [21] Y. Wang, Algebraic Differential Equations and Nonlinear Control Systems, Doctoral Dissertation, Rutgers University (1990). [22] Y. Wang and E. D. Sontag, Algebraic differential equations and rational control systems, SIAM J. Control Optim. 30 (1992) 11261149.
8007
FrC01.4
A Unified Approach to Generating Series for Nonlinear Cascade Systems W. Steven Gray
Abstract— The goal of the paper is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input-output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of Fliess operators or memoryless maps provided that the composite system is well defined and resides in one of these two classes.
I. I NTRODUCTION In many applications, input-output systems are interconnected to form more complex systems. Describing the nature of the composite system and providing some explicit parametrization for it are generally nontrivial problems when the subsystems are nonlinear. In this paper, cascade connections involving two classes of analytic nonlinear inputoutput systems are considered. The first class of systems are those that can be represented by memoryless maps of the form f : Rm → Rℓ , which are real analytic on some domain of interest. Thus, each f can be written in terms of a Taylor series whose coefficients are associated with a formal power series c over a commutative alphabet. Such a map is denoted by fc , and c will be called its generating series. The second class of systems under consideration are those that can be represented as Fliess operators [6]–[8]. This class includes any input-output system that can be written as a Volterra operator having analytic kernels (including the linear, time-invariant case). Such an operator Fd can be expressed in terms of a generating series d, which is a formal power series over a noncommutative alphabet. In general, Fliess operators are not memoryless except for the trivial case where their generating series corresponds to a constant polynomial d = y0 ∈ Rℓ , in which case Fd [u] = y0 for every input u. The specific goal of the paper is to describe in a unified fashion the generating series for the cascade connection of any two systems from one or both of these system classes as shown in Fig. 1. It will be shown that a single general definition of a composition product involving formal power series can be formulated to produce the generating series for any possible cascade connection, as long as the composite system is well defined and resides in one of these two classes. Certain elements of this problem are well understood by other means. For example, the composition of analytic maps in one variable is introduced in most classic books on power series, e.g., [17]. More advanced versions of the problem are addressed in [2], [9]. The interconnection of memoryless maps (not necessarily analytic) with linear time-invariant dynamical systems, so called Wiener and Hammerstein systems, are analyzed in [13], [14], [19], [20]. The author is affiliated with the Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, USA.
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
u
fd
v
fc
y
(a) cascade of memoryless maps
u
Fd
v
fc
y
Fc
y
(b) Wiener-Fliess system
u
fd
v
(c) Hammerstein-Fliess system
u
Fd
v
Fc
y
(d) cascade of Fliess operators Fig. 1.
Cascade systems under consideration
The composition of Fliess operators was considered in [3], [4], [10], [18]. The primary interest here, however, is in showing that all of these interconnections have a common algebraic/combinatoric source. From this vantage point it is easier to see how certain cascade connections fit easily into existing theoretical frameworks, while others are more problematic. For example, in the case of a Wiener system, replacing the linear dynamical system with an arbitrary Fliess operator to produce a Wiener-Fliess system does not require any additional mathematical machinery. It can be viewed as a generalization of existing results for the case when fc is a polynomial [6], [21], [22]. On the other hand, a Hammerstein-Fliess system, as shown in Fig. 1(c), can easily fall outside the domain of the theory presented here without significant restrictions. The paper is organized as follows. In the first section some preliminaries are presented to properly frame the problems under consideration. In Section III the general notion of a composition product is introduced along with some specific examples and conditions under which it is well defined. In the subsequent section, the concept is applied to the
8002
FrC01.4 four cascade connections shown in Fig. 1. The final section summarizes the main contributions of the paper. II. P RELIMINARIES A finite nonempty set of noncommuting symbols X = {x0 , x1 , . . . , xm } is called an alphabet. Each element of X is called a letter, and any finite string of letters from X, η = xi1 · · · xik , is called a word over X. The length of η, |η|, is the number of letters in η, while |η|xi is the number of times the letter xi appears in η. The set of all words with length k will be denoted by X k . The set of all words including the empty word, ∅, will be denoted by X ∗ . It forms a monoid under catenation. A language is any subset of X ∗ . Any mapping c : X ∗ → Rℓ is called a formal power series. The value of c at η ∈ X ∗ is written asP(c, η). Typically, c is represented as the formal sum c = η∈X ∗ (c, η)η. The coefficient (c, ∅) is referred to as the constant term, and c is called proper when this coefficient is zero. The support of c is the language supp(c) = {η ∈ X ∗ : (c, η) 6= 0}. The order of c is defined as min{|η| : η ∈ supp(c)} ord(c) = ∞
: :
c 6= 0 c = 0.
η∈X
where X
η=ξν
(c, ξ)(d, ν)
t0
where xi ∈ X, η¯ ∈ X ∗ , and u0 = 1. The input-output operator corresponding to c is the Fliess operator X (c, η) Eη [u](t, t0 ). Fc [u](t) = η∈X ∗
The collection of all formal power series over X is denoted by Rℓ hhXii. It forms an associative R-algebra under the (Cauchy) catenation product and a commutative R-algebra under the shuffle product, designated by ⊔⊔ . Rℓ [[X]] denotes the set of all formal power series when the letters of X commute. In this situation it is more natural to impose the ordering x0 < x1 < · · · < xm on X and then assume that the support of c is only on ordered words, specifically, words of the form η = xi1 · · · xik , where ij ≤ ij+1 for all j = 1, 2, . . . , k−1. For c, d ∈ R [[X]] an alternative definition of the catenation product is preferred, namely, X η (cd, η) , cd = η! ∗ (cd, η) =
|(c, η˜)| ≤ KM |˜η| |˜ η |!, where |z| := max{|zi | : 1 ≤ i ≤ ℓ} ˜ ∗. when z ∈ Rℓ , for fixed constants K, M > 0 and all η˜ ∈ X Such a power series is called locally convergent, and the ˜ If instead set of all such series is denoted by RℓLC [[X]]. c ∈ Rℓ hhXii, one can formally associate a causal m-input, ℓ-output operator, Fc , in the following manner. Let p ≥ 1 and t0 < t1 be given. For a measurable function u : [t0 , t1 ] → 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 [t0 , t1 ]. Let Lm p [t0 , t1 ] denote the set of all measurable functions defined on [t0 , t1 ] having a finite k · kp norm and Bpm (R)[t0 , t1 ] := {u ∈ Lm p [t0 , t1 ] : kukp ≤ R}. Define recursively for each η ∈ X ∗ the map Eη : Lm 1 [t0 , t1 ] → C[t0 , t1 ] by setting E∅ [u] = 1 and letting Z t ui (τ )Eη¯[u](τ, t0 ) dτ, Exi η¯[u](t, t0 ) =
η! ξ!ν!
and η! := |η|x0 ! |η|x1 ! · · · |η|xm !. For a single letter alphabet, this yields what is commonly called binomial convolution i X i (c, xj )(d, xi−j ). (cd, xi ) = j j=0 So the catenation product in the general commutative case will be referred to as the multinomial catenation product. Let X = {x0 , x1 , . . . , xm } be a noncommutative alphabet ˜ = {˜ and X x1 , x ˜2 . . . , x ˜m } a second alphabet, distinct from ˜ one can associate X and commutative. For each c ∈ Rℓ [[X]] a formal function X z η˜ (c, η˜) , fc (z1 , z2 , . . . , zm ) = η˜! ∗ ˜ η˜∈X
˜ik . If fc is analytic ˜i1 · · · x where z η˜ := zi1 zi2 · · · zik if η˜ = x then its coefficients must satisfy a Cauchy growth condition
If |(c, η)| ≤ KM |η| |η|! for all η ∈ X ∗ , then Fc constitutes a well defined operator from Bpm (R)[t0 , t0 + T ] into Bqℓ (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 [11]. The set of all locally convergent series in this setting is denoted by RℓLC hhXii. III. C OMPOSITION P RODUCTS In this section a class of formal power series products called composition products is considered. Each specific product introduced comes from essentially the same con˜ = struction process. Let X = {x0 , x1 , . . . , xm } and X {˜ x0 , x ˜1 , . . . , x ˜m ˜ } be two arbitrary alphabets and consider the ˜ ˜ formal power series c ∈ Rℓ hhXii and d ∈ Rℓ hhXii. Assume that RhhXii is endowed with an associative R-algebra having product 2 and multiplicative identity element 1. Definition 1: A composition product of c and d is any binary operation ˜ ˜ Rℓ hhXii × Rℓ hhXii
(c, d)
˜
→ Rℓ hhXii X (c, η˜) η˜ ◦ d, 7→ c ◦ d = ˜∗ η˜∈X
where η˜ ◦ d is the unique extension of ˜∗
x ˜i ◦ d = ρi (d), i = 0, 1, . . . , m ˜
to X given by (˜ xik x ˜ik−1 · · · x ˜i1 ) ◦ d = ρik (d)2ρik−1 (d)2 · · · 2ρi1 (d) with ρi : Rℓ hhXii → RhhXii such that ρi (∅) = 1, i = 0, 1, . . . , m ˜ and ∅ ◦ d = 1. This type of construction arises in many forms when systems are interconnected to produce new systems. The particular form of the composition product depends precisely on the nature of the systems being connected. Whenever possible, the same notation will be used for any given composition product, and the specific definition will be evident from the context. It is easily verified that any composition product
8003
FrC01.4 is in general R-linear in its left argument but not its right argument. It should also be mentioned that a composition product can be viewed in terms of transductions [5], [16]. ˜ X)-transduction is any formal power series t ˜ ∈ An (X, XX ˜ [RhhXii]hhXii. For a given d ∈ Rℓ hhXii one can define the ˜ X)-transduction (X, X X (˜ η ◦ d) η˜. (tX X˜ , η˜) η˜ = tX X˜ = ˜∗ η˜∈X
˜∗ η˜∈X
X
X
˜ = X, Example 3: Suppose X = {x0 , x1 , x2 , . . . , xm }, X ℓ m c ∈ R hhXii and d ∈ R hhXii. Define their composition as in (1)-(2) with d0 = 1. This type of composition product describes a Hammerstein-FliessPsystem, Fc ◦ fd = Fc◦d , m provided that fd (z1 , . . . , zm ) = j=0 (d, xj )zj with z0 = 1. Otherwise, as explained in the next section, Fc ◦ fd has no Fliess operator representation. Example 4: Using the same setup as in the previous example, define the family of mappings
In which case, c◦d
=
˜∗ η˜∈X
(c, η˜) η˜ ◦ d =
= (c, tX X˜ ).
Dxi
(c, η˜)(tX X˜ , η˜)
˜∗ η˜∈X
Before pursuing any technical issues, such as conditions under which a composition product is well defined, some specific examples are introduced. ˜ = Example 1: Suppose X = {x1 , x2 , . . . , xm }, X ℓ˜ ˜ m ˜ {˜ x1 , x ˜2 , . . . , x ˜m ˜ }, c ∈ R hhXii and d ∈ R hhXii. Let di denote the i-th component series of d, that is, (di , ξ) = (d, ξ)i for every ξ ∈ X ∗ . Consider the composition product defined by letting ρi (d) = di , i = 1, . . . , m, ˜ and (˜ xi1 x ˜i2 · · · x ˜ik ) ◦ d = di1 di2 · · · dik =: dη˜ {z } |
(1)
(xik xik−1 · · · xi1 ) ◦ d = Dxik Dxik−1 · · · Dxi1 (1) = Dη (1). {z } | η
In which case,
c◦d=
with d∅ = 1. Therefore,
X
(c, η˜) dη˜.
(2)
˜∗ η˜∈X
˜ commute, then the composition product If the letters in X is written in the exponential form X dη˜ (c, η˜) . c◦d= η˜! ∗ ˜ η˜∈X
If, in addition, the letters in the alphabet X commute, then the power dη˜ is taken to be the multinomial catenation power having coefficients X η! , η ∈ X ∗. (dη˜, η) = (di1 , η1 ) · · · (dik , ηk ) η ! · 1 · · ηk ! η ···η =η 1
k
It is shown in the next section that this type of composition product describes function composition, fc ◦ fd = fc◦d . ˜ = Example 2: Let X = {x0 , x1 , x2 , . . . , xm }, X ˜ and d ∈ Rn hhXii. If ρi (d) = {˜ x1 , x ˜2 , . . . , x ˜n }, c ∈ Rℓ [[X]] di , i = 1, . . . , n, and ˜ik ) ◦ d = di1 ˜i2 · · · x (˜ xi1 x | {z }
⊔⊔
di2
⊔⊔
···
⊔⊔
dik =: d ⊔⊔ η˜
η˜
with d ⊔⊔ ∅ = 1 then
c◦d=
X
(c, η) Dη (1).
η∈X ∗
This type of composition product describes the composition of two Fliess operators, Fc ◦ Fd = Fc◦d . A variation of this product, where instead Dxi
: RhhXii → RhhXii : e 7→ xi e + x0 (di ⊔⊔ e)
for i = 1, 2, . . . , m (the i = 0 case is unchanged), can be used to describe the feedback connection of two Fliess operators [10], [12]. The well definedness of a composition product is considered next. The following two theorems will cover all the examples described above. Theorem 1: Consider a composition product defined by (ρ, 2), where 1) ord(ρi (d)) > ord(d), i = 0, 1, . . . , m, ˜ d ∈ Rℓ hhXii ′ ′ 2) ord(d2d ) = ord(d) + ord(d ), d, d′ ∈ RhhXii. ˜ ˜ and d ∈ Rℓ hhXii the composition c◦d For any c ∈ Rℓ hhXii ˜
is a well defined series in Rℓ hhXii. Proof: It suffices to show that the family of formal power series {˜ η ◦ d}η˜∈X˜ ∗ is locally finite, and hence, summable [1]. For a fixed d ∈ Rℓ hhXii defined the integers ri = ord(ρi (d))−ord(d) > 0, i = 0, 1, . . . , m, ˜ and r = mini ri > ˜ ∗: 0. Then given any nonempty word η˜ ∈ X ord(˜ η ◦ d)
d ⊔⊔ η˜ . (c, η˜) η˜! ∗
X
˜ η˜∈X
RhhXii → RhhXii e 7→ x0 (di ⊔⊔ e),
i = 0, 1, . . . , m with d0 = 1. Assume D∅ is the identity map on RhhXii. Such maps can be composed in the obvious way so that Dxi xj := Dxi Dxj provides an R-algebra which is isomorphic to a subalgebra of the usual R-algebra on RhhXii under the catenation product. In this setting, consider a formal power series composition product defined by letting ρi (d) = Dxi , i = 0, 1, . . . , m, and
η˜
c◦d=
: :
=
˜i1 ◦d) ˜ik−1 · · · x ord(˜ xik x | {z } η˜
=
This type of composition product describes a Wiener-Fliess system, fc ◦ Fd = Fc◦d .
8004
=
ord(ρik (d)2ρik−1 (d)2 · · · 2ρi1 (d))
k X j=1
ord(ρij (d))
FrC01.4
=
k X
ord(d) + rij
j=1
≥
|˜ η | (ord(d) + r).
Since ord(d) + r > 0, ord(˜ η ◦ d) increases at least proportionally as the length of η˜ increases. So for a fixed ξ ∈ X ∗ the set ˜ ∗ : (˜ Id (ξ) := {˜ η∈X η ◦ d, ξ) 6= 0} must be finite since (˜ η ◦ d, ξ) = 0 when the length of η˜ is such that
A. Cascades of Memoryless Maps In the context of Example 1, consider the following theorem. Theorem 3: Suppose U and V are two neighborhoods of ˜ ˜ ˜ the origin. Let fc : U ⊂ Rm → Rℓ and fd : V ⊂ Rm → Rm be real analytic functions with fd (V ) ⊂ U and having Taylor series about z˜ = 0 and z = 0: X z˜η˜ (c, η˜) fc (˜ z1 , z˜2 , . . . , z˜m ˜) = η˜! ∗ ˜ η˜∈X
fd (z1 , z2 , . . . , zm ) =
X
(d, η)
η∈X ∗
|˜ η | (ord(d) + r) > |ξ| . In which case, the family of series in question is locally finite. Theorem 2: Consider a composition product defined by (ρ, 2), where 1) ord(ρi (d)) ≥ ord(d), i = 0, 1, . . . , m, ˜ d ∈ Rℓ hhXii ′ ′ 2) ord(d2d ) = ord(d) + ord(d ), d, d′ ∈ RhhXii. ˜ ˜ and proper d ∈ Rℓ hhXii the For any c ∈ Rℓ hhXii
zη , η!
˜ ˜ ˜ and proper d ∈ Rm respectively, with c ∈ RℓLC [[X]] LC [[X]]. Then fc ◦ fd = fc◦d . Proof: The composite function fc ◦fd corresponds to setting z˜i = fd,i (z), i = 1, 2, . . . , m. ˜ Since d is proper, fd (0) = 0. Thus, by direct substitution
fc ◦ fd (z) =
˜
composition c ◦ d is a well defined series in Rℓ hhXii. Proof: Following the same logic as in the previous theorem, one can conclude here that
=
ord(˜ η ◦ d) ≥ |˜ η | ord(d).
=
Since d is proper, ord(d) > 0. In which case, Id (ξ) is finite since (˜ η ◦ d, ξ) = 0 when the length of η˜ is such that |˜ η | ord(d) > |ξ| . This proves the theorem.
=
Example 5: Suppose ρi (d) = di as in Examples 1-3. Clearly, ord(ρi (d)) = ord(di ) ≥ ord(d), so in these cases, Theorem 2 applies. Therefore, the respective composition products are well defined if d is proper. Example 6: Consider the composition product defined in Example 4. Observe that for any e ∈ RhhXii
=
ord(Dxi (e)) = ord(di ) + ord(e) + 1. So it makes sense to define ord(Dxi ) = ord(di )+1. It then follows that
=
ord(ρi (d)) = ord(Dxi ) = ord(di ) + 1 > ord(d) =
for i = 0, 1, . . . , m. In which case, Theorem 1 applies and c ◦ d is well defined everywhere on Rℓ hhXii × Rm hhXii.
z˜ (c, η˜) η˜! ˜∗ η˜∈X
X
η∈X ∗
IV. N ONLINEAR C ASCADE S YSTEMS
z˜=fd (z)
fd,i1 (z) · · · fd,ik (z) (c, η˜) η˜! ˜∗ η ˜∈X X 1 X z η1 (c, η˜) (di1 , η1 ) ··· η˜! η1 ! ˜∗ η1 ∈X ∗ η ˜∈X ηk X z (dik , ηk ) ηk ! ∗ ηk ∈X X X 1 (di1 , η1 ) · · · (dik , ηk ) · (c, η˜) η˜! ˜∗ η1 ,...,ηk ∈X ∗ η ˜∈X z η1 · · · z ηk η1 ! · · · ηk ! " X X 1 X (c, η˜) (di1 , η1 ) · · · (dik , ηk ) · η˜! ∗ η ···η =η ∗ ˜ η∈X 1 k η ˜∈X η z η! η1 ! · · · ηk ! η! X zη 1 X η˜ (d , η) (c, η˜) η˜! η! ˜∗ η∈X ∗ η ˜∈X X X 1 zη (c, η˜) (dη˜, η) η˜! η! ∗ ∗ X
η∈X
=
η˜
X
˜ η˜∈X
(c ◦ d, η)
zη η!
= fc◦d (z).
In this section the generating series for the four cascade connections shown in Figure 1 are computed directly and then related to the composition products in Examples 1-4. Local convergence of the resulting series is also considered.
In which case, fc ◦ fd = fc◦d as claimed. It is well known that the composition of two analytic functions is again analytic, therefore c ◦ d is a locally
8005
FrC01.4 convergent series. In the case where fc and fd are merely formal functions, c ◦ d is the generating series for the formal function fc ◦ fd . B. Wiener-Fliess Systems In the context of Example 2, consider the following theorem. Theorem 4: Given a Fliess operator Fd , d ∈ Rn hhXii and a formal function fc : Rn → Rℓ with generating series ˜ at z = (d, ∅), that is, c ∈ Rℓ [[X]] fc (z) =
X
(c, η˜)
˜∗ η˜∈X
(z − (d, ∅))η˜ , η˜!
X
(c, η˜)
˜∗ η˜∈X
˜ η˜∈X
Finally, using Stirling’s approximation, the asymptotic approximation 1 2ℓ ∼ √ 4ℓ , ℓ ≥ 1 ℓ−1 πℓ holds, and, in particular, 2ℓ ≤ 4ℓ , ℓ ≥ 1. ℓ−1
the cascade connection fc ◦ Fd has the generating series c◦d=
In addition, the following identities hold: ℓ X ℓ+k−1 2ℓ = , ℓ≥1 k−1 ℓ−1 k=1 X k! = nk , k ≥ 0. η ˜ ! k
(d − (d, ∅)) ⊔⊔ η˜ . η˜!
That is, fc ◦ Fd = Fc◦d . Proof: The proof follows from basic properties of the shuffle product. Define the proper series d˜ := d−(d, ∅), and observe that X (z − (d, ∅))η˜ (c, η˜) fc ◦ Fd [u] = η˜! z=Fd [u] ∗
Now the proof. The case where ν = ∅ is trivial. For any nonempty word ν ∈ X ∗ , the properness of d˜ and the assumption that nKd Mc ≥ 1 gives |(c ◦ d, ν)| X ˜ ⊔⊔ η˜, ν) ( d (c, η˜) = η˜! η˜∈X˜ ∗
=
X
(c, η˜)
˜∗ η˜∈X
=
≤
(Fd [u] − F(d,∅) [u])η˜ η˜!
X (c, η˜) (Fd˜[u])η˜ η ˜ ! ∗
˜ η˜∈X
=
X (c, η˜) F ˜ [u]Fd˜i [u] · · · Fd˜i [u] 1 k−1 η˜! dik ∗
X (c, η˜) F˜ η˜! dik ∗
˜ η˜∈X
=
⊔⊔
|ν|
Kc Mck k!
≤ Kc (4nKd Mc Md )|ν| |ν|!.
˜ η˜∈X
=
|ν| X X
Kdk Md (|ν| + k − 1)! η˜! (k − 1)! ˜k k=1 η˜∈X |ν| X |ν| + k − 1 |ν| Md |ν|! = Kc (nKd Mc )k k−1 k=1 2 |ν| (nKd Mc Md )|ν| |ν|! ≤ Kc |ν| − 1
˜ η˜∈X
This completes the proof.
d˜ik−1 ··· ⊔⊔ d˜i1 [u]
Example 7: Consider an analytic state space system
X (c, η˜) F ˜ ⊔⊔ η˜ [u] η˜! d ∗
z˙
= g0 (z) +
˜ η˜∈X
m X
gi (z) ui , z(t0 ) = z0
(3)
i=1
= Fc◦d [u].
y
= h(z).
(4)
Using Theorem 2, the properness of d˜ ensures that c ◦ d is a well defined series in Rℓ hhXii.
It is well known that (3) has the solution z = Fcz [u], where
Convergence properties are considered in the next theorem. ˜ and d ∈ Rn hhXii Theorem 5: Suppose c ∈ RℓLC [[X]] LC with growth constants Kc , Mc > 0 and Kd , Md > 0, respectively. Then c ◦ d ∈ RℓLC hhXii. Specifically, if nKd Mc ≥ 1 (which is no loss of generality) then
[6], [15]. Here the iterated Lie derivatives with respect to the vector fields gi are written as
|(c ◦ d, ν)| ≤ Kc (4nKd Mc Md )|ν| |ν|!, ∀ν ∈ X ∗ .
Proof: The proof uses several facts. First, a direct extension of a growth condition for shuffle products appearing in [21] to shuffle powers is ⊔⊔ k |ν| (|ν| + k − 1)! (d , ν) ≤ Kdk Md , ∀ν ∈ X ∗ , k ≥ 1. (k − 1)!
(cz , η) = Lgη I(z0 ), ∀η ∈ X ∗
Lgη = Lgi1 · · · Lgik , η = xik · · · xi1 ∈ X ∗ with L∅ := I being the identity map, i.e., I(z) = z. The corresponding Wiener-Fliess system is then y = h(z) = h(Fcz [u]) = Fch ◦cz [u], where ch denotes the generating series for h. From Fliess’s fundamental formula (ch ◦ cz , η)
8006
= =
Lgη (h ◦ I)(z0 ) Lgη h(z0 )
FrC01.4 =:
(c, η)
⊔⊔
for all η ∈ X ∗ . Therefore, y = Fc [u], that is, the state space system (g, h, z0 ) realizes the input-output system Fc . C. Hammerstein-Fliess Systems First an example is presented to justify the restriction introduce on the function fd in Example 3. Example 8: A correlator corresponds to a Hammerstein system where Fc = Ex1 and fd (u) = u2 . By direct substitution Z t u2 (τ ) dτ Fc ◦ fd [u](t) = 0 Z t Z τ2 = δ(τ2 − τ1 )u(τ2 )u(τ1 ) dτ1 dτ2 . 0
0
When viewed as a second order Volterra operator, it is clear that this system does not have a Fliess operator realization since the kernel function w2 (τ2 , τ1 ) = δ(τ2 − τ1 ) is not analytic.
Now consider the following theorem in the context of Example 3. Theorem 6: Given a Fliess operator Fc , c ∈ Rm hhXii, m and a function fd : Rm → RP with generating series d ∈ m R hhXii of the form d = xj ∈X (d, xj )xj , the cascade connection Fc ◦ fd has the generating series X (c, η)dη , c◦d= η∈X ∗
where d0 = 1. That is, Fc ◦ fd = Fc◦d . Proof: With u0 = 1 observe that for any xi ∈ X Z t X (di , xj ) uj (τ ) dτ Exi [fd (u)](t, t0 ) = t0 x ∈X j
X
=
(di , xj ) Exj [u](t, t0 )
xj ∈X
= Fdxi [u](t). Inductively, it is easy to show that Eη [fd (u)] = Fdη [u] for any η ∈ X ∗ . Therefore, X Fc ◦ fd [u] = (c, η) Eη [fd (u)] η∈X ∗
=
X
(c, η) Fdη [u]
η∈X ∗
= Fc◦d [u]. The properness of d ensures that c ◦ d is well defined. Since d is a polynomial, it is straightforward to show that if c is locally convergent then so is c ◦ d. D. Cascades of Fliess Operators In the context of Example 4, the composition product of a word η ∈ X ∗ , written generically in the form n
η = xn0 k xik x0 k−1 xik−1 · · · xn0 1 xi1 xn0 0 ,
where ij 6= 0 for j = 1, . . . , k, and a series d ∈ Rm hhXii was defined in [3], [4] as η◦d
= xn0 k +1 [dik
n
⊔⊔
x0 k−1
+1
[dik−1
⊔⊔
···
x0n1 +1 [di1
⊔⊔
xn0 0 ] · · · ]].
It is easy to verify that this composition is identical to Dη (1), and thus, this original notion of series composition is equivalent to the one defined here for Fliess operator composition. The following theorem is included for completeness. Theorem 7: [3], [10] For any c ∈ Rℓ hhXii and d ∈ m R hhXii, the identity Fc ◦Fd = Fc◦d is satisfied. In addition, if c and d are locally convergent then c ◦ d is also locally convergent. V. C ONCLUSIONS A unified treatment of generating series for cascade connections of analytic input-output systems was presented. In particular, it was shown that a single general definition of a composition product can be formulated so as to describe the generating series for any possible cascade connection, as long the composite system resides in one of the two systems classes of interest: memoryless systems and inputoutput systems representable as Fliess operators. R EFERENCES [1] J. Berstel and C. Reutenauer, Rational Series and Their Languages, Springer-Verlag, Berlin, 1988. [2] J. Chaumat and A.-M. Chollet, On composite formal power series, Trans. AMS 353 (2001) 1691-1703. [3] A. Ferfera, Combinatoire du Mono¨ıde Libre Appliqu´ee a` la Composition et aux Variations de Certaines Fonctionnelles Issues de la Th´eorie des Syst`emes, Doctoral Dissertation, University of Bordeaux I, 1979. [4] , Combinatoire du mono¨ıde libre et composition de certains syst`emes non lin´eaires, Ast´erisque 75-76 (1980) 87-93. [5] M. Fliess, Transductions de s´eries formelles, Discrete Math. 10 (1974) 57-74. [6] , Fonctionnelles causales non lin´eaires et ind´etermin´ees non commutatives, Bull. Soc. Math. France 109 (1981) 3-40. , R´ealisation locale des syst`emes non lin´eaires, alg`ebres de [7] Lie filtr´ees transitives et s´eries g´en´eratrices non commutatives, Invent. Math. 71 (1983) 521-537. [8] M. Fliess, M. Lamnabhi and F. Lamnabhi-Lagarrigue, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits Systems 30 (1983) 554-570. [9] X.-X. Gan and D. Knox, On composition of formal power series, Int. J. Math. Math. Sci 30 (2002) 761-770. [10] W. S. Gray and Y. Li, Generating series for interconnected analytic nonlinear systems, SIAM J. Control Optim. 44 (2005) 646-672. [11] W. S. Gray and Y. Wang, Fliess operators on Lp spaces: convergence and continuity, Systems Control Lett. 46 (2002) 67-74. , Formal Fliess operators with applications to feedback intercon[12] nections, Proc. 18th Inter. Symp. Mathematical Theory of Networks and Systems, Blacksburg, Virginia, 2008. [13] W. Greblicki, Continuous-time Hammerstein system identification, IEEE Trans. Automat. Contr. 45 (2000) 1232-1236. [14] , Continuous-time Hammerstein system identification from sampled data, IEEE Trans. Automat. Contr. 51 (2006) 1195-1200. [15] A. Isidori, Nonlinear Control Systems, 3rd Ed., Springer-Verlag, London, 1995. [16] G. Jacob, Sur un th´eor`eme de Shamir, Inform. and Control 27 (1975) 218-261. [17] K. Knopp, Infinite Sequences and Series, Dover Publications, Inc., New York, 1956. [18] Y. Li and W. S. Gray, The formal Laplace-Borel transform of Fliess operators and the composition product, Int. J. Math. Math. Sci. 2006 (2006) Article ID 34217. [19] W. J. Rugh, Nonlinear System Theory, The Volterra/Wiener Approach, The Johns Hopkins University Press, Baltimore, Maryland, 1981. [20] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, John Wiley, New York, 1980. [21] Y. Wang, Algebraic Differential Equations and Nonlinear Control Systems, Doctoral Dissertation, Rutgers University (1990). [22] Y. Wang and E. D. Sontag, Algebraic differential equations and rational control systems, SIAM J. Control Optim. 30 (1992) 11261149.
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