SISO Output Affine Feedback Transformation Group and Its Faá di Bruno Hopf Algebra

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1 Old ominion University OU igital Commons Electrical & Computer Engineering Faculty Publications Electrical & Computer Engineering 27 SISO Output Affine Feedback Transformation Group and Its Faá di Bruno Hopf Algebra W. Steven Gray Old ominion University, Kurusch Ebrahimi-Fard Follow this and additional works at: Part of the Algebra Commons, and the Electrical and Computer Engineering Commons Repository Citation Gray, W. Steven and Ebrahimi-Fard, Kurusch, "SISO Output Affine Feedback Transformation Group and Its Faá di Bruno Hopf Algebra" (27). Electrical & Computer Engineering Faculty Publications Original Publication Citation Gray, W. S., & Ebrahimi-Fard, K. (27). SISO output affine feedback transformation group and its Faà di Bruno Hopf algebra. SIAM Journal on Control & Optimization, 55(2), doi:.37/ This Article is brought to you for free and open access by the Electrical & Computer Engineering at OU igital Commons. It has been accepted for inclusion in Electrical & Computer Engineering Faculty Publications by an authorized administrator of OU igital Commons. For more information, please contact

2 SIAM J. CONTROL OPTIM. Vol. 55, No. 2, pp c 27 Society for Industrial and Applied Mathematics SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP AN ITS FAÀ I BRUNO HOPF ALGEBRA W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR Abstract. The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen Fliess functional expansions or Fliess operators and are independent of the existence of any state space models. This interconnection, called an output affine feedback connection, is distinguished from conventional output feedback by the presence of a multiplier in an outer loop. Once this transformation group is established, three basic questions are addressed. How can this transformation group be used to provide an explicit Fliess operator representation of such a closedloop system? Is it possible to use this feedback scheme to do system inversion purely in an inputoutput setting? In particular, can feedback input-output linearization be posed and solved entirely in this framework, i.e., without the need for any state space realization? Last, what can be said about feedback invariants under this transformation group? A final objective of the paper is to describe the Lie algebra of infinitesimal characters associated with the group in terms of a pre-lie product. Key words. nonlinear control systems, Chen Fliess series, Hopf algebras, transformation groups AMS subject classifications. 93C, 93B8, 6T3 OI..37/ Introduction. Feedback transformation groups have been used extensively in control theory since its inception to explain the way that feedback can alter the nature of a system. The early work of Brockett and Krishnaprasad in the case of linear systems [3, 4] and Brockett, Jakubczyk, Respondek, and many others in the context of nonlinear state space systems [2, 25, 26, 37] has been important, for example, in identifying feedback invariants. The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen Fliess functional expansions or Fliess operators and are independent of the existence of any state space models [, 2]. This interconnection, called an output affine feedback connection, is shown in Figure. The presence of a multiplier in an outer loop distinguishes this structure from conventional output feedback. Once this transformation group is established, three basic questions are addressed. How can this transformation group be used to provide an explicit Fliess operator representation of such a closed-loop system? Is it possible to use this feedback scheme to do system inversion purely in an input-output setting? In particular, can feedback input-output linearization be posed and solved entirely in this framework, i.e., without the need for any state space realization? Last, what can be said about feedback invariants under this transformation group? A final objective of the paper is to describe the Lie algebra of infinitesimal Received by the editors November 25, 24; accepted for publication (in revised form) ecember 2, 26; published electronically March 28, 27. A portion of this paper appeared in preliminary form in Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, Funding: The first author was supported by grant SEV-2-87 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. This research was also supported by a grant from the BBVA Foundation. Old ominion University, Norfolk, VA (sgray@odu.edu). Norwegian University of Science and Technology NTNU, 749 Trondheim, Norway (kurusch.ebrahimi-fard@math.ntnu.no). On leave from UHA, Mulhouse, France. 885

3 886 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR v + u F c y F d R F d L Fig.. Output affine feedback connection of Fliess operators. characters associated with this new transformation group and show that it is induced by a pre-lie product analogous to what was found for the output feedback transformation group in [3, 4]. This fact may give deeper insight into the combinatorial aspects of the theory and group invariants (see [2]), but this aspect of the problem will be deferred to a later publication. The basic approach to solving these problems is to use an operator algebra encoded by various products of formal power series. This is somewhat analogous to the familiar way in which interconnections of linear time-invariant systems are analyzed using only transfer functions and the algebra of rational functions. The feedback connection in general requires the use of a certain inverse operator that is naturally described in terms of a group. To do any calculations explicitly and efficiently here, it is necessary to present an associated Hopf algebra induced by the group product and its inverse. It has been known for a long time in some physics communities that Hopf algebras provide a natural computational framework for working with groups [5,, 5]. For example, the group inverse can be computed explicitly in terms of the antipode, a particular antihomomorphism of the Hopf algebra. If the latter is both connected and graded, then this antipode has a recursive description which is ideal for software implementation. As will be seen, the Hopf algebra of interest here is directly related to a composition of Fliess operators. The inversion formula that it provides is a kind of functional analogue of Lagrange s formula for the inversion of smooth functions in terms of their Taylor series coefficients. This classical formula has a combinatorial interpretation in terms of a connected graded Hopf algebra [27]. In particular, the coproduct of this Hopf algebra is defined in terms of the classical Faà di Bruno formula which gives the Taylor series coefficients for the composition of two sufficiently well behaved functions. Therefore, this Hopf algebra is often referred to as a Faà di Bruno type Hopf algebra [, 5]. The Hopf algebra presented in this paper is also of Faà di Bruno type and is best viewed as a generalization of existing work on output feedback (corresponding to setting F dl [y] = for all y in Figure ) by the authors and others appearing in [7, 3, 4, 6, 7, 8, 2, 22, 38]. The focus here will be on the single-input single-output (SISO) case as the multivariable case introduces additional technical issues that are nontrivial and require more advanced concepts from Hopf algebra theory. The paper is organized as follows. In the next section, some preliminary concepts concerning Fliess operators, ultrametric spaces, and Hopf algebras are briefly outlined. In section 3, the output affine feedback transformation group is described. The problem of determining a Fliess operator representation of the output affine closedloop system is then solved. The Hopf algebra is next developed in section 4 and subsequently applied in the following section to address the remaining two problems

4 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 887 concerning system inversion/feedback linearization and feedback invariants. In section 6, the Lie algebra of infinitesimal characters is described. 2. Preliminaries. A finite nonempty set of noncommuting symbols X = {x, x,..., x m } is called an alphabet. Each element of X is called a letter, and any finite sequence of letters from X, η = x i x ik, is called a word over X. The length of the word η, η, is given by the number of letters it contains. The set of all words with length k is denoted by X k. The set of all words including the empty word,, is designated by X, while X + := X { }. The set X forms a monoid under catenation. The set ηx comprises all words with the prefix η X. For any fixed integer l, a mapping c : X R l is called a formal power series. The value of c at η X is written as (c, η) R l and called the coefficient of η in c. Typically, c is represented as the formal sum c = η X (c, η)η. If the constant term (c, ) =, then c is said to be proper. The support of c, supp(c), is the set of all words having nonzero coefficients. The order of c, ord(c), is the length of the shortest word in its support (ord() := ). The collection of all formal power series over X is denoted by R l X. It forms an associative R-algebra under the catenation product and a commutative and associative R-algebra under the shuffle product, denoted here by the shuffle symbol. The latter is the R-bilinear extension of the shuffle product of two words, which is defined inductively by (2.) (x i η) (x j ξ) = x i (η (x j ξ)) + x j ((x i η) ξ) with η = η = η for all η, ξ X and x i, x j X []. 2.. Fliess operators and their interconnections. One can formally associate with any series c R l X a causal m-input, l-output operator, F c, in the following manner. Let p and t < t be given. For a Lebesgue measurable function u : [t, t ] R m, define u p = max{ u i p : i m}, where u i p is the usual L p -norm for a measurable real-valued function, u i, defined on [t, t ]. Let L m p [t, t ] denote the set of all measurable functions defined on [t, t ] having a finite p norm and Bp m (R)[t, t ] := {u L m p [t, t ] : u p R}. Assume C[t, t ] is the subset of continuous functions in L m [t, t ]. efine inductively for each word η X the map E η : L m [t, t ] C[t, t ] by setting E [u] = and letting E xi η[u](t, t ) = t t u i (τ)e η [u](τ, t ) dτ, where x i X, η X, and u =. The input-output operator corresponding to c = η X (c, η)η Rl X is the Fliess operator F c [u](t) = η X (c, η) E η [u](t, t ) [, 2]. If there exist real numbers K c, M c > such that (2.2) (c, η) K c M η c η! η X, then F c constitutes a well defined mapping from B m p (R)[t, t +T ] into B l q(s)[t, t +T ] for sufficiently small R, T > and some S >, where the numbers p, q [, ] are For notational convenience, p = (p, ) R X is often abbreviated as p = (p, ).

5 888 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR conjugate exponents, i.e., /p + /q = [2]. (Here, z := max i z i when z R l.) The set of all such locally convergent series is denoted by R l LC X. A Fliess operator F c defined on Bp m (R)[t, t + T ] is said to be realizable when there exists a state space model consisting of n differential equations and l output functions (2.3a) (2.3b) m ż = g (z) + g i (z) u i, z(t ) = z, i= y j = h j (z), j =, 2,..., l, where each g i is an analytic vector field expressed in local coordinates on some neighborhood W of z, and each output function h j is an analytic function on W such that (2.3a) has a well defined solution z(t), t [t, t + T ] on W for any given input u B m p (R)[t, t + T ], and y j (t) = F cj [u](t) = h j (z(t)), t [t, t + T ], j =, 2,..., l [, 2, 2, 24, 32]. It can be shown that for any word η = x ik x i X (2.4) (c j, η) = L gη h j (z ) := L gi L gik h j (z ), where L gi h j is the Lie derivative of h j with respect to g i. For any c R l X, the R-linear mapping H c : R X R l X uniquely specified by (H c (η), ξ) = (c, ξη), ξ, η X is called the Hankel mapping of c. The series c is said to have Lie rank n when the range space of H c restricted to the R-vector space of Lie polynomials over X, i.e., the free Lie algebra L(X), has dimension n. It is well known that F c is realizable if and only if c R l LC X has finite Lie rank [2, 24, 32]. When convergent Fliess operators are connected in a parallel-sum configuration, it is elementary to show that the composite system satisfies F c +F d = F c+d. It was shown in [] for the parallel-product connection that F c F d = F c d. If F c and F d with c R l LC X and d Rm LC X are interconnected in a cascade manner, the composite system F c F d has the Fliess operator representation F c d, where c d denotes the composition product of c and d as described in [9]. This product is associative and R-linear in its left argument c. In the event that two convergent Fliess operators are interconnected to form a feedback system, the closed-loop system has a convergent Fliess operator representation whose generating series is the output feedback product of c and d, denoted by c@d [7, 38]. Consider, for example, the SISO case where the alphabet X = {x, x } and l =. efine the set of unital Fliess operators F = {I + F c : c R LC X }, where I denotes the identity map, i.e., I[u] = u. It is convenient to introduce the symbol as the (fictitious) generating series for I. That is, F := I such that I + F c := F +c = F c with c := + c. The set of all such generating series for F is denoted by R LC X with the augmented alphabet X := {} X. It has been proved that F forms a group under the composition F c F d = (I + F c ) (I + F d ) = F c d, where c d := + d + c d, and denotes the modified composition product [9]. 2 The unit element of this group is the identity map I. Of central importance is the so-called coordinate algebra of functions, H, on the corresponding group (R LC X,, ). In this setting the group multiplication is dualized to a comultiplication on H, which is captured through the notion of a Hopf algebra. In fact, the compositional nature of the product on F turns H into a Faà di Bruno type Hopf algebra. Thus, the group (composition) inverse in (R LC X,, ), 2 The same symbol will be used for composition on R X and R X. As elements in these two sets have a distinct notation, i.e., c versus c, respectively, it will always be clear which product is at play.

6 c SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 889 =: +c, can be computed efficiently by a recursive algorithm [7]. This inverse is also key in describing the output feedback product as shown in the following theorem. Theorem 2. (see [7, 38]). For any c, d R LC X it follows that c@d = c ( d c) with c@d R LC X. In addition, c@d = c (d (c@d)). It is known that the set of locally convergent generating series is closed under addition, the shuffle product [39], the composition product [9], the modified composition product [28], and the feedback product [7, 38]. Therefore, all the corresponding elementary interconnections of Fliess operators preserve local convergence. Finally, the algebraic theory described above remains largely intact for formal power series that do not necessarily satisfy convergence condition (2.2). In place of a convergent Fliess operator one can introduce a formal Fliess operator which maps formal functions to formal functions [22]. The motivation for this is the observation that if c R l LC X and u is analytic with a convergent Taylor series representation u = n (c u, x n )(t t ) n /n! = n (c u, x n )E x n [u](t, t ), then y = F c [u] is also analytic with Taylor series coefficients given by c y = c c u so that c u R m LC [[X ]] and c y R l LC [[X ]] with X := {x }. But when c is not locally convergent, the mapping R m [[X ]] R l LC [[X ]] : c u c y = c c u is still well defined. It can be viewed as the formal version of F c. The set R l X, which also forms a group under the product c d, is the corresponding set of generating series for these formal operators Shuffle product operations on ultrametric spaces. Given a set S, a function U : S S R is called an ultrametric if it satisfies the following properties for all s, s, s S: i. U (s, s ). ii. U (s, s ) = if and only if s = s. iii. U (s, s ) = U (s, s). iv. U (s, s ) max{u (s, s ), U (s, s )}. The pair (S, U ) is referred to as an ultrametric space. In the event that property iv is replaced with the triangle inequality, (S, U ) reduces to the usual definition of a metric space. It is not difficult to show for any fixed < σ < that the R-vector space R X with the distance between two series defined as dist(c, d) = σ ord(c d) is a complete ultrametric space []. In this section two key lemmas are presented which describe how the distance between two series is altered by operations involving the shuffle product defined in (2.). These results are employed in section 3 to prove the existence of a group inverse using ultrametric contractions on (R X, dist). Lemma 2.2. For any series c i, d i R X, i =, 2, dist(c d, c 2 d 2 ) max(σ ord(c) dist(d, d 2 ), σ ord(d2) dist(c, c 2 )). Proof. First observe that dist(c i d, c i d 2 ) = σ ord(ci (d d2)) = σ ord(ci)+ord(d d2) = σ ord(ci) dist(d, d 2 ). Therefore, it follows that dist(c d, c 2 d 2 ) max(dist(c d, c d 2 ), dist(c d 2, c 2 d 2 )) = max(σ ord(c) dist(d, d 2 ), σ ord(d2) dist(c, c 2 )). Corollary 2.3. For a fixed c R X, the mapping d c d is an ultrametric contraction if c is proper and an isometry on (R X, dist) otherwise.

7 89 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR Theorem 2.4 (see [8]). The set of nonproper series R np X R X is a group under the shuffle product. In particular, the shuffle inverse of any such series c R np X is c = ((c, )( c )) := (c, ) (c ), where c := c/(c, ) is proper and (c ) := k (c ) k. Lemma 2.5. The shuffle inverse is an isometry for any c, d R np X having identical constant terms. Proof. For any c, d R np X with (c, ) = (d, ) observe that ( ) ord(c d ) = ord (c ) k (d ) k = ord(c d ) = ord(c d), and hence the lemma is proved. k= 2.3. Hopf algebras. Some definitions and basic facts concerning Hopf algebras are summarized in this section for later use. The treatment is based largely on [, 23, 29, 34, 36]. In the following, a unital R-algebra refers to a vector space A over the base field R with associative product m A : A A A and unit map (element) e A : R A (e A () = A ). A counital coalgebra over R consists of a triple (C, C, ε C ). The coproduct C : C C C is coassociative, that is, (id C C ) C = ( C id C ) C, where id C is the identity map on C and ε C : C R denotes the counit (augmentation) map. A coalgebra is cocommutative if τ C C = C, where τ C : C C C C is the flip map, τ C (x y) = y x. A bialgebra B is both a unital algebra and a counital coalgebra together with compatibility relations, such as both the algebra product, xy := m B (x y), and the unit map, e B : R B, are coalgebra morphisms. This provides, for example, the identity B (xy) = B (x) B (y). The unit of B is denoted by B = e B (). A bialgebra is called graded if there are R-vector subspaces B n, n N such that B = n B n with m B (B k B l ) B k+l and B B n k+l=n B k B l. Elements x B n are given a degree deg(x) = n. Moreover, B is called connected if the R-vector subspace B = R B. efine B + = n> B n. In a connected graded bialgebra, the coproduct for any x B n is of the form B (x) = x B + B x + B(x) B k B l, k+l=n where B (x) := B(x) x B B x k+l=n B k B l is the reduced coproduct. k,l> Note that in the following the use of subscripts on the structure maps is limited to the cases where there is potential for notational confusion. Suppose B is a bialgebra and A is an R-algebra with product m A : A A A and unit map e A, e.g., A = R or A = B. The vector space L(B, A) of linear maps from the bialgebra B to A together with the convolution product (2.5) Φ Ψ := m A (Φ Ψ) : B A, where Φ, Ψ L(B, A), is an associative algebra with unit ι := e A ε. A Hopf algebra H is a bialgebra together with a particular R-linear map called an antipode S : H H which satisfies the Hopf algebra axioms [34, 36] and has the property of being an antihomomorphism for both the algebra and the coalgebra structures, i.e., S(xy) = S(y)S(x) and S = (S S) τ. When A = H, the necessarily unique antipode S L(H, H) is the inverse of the identity map id : H H with respect to the convolution product, that is, (2.6) S id = id S := m (S id) = e ε.

8 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 89 Group-like and primitive elements in a Hopf algebra satisfy (x) = x x and (y) = y + y, respectively. Note that if x, x 2 are primitive in H, then [x, x 2 ] := m H (x x 2 ) m H (x 2 x ) is primitive as well. In particular, the set of primitive elements P (H) H is a Lie algebra. Another important observation is that a connected graded bialgebra H = n H n is always a connected graded Hopf algebra. Let H = n H n be a connected graded commutative bialgebra, and suppose A is a commutative unital algebra. Any Φ L(H, A) is called a character if Φ() = A and (2.7) Φ(xy) = Φ(x)Φ(y) for all x, y H. The set of characters is denoted by G A L(H, A) and forms a group with respect to the convolution product (2.5). The neutral element ι := e A ε in G A is given by ι() = A and ι(x) = for elements x in the augmentation ideal Ker(ε) = H + = n> H n. The inverse of Φ G A is given by composition with the antipode (2.8) Φ = Φ S, which follows from (2.6) and the fact that Φ is an algebra morphism. Specifically, Φ (Φ S) = Φ (id S) = Φ (S id) = (Φ S) Φ = Φ e ε = e ε. An infinitesimal character with values in A is a linear map ξ L(H, A) such that for x, y H +, ξ(m H (x y)) =, which implies ξ() =. The linear space of infinitesimal characters g A L(H, A) forms a Lie algebra with respect to the Lie bracket defined in terms of the convolution product (2.5). For α g A and any x H n the exponential exp (α)(x) := j j! α j (x) is a finite sum terminating at j = n. k Proposition 2.6. exp restricts to a bijection from g A onto G A. For details see references [8, ]. The inverse of exp is given by log (ι + γ)(x) = ( ) k k γ k (x), where again the sum terminates at k = n for any x H n. Finally, a brief description of the relationship between the group G A L(H, A) and the Hopf algebra H is given. The reader is referred to the paper of Frabetti and Manchon [5] for details and additional references. By definition, elements in G A map all of H into the commutative unital algebra A. However, any element x H can also be seen as an A-valued function on G A. Indeed, let Φ G A ; then x(φ) := Φ(x) A and the usual pointwise product of functions (xy)(φ) = x(φ)y(φ) follows from (2.7) since Φ G A. The definition of the convolution product (2.5) in terms of the coproduct of H implies a natural coproduct on functions x H, that is, (x)(φ, Ψ) := (Φ Ψ)(x) A. Similarly, the inverse of G A as well as its unit correspond naturally to the antipode and counit map on H, respectively. This reversed perspective on the relationship between H and its group of characters G A allows one to interpret H as the (Hopf) algebra of coordinate functions of the group G A. More precisely, H contains the representative functions over G A. The reader is directed to Cartier s work [5] for a comprehensive review of this topic. In this paper, the starting point will be a particular group whose product, unit, and inverse are used to identify its Hopf algebra of coordinate functions. 3. Output affine feedback transformation group. In general, if G is a group and S a given set, then G is said to act as a transformation group on the right of S if there exists a mapping φ : S G S : (h, g) hg such that

9 892 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR i. h = h, where is the identity element of G; ii. h(g g 2 ) = (hg )g 2 for all g, g 2 G. The action is said to be free if hg = h implies that g =. It will be evident in what follows that it is most convenient to redefine R X := R X R X and to identify the output affine feedback transformation group with those elements of R X which are invertible in some sense. Each tuple c := (c L, c R ) R LC X := R LC X R LC X gives rise to an operator of the form F c [u] := F cl [u]u+f cr [u], and by design the group product c d for elements in R LC X will satisfy the identity F c F d = F c d when c d R LC X. In addition, a second product c d is defined which describes the action of the group R X on R X so that F c F d = F c d when the series involved are all locally convergent. This product, referred to as the mixed composition product, plays a central role in the paper, so it will be the starting point for the section. Once these concepts are in place and the transformation group is established, the affine extension of the output feedback product in Theorem 2. will be given. Henceforth, the focus is on the SISO case. For a fixed d = (d L, d R ) R X, the mixed composition product is defined in terms of an algebra morphism φ d which takes words in X to a composition of simple vector space endomorphisms involving only the shuffle product and catenation with a single letter x or x. This sets up an inductive definition which provides some insight into the core combinatorial structures at play. The reader is referred in particular to [3, 4] for a broader theoretical view behind this approach. efinition 3.. Let d = (d L, d R ) R X. efine the mixed composition product mapping R X R X into R X as (3.) c d = φ d (c)() = (c, η) φ d (η)(), η X where φ d is the continuous (in the ultrametric sense) algebra homomorphism from R X to End(R X ) uniquely specified by φ d (x i η) = φ d (x i ) φ d (η) with (3.2) φ d (x )(e) = x e, φ d (x )(e) = x (d L e) + x (d R e) for any e R X, and where φ d ( ) denotes the identity map on R X. Example 3.2. Suppose c = 2x + x x and d = (d L, d R ) = (3x, x 2 ). Then c d = 2φ d (x )() + φ d (x ) φ d (x )(), where φ d (x )(e) = x e and φ d (x )(e) = x (3x e)+x ( x 2 e). Therefore, c d = 2x + 3x 2 x + 3x x x x x 2 x x x x x x 2 x 2. The modified composition product referred to in section 2 and used in earlier work on output feedback [7, 8, 9, 28, 38] corresponds here to the special case where d L =. Some fundamental properties of the mixed composition product are given next. Lemma 3.3. The mixed composition product (3.) () is left R-linear; (2) satisfies c (, ) = c; (3) satisfies c d = k R for any fixed d if and only if c = k; (4) satisfies (x c) d = x (c d ) and (x c) d = x (d L (c d )) + x (d R (c d )); (5) distributes to the left over the shuffle product.

10 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 893 Proof. () This fact follows directly from the definition of the mixed composition product. (2) The claim is immediate since φ (,) (η)() = η. (3) The only nontrivial assertion is that c d = k implies c = k. This claim is best handled in a Hopf algebra setting. So the proof is deferred until section 4. (4) Observe (x c) d = φ d (x c)() = φ d (x ) φ d (c)() = x (c d ), (x c) d = φ d (x c)() = φ d (x ) φ d (c)() = x (d L (c d )) + x (d R (c d )). (5) For any e R X, one can define a shuffle product on End(R X ) via φ e (x i η) φ e (x j ξ) = φ e (x i ) [φ e (η) φ e (x j ξ)] + φ e (x j ) [φ e (x i η) φ e (ξ)]. Therefore, φ e acts as an algebra map between the shuffle algebra on R X and the shuffle algebra on End(R X ). That is, φ e (c d) = φ e (c) φ e (d). Hence, (c d) e = φ e (c d)() = φ e (c)() φ e (d)() = (c e ) (d e ). It is easily checked that dist(c, d ) := max(dist(c L, d L ), dist(c R, d R )) is an ultrametric on R X, and R X is complete. 3 The following lemma states that the mixed composition product reduces ultrametric distance in a certain sense. Lemma 3.4. For any c R X and d,, d,2 R X it follows that where c = (c, ) + c with c proper. dist(c d,, c d,2 ) σ ord(c ) dist(d,, d,2 ), Proof. For a fixed d R, consider the map d L c (d L, d R ). Likewise, for a fixed d L there is a companion map d R c (d L, d R ). It is first shown that on the ultrametric space (R X, dist), (3.3a) (3.3b) dist(c (d L,, d R ), c (d L,2, d R )) σ ord(c ) dist(d L,, d L,2 ), dist(c (d L, d R, ), c (d L, d R,2 )) σ ord(c ) dist(d R,, d R,2 ). Inequality (3.3a) is clearly true when d L, = d L,2. When d L, d L,2, it is first necessary to show that for any η X (3.4) ord(φ d (η)() φ d2 (η)()) η + ord(d L, d L,2 ). The proof is by induction on the length of η. The claim is trivial when η is empty or a single letter. Assume the inequality holds for words up to length k. For any x η with η X k, inequality (3.4) follows directly from the induction hypothesis. The case for x η is handled as follows: ord(φ d (x η)() φ d2 (x η)()) = ord(x [d L, φ d (η)() d L,2 φ d2 (η)()] + x [d R (φ d (η)() φ d2 (η)())]) = ord(x [(d L, d L,2 ) φ d (η)() + d L,2 (φ d (η)() φ d2 (η)())] + x [d R (φ d (η)() φ d2 (η)())]) 3 Using dist for both the ultrametric on R X and R X should cause minimal confusion since their arguments are distinct.

11 894 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR + min(ord([d L, d L,2 ] φ d (η)()), ord(d L,2 [φ d (η)() φ d2 (η)()]), ord(d R [φ d (η)() φ d2 (η)()])) + min(ord(d L, d L,2 ) + η, ord(φ d (η)() φ d2 (η)())) = η + + ord(d L, d L,2 ). Therefore, (3.4) holds for any η X. The inequality (3.3a) is now derived. Observe dist(c (d L,, d R ), c (d L,2, d R )) = dist(c (d L,, d R ), c (d L,2, d R )) = σ ord( η (c,η)[φ d (η)() φ d2 (η)()]) σ min η supp(c ) ord(φ d (η)() φ d2 (η)()) σ min η supp(c ) η +ord(d L, d L,2 ) = σ ord(c ) dist(d L,, d L,2 ). The proof for (3.3b) is completely analogous. The final step of the proof is to employ the ultrametric property iv in conjunction with (3.3). Observe dist(c d,, c d,2 ) = dist(c (d L,, d R, ), c (d L,2, d R,2 )) max(dist(c (d L,, d R, ), c (d L,2, d R, )), dist(c (d L,2, d R, ), c (d L,2, d R,2 ))) σ ord(c ) max(dist(d L,, d L,2 ), dist(d R,, d R,2 )) = σ ord(c ) dist(d,, d,2 ). The group product on R X is now defined in terms of the mixed composition product. Its basic properties are given in the subsequent lemma. efinition 3.5. The composition product on R X is defined as c d = ((c L d ) d L, (c L d ) d R + c R d ). Lemma 3.6. The composition product on R X () is left R-linear; (2) satisfies (c d ) e = c (d e ) (mixed associativity); (3) is associative. Proof. () This property follows from the left linearity of the mixed composition product. (2) In light of the first item, it is sufficient to prove the claim only for c = η X k, k. The cases k = and k = are trivial. Assume the claim holds up to some fixed k. Then via Lemma 3.3(4) and the induction hypothesis it follows that ((x η) d ) e = (x (η d )) e = x ((η d ) e ) = x (η (d e )) = (x η) (d e ). In a similar fashion, apply the properties in Lemma 3.3(), (4), and (5) to get ((x η) d ) e = [x (d L (η d )) + x (d R (η d ))] e = [x (d L (η d ))] e + [x (d R (η d ))] e = x [e L ((d L (η d )) e )] + x [e R ((d L (η d )) e )] + x [(d R (η d )) e ]

12 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 895 = x e L (d L e ) ((η d ) e ) }{{} (d e ) L + x ((d L e ) e R + d R e ) ((η d ) e ). }{{} (d e ) R Now employ the induction hypothesis so that ((x η) d ) e = x [(d e ) L (η (d e ))] + x [(d e ) R (η (d e ))] = (x η) (d e ). Therefore, the claim holds for all η X, and the identity is proved. (3) First apply efinition 3.5 twice and Lemma 3.3() and (5) to get (c d ) e = ((c L d ) d L, (c L d ) d R + c R d ) e = ([((c L d ) d L ) e ] e L, [((c L d ) d L ) e ] e R + [(c L d ) d R + c R d ] e ) = ([(c L d ) e ] [d L e ] e L, [(c L d ) e ] [d L e ] e R + ((c L d ) e ) (d R e ) + (c R d ) e ). Now apply the mixed associativity property from the previous item and then recombine terms according to efinition 3.5 so that (c d ) e = ([c L (d e )] [d L e ] e L }{{} (d e ) L, [c L (d e )] and the lemma is proved. [d L e ] e R + (c L (d e )) (d R e ) + c R (d e )) = ([c L (d e )] (d e ) L, [c L (d e )] [(d L e ) e R + d R e }{{} (d e ) R ] + c R (d e )) = ((c (d e )) L, (c (d e )) R ) = c (d e ), The Fliess operator interpretations of the two formal power series products described above are given in the next theorem. Theorem 3.7. For any c R LC X and c, d R LC X such that c d R LC X and c d R LC X, the following identities hold: () F c F d = F c d, (2) F c F d = F c d. Proof. () It is sufficient to prove the claim for c = η X. This is done by induction on the length of η. The case for the empty word is trivial. Assume the identity holds for words η X k up to some fixed length k. Then t E xη F d [u](t) = E η [F d [u]](τ, t ) dτ = F η d [u](τ) dτ t t = F x(η d )[u](t) = F (xη) d [u](t). t

13 896 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR Similarly, E xη F d [u](t) = t t F η d [u](τ)[f dl [u](τ)u(τ) + F dr [u](τ)] dτ t Hence, the claim holds for all η X. (2) Observe = F dl (η d )[u](τ)u(τ) + F (η d ) d R [u](τ) dτ t = F x(d L (η d ))+x ((η d ) d R )[u](t) = F (xη) d [u](t). F c F d [u] = (F cl [u]u + F cr [u]) (F dl [u]u + F dr [u]) = F cl [F d [u]]f dl [u]u + F cl [F d [u]]f dr [u] + F cr [F d [u]] = F (cl d ) d L [u]u + F (cl d ) d R [u] + F cr d [u] = F (c d ) L [u]u + F (c d ) R [u] = F c d [u]. Let R np X denote the subset of R X whose left series are nonproper. Theorem 3.8. The set (R np X,, (, )) is a group. Proof. From Lemma 3.3(2), it follows directly that c (, ) = c. Using the identities c = and c = from Lemma 3.3(3), it is straightforward to see that (, ) c = c. Associativity of the group product was established in Lemma 3.6(3). So the only open issue is the existence of inverses. Suppose c is fixed, and one seeks a right inverse c := (c L, c R ), that is, c c = (, ). 4 Then it follows directly from Theorem 2.4 and efinition 3.5 that (3.5a) (3.5b) c L c R It is first shown that the mapping = (c L (c L, c = c L R )), (c R (c L, c R )). S R :(e L, e R ) ((c L (e L, e R )), e L (c R (e L, e R ))) is an ultrametric contraction on R X and therefore has a unique fixed point, which by design is a right inverse, c. Note that for any e it follows that (S R (e L, e R ) L, ) = (c L, ). Thus, the fixed point will always be in the group. Then it is shown that this same series is also a left inverse, that is, c c = (, ), or equivalently, (3.6a) (3.6b) c L = (c L (c L, c R )), c R = c L (c R (c L, c R )). To establish the first claim, observe via Corollary 2.3 and Lemma 2.5 that for any e, ē R X dist(s R (e ), S R (ē )) = max(dist((c L (e L, e R )), (c L (ē L, ē R )) ), dist( e L (c R (e L, e R )), ē L (c R (ē L, ē R )))) max(dist(c L (e L, e R ), c L (ē L, ē R )), dist(c R (e L, e R ), c R (ē L, ē R ))). 4 The notation c, c R X, is used in this situation instead of c to distinguish it from c.

14 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 897 Thus, from Lemma 3.4 it follows that dist(s R (e ), S R (ē )) max(σ ord(c L ) dist((e L, e R ), (ē L, ē R )), σ ord(c R ) dist((e L, e R ), (ē L, ē R ))) σ dist(e, ē ). To address the second claim, suppose c satisfies (3.5a). Then it follows that (c L c ) c L =, (c L c ) (c L Using Lemma 3.6(2) and Lemma 3.3(5) gives (c L c ) ((c L (c L (c L (c c )) =. c ) c )) =, c )) c =. Applying Lemma 3.3(3) then yields c L (c L c )=, and thus, c L =(c L c ), which is (3.6a). If, in addition, c also satisfies (3.5b), then substituting (3.5a) into (3.5b) gives c R = (c L c ) (c R c ). Therefore, in a similar fashion, (c L c ) c = c R c ( c L (c R (c R + c L (c R R c )) c c )) c =., = c R c, Once again applying Lemma 3.3(3) gives c R +c L (c R c ) =, which is equivalent to (3.6b). The following theorem is the first main result of this section. The issue of freeness will be addressed in section 5. Theorem 3.9. The group (R np X,, (, )) acts as a right transformation group on R X via the action c d. Proof. See Lemma 3.3(2) and Lemma 3.6(2). The second main result of the section is a formula for the group element e which describes the output affine feedback connection shown in Figure. This defines the output affine feedback product. Theorem 3.. Let c R LC X and d = (d L, d R ) R LC X. Then the generating series for the output affine feedback connected system is c@d := c e, where (3.7) e = ((d L c), (d L c) (d R c)), provided that e R LC X and c@d R LC X. In addition, c@d satisfies the fixed-point equation (3.8) c@d = c (d L (c@d ), d R (c@d )). Proof. From Figure it is clear that u = F dl [y]v + F dr [y] = F dl c[u]v + F dr c[u]. Therefore, using the nonproperness of d L c, and the fact that the shuffle product, shuffle inverse, and composition product on R LC X all preserve local convergence, it follows that u F dr c[u] = F dl c[u]v,

15 898 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR F (dl c) [u](u F d R c[u]) = v, F ((dl c), (d L c) (d R c))[u] = v. It is easily verified that e := ((d L c), (d L c) (d R c)) R np X R LC X, so if its inverse is locally convergent, then u = F e [v], in which case, y = F c [u] = F c e [v] = F c@d [v]. Since generating series are always unique, this implies identity (3.7). To determine the fixed-point equation that c@d must satisfy, observe that the output affine feedback equation is y = F c [F dl [y]v + F dr [y]]. So if y = F c@d [v], then y = F c [F dl [F c@d [v]]v + F dr [F c@d [v]]] = F c (dl (c@d ),d R (c@d ))[v]. Again, since generating series are unique, (3.8) must hold. Example 3.. Consider the feedback system shown in Figure with d L =. As discussed in the introduction, this corresponds to conventional output feedback and was the main object of study in [6, 7]. In this case, F c [u] = u + F cr [u] when c R R LC X, and (3.7) (3.8) reduce to their expected forms e = (, (d R c)) and (c@d ) = c (, d R (c@d )), respectively, in Theorem 2.. Example 3.2. Consider now the case where d R = in Figure, i.e., pure multiplicative feedback as used, for example, in phase-locked loops. This problem lies entirely outside the framework of earlier work by the authors. In this case, (3.7) (3.8) reduce to e = ((d L c), ) and c@d = c (d L (c@d ), ), respectively. A specific example exercising these formulas is given in Example 4. after the Hopf algebra machinery is developed in the next section for computing the composition inverse. 4. Hopf algebra of coordinate functions for R np X. In this section the Hopf algebra of coordinate functions for the group R np X is described. This provides an explicit computational framework in which to compute group inverses and thus to evaluate the output affine feedback product as described in the previous section. It is first necessary to restrict the setup to the subset of series having the form c = ( + c L, c R), where c L is proper. From a control theory point of view, there is no loss of generality since the generating series of any Fliess operator y = F c [u] can assume this form by simply rescaling y. So abusing the notation, in this section R np X will be used to denote exclusively this subset. The strategy is to first introduce a connected graded algebra and then a compatible coalgebra in terms of the composition products. The connectedness property ensures that this bialgebra is a Hopf algebra. The section culminates by providing a purely inductive formula for the antipode of this Hopf algebra, and some illustrative examples are given. For any word η X define the left and right coordinate functions as (4.) b η : R np X R : c (c L, η), a η : R np X R : c (c R, η), respectively. 5 Let V denote the R-vector space spanned by these maps. efine the corresponding free commutative algebra, H, with product µ : h η h ξ h η hξ, h, h {a, b} and unit which maps every c R np X to. Consequently, definition (4.) is extended multiplicatively, that is, for h, h {a, b} the product h η hξ corresponds to the pointwise product of coordinate functions with respect to c = (c L, c R ) R np X (4.2) h η hξ (c ) := h η (c ) h ξ (c ). 5 This terminology will be justified later.

16 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 899 The degree of a coordinate function is taken as deg(b η ) = 2 η x + η x, deg(a η ) = 2 η x + η x +, and deg() =. Therefore, V is a connected graded vector space, that is, V = n V n with V n denoting the span of all coordinate functions of degree n and V = R. Let V + = n V n. Similarly, H has the connected graduation H = n H n with H = R. Three coproducts are now introduced. The first coproduct is used to define the Hopf algebra on H. The remaining two products provide an inductive method in which to compute it. Recalling that (c d ) L = (c L d ) d L and (c d ) R = (c L d ) d R + c R d, define coproduct for any b η, a η V + such that (4.3a) (4.3b) b η (c, d ) = b η (c d ) = ((c L d ) d L, η), a η (c, d ) = a η (c d ) = ((c L d ) d R, η) + (c R d, η). The coassociativity of follows directly from the associativity of the group product on R np X. The second coproduct is (V + ) V + V +, which is defined for any h, h V + by (4.4a) (4.4b) h h =h h, h θ k =(θ k + θ k ) h, where θ k denotes the endomorphism on V + specified by θ k h η = h xk η for k =,. Clearly this coproduct can be computed recursively. Finally, consider for any η X the coproduct (V + ) V + H, where (4.5) bη (c, d ) = (c L d, η), aη (c, d ) = (c R d, η). In either case, using the notation of Sweedler [36], h η (c, d ) = h η() (c ) h η(2) (d ) = h η() h η(2) (c, d ), where h η() V + and h η(2) H. The sums are taken over all the terms that appear in the respective composition in (4.5), and the specific nature of the factors h η() and h η(2) is not important here. Like the coproduct, this coproduct can also be computed inductively as described next. Lemma 4.. The following identities hold: () h = h, (2) θ = (θ µ) ( id) b, (3) θ = (θ id) + (θ µ) ( id) a, where id denotes the identity map on H. Proof. () Assume h = a and write c R = x c R + x c R + (c R, ) with c R, c R R X. Then from Lemma 3.3() and (4) it follows that c R d = (x c R) d + (x c R) d + (c R, ) d = x (c R d ) + x (d L (c R d )) + x (d R (c R d )) + (c R, ). Therefore, a (c, d ) = (c R d, ) = (c R, ) = (a )(c, d ). A similar argument holds when h = b.

17 9 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR (2) If h = a, then ( θ i )a η (c, d ) = a xiη(c, d ) = (c R d, x i η) = (x i [x (c R d ) + x (d L (c R d )) + x (d R (c R d ))], η) = ( i [c R d + d R (c R d )] + i [d L (c R d )], η), where x i ( ) is the R-linear left-shift operator specified by x i (η) = η when η = x i η with η X and zero otherwise, and xy is the indicator function. So xy = when x = y and zero otherwise. Letting c = (c L, c R ), it follows that ( θ )a η (c, d ) = (d L (c R d ), η) = (c R d, ξ)(d L, ν)(ξ ν, η) ξ,ν X = aξ (c, d )b ν (d )(ξ ν, η) ξ,ν X = aξ() a ξ(2) (c, d )b ν (d )(ξ ν, η) ξ,ν X = ξ,ν X = (θ id) θ (a ξ() ) a ξ(2) (c, d )b ν (d )(ξ ν, η) ξ,ν X aξ b ν (c, d, d )(ξ ν, η) = (θ µ) ( id) b a η (c, d ). The proof when h = b is perfectly analogous. (3) If h = a, then ( θ )a η (c, d ) = (c R d, η) + (d R (c R d ), η). From this point, the method of proof is exactly the same as that in part (2) modulo the fact that a is used here due to the presence of d R instead of d L in the shuffle product. Example 4.2. Applying the identities in Lemma 4. gives the first few coproduct terms ordered by degree n b (h = b), n a (h = a): n b, n a =, : h = h, n b, n a =, 2 : h x = h x, n b, n a = 2, 3 : h x = h x + h x a, n b, n a = 2, 3 : h x 2 = h x 2 + h x b x, n b, n a = 3, 4 : h xx = h xx + h x a x + h x 2 a, n b, n a = 3, 4 : h xx = h xx + h x b x + h x 2 a, n b, n a = 3, 4 : h x 3 = h x 3 + 3h x 2 b x + h x b x 2, n b, n a = 4, 5 : h x 2 = h x 2 + h x a x + h xx a + h xx a + h x 2 (a ) 2. The next lemma provides a grading for the coproduct, which is clearly evident in the example above. Its particular form, namely, having V on the left, implies that H is a right-sided Hopf algebra [29, 3], a property that is essential in section 6.

18 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 9 Lemma 4.3. For any h η V n (4.6) hη j+k=n V j H k =: (V H) n. Proof. The following facts are essential: () deg(θ h) = deg(h) +, (2) deg(θ h) = deg(h) + 2, (3) h h (V V ) deg(h)+ ha. The proof is via induction on the length of η. When η = then clearly h = h V n H, where n = deg(h ) {, } (noting that b ). Assume now that (4.6) holds for words up to length η. Let n = deg(h η ). There are two ways to increase the length of η. First consider h xη. From item () above deg(h xη) = n +. Now apply item (3), the induction hypothesis, and Lemma 4. in that order: b h η (V V ) n, ( id) b h η (V H V ) n, (θ µ) ( n id) b h η V j+ H k, j+k= h xη (V H) n+, which proves the assertion. Consider next h xη. From item (2) above deg(h xη) = n + 2. In this case, repeat the first two steps of the previous case and apply item () to get (θ µ) ( id) a h η n+ j+k= V j+ H k (V H) n+2. In addition, from the induction hypothesis and item (2) it follows that (θ id) h η n j+k= V j+2 H k (V H) n+2. Thus, applying Lemma 4., h xη (V H) n+2, which again proves the assertion and completes the proof. The following lemma is of central importance as it shows how to compute the Hopf algebra coproduct in terms of and. Lemma 4.4. The following identities hold: () b η = (id µ) ( id) b b η, (2) a η = (id µ) ( id) a b η + a η. Proof. () Observe b η (c, d ) = (c L d, ξ)(d L, ν)(ξ ν, η) ξ,ν X = bξ (c, d )b ν (d )(ξ ν, η) ξ,ν X

19 92 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR = ( id) b b η (c, d, d ) = (id µ) ( id) b b η (c, d ). (2) In a similar fashion a η (c, d ) = ( id) a b η (c, d, d ) + a η (c, d ) = [(id µ) ( id) a b η + a η ](c, d ). Example 4.5. Applying the identities in Lemma 4.4 gives the first few reduced coproduct terms, namely, h η := h η h η h η : n = : b x =, n = 2 : b x = b x a, n = 2 : b x 2 = 3b x b x, n = 3 : b xx = b x b x + b x b x + b x a x + b x b x a + b x 2 a, n = 3 : b xx = b x b x + 2b x b x + b x b x a + b x 2 a, n = 3 : b x 3 = 6b x 2 b x + 4b x b x 2 + 3b x (b x ) 2, n = 4 : b x 2 = 2b x b x + b x a x + 2b x b x a + b xx a + b xx a n = : a =, + b x 2 (a ) 2, n = 2 : a x = b x a, n = 3 : a x = b x a + a x a + b x (a ) 2, n = 3 : a x 2 = a x b x + 2b x a x + b x b x a + b x 2 a, n = 4 : a xx = b x a x + b x a x + a x a x + b xx a + a x 2 a + 2b x a x a + b x 2 (a ) 2, n = 4 : a xx = a x b x + b x a x + b x a x + b x b x a + b xx a + a x 2 a + b x a x a + b x 2 (a ) 2, n = 4 : a x 3 = b x 3 a + 3b x 2 b x a + b x b x 2 a + 3b x 2 a x + 3b x b x a x + 3b x a x 2 + 3a x 2 b x + a x b x 2, n = 5 : a x 2 = 2b x a x + a x a x + b x 2 a + a xx a + a xx a + 3b x a x a + b xx (a ) 2 + b xx (a ) 2 + a x 2 (a ) 2 + b x 2 (a ) 3. The main result of this section is given next, namely, that the commutative unital algebra (H, µ) together with the coproduct is a connected graded bialgebra and thus has an antipode S. Before stating the actual theorem, some additional motivation is given concerning the nature of the antipode. Observe that for any c R np X one can naturally associate a Hopf algebra character Φ c L(H, R) by letting Φ c : b η (c L, η), Φ c : a η (c R, η), and Φ c () =. Note that (4.2) implies Φ c (h η hξ ) = Φ c (h η )Φ c ( h ξ ) for any h, h {a, b} and η, ξ X. Since the coproduct of H has been constructed to satisfy (4.3), the following simple calculation shows that for all c, d R np X, the convolution product of Φ c, Φ d L(H, R) corresponds to composition in R np X : (Φ c Φ d )(h η ) := m R (Φ c Φ d ) (h η ) = h η (c d ) = (c d, η).

20 SISO OUTPUT AFFINE FEEBACK TRANSFORMATION GROUP 93 (See [7, Lemma 2] for a similar calculation.) In light of (2.8), it then follows that (4.7) h η (c ) = (Sh η)(c ) η X, h {b, a}. Thus, the antipode provides an explicit way to compute the group inverse. Theorem 4.6. (H, µ, ) is a connected graded commutative unital Hopf algebra. Proof. From the development above, it is clear that (H, µ, ) is a connected bialgebra with unit and a counit ε defined by ε(a η ) = for all η X, ε(b η ) = for all nonempty η X, and ε() =. Here it is shown that this bialgebra is also graded and thus is automatically a Hopf algebra, i.e., has a well defined antipode, S []. It only remains to be shown for any n that H n (H H) n. It is well known if h V n, then h h (V V ) n. Therefore, it follows directly from Lemmas 4.3 and 4.4 that h (V H) n. Thus, via the identity (a i ηa j ξ ) = ai η a j ξ, it must hold that H n (H H) n, n. The next theorem says that the antipode of any connected graded Hopf algebra can be computed in a recursive manner once the coproduct is computed. Theorem 4.7 (see []). The antipode, S, of any connected graded Hopf algebra (H, µ, ) can be computed for any a H k, k, by Sa = a (Sa () )a (2) = a a () Sa (2), where the reduced coproduct is a = a a a = a () a (2). As noted earlier, the coproducts and can be computed recursively, and is computed directly in terms of and. So in fact the antipode of H can be computed in a fully recursive manner as described next. Theorem 4.8. The antipode, S, of any h η V + can be computed by the following algorithm: i. Recursively compute h via (4.4). ii. Recursively compute via Lemma 4.. iii. Compute via Lemma 4.4. iv. Recursively compute S via Theorem 4.7. Example 4.9. The first few antipode terms computed via Theorem 4.8 are n = : Sb x = b x, n = 2 : Sb x = b x + b x a, n = 2 : Sb x 2 = b x 2 + 3(b x ) 2, n = 3 : Sb xx = b xx + b x 2 a 3(b x ) 2 a + 2b x b x + b x a x, n = 3 : Sb xx = b xx + b x 2 a + 3b x b x 3(b x ) 2 a, n = 3 : Sb x 3 = b x 3 + b x b x 2 5(b x ) 3, n = 4 : Sb x 2 = b x 2 + b xx a + b xx a b x 2 (a ) 2 + 3(b x ) 2 (a ) 2 + 2(b x ) 2 5b x b x a b x a a x + b x a x, n = : Sa = a, n = 2 : Sa x = a x + b x a,

21 94 W. STEVEN GRAY AN KURUSCH EBRAHIMI-FAR n = 3 : Sa x = a x + b x a b x (a ) 2 + a a x, n = 3 : Sa x 2 = a x 2 + b x 2 a 3(b x ) 2 a + 3b x a x, n = 4 : Sa xx = a xx + b xx a 2b x b x a 4b x a a x + 3(b x ) 2 (a ) 2 b x 2 (a ) 2 + (a x ) 2 + a a x 2 + b x a x + b x a x, n = 4 : Sa xx = a xx + b xx a + a a x 2 3b x b x a 3b x a a x b x 2 (a ) 2 + 3(b x ) 2 (a ) 2 + 2b x a x + b x a x, n = 4 : Sa x 3 = a x 3 + b x 3 a b x b x 2 a + 5(b x ) 3 a + 4b x 2 a x 5(b x ) 2 a x + 6b x a x 2, n = 5 : Sa x 2 = a x 2 + b x 2 a 2(b x ) 2 a + a a xx + a a xx 3b x a x a 3b x a x a a (a x ) 2 b xx (a ) 2 b xx (a ) 2 + 5b x b x (a ) 2 (a ) 2 a x 2 + 4b x (a ) 2 a x + b x 2 (a ) 3 3(b x ) 2 (a ) 3 + 2b x a x + a x a x. Example 4.. Reconsider the multiplicative feedback system described in Example 3.2. Suppose c = k k! xk = k x k = ( x ) and F d = I (unity feedback). In this case, (3.8) reduces to (c@d ) = c (d L c@d, ) so that c@d = c (c, ) = (c, ) = ( x, ). The composition inverse can be computed directly from (4.7) and the antipode formulas above. Namely, taking into account only the antipode terms containing powers of the coordinate function b x (all others will yield zero), it follows that c@d = ( x, ) = + ηsb η ( x ), ηsa η () η X + η X ([ ] ) = + x Sb x + x 2 Sb x 2 + x 3 Sb x 3 + ( x ), ([ = ( + x ( b x ( x )) + x 2 b x 2 + 3(b x ) 2] ( x ) ([ + x 3 b x 3 + b x b x 2 5(b x ) 3] ) ) ( x ), = ( + x + 3x 2 + 5x , ). The deferred proof from section 3 is presented next. Proof of Lemma 3.3(3). The assertion is that c d = k implies c = k. The proof is by induction on the grading of H. If c d = k, then clearly k = a (c d ) = a (c, d ) = a (c ) assuming without loss of generality that c = (, c). Therefore, (c, ) = k. Similarly, it follows that = a x (c d ) = a x (c, d ) = a x (c ). Thus, (c, x ) =. Now suppose a η (c ) = for all a η H n up to some fixed n 2. Then for any x j X = a xjη(c, d ) = a xjη(c ) + a xjη()(c ) a xjη(2)(d ), a xj η(2) where in general a xjη() a. Therefore, a xjη(c ) =, or equivalently, (c, x j η) =. Thus, c = k. The section is concluded by some dimensional analysis of the grading of V and H. This information could be useful in determining the complexity of the antipode recursion as a function of the degree of the coordinate functions, but that topic is beyond