Formal Series Approach to Multiplicative Dynamic Feedback Connection]A Formal Power Series Approach to Multiplicative Dynamic Feedback Connection

Venkatesh G. S.
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway
Kurusch Ebrahimi-Fard
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway

The goal of the paper is multi-fold. The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen–Fliess series description is in multiplicative output feedback connection with another system given in Chen–Fliess series description. Further, the objective extends in showing that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the dual Hopf algebras corresponding to the shuffle group and the multiplicative feedback group.

§ INTRODUCTION

The objective of the document is two fold and works with the Chen–Fliess functional series [Fliess, 1981]. There is no need that these input-output systems have a state space realization and thus, the results presented here are independent of any state space embedding when a realization is possible [Fliess, 1983]. Firstly, let $F_c$ and $F_d$ be two nonlinear input-output systems represented by Chen–Fliess series. It was shown in [Gray & Li, 2005] that the additive feedback interconnection of two such systems result in a Chen–Fliess series description for the closed-loop system. The convergence of the closed-loop system was characterized in [Thitsa & Gray, 2012]. An efficient computation of the generating series for closed-loop system is facilitated through a combinatorial Hopf algebra [Gray, et al., 2014, Foissy, 2015, Duffaut Espinosa, et al., 2016]. The feedback product formula and its computation were used to solve system inversion problems [Gray, et al., 2014] and trajectory generation problems [Duffaut Espinosa & Gray, 2017].

However, when the nature of interconnection becomes multiplicative feedback, the similar set of questions persist in general. It is known that, in single-input single-output (SISO) setting, the closed-loop system in the affine feedback case (of which multiplicative feedback is a special case) has a Chen–Fliess series description and the computation of feedback formula is facilitated through a combinatorial Hopf algebra <cit.>. The present document, in one part, shows that even in multi-input multi-output (MIMO) setting the closed-loop system under multiplicative feedback has a Chen–Fliess series representation and provides an explicit expression of the closed-loop generating series which will be called as multiplicative dynamic feedback product . Furthermore, it will be shown that this feedback product has a natural interpretation as a transformation group acting on the plant. The algorithmic framework for the computation of the multiplicative dynamic feedback product formula for a general MIMO case is devised using the dual Hopf algebras corresponding to the shuffle product and to the multiplicative dynamic output feedback group. The characterization of convergence of the Chen–Fliess series for the closed-loop system is deferred for future work.

The paper is organized as follows. The next section provides a summary of the concepts related to non-commutative formal power series, Hopf algebra,  Chen–Fliess series and their interconnections. The Section <ref> builds the pivotal multiplicative dynamic output feedback group. The Hopf algebra construction corresponding to the shuffle group is drafted in Section <ref>.  Section <ref> is where the multiplicative dynamic feedback connection is analyzed. The invariance of relative degree under multiplicative output feedback is asserted in Section <ref>. The framework for computing the feedback product is devised in Section <ref> and is demonstrated using examples. The conclusions of the paper and directions for future work is given in the last section.

§ PRELIMINARIES: FORMAL POWER SERIES

A finite nonempty set of noncommuting symbols $X=\{ x_0,x_1,\ldots,x_m\}$ is called an alphabet. Each element of $X$ is called a letter. Any finite sequence, $\eta=x_{i_1}\cdots x_{i_k}$, of letters from $X$ is called a word over $X$ and its length is $\abs{\eta}=k$. The set $X^\ast$ of all words includes the empty word, denoted $\emptyset \in X^\ast$ and $X^+:=X^\ast\backslash\emptyset$, and forms a monoid under catenation. Any mapping $c: X^\ast\rightarrow\re^\ell$ is called a formal power series. The value of $c$ at $\eta\in X^\ast$ is denoted by $(c,\eta)$ and called the coefficient of $\eta$ in $c$. Normally, $c$ is written as a formal sum $c=\sum_{\eta\in X^\ast}(c,\eta)\eta.$  A series $c$ is proper when the coefficient $(c,\emptyset)=0$ else it is a non-proper series. The support of $c$ is the set $\supp(c)$ containing all words having nonzero coefficients. The order of $c$, denoted $\orde(c)$, is the length of the minimal length word in its support. The collection of all formal power series over $X$ is denoted by $\allseriesell$. The $i^{th}$ component of a vector $v \in \re^{\ell}$ is denoted by $v_i$ and consequently the $i^{th}$ component of a series $c \in \allseriesell$ is denoted by $c_i$ viz. $\left(c_i,\eta\right) = \left(c,\eta\right)_i$.

A series $c^{\prime} \in \allseries{\ell}$ is called a subseries of $c \in \allseries{\ell}$ if there exists another series $c^{\prime\prime} \in \allseries{\ell}$ such that the intersection $\supp\left(c^{\prime}\right) \cap \supp\left(c^{\prime\prime}\right)$ is empty and the series $c$ can be decomposed as $c = c^{\prime} + c^{\prime\prime}$.

Let $c \in \allseries{\ell}$, then the natural part of the series $c$ is the subseries denoted by $c_N$ such that $c = c_N + c_F$ and $\supp\left(c_F\right) \subseteq X^{\ast}\setminus\{x_0^k : k \in \nat_0\}$. The subseries $c_F$ is called as forced part of the series $c$.

Definition <ref> asserts that the forced part $c_F$ of a series $c$ should not contain any word formed by the letter $x_0$ alone, including the empty word $\emptyset$. For the remainder of the document, $\re^{\ell}$ is given the structure of a unital commutative ring under Hadamard or pointwise product viz. $\left(xy\right)_i = x_iy_i$ with $\doubleone = \left[1\,1\cdots1\right]^{t} \in \re^{\ell}$ being the multiplicative unit. Formal power series $\allseriesell$ form a $\re^{\ell}$-module and the submodule of all proper series in $\allseriesell$ is denoted by $\allproperseries{\ell}$, while the subset of non-proper series is denoted by $\allnproperseries{\ell}$.

A series $c \in \allseries{\ell}$ is called purely improper if $c_i$ is non-proper $\forall i = 1,\ldots,\ell$. The subset of all purely improper series in $\allseriesell$ is denoted by $\allpiseries{\ell}$.

Observe that $\allpiseries{\ell} \subsetneq \allnproperseries{\ell}$ if $\ell > 1$, otherwise $\allpiseries{} = \allnproperseries{}$.

§.§ Shuffle Product

The shuffle product $\alpha \shuffle \beta$ of two words is a bilinear product on the linear span of words, which can be uniquely specified iteratively
where  $\eta,\xi\in X^\ast$ and $x_i,x_j\in X$. See for instance [Fliess, 1981]. The shuffle product of two series, $\left(c,d\right) \mapsto c\shuffle d$ is defined as

\begin{align*}
(c\shuffle d,\eta) = \sum_{\substack{\zeta,\nu \in X^{\ast}\\ \eta \in \supp(\zeta \shuffle \nu)}} \left(c,\zeta\right) \left(d,\nu\right).
\end{align*}

We define for any $x_i,x_j \in X$ and any word $\eta \in X^{\ast}$

\eta,  & i=j\\
0, 	& {\text{else}}
\end{cases}

The following proposition is vital in understanding the bialgebra and Hopf algebra devised in Sections <ref> and <ref>.

If $c,d \in \allseries{\ell}$, then $\forall x_i \in X$

\begin{align*}
x_i^{-1}\left(c\shuffle d\right) &= \left(x_i^{-1}\left(c\right)\shuffle d\right) + \left(c\shuffle x_i^{-1}\left(d\right)\right).
\end{align*}

Note that $\allseries{\ell}$ forms an associative and commutative $\re^{\ell}$-algebra under the shuffle product. If $d \in \allpiseries{\ell}$, then shuffle inverse of $d$, denoted by $d^{\shuffle -1}$ is defined as

\begin{align*}
d^{\shuffle -1}_i = \left(d_i,\emptyset\right)^{-1}\left(\sum_{k \in \nat_0} \left(d_i^{\prime}\right)^{\shuffle k}\right),
\end{align*}

where $d_i^{\prime} = 1 - \left(d_i/\left(d_i,\emptyset\right)\right)$. Hence, $\allpiseries{\ell}$ forms an Abelian group under the shuffle product with $\doubleone$ as the identity element.

Let $X = \{x_0,x_1\}$ and $c \in \allseries{}$ described as $c = 1-x_1$. Then the shuffle inverse is computed as:

\begin{align*}
c^{\shuffle -1} &= \sum_{k \in \nat_0} \left(1-\left(1-x_1\right)\right)^{\shuffle k} \\
&= \sum_{k \in \nat_0} x_1^{\shuffle k} = \sum_{k \in \nat_0} k!x_1^k.
\end{align*}

Therefore, $c^{\shuffle -1} = 1 + x_1 + 2x_1^2 + 6x_1^3 + \cdots + n!x_1^n + \cdots$.

Observe that $\left(c\shuffle d, \emptyset\right) = \left(c,\emptyset\right) \left(d, \emptyset\right)$. Hence, the set

\begin{align*}
M_{\tiny{\shuffle}} = \lbrace \doubleone + c \: : c \in \allproperseries{n} \rbrace,
\end{align*}

where $c $ is a proper series in $\allseries{n}$, forms a subgroup of the shuffle group.  The group $M_{\tiny{\shuffle}}$ is vital in the design of a computational framework of multiplicative dynamic feedback product as explained in Section <ref>.

The set $\allseriesell$ is endowed with ultrametric structure where the metric $\kappa$ is defined as
κ(c,d) = σ^(c-d),
for $c,d \in \allseriesell$ and $\sigma \in \: ]0,1[$. For brevity, $\kappa(c,0)$ is written as $\kappa(c)$, and $\kappa(c,d) = \kappa(c-d)$. The ultrametric space $(\allseriesell,\kappa)$ is Cauchy complete [Berstel & Reutenauer, 1988]. The following definition of contraction maps between metric spaces will be useful.

Given metric spaces $(E,d)$ and $(E^{\prime},d^{\prime})$, a map $f : E \longrightarrow E^{\prime}$  is said to be a strong contraction map if $\forall s,t \in E$, it satisfies the condition $d^{\prime}(f(s),f(t)) \leq \alpha d(s,t)$ where $\alpha \in [0,1[$. If $\alpha = 1$, then the map $f$ is said to be a weak contraction map or a non-expansive map.

§ BIALGEBRA AND HOPF ALGEBRA: PRELIMINARIES

The goal is to provide the definitions of algebraic structures such as algebra, coalgebra, bialgebra and Hopf algebra [Abe, 2004, Sweedler, 1969]. We let $K$ be a commutative ring with identity $1_K$.

§.§ Algebra

The definition of an algebra can be facilitated through the category of modules. It allows to define the concept of a coalgebra (the dual notion) with ease.

An algebra over $K$ is a $K$-module $\mathscr{A}$ along with the morphisms of $K$-modules $\HAprod : \mathscr{A} \otimes \mathscr{A} \longrightarrow \mathscr{A}$, called the multiplication or product map, and $\eta : K \longrightarrow \mathscr{A}$, called the unit map, such that the following diagrams are commutative.

\begin{equation}\label{fig:arr_diag_alg}
\begin{array}{cc}
\xymatrix@C=5pc@R=4pc{
\mathscr{A}\otimes\mathscr{A}\otimes\mathscr{A}\ar[r]^{{\makebox{\tiny\textbf{m}}}\otimes \mathbf{id}_{\mathscr{A}}}\ar[d]_{\mathbf{id}_{\mathscr{A}}\otimes {\makebox{\tiny\textbf{m}}}} & \mathscr{A}\otimes\mathscr{A}\ar[d]^{{\makebox{\tiny\textbf{m}}}}\\
\mathscr{A}\otimes\mathscr{A}\ar[r]^{\makebox{\tiny\textbf{m}}} & \mathscr{A}}
& \hspace{1 in}
\xymatrix@C=5pc@R=4pc{
K\otimes \mathscr{A}\ar[r]^{\eta \otimes \mathbf{id}_{\mathscr{A}}}\ar[rd]_{\cong} &\mathscr{A}\otimes\mathscr{A}\ar[d]^{\makebox{\tiny\textbf{m}}} \\
\empty & \mathscr{A} \\
\mathscr{A}\otimes K\ar[ru]^{\cong}\ar[r]^{\mathbf{id}_{\mathscr{A}}\otimes \eta} & \mathscr{A}\otimes\mathscr{A}\ar[u]^{\makebox{\tiny\textbf{m}}}
\end{array}
\end{equation}

The tuple $\left(\mathscr{A},\HAprod,\eta\right)$ is called a $K$-algebra.

The commutative diagrams fig:arr_diag_alg mean that a $K$-algebra $\mathscr{A}$ must satisfy the following properties:

* The product map $\HAprod$ must be associative.

* The scalar multiplication through the $\eta$ map must have a unit.

The concept of a $K$-algebra morphism is defined next.

Let $\left(\mathscr{A},\HAprod,\eta\right)$, $\left(\mathscr{A'},\HAprod',\eta'\right)$ be $K$-algebras. A map $f: \mathscr{A} \longrightarrow \mathscr{A'}$ is called a $K$-algebra morphism provided the following diagrams commute.

\begin{equation*}
\begin{array}{cc}
\xymatrix@C=5pc@R=4pc{
\mathscr{A}\otimes\mathscr{A}\ar[r]^{{\makebox{\tiny\textbf{m}}}}\ar[d]^{f\otimes f} & \mathscr{A}\ar[d]^{f} \\
\mathscr{A'}\otimes\mathscr{A'}\ar[r]^{{\makebox{\tiny\textbf{m}}}'} & \mathscr{A'}
\hspace*{1 in}
\xymatrix@C=5pc@R=4pc{
K\ar[rr]^{\eta}\ar[rd]^{\eta'} & \empty & \mathscr{A}\ar[ld]^{f} \\
\empty & \mathscr{A'} & \empty
\end{array}
\end{equation*}

Let $P$ and $Q$ be modules over $K$. The twisting morphism $\tau$ of $K$-modules is $\tau : P \otimes Q \longrightarrow Q \otimes P$ with

\begin{align*}
\tau (p\otimes q) =  q \otimes p \quad \forall \; q \in Q, p \in P.
\end{align*}

A $K$-algebra $\mathscr{A}$ is commutative if and only if the following diagram commutes.

\begin{align*}
\xymatrix@C=5pc@R=4pc{
\mathscr{A}\otimes\mathscr{A}\ar[d]_{\tau}\ar[rd]^{\tiny\HAprod} & \empty \\
\mathscr{A}\otimes\mathscr{A}\ar[r]^{\tiny\HAprod} & \mathscr{A}
\end{align*}

A $K$-algebra $\mathscr{A}$ is a graded algebra if the underlying $K$-module structure is graded viz. $\mathscr{A} = \bigoplus_{n \in \nat_{0}} \mathscr{A}_n$, where $\mathscr{A}_n$ is a $K$-module for all $n \in \nat_{0}$ such that $\HAprod\left(\mathscr{A}_m \otimes \mathscr{A}_n\right) \subseteq \mathscr{A}_{m+n}$, for all $m,n \in \nat_{0}$. The graded $K$-algebra is connected if $\eta : K \longrightarrow \mathscr{A}_{0}$ is a $K$-algebra isomorphism.

§.§ Coalgebra

The notion of a $K$-coalgebra is a categorical structure dual to that of a $K$-algebra.

A $K$-coalgebra $\mathscr{C}$ is a $K$-module with the $K$-module morphisms $\Delta : \mathscr{C} \longrightarrow \mathscr{C}\otimes\mathscr{C}$, called the comultiplication or coproduct map, and $\epsilon : \mathscr{C} \longrightarrow K$, called the counit map, such that the following diagrams commute.

\begin{equation}\label{fig:arr_diag_coalg}
\begin{array}{cc}
\xymatrix@C=5pc@R=4pc{
\mathscr{C}\ar[r]^{\Delta}\ar[d]_{\Delta} & \mathscr{C}\otimes\mathscr{C}\ar[d]^{\Delta\otimes \mathbf{id}_{\mathscr{C}}} \\
\mathscr{C}\otimes\mathscr{C}\ar[r]^{\mathbf{id}_{\mathscr{C}}\otimes\Delta} & \mathscr{C}\otimes\mathscr{C}\otimes\mathscr{C}
\xymatrix@C=5pc@R=4pc{
\mathscr{C}\otimes \mathscr{C}\ar[r]^{\epsilon \otimes \mathbf{id}_{\mathscr{C}}} &K\otimes\mathscr{C}\ar[d]_{\cong} \\
\empty & \mathscr{C}\ar[lu]^{\Delta}\ar[ld]^{\Delta} \\
\mathscr{C}\otimes \mathscr{C}\ar[r]^{\mathbf{id}_{\mathscr{C}}\otimes \epsilon} & \mathscr{C}\otimes K\ar[u]^{\cong}
\end{array}
\end{equation}

The tuple $\left(\mathscr{C},\Delta,\epsilon\right)$ is called a $K$-coalgebra.

The commutative diagrams fig:arr_diag_coalg imply that a $K$-coalgebra $\mathscr{C}$ must satisfy the following properties:

* The coproduct map $\Delta$ must be coassociative.

* The counit map $\epsilon$ is the categorical dual to the unit map $\eta$ for a $K$-algebra.

The coalgebra $\mathscr{C}$ is called cocommutative if the following diagram commutes,

\begin{align*}
\xymatrix@C=5pc@R=4pc{
\mathscr{C}\ar[d]_{\Delta}\ar[rd]^{\Delta} & \empty \\
\mathscr{C}\otimes\mathscr{C}\ar[r]^{\tau} & \mathscr{C}\otimes\mathscr{C}
\end{align*}

where $\tau$ is the twisting morphism given in Definition <ref>. Sweedler's notation is very useful in representing the coproduct map and is adopted in Sections <ref> and <ref>.

[Sweedler, 1969]. Given the $K$-coalgebra tuple $(\mathscr{C},\nabla,\epsilon)$ and an element $c \in \mathscr{C}$, then the Sweedler notation for the coproduct

\begin{equation*}
\Delta(c) = \sum_{\left(c\right)} c_{\left(1\right)} \otimes c_{\left(2\right)} ,
\end{equation*}

where $c_{\left(1\right)}, c_{\left(2\right)} \in \mathscr{C}$ are the components of the tensors resulting from the coproduct of $c$.

Next, the definition of a $K$-coalgebra morphism is given.

Let $\left(\mathscr{C},\Delta,\epsilon\right)$, $\left(\mathscr{C'},\Delta',\epsilon'\right)$ be $K$-coalgebras. A map $f : \mathscr{C} \longrightarrow \mathscr{C'}$ is called a $K$-coalgebra morphism provided the following diagrams commute.

\begin{equation*}
\begin{array}{cc}
\xymatrix@C=5pc@R=4pc{
\mathscr{C}\ar[r]^{\Delta}\ar[d]^{f} & \mathscr{C}\otimes\mathscr{C}\ar[d]^{f\otimes f} \\
\mathscr{C'}\ar[r]^{\Delta'} & \mathscr{C'}\otimes\mathscr{C'}
\hspace*{1 in}
\xymatrix@C=5pc@R=4pc{
\mathscr{C}\ar[rr]^{\epsilon}\ar[rd]^{f} & \empty & K \\
\empty & \mathscr{C'}\ar[ru]^{\epsilon'} & \empty
\end{array}
\end{equation*}

§.§ Bialgebra

The bialgebra structure over a commutative ring is fundamental for defining a Hopf algebra. A bialgebra is an amalgamation of the algebra and coalgebra structures such that both are compatible with each other.

A bialgebra $H$ over $K$ is a tuple $\left(H,\HAprod,\eta,\Delta,\epsilon\right)$ such that

* $H$ is a $K$-module.

* $\left(H,\HAprod,\eta\right)$ is a $K$-algebra, where $\HAprod$ and $\eta$ are the product and unit maps, respectively.

* $\left(H,\Delta,\epsilon\right)$ is a $K$-coalgebra, where $\Delta$ and $\epsilon$ are the coproduct and counit maps, respectively.
such that the following diagrams commute.

\begin{equation}\label{comp1_dia}
\xymatrix@C=5pc@R=4pc{
H\otimes H\ar[r]^{\tiny\HAprod}\ar[d]_{\Delta\otimes\Delta} & H\ar[r]^{\Delta} & H\otimes H \\
H\otimes H\otimes H\otimes H \ar[rr]^{\mathbf{id}_{H}\otimes\tau\otimes \mathbf{id}_{H}} &\empty & H\otimes H\otimes H\otimes H\ar[u]_{\tiny\HAprod\otimes\tiny\HAprod}
\end{equation}

\begin{equation}\label{comp2_dia}
\xymatrix@C=5pc@R=4pc{
H\otimes H\ar[r]^{\tiny\HAprod}\ar[rd]_{\epsilon\otimes\epsilon} & H\ar[d]^{\epsilon} \\
\empty & K\cong K\otimes K \ar[ld]_{\eta \otimes \eta}\ar[d]^{\eta} \\
H \otimes H & H\ar[l]_{\Delta}
\end{equation}

\begin{equation}\label{comp3_dia}
\xymatrix@C=5pc@R=4pc{
\empty & H\ar[rd]^{\epsilon} & \empty \\
K\ar[ru]^{\eta}\ar[rr]^{\tiny\id_{K}} & \empty & K
\end{equation}

The diagrams comp1_dia and  comp2_dia state that the product map $\HAprod$ and the unit map $\eta$ are $K$-coalgebra morphisms, while the coproduct map $\Delta$ and the counit map $\epsilon$ are $K$-algebra morphisms. Diagram comp3_dia describes that the unit map $\eta$ is a section of the counit map $\epsilon$ in the category of $K$-modules.

§.§ Hopf Algebra

Hopf algebras are an important class of bialgebras. A Hopf algebra is a bialgebra equipped with a particular $K$-linear map called  antipode.

A Hopf algebra $H$ over $K$ is a tuple $\left(H,\HAprod,\eta,\Delta,\epsilon,S\right)$ such that the following conditions are satisfied:

* $\left(H,\HAprod,\eta,\Delta,\epsilon\right)$ is a $K$-bialgebra.

* $S : H \longrightarrow H$ is a $K$-linear map such that the following diagram commutes.

\begin{equation}\label{antipode_arr_diag}
\xymatrix@C=5pc@R=4pc{
\empty & H\otimes H \ar[rr]^{\mathbf{id}_{H}\otimes S} & \empty & H\otimes H\ar[rd]^{\tiny\HAprod} & \empty \\
H\ar[rr]^{\epsilon}\ar[ru]^{\Delta}\ar[rd]^{\Delta} & \empty & K\ar[rr]^{\eta} & \empty & H \\
\empty & H\otimes H \ar[rr]^{S\otimes \mathbf{id}_{H}} & \empty & H\otimes H\ar[ru]^{\tiny\HAprod} & \empty
\end{equation}

An element $a \in H$ is called group-like if $\Delta(a) = a \otimes a$ and thus $a \notin \ker(\epsilon)$, where $\ker(.)$ represents the kernel of a $K$-module map. A graded Hopf algebra $H = \bigoplus_{n \in \nat_0} H_n$ is connected if and only if $H_0 \cong K\eta(1_K)$ as $K$-modules. Equivalently, a graded Hopf algebra $H$ is connected if and only if $H^{+} := \bigoplus_{k \geq 1} H_k$ is isomorphic to $\ker(\epsilon)$ as $K$-modules viz. $\eta \circ \epsilon = \id_{H_0}$ and zero otherwise. For simplicity denote $\HAprod\left(a,b\right) := ab$, for all $a,b, \in H$. Using Sweedler's notation, diagram antipode_arr_diag implies that for all $c \in H$,

\begin{align*}
\sum_{\left(c\right)} S\left(c_{\left(1\right)}\right) c_{\left(2\right)}
= \sum_{\left(c\right)} c_{\left(1\right)}S\left(c_{\left(2\right)}\right)
= \epsilon\left(c\right)1_{H} \;,
\end{align*}

where $1_{H}$ is the multiplicative unit of the Hopf algebra $H$. The computation of the antipode of an element $c$ becomes easier when the algebra structure of $H$ is graded and connected.

If the Hopf algebra $H$ is graded and connected, then the antipode can be computed for any $a \in H^+ := \bigoplus_{k \geq 1}H_k$ as

\begin{align*}
S(a) &= -a - \sum a^{\prime}_{(1)}S(a^{\prime}_{(2)}),
\end{align*}

where the summation is taken over all components of the reduced coproduct $\Delta^{\prime}$ defined as:

\begin{align*}
\Delta^{\prime}\left(a\right)
&:= \Delta\left(a\right) - a \otimes \eta\left(1_{K}\right) - \eta\left(1_K\right) \otimes a.
\end{align*}

§ UNSHUFFLE HOPF ALGEBRA AND ITS COACTION

The goal of this section is to explain and illustrate the computational framework to compute the shuffle product of two series and the shuffle inverse using the coordinate maps of the series. The framework is well-developed in the literature  [Foissy, 2015] and was utilized in study of interconnections of Chen–Fliess series [Venkatesh & Gray, 2022, Venkatesh & Gray, 2021, Venkatesh & Gray , 2020, Gray, et al., 2014, Gray, et al., 2014].

§.§ Unshuffle Hopf Algebra

We construct a dual Hopf algebra reflecting the group structure of $M_{\shuffle}$ as defined in Section <ref>. The antipode constructed in the Hopf algebra provides a framework for computing the shuffle inverse of purely improper series $c$.

Let the set $W_b \subset \allseries{m}^{\ast}$ (dual module of $\allseries{m}$) be defined as the collection of coordinate maps:

\begin{align*}
W_b &= \{ a_{\eta} \: : a_{\eta}(c) = (c,\eta),\ \eta \in \allwords, c \in \allseries{m}\}.
\end{align*}

Define $W$ to be the free $\re^m$-module spanned by the set $W_b$. Let $H_{\shuffle}$ denote the reduced symmetric algebra generated by the module $W$. The $\re^m$-algebra $H_{\shuffle}$ can equivalently be seen as the polynomial algebra of coordinate maps (corresponding to non-empty words) of $\allseries{m}$. The unit map $\xi : \re^m \longrightarrow H_{\shuffle}$ is defined by $\xi(\doubleone) = a_{\emptyset}$. Observe that $a_{\emptyset}: c \mapsto \doubleone$, for all $c \in M_{\shuffle}$. By construction, $H_{\shuffle}$ is an $\re^m$-associative, commutative and unital algebra with addition and scalar multiplication defined, respectively, as

\begin{align*}
(a_{\eta} + a_{\zeta})(c)
&= a_{\eta}(c) + a_{\zeta}(c)\\
(k a_{\eta})(c)
&= k (a_{\eta}(c)),
\end{align*}

where $c \in \allseries{m}$ and $k \in \re^m$, and product

\HAprod(a_{\eta},a_{\zeta})(c) = a_{\eta}(c).a_{\zeta}(c),

for $c \in M_{\shuffle}$. Then $H_{\shuffle}$ is equipped with a coproduct $\hat{\Delta}_{\shuffle} : H_{\shuffle} \longrightarrow H_{\shuffle} \bigotimes H_{\shuffle}$ such that $\hat{\Delta}_{\shuffle} a_{\eta}(c,d) = \left(c\shuffle d, \eta\right)$, for all $c,d \in M_{\shuffle}$ and $\eta \in X^{*}$. The counit map $\epsilon : H_{\shuffle} \longrightarrow \re^m$ is defined as

\begin{align*}
\epsilon(h) = \begin{cases}
\doubleone \; : & \!\!\! h = a_{\emptyset} \\
0\; : & \!\!\!\text{otherwise}.
\end{cases}
\end{align*}

Since the shuffle product is associative and commutative, thus dually the coproduct $\hat{\Delta}_{\shuffle}$ is coassociative and cocommutative. Therefore, $(H_{\shuffle},\HAprod,\xi,\hat{\Delta}_{\shuffle},\epsilon)$ forms a $\re^m$-bialgebra. The following lemma is vital in the framework for computing both shuffle product and dynamic feedback group product. Define a collection of linear endomorphisms $\{\theta_i\}_{i=0}^m$ on $W$

\begin{align*}
\theta_i : W &\longrightarrow W \\
a_{\eta} &\longmapsto a_{x_i\eta},
\end{align*}

for all $x_i \in X$, $\eta \in \allwords$. Thus $\theta_i\left(a_{\eta}\right) \left(c\right) = a_{\eta}\left(x_i^{-1}\left(c\right)\right)$.

The coproduct $\hat{\Delta}_{\shuffle}$ can be recursively constructed as defined in the following proposition.

[Foissy, 2015] On the module $W$

\begin{align*}
\hat{\Delta}_{\shuffle} \circ \theta_k
= \left(\theta_k \otimes \id + \id \otimes \theta_k\right) \circ \hat{\Delta}_{\shuffle},
\end{align*}

for all $i = 1,2,\ldots,m$ and $k = 0,1,\ldots,m$ with base case being $\hat{\Delta}_{\shuffle}a_{\emptyset} = a_{\emptyset}\otimes a_{\emptyset}$.

Proposition <ref> infers that the maps $\theta_i$, for $i= 1,2,\ldots,m$, are coderivations on the underlying coalgebra of $H_{\shuffle}$.

We note that the unshuffle coproduct $\hat{\Delta}_{\shuffle}$ was utilized in the design of an algorithmic framework for computation of Wiener-Fliess composition product and subsequently additive static feedback product [Venkatesh & Gray, 2021, Venkatesh & Gray, 2022, Venkatesh, 2021] and also in the computation of shuffle-rational series from its representation [Venkatesh & Gray , 2020, Venkatesh, 2021]. Moreover, the unshuffle coproduct was also crucial in the computational framework for the multivariate additive output feedback [Gray, et al., 2014, Gray, et al., 2014] and for SISO affine output feedback <cit.>.

Let $\{\pi^i\}_{i=1}^m$ be the collection of co-ordinate projection maps on the module $W$ defined as

\begin{align*}
a_{\eta}^i(c) := \pi^i(a_\eta) (c) = (c,\eta)_i = (c_i,\eta),
\end{align*}

for all $\eta \in \allwords$. Thus, define the following notation

\begin{align*}
\hat{\Delta}_{\shuffle}^j a_{\eta}^i := (\pi^i \otimes \pi^j) \circ \hat{\Delta}_{\shuffle}a_{\eta}.
\end{align*}

Note that the projection maps $\{\pi^i\}_{i=1}^m$ commute with the maps $\{\theta_j\}_{j=0}^m$ viz. $\theta_i\left(a_{\eta}^j\right) = a_{x_i\eta}^j$. The significance of these notations are well-reflected in the computational framework in Section <ref>. The following example is to demonstrate the result of Proposition <ref> for a few words.

A few examples of the computation of deshuffle coproduct $\hat{\Delta}_{\shuffle}$ on $W$ (akin to Example <ref>) using Proposition <ref>  are given as follows(indices $i = 1,2,\ldots,m$ and $k,s = 0,1,\ldots,m$):

\begin{align*}
\hat{\Delta}_{\shuffle}^ja_{x_k}^i &= a_{x_k}^i \otimes a_{\emptyset}^j + a_{\emptyset}^i \otimes a_{x_k}^j. \\
\hat{\Delta}_{\shuffle}^ja_{x_kx_k}^i &= a_{x_kx_k}^i \otimes a_{\emptyset}^j + 2 a_{x_k}^i \otimes a_{x_k}^j + a_{\emptyset}^i \otimes a_{x_kx_k}^j. \\
\hat{\Delta}_{\shuffle}^ja_{x_kx_s}^i &= a_{x_kx_s}^i \otimes a_{\emptyset}^j + a_{x_k}^i \otimes a_{x_s}^j + a_{x_s}^i \otimes a_{x_k}^j + a_{\emptyset}^i \otimes a_{x_kx_s}^j.
\end{align*}

The connected $\re^m$-bialgebra $H_{\shuffle}$ is endowed with an  antipode map $S_{\shuffle}$ given as:

\begin{align*}
S_{\shuffle} :~&H_{\shuffle} \longrightarrow H_{\shuffle} \\
&a_{\eta} \mapsto S_{\shuffle}a_{\eta}
\end{align*}

such that $S_{\shuffle}a_{\eta}\left(c\right) = \left(c^{\shuffle -1}, \eta \right)$, for $\eta \in X^{*}$, $c \in M_{\shuffle}$.

§.§ Gradation of Bialgebra $H_{\shuffle}$

The Hopf algebra $H_{\shuffle}$ can be equipped with a grading such that it is connected and all its homogeneous components are finite-dimensional.

Given $\eta \in X^{+}$, define the degree of $a_\eta$ as $\deg\left(a_\eta\right) = \abs{\eta}$.

* Define gradation on the $\re^m$-module $W$ viz.

\begin{align*}
W = \bigoplus_{k \geq 1} W_k,
\end{align*}

where $W_k$ is the free $\re^m$-module spanned by the $a_{\eta}$ of $\deg\left(a_{\eta}\right) = k$.

* The gradation on the module $W$ induces a graded structure on the algebra $H_{\shuffle}$ as

\begin{align*}
H_{\shuffle} = \bigoplus_{n \in \nat_{0}} \hat{H}_n,
\end{align*}

with $\hat{H}_0 \cong \re^m$ in the category of $\re^m$-modules.

The following proposition asserts that the above gradation is connected and all its homogeneous components are finite-dimensional.

Given the gradation for the Hopf algebra $H_{\shuffle}$,

* $H_{\shuffle}$ is a graded and connected Hopf algebra viz.

\begin{align*}
\hat{\Delta}_{\shuffle}\left(\hat{H}_n\right) \subseteq \bigoplus_{\substack{i+j = n \\ i,j \geq 0}} \hat{H}_i \otimes \hat{H}_j.
\end{align*}

* For all $k$: define $w_k = \dim\left(W_k\right)$ and $F_W = \sum_{k \geq 1} w_k Z^k$ is the geometric series given by

\begin{align*}
F_W = \dfrac{1}{1-mZ} \, ,
\end{align*}

where $m = \abs{X}$ and for all $k \geq 1$:

\begin{align*}
w_k = \dim\left(W_k\right) = m^k.
\end{align*}

* Define $F_{\hat{H}} = \sum_{n \geq 1} h_nZ^n$ where $h_n = \dim (\hat{H}_n)$ then

\begin{align*}
F_{\hat{H}} = \prod_{k = 1}^{\infty} \dfrac{1}{\left(1-Z^k\right)^{w_k}}.
\end{align*}

* The Hopf algebra $H_{\shuffle}$ follows from the fact that if $\gamma (\neq \eta,\zeta) \in \supp(\eta \shuffle \zeta)$ then

\begin{align*}
\deg\left(\gamma\right)= \abs{\gamma} = \abs{\eta} + \abs{\zeta} = \deg\left(\eta\right) + \deg\left(\zeta\right),
\end{align*}

for all $\eta,\zeta,\gamma \in \allwords$.

* Define the formal power series

\begin{align*}
F(Z_0,Z_1,\dots,Z_m) &= \sum_{k \geq 1}\sum_{\substack{i_0,i_1,\ldots,i_m \geq 0 \\
i_0+i_1+\cdots+i_m = k}} \#\{\eta : \abs{\eta}_{x_j} = i_j \, \forall\, j = 0,1,2,\ldots,m\}Z_0^{i_0}Z_1^{i_1}\cdots Z_m^{i_m} \\
&= \dfrac{\left(Z_0 + Z_1 + \cdots + Z_m\right)}{1- \left(Z_0 + Z_1 + \cdots + Z_m\right)}.
\end{align*}

Since each letter contributes equally to the degree (viz. length), thus

\begin{align*}
F_W = F(Z,Z,\ldots,Z) = \dfrac{mZ}{1-mZ}.
\end{align*}

* The proposition follows from the item $2$ as $\hat{H}$ is the symmetric algebra generated by the $\re^m$-module $W$.

Dimensions of the homogeneous components of module $W$ and $H_{\shuffle}$ (when $m=2$)

2||c|k     0     1     2     3     4     5     6     7     8     9     10
2||c|$\dim\left(W_k\right)$     1     2     4     8     16     32     64     128     256     512     1024
2||c|$\dim(\hat{H}_k)$     1     2     7     20     59     162     449     1200     3194     8348     21646

The dimensions of the homogeneous components of the graded module $W$ (up to $k = 10$) and the graded algebra $H_{\shuffle}$ for $m= 2$ viz when $X = \{x_0,x_1\}$ is tabulated in Table $1$.

The sequence $\{\dim(\hat{H}_k)\}_{k \in \nat_0}$ is the sequence $A034899$ in [OEIS, 2022] which corresponds to the number of multisets of binary words of total length $n$.

§.§ Coaction of $H_{\shuffle}$

The subsection explains the coaction of the Hopf algebra $H_{\shuffle}$ (<ref>) on the algebra of coordinate functions. It is utilized subsequently to develop an algorithm to compute the multiplicative mixed composition product explained in Section <ref> and dynamic feedback product as defined in Theorem <ref>. Let $W$ to be the $\re^m$-module as described in Section <ref>. Let $S^{+}\left(W\right)$ denote the reduced symmetric algebra generated by the module $W$. The non-unital $\re^m$-algebra $S^{+}(W)$ are equivalently the polynomials without constant term of coordinate maps of $\allseries{m}$.  By construction $S^{+}(W)$ has a non-unital $\re^m$-associative, commutative algebra structure with addition, scalar multiplication and product defined, respectively, as

\begin{align*}
(a_{\eta} + a_{\zeta})(c) &= a_{\eta}(c) + a_{\zeta}(c)\\
(k a_{\eta})(c) &= k (a_{\eta}(c))
\end{align*}

where $c \in \allseries{m}$, and

\HAprod(a_{\eta},a_{\zeta})(c) = a_{\eta}(c).a_{\zeta}(c),

where $c \in M_\shuffle$. The $\re^m$-algebra $S^{+}(W)$ is isomorphic to the algebra structure of $H_{\shuffle}$ with forgetting of the unit map $\xi$.  The right coaction map $\rho_{\shuffle} : S^{+}\left(W\right) \longrightarrow S^{+}\left(W\right) \otimes H_{\shuffle}$ is recursively defined on the module $V$ as given by the following proposition.

For all $i = 0,1,2,\ldots,m:$

\begin{align*}
\rho_{\shuffle}\circ \theta_i = \left(\theta_i \otimes \id + \id \otimes \theta_i\right) \circ \rho_{\shuffle},
\end{align*}

with base case being $\rho_{\shuffle}a_{\emptyset} = a_{\emptyset}\otimes a_{\emptyset}$.

Proposition <ref> might appear as repetition of Proposition <ref>. It is vital to note that Proposition <ref> is for defining the coproduct of Hopf algebra $H_{\shuffle}$, where $a_{\emptyset}$ is the unit element.
Observe that,

\begin{align*}
\rho_{\shuffle}a_{\eta}^i(c,d) = a_{\eta}^i(c\shuffle d),
\end{align*}

where $c \in \allseries{}$ (not necessarily in $M_{\shuffle}$) and $d \in M_{\shuffle}$. The coaction $\rho_{\shuffle}$ thus is a corepresentation of the Hopf algebra $H_{\shuffle}$ on the algebra $S^{+}\left(W\right)$ or equivalently, $\rho_{\shuffle}$ makes $S^{+}\left(W\right)$, a $H_{\shuffle}$-algebra. Let $\{\pi^i\}_{i=1}^m$ be the collection of co-ordinate projection maps on the module $W$ defined as

\begin{align*}
a_{\eta}^i(c) := \pi^i(a_\eta) (c) = (c,\eta)_i = (c_i,\eta),
\end{align*}

for all $\eta \in \allwords$ and thus the following notation is well-defined,

\begin{align*}
\rho_{\shuffle}^j a_{\eta}^i := (\pi^i \otimes \pi^j) \circ \rho_{\shuffle}a_{\eta}.
\end{align*}

These notations are very much utilized in developing computational framework for the multiplicative mixed composition product as discussed in Section <ref>.

If $n \in \nat_0$, then for all $i = 0,1,2,\ldots,m$ and $j,k = 1,2,\dots,m$ (defining $x_j^0 := \emptyset$):

\begin{align*}
\rho_{\shuffle}^ja_{{x_i}^n}^k = \sum_{r = 0}^n \binom{n}{r}a_{{x_i}^r}^k \otimes a_{{x_i}^{n-r}}^j \; .
\end{align*}

The statement is proved by induction on $n \in \nat_0$. The base case ($n=0$) follows from Proposition <ref>. Assume the statement is true for $n = p-1$, then

\begin{align*}
\rho_{\shuffle}^j a_{{x_i}^p}^k
&= \rho_{\shuffle}^j \circ \theta_i a_{{x_i}^{p-1}}^k \\
&= \left(\theta_i \otimes \id + \id \otimes \theta_i\right) \circ \Delta_{\shuffle}^ja_{{x_i}^{p-1}}^k.
\end{align*}

Using the induction hypothesis,

\begin{align*}
\rho_{\shuffle}^j a_{{x_i}^p}^k
&= \left(\theta_i \otimes \id + \id \otimes \theta_i\right)  \left(\sum_{r = 0}^{p-1} \binom{p-1}{r}a_{{x_i}^r}^k \otimes a_{{x_i}^{{p-1}-r}}^j\right) \\
&= \sum_{r=1}^{p}\binom{p-1}{r-1} a_{{x_i}^r}^k \otimes a_{{x_i}^{p-r}}^j + \sum_{r=0}^{p-1}\binom{p-1}{r} a_{{x_i}^r}^k \otimes a_{{x_i}^{p-r}}^j.		\\
&= \sum_{r = 0}^p \binom{n}{r}a_{{x_i}^r}^k \otimes a_{{x_i}^{p-r}}^j.
\end{align*}

Since the $S^{+}\left(W\right)$ and $H_{\shuffle}$ are isomorphic as $\re^m$-modules, the following lemma states the coaction of $H_{\shuffle}$ on $S^{+}\left(W\right)$ and the unshuffle coproduct coincide when the evaluation of coordinate maps are restricted to the group $M_{\shuffle}$.

Given $c,d \in M_{\shuffle}$, $\eta \in \allwords$ and $i = 1,2,\ldots,m$,

\begin{align*}
\hat{\Delta}_{\shuffle}a_{\eta}\left(c,d\right) = \left(c\shuffle d, \eta\right) = \rho_{\shuffle} a_{\eta}\left(c,d\right),
\end{align*}

where $c,d \in M_{\shuffle}$ and $\hat{\Delta}_{\shuffle}^i$ is the coproduct from the bialgebra $H_{\shuffle}$ constructed in Section <ref>.

A few examples of the computation of the coaction map $\rho_{\shuffle}$ on $W$ using Proposition <ref>  are given as follows(indices $i,j = 1,2,\ldots,m$ and $k,s = 0,1,\ldots,m$):

\begin{align*}
\Delta_{\shuffle}^ja_{\emptyset}^i &= a_{\emptyset}^i \otimes a_{\emptyset}^j. \\
\Delta_{\shuffle}^ja_{x_k}^i &= a_{x_i}^i \otimes a_{\emptyset}^j + a_{\emptyset}^i \otimes a_{x_i}^j. \\
\Delta_{\shuffle}^ja_{x_kx_k}^i &= a_{x_kx_k}^i \otimes a_{\emptyset}^j + 2 a_{x_k}^i \otimes a_{x_k}^j + a_{\emptyset}^i \otimes a_{x_kx_k}^j. \\
\Delta_{\shuffle}^ja_{x_kx_s}^i &= a_{x_kx_s}^i \otimes a_{\emptyset}^j + a_{x_k}^i \otimes a_{x_s}^j + a_{x_s}^i \otimes a_{x_k}^j + a_{\emptyset}^i \otimes a_{x_kx_s}^j.
\end{align*}

The following example illustrates the application of the deshuffle coproduct $\Delta_{\shuffle}$ in the computation of the shuffle product of two series.

Let $X = \{x_0,x_1\}$ and $c,d \in \allseries{2}$ described as

\begin{align*}
c = \begin{bmatrix}
1 + x_1 + x_1^2 + x_1^3 + \cdots  \\
x_0 + x_0x_1 + x_1^{100}
\end{bmatrix}  \quad  \& \quad
d  = \begin{bmatrix}
1 + x_0^2 + \exp\left(x_1\right)\\
1 + x_0^2x_1
\end{bmatrix},
\end{align*}

where $\exp(.)$ is the standard exponential function expressed in its Taylor series. Note that $c \, \notin\, M_{\shuffle}$ but $d \in M_{\shuffle}$. The coefficient of $x_0x_1^2$ in series $c_2 \shuffle d_1$ can be computed as:

\begin{align*}
\left(c_2\shuffle d_1,x_0x_1^2\right)
&= \Delta_{\shuffle}^1a_{x_0x_1^2}^2\left(c,d\right)
= (\pi^2 \otimes \pi^1)\circ \Delta_{\shuffle}a_{x_0x_1^2}\left(c,d\right) \\
&= \Delta_{\shuffle}^1 \circ \theta_0 a_{x_1^2}\left(c,d\right).
\end{align*}

Using Proposition <ref>,

\begin{align*}
\left(c_2\shuffle d_1,x_0x_1^2\right) &= \left(\theta_0 \otimes \id
+ \id\otimes \theta_0\right) \circ \Delta_{\shuffle}^1a_{x_1^2}^2\left(c,d\right).
\end{align*}

Using Corollary <ref>,

\begin{align*}
\left(c_2\shuffle d_1,x_0x_1^2\right)
&= \left(\theta_0 \otimes \id + \id \otimes \theta_0\right) \circ \left(a_{{x_1}^2}^2\otimes a_{\emptyset}^1 + 2a_{x_1}^2 \otimes a_{x_1}^1 + a_{\emptyset}^2 \otimes a_{{x_1}^2}^1\right) \left(c,d\right) \\
&= \big(a_{x_0{x_1}^2}^2\otimes a_{\emptyset}^1 + 2a_{x_0x_1}^2 \otimes a_{x_1}^1 + a_{x_0}^2 \otimes a_{{x_1}^2}^1 + a_{{x_1}^2}^2\otimes a_{x_0}^1 + \\
& \quad 2a_{x_1}^2 \otimes a_{x_0x_1}^1 + a_{\emptyset}^2 \otimes a_{x_0{x_1}^2}^1\big) \left(c,d\right) \\
&= (0)(1) + 2(1)(1) + (1)(0.5) + (0)(0) + 2(0)(0) + (0)(0) = 2.5.
\end{align*}

Therefore $\left(c_2\shuffle d_1,x_0x_1^2\right) = 2.5$.

§ CHEN–FLIESS SERIES AND ITS INTERCONNECTIONS

The objective of the section is to describe Chen–Fliess series and the necessary non-recursive interconnections of Chen–Fliess series to understand the results about the multiplicative dynamic feedback product in Section <ref>.

§.§ Chen–Fliess Series

Let $\mathfrak{p}\ge 1$ and $t_0 < t_1$ be given. For a Lebesgue measurable function $u: [t_0,t_1] \rightarrow\re^m$, define $\norm{u}_{\mathfrak{p}}=\max\{\norm{u_i}_{\mathfrak{p}}: \ 1 \le i \le m\}$, where $\norm{u_i}_{\mathfrak{p}}$ is the usual $L_{\mathfrak{p}}$-norm for a measurable real-valued function,
$u_i$, defined on $[t_0,t_1]$.  Let $L^m_{\mathfrak{p}}[t_0,t_1]$ denote the set of all measurable functions defined on $[t_0,t_1]$ having a finite $\norm{\cdot}_{\mathfrak{p}}$ norm and $B_{\mathfrak{p}}^m(R)[t_0,t_1]:=\{u \in L_{\mathfrak{p}}^m[t_0,t_1]:\norm{u}_{\mathfrak{p}}\leq R\}$. Given any series $c\in\allseriesell$, the corresponding
Chen–Fliess series is
F_c[u](t) = ∑_η∈X^∗ (c,η) F_η[u](t,t_0),
where $E_\emptyset[u]=1$ and
F_x_iη̅[u](t,t_0) =
∫_t_0^tu_i(τ)F_η̅[u](τ,t_0) dτwith $x_i\in X$, $\bar{\eta}\in X^{\ast}$, and $u_0=1$ [Fliess, 1981].
If there exist constants $K,M>0$ such that

\begin{align}
\abs{(c_i,\eta)}\leq K M^{\abs{\eta}} \abs{\eta}!,\;\; \forall \eta\in X^\ast, \; \forall i = 1,\ldots,\ell\;,\label{eq:Gevrey-growth-condition}
\end{align}

then $F_c$ constitutes a well-defined mapping from
$B_{\mathfrak p}^m(R)[t_0,$ $t_0+T]$ into $B_{\mathfrak q}^{\ell}(S)[t_0, \, t_0+T]$ for sufficiently small $R,T >0$, where the numbers $\mathfrak{p},\mathfrak{q}\in[1,\infty]$ are conjugate exponents, i.e., $1/\mathfrak{p}+1/\mathfrak{q}=1$ [Gray & Wang, 2002]. This map is referred to as a Fliess operator. A series $c \in \allseriesell$ obeying the growth condition in  eq:Gevrey-growth-condition is called a locally convergent generating series. The set of all locally convergent generating series is denoted by $\allseriesellLC$. The supremum of the set of all $\max\{R,T\}$ for which a Fliess operator $F_c$ is a well-defined mapping from $B_{\mathfrak p}^m(R)[t_0,$ $t_0+T]$ into $B_{\mathfrak q}^{\ell}(S)[t_0, \, t_0+T]$ is called the radius of convergence of the Fliess operator $F_c$ and is denoted by $\rho\left(F_c\right)$. A Fliess operator $F_c$ is called locally convergent if $\rho\left(F_c\right) > 0$. If there exist constants  $K,M >0$ and $\gamma \in [0,1[$ such that

\begin{align}
\abs{(c_i,\eta)}\leq K M^{\abs{\eta}} \left(\abs{\eta}!\right)^{\gamma},\;\; \forall \eta\in X^\ast,\; \forall i = 1,\ldots,\ell\;, \label{eq:GC_condition}
\end{align}

then $F_c$ constitutes a well defined mapping from $B_{\mathfrak p}^m(R)[t_0,$ $t_0+T]$ into $B_{\mathfrak q}^{\ell}(S)[t_0, \, t_0+T]$ for all $R,T >0$ [Winter-Arboleda, 2019, Winter-Arboleda, et al., 2015]. The infimum of all the $\gamma \in [0,1[$ such that eq:GC_condition is satisfied for a series $c \in \allseries{\ell}$ is called the Gevrey order of the series $c$.

A series $c \in \allseries{\ell}$ obeying the growth condition in eq:GC_condition is called a globally convergent series. The set of all globally convergent series in $\allseries{\ell}$ is denoted as $\allseriesGC{\ell}$. A Fliess operator $F_c$ is globally convergent if and only if there exists no real number $M > 0$ such that $\rho\left(F_c\right) < M$. Observe that a noncommutative polynomial $\allpoly$ is a globally convergent series with Gevrey degree $0$. As described above, a series $c \in \allseriesGC{\ell}$ is only a sufficient condition for the corresponding Fliess operator $F_c$ to be globally convergent. Necessary conditions are well-detailed in the literature [Winter-Arboleda, 2019, Venkatesh, 2021]. In the absence of any convergence criterion, eq:Fliess-operator-defined only defines an operator in a formal sense.

§.§ Interconnections of Chen–Fliess Series: Parallel and Cascade Connections

Given Chen–Fliess series $F_c$ and $F_d$, where $c,d\in\allseriesell$, the parallel and product connections satisfy $F_c+F_d=F_{c+d}$ and $F_cF_d=F_{c\shuffle d}$,
respectively [Ree, 1958, Fliess, 1981]. The parallel and product connections preserve local convergence and hence the interconnected systems has a Fliess operator representation [Thitsa & Gray, 2012, Venkatesh, 2021]. When Chen–Fliess series $F_c$ and $F_d$ with $c\in\allseriesXpri{k}$ and $d\in\allseries{\ell}$ are interconnected in a cascade fashion, where $\lvert X^{\prime} \rvert = \ell + 1$, the composite system $F_c\circ F_d$ has a Chen–Fliess series representation $F_{c\circ d}$, where
the composition product of $c$ and $d$ is given by

c ∘d=∑_η∈X^'∗ (c,η) ψ_d(η)(1)

[Ferfera, 1979, Ferfera, 1980]. Here $\mbf{1}$ denotes the monomial $1\emptyset$, and $\psi_d$ is the continuous (in the ultrametric sense) algebra homomorphism from $\allseriesXpri{}$ to the set of vector space endomorphisms on $\allseries{}$, $\Endallseries$, uniquely specified by

\psi_d(x_i^{\prime}\eta)=\psi_d(x_i^{\prime})\circ \psi_d(\eta)

$ %\label{eq:psi-d-on-words}
\psi_d(x_i^{\prime})(e)= x_0(d_i\shuffle e),
for any $e\in\allseries{}$, and where $d_i$ is the $i$-th component series of $d$
($d_0:=\mbf{1}$). By definition, $\psi_d(\emptyset)$ is the identity map on $\allseries{}$. The cascade interconnection preserves local convergence and thus the composite has a Fliess operator representation [Thitsa & Gray, 2012]. The linearity of the composition product in the left argument is evident form the definition. However, the following theorem states that the composition product distributes over the shuffle product from the right.

[Gray & Li, 2005] Let $c,d \in \allseriesXpri{k}$ and $e \in \allseries{\ell}$, such that $\abs{X^{\prime}} = \ell + 1$, then $\left(c\shuffle d\right)\circ e = \left(c \circ e\right) \shuffle \left(d \circ e\right)$.

Given a series $e \in \allseries{\ell}$, define a map $\Upsilon_e : \allseriesXpri{k} \longrightarrow \allseries{k}$ defined as $c \mapsto c \circ e$. Theorem <ref> infers that $\Upsilon_e$ is an $\re$-algebra homomorphism from the shuffle algebra of $\allseriesXpri{k}$ to the shuffle algebra of $\allseries{\ell}$. The composition product preserves the purely improper property of the left argument which is stated in the following theorem.
If $c \in \allseriesXpri{k}$ and $d \in \allseries{\ell}$ such that $\abs{X^{\prime}} = \ell +1$, then $\left(c\circ d, \emptyset\right) = \left(c,\emptyset\right)$. Hence, if $c \in \allpiseriesXpri{k}$ then $c\circ d \in \allpiseries{k}$ and vice-versa. Similarly if $c$ is a proper series then $c \circ d$ is also a proper series and vice-versa.

The proof follows immediately from eq:c-circ-d.

The composition product is a strong contraction map with respect to its right argument in the ultrametric topology and is stated in the following theorem.
[Gray & Li, 2005] Let $c \in \allseriesXpri{k}$ and $d,e \in \allseries{\ell}$, such that $\abs{X^{\prime}} = \ell + 1$, then $\kappa\left(c\circ d, c\circ e\right) \leq \sigma \kappa\left(d,e\right)$ where $\sigma \in [0,1[$.

§.§ Cascading of Chen–Fliess with Multiplicative Feedforward of Input
The cascade interconnection of a Chen–Fliess series $F_c$ and $F_{d}$ along with the multiplicative feedforward of the input, as shown in Figure <ref>, arises primarily in the analysis of multiplicative feedback interconnection discussed in Section <ref>. A semblance of such an interconnection has appeared in Definition $3.1$ of  <cit.>, without being explicit and limited to the SISO case. With respect to Figure <ref>, the map $u \mapsto y$ viz. $y = F_c[u.F_d[u]]$ has Chen–Fliess series  representation denoted by $F_{c \lhook d}$, where $c \lhook d$ denotes the multiplicative mixed composition product of $c \in \allseries{p}$ and $d \in \allseries{m}$ defined as

\begin{align}\label{eqn:mixprocomp_def}
c \lhook d
= \sum_{\eta\in X^\ast} \left(c,\eta\right) \eta \lhook d
:= \sum_{\eta\in X^\ast} \left(c,\eta\right) \bar{\phi}_d \left(\eta\right)\left(\mbf{1}\right).
\end{align}

Here, $\bar{\phi}_d : \allseries{} \longrightarrow \Endallseries$ is an $\re$-algebra homomorphism such that
\bar{\phi}_d(x_0)(e) = x_0e

\bar{\phi}_d(x_i)(e) = x_i(d_i \shuffle e).

Recall that $\allseries{}$ is an $\re$-algebra under Cauchy product and $\Endallseries$. The multiplicative mixed composition defined in (<ref>) asserts that, for all $\eta \in \allwords$ and $d \in \allseries{m}$,

\begin{align*}
\emptyset \lhook d &= \emptyset\\
x_0\eta \lhook d &= x_0\left(\eta \lhook d\right)\\
x_i\eta \lhook d &= x_i\left(d_i \shuffle (\eta \lhook d)\right) \quad \qquad \forall \, i = 1,2,\ldots,m.
\end{align*}

For later reference, we summarise the properties of (<ref>) in the following

The multiplicative mixed composition product (<ref>)  is linear in its left argument and $\left(c\lhook d,\emptyset\right) = \left(c,\emptyset\right)$, for all $c \in \allseries{p}$ and $d \in \allseries{m}$.
The following results are already known in the single-input single-output (SISO) setting. However, their multi-input multi-output (MIMO) extensions are straightforward and to avoid reiteration of the proofs, only the statements are provided in this document. The foremost of the theorems asserts that the multiplicative mixed composition product distributes over shuffle product from the right.

Cascade connection of Chen–Fliess $F_d$ with $F_c$ along with multiplicative feedforward of input

<cit.> Let $c,d \in \allseries{p}$ and $e \in \allseries{m}$, then $\left(c \shuffle d\right) \lhook e = \left(c \lhook e\right) \shuffle \left(d \lhook e\right)$.

The inference of Theorem <ref> is that for any $e \in \allseries{m}$, the map $\Gamma_e : \allseries{p} \longrightarrow \allseries{p}$ given by $d \mapsto d \lhook e$ is an $\re$-algebra endomorphism on the shuffle algebra $\allseries{p}$. The next lemma is essential in proving that multiplicative mixed composition product is a strong contraction map in its right argument in the ultrametric topology.

<cit.> Let $\eta \in \allwords$ and $d, e \in \allseries{m}$, then $\kappa\left(\eta \lhook d, \eta \lhook e\right) \leq \sigma^{\abs{\eta}} \kappa\left(d,e\right)$ where $\sigma \in [0,1[$.

The following theorem states the strong contraction property of the multiplicative mixed composition product which is an essential result in Section <ref>.

Let $d,e \in \allseries{m}$ and $c \in \allseries{p}$, then $\kappa\left(c \lhook d, c \lhook e\right) \leq \sigma^{\ord\left(c^{\prime}\right)}\kappa\left(d,e\right)$, where $c^{\prime} = c - \left(c,\emptyset\right)$, the proper part of $c$.

Since $\ord\left(c^{\prime}\right) \geq 1$ and $\sigma \in ]0,1[$, then from Theorem <ref>, the map $\bar{\Gamma}_c : e \mapsto c \lhook e$ is a strong contraction map in the ultrametric topology. The following lemma is essential in proving the mixed associativity of the composition and multiplicative mixed composition product. The result, along with Theorem <ref> can be inferred in the SISO setting from Lemma $3.6$ in <cit.>, and its extension to the MIMO case is straightforward.

Let $X^{\prime} = \{x'_0,\ldots,x'_p\}$ and $\eta \in {X^{\prime}}^{\ast}$. Let $d \in \allseries{p}$ and $e \in \allseries{m}$, then $\eta \circ \left(d \lhook e\right) = \left(\eta \circ d\right)\lhook e$.

The following theorem states that the  composition product and multiplicative mixed composition product are associative in combination.

Let $X^{\prime} = \{x'_0,\ldots,x'_p\}$ and $c \in \allseriesXpri{q}$. Let $d \in \allseries{p}$ and $e \in \allseries{m}$, then $c \circ \left(d \lhook e\right) = \left(c \circ d\right)\lhook e$.

§.§ Multiplicative Dynamic Output Feedback Group

The dynamic multiplicative feedback group plays a vital role in computation of the multiplicative dynamic feedback formula, as well as in assessing the feedback as a group action in Section <ref>. Indeed, consider the cascade interconnection of two Chen–Fliess series $F_c$ and $F_d$ along with their multiplicative feedforward of inputs displayed in Figure <ref>, where $c,d \in \allseries{m}$. The input-output relation of the composite system, $u \mapsto y$ is $u.F_d[u]F_c[u.F_d[u]]$ and can be represented by Chen–Fliess series as follows. Consider

u.F_{c \star d}[u]:=u.F_d[u]F_c[u.F_d[u]],

where the multiplicative composition product of $c$ and $d$ is defined as

\begin{align}\label{eqn:grp_prod}
c \star d = d \shuffle \left(c \lhook d \right).
\end{align}

Cascade connection of Chen–Fliess $F_d$ with $F_c$ along with multiplicative feedforward of their inputs.

The following theorems appeared in <cit.> in the SISO setting. We underline that the latter restriction is not essential, that is, the statements along with the proofs naturally extend to the MIMO setting.
Let $c,d,e \in \allseries{m}$, then, $\left(c \star d\right)\star e = c \star \left(d \star e \right)$.
Observe that eqn:grp_prod and Theorem <ref> infer that $\allseries{m}$ forms a non-commutative monoid under multiplicative composition product, with the identity element $\doubleone$. The following theorem states that the multiplicative mixed composition product is a right action on $\allseries{q}$ by the monoid $\left(\allseries{m},\star, \doubleone\right)$.

Let $c \in \allseries{q}$ and $d,e \in \allseries{m}$, then $\left(c \lhook d\right)\lhook e = c \lhook \left(d \star e\right)$.

The prominent question is to find the invertible elements of the monoid $\left(\allseries{m},\star\right)$ and the motivation to find the unit elements of the monoid shall be evident in Section <ref>. Let $d,e \in \allpiseries{m}$ and suppose

\begin{align*}
d \star e &= \doubleone.
\end{align*}

Observe that $d \in \allpiseries{m}$ implies $\left(d \lhook e\right) \in \allpiseries{m}$ and using Theorem <ref>,

\begin{align*}
e = \left(d \lhook e \right)^{\shuffle -1} = d^{\shuffle -1} \lhook e.
\end{align*}

Hence, for $e$ to be right inverse of $d$, the purely improper series $e$ has to satisfy the fixed point equation

\begin{align}\label{eqn:fixed_point_grp_eqn_rinv}
e = d^{\shuffle -1} \lhook e
\end{align}

Observe from Theorem <ref> that the map $e \mapsto d^{\shuffle -1} \lhook e$ is a strong contraction in the ultrametric space inferring that eqn:fixed_point_grp_eqn_rinv has a unique fixed point. Suppose $e$ is the left inverse of $d$ viz. $e \star d$, then a similar procedure shows that $e$ has to satisfy the equation

\begin{align}\label{eqn:grp_eqn_linv}
d = e^{\shuffle -1} \lhook d
\end{align}

Note that if $e$ is a solution of eqn:fixed_point_grp_eqn_rinv, then $e$ satisfies eqn:grp_eqn_linv and also the converse holds true. Hence, $e$ is the unique inverse of $d$ and is given the notation $d^{\star -1}$ for $d \in \allpiseries{m}$. Thus, $\allpiseries{m}$ forms a group under multiplicative composition product, $\star$, and is termed as the multiplicative dynamic output feedback group and is formally stated in the following theorem.

$\left( \allpiseries{m},\star\right)$ forms a group with the identity element $\doubleone$.

It is worth noting that <cit.> proved Theorem <ref> for one-dimensional case viz. $m=1$. In light of Theorem <ref>, Theorem <ref> and eqn:grp_prod one obtains the following relations for $c \in \allpiseries{m}$:

\begin{align}\label{eqn:inverse_relations}
c^{\star -1} &= c^{\shuffle -1} \lhook c^{\star -1}\\
\left(c^{\star -1}\right)^{\shuffle -1} &= c \lhook c^{\star -1}. \nonumber
\end{align}

The following lemma is essential in defining a subgroup of the multiplicative dynamic output feedback group upon which the computational framework for the multiplicative feedback products is discussed in Section <ref>.

Let $c,d \in \allpiseries{m}$, then $\left(c\star d, \emptyset\right) = \left(c,\emptyset\right)\left(d,\emptyset\right)$.

Observe from eqn:grp_prod that,

\begin{align*}
\left(c\star d, \emptyset\right) &= \left(d \shuffle \left(c\lhook d\right),\emptyset\right) \\
&= \left(c\lhook d,\emptyset\right) \left(d,\emptyset\right)
\end{align*}

Since $\left(c\lhook d, \emptyset \right) = \left(c,\emptyset\right)$,

\begin{align*}
\left(c\star d, \emptyset\right) &= \left(c,\emptyset\right)\left(d,\emptyset\right).
\end{align*}

Lemma <ref> thus proves that the set of all series which are of the form $\doubleone + c$, where $c$ is a proper series, forms a subgroup of the multiplicative dynamic feedback group, which is stated in the following theorem.

Let $M = \lbrace \doubleone + c \: : c \in \allproperseries{m}\rbrace$, then $\left( M, \star, \doubleone\right)$ forms a subgroup of the multiplicative dynamic feedback group.

The algorithmic framework for the computation of multiplicative feedback products is fundamentally based on the subgroup $M$ as asserted in Theorem <ref>. The group $M$ is isomorphic to the character group of the Hopf algebra $H$ which is used for computation of feedback and the framework is explained in detail in Section <ref>.

§ CHEN–FLIESS SERIES UNDER MULTIPLICATIVE DYNAMIC OUTPUT FEEDBACK

Let $F_c$ be a Chen–Fliess series with a generating series $c \in \allseries{q}$. Assume it is interconnected with a Chen–Fliess series $F_d$ with a purely improper generating series $d \in \allpiseriesXpri{m}$, as shown in Figure <ref>. Note that, $\abs{X} = m+1$ and $\abs{X^{\prime}} = q+1$. The primary goal of this section is to show that the closed-loop system has a Chen–Fliess series representation, say $y = F_e[v]$, where $e \in \allseries{q}$. If this is the case, then necessarily

\begin{align*}
y &= F_e[v] = F_c[u] = F_c[vF_d[y]] \\
&= F_c[vF_d[F_e[v]]] = F_c[vF_{d\circ e}[v]]\\
&= F_{c \lhook \left(d \circ e\right)}[v]
\end{align*}

for any admissible input $v$. Therefore, the series $e$ has to satisfy the fixed point equation

\begin{align}\label{eqn:dyn_feedb_fixed_point_eqn}
e = c \lhook \left(d\circ e\right).
\end{align}

Observe that, in light of Theorem <ref> and Theorem <ref> the map $e \mapsto c \lhook \left(d \circ e \right)$ is a strong contraction map in the ultrametric space and thus eqn:dyn_feedb_fixed_point_eqn has a unique fixed point. The following thoerem establishes the first main result of this section, which follows immediately.

The series $c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1} \in \allseries{q}$ is the unique fixed point of the map  $e \mapsto c \lhook \left(d \circ e \right)$.
If $e := c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1}$, then

\begin{align*}
c \lhook \left(d \circ e \right) &= c \lhook \left[d \circ \left(c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1}\right)\right] \\
\end{align*}

Using Theorem <ref> and then Theorem <ref>,

\begin{align*}
c \lhook \left(d \circ e \right) &= c \lhook \left[\left(d\circ c\right) \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1} \right]\\
&= c \lhook \left[\left(d \circ c \right)^{\shuffle -1} \lhook \left(d^{\shuffle-1}\circ c\right)^{\star -1}\right]^{\shuffle -1}.
\end{align*}

Using Theorem <ref>,

\begin{align*}
c \lhook \left(d \circ e \right) &= c \lhook \left[\left(d^{\shuffle -1}\circ c\right) \lhook \left(d^{\shuffle-1}\circ c\right)^{\star -1}\right]^{\shuffle -1}.
\end{align*}

Using the relations eqn:inverse_relations,

\begin{align*}
c \lhook \left(d \circ e \right) &= c \lhook \left[\left(\left(d^{\shuffle -1}\circ c\right)^{\star -1}\right)^{\shuffle -1}\right]^{\shuffle -1} \\
&= c \lhook \left(d^{\shuffle -1} \circ c\right)^{\star -1} = e.
\end{align*}

Given a series $c \in \allseries{q}$ and a purely improper series $d \in \allpiseriesXpri{m}$ (such that $\abs{X} = m+1$ and $\abs{X^{\prime}} = q+1$), then the generating series for the closed-loop system in Figure <ref> is given by the multiplicative dynamic feedback product $c\check{@}d := c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1}$.

The notion that feedback can described mathematically as a transformation group acting on the plant is well established in control theory [Brockett, 1978]. The following theorem describes the situation in the present context.

The multiplicative dynamic feedback product is a right group action by the multiplicative group $\left(\allpiseriesXpri{m},\shuffle,\doubleone\right)$ on the set $\allseries{q}$, where $\abs{X} = m+1$ and $\abs{X^{\prime}} = q+1$.

Chen–Fliess series $F_c$ in multiplicative output feedback with Chen-Flies series $F_d$

Let $c \in \allseries{q}$. Observe that from Theorem <ref>,

\begin{align*}
c \check{@}\doubleone &= c\lhook \left(\doubleone^{\shuffle -1} \circ c\right)^{\star-1}\\
&= c \lhook \doubleone = c.
\end{align*}

Let $d_1,d_2 \in \allpiseriesXpri{m}$. It needs to be proven that $\left(c\check{@}d_1\right)\check{@}d_2 = c\check{@}\left(d_1 \shuffle d_2\right)$.
From Theorem <ref>, observe that

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= \left(c\check{@}d_1\right) \lhook \left(d_2^{\shuffle -1}\circ \left(c \check{@}d_1\right)\right)^{\star -1} \\
&= \left(c \lhook \left(d_1^{\shuffle -1} \circ c\right)^{\star -1}\right) \lhook \left(d_2^{\shuffle -1}\circ \left(c \lhook \left(d_1^{\shuffle -1}\circ c\right)^{\star -1}\right)\right)^{\star -1}.
\end{align*}

Applying Theorem <ref>,

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= \left(c \lhook \left(d_1^{\shuffle -1} \circ c\right)^{\star -1}\right) \lhook \left(\left(d_2^{\shuffle -1}\circ c\right)\lhook \left(d_1^{\shuffle -1}\circ c\right)^{\star -1}\right)^{\star -1}.
\end{align*}

Applying Theorem <ref> and fact that the group inverse is anti-homomorphism with respect to the group product,

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= c \lhook \Big[ \left(d_1^{\shuffle -1}\circ c\right)^{\star -1} \star \left(\left(d_2^{\shuffle -1}\circ c\right)\lhook \left(d_1^{\shuffle -1}\circ c\right)^{\star -1}\right)^{\star -1} \Big] \\
&=c \lhook \Big[\left(\left(d_2^{\shuffle -1}\circ c\right) \lhook \left(d_1^{\shuffle -1}\circ c\right)^{\star -1}\right) \star \left(d_1^{\shuffle -1}\circ c\right)\Big]^{\star -1}.
\end{align*}

Applying eqn:grp_prod,

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= c \lhook \Bigg[\left(d_1^{\shuffle -1} \circ c\right) \shuffle
\Bigg(\Big(\left(d_{2}^{\shuffle -1}\circ c\right) \lhook \left(d_{1}^{\shuffle -1}\circ c\right)^{\star -1}\Big) \lhook \left(d_1^{\shuffle -1}\circ c\right)\Bigg)\Bigg]^{\star-1}.
\end{align*}

Using Theorem <ref>,

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= c \lhook \Bigg[\left(d_1^{\shuffle -1} \circ c\right) \shuffle
\Bigg(\left(d_{2}^{\shuffle -1}\circ c\right) \lhook \left(\left(d_1^{\shuffle -1}\circ c\right)^{\star -1} \star \left(d_1^{\shuffle -1}\circ c\right)\right)\Bigg)\Bigg]^{\star -1} \\
&= c \lhook \left(\left(d_{1}^{\shuffle -1}\circ c\right) \shuffle \left(\left(d_2^{\shuffle -1} \circ c\right) \lhook \doubleone\right)\right)^{\star -1} \\
&= c \lhook \left(\left(d_{1}^{\shuffle -1}\circ c\right) \shuffle \left(d_{2}^{\shuffle -1}\circ c\right)\right)^{\star -1}.
\end{align*}

In light of Theorem <ref>,

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= c \lhook \left(\left(d_1^{\shuffle -1}\shuffle d_2^{\shuffle -1}\right)\circ c\right)^{\star -1} \\
&= c \lhook \left(\left(d_1\shuffle d_2\right)^{\shuffle -1}\circ c\right)^{\star -1}.
\end{align*}

\begin{align*}
\left(c\check{@}d_1\right)\check{@}d_2 &= c \check{@}\left(d_1 \shuffle d_2\right).
\end{align*}

It is worth noting that for the additive dynamic feedback product the transformation group is the additive group $(\allseriesXpri{m},+,0)$ while here $(\allpiseriesXpri{m},\shuffle,\doubleone)$ plays the role.

§ INVARIANCE OF CLASS AND RELATIVE DEGREE UNDER MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION

The notion of relative degree of a plant is very essential and prime in the studies of feedback linearization [Isidori, 1995], flatness and system inversion etc. The existence and quantification of relative degree of a interconnection of systems is vital in systems theory. The notion of class and relative degree of a SISO Chen–Fliess series is equivalently characterized by the notion of relative degree of its generating series and the definition was furnished in [Gray, et al., 2014, Gray & Venkatesh, 2019] and the existence and quantification of relative degree of interconnected system of Chen–Fliess series was described in [Gray & Venkatesh, 2019, Venkatesh, 2021]. In addition, this definition of relative degree is consistent with the classical definition whenever $y=F_c[u]$ has an input-affine analytic state space realization <cit.>.  Let $X = \{ x_0,x_1\}$ and the following definition explains the concept of a class, a weaker notion than the relative degree of a series in $\allseries{}$.

[Gray & Venkatesh, 2019]
A series $c\in\allseries{}$ is said to be of r-class, denoted by $\mathscr{C}(c) = r$, if $\supp(c_F) \subseteq x_{0}^{r-1}X^{+}$ and $\supp(c_F) \nsubseteq x_{0}^{r}X^{+}$. By definition, let $\mathscr{C}(c) = \infty$ if $c_F = 0$.

The notion of class is universal and is versed in the following theorem.

[Gray & Venkatesh, 2019] Every series $c\in \allseries{}$ has a class.

Definition <ref> of class is illustrated in the following example.

Let $c = 1 + x_0x_1^{2} + x_{0}^{2}x_{1}$, so that $c_F = x_0x_1^{2} + x_{0}^{2}x_{1}$.
Observe that $\supp(c_F) \subseteq x_{0}X^{+}$ but $\supp(c_F) \nsubseteq x_{0}^{2}X^{+}$. Thus, $\mathscr{C}(c) = 2$.

The following lemma is essential in the proof of quantification of class for the multiplicative mixed composition product.

Let $c,c^{\prime},d \in \allseries{m}$ such that $\supp\left(c^{\prime}\right) \not\subseteq x_0\allwords$. Then the following statements are true:

* $x_0^k \lhook d = x_0^k \; \;\forall k \in \nat_0$.
* $c_N \lhook d = c_N$ where $c_N$ is the natural part of the series $c$.
* $\supp\left(c^{\prime}\lhook d\right) \not\subseteq x_0\allwords$.

* The proof is by induction on $k \in \nat_0$. The base case being $k=0$ is true viz $\emptyset \lhook d = \emptyset$ from  eqn:mixprocomp_def. Assume the proposition is true for $k = n-1$, then using  eqn:mixprocomp_def

\begin{align*}
x_0^n \lhook d = x_0\left(x_0^{n-1}\lhook d\right) = x_0\left(x_0^{n-1}\right) = x_0^n.
\end{align*}

Hence proved by induction on $\nat_0$.

* Observe that from Definition <ref>, $\supp\left(c_N\right) \subseteq \{x_0^k : k \in \nat_0\}$. Thus, using the previous statement $(1)$ and Theorem <ref> it follows that
$c_N \lhook d = c_N$.

* Since $\supp\left(c^{\prime}\right) \not \subseteq x_0\allwords$, there exists a word $x_i\eta \in \supp\left(c^{\prime}\right)$ where $x_i \neq x_0$ and $\eta\in \allwords$. Using eqn:mixprocomp_def,

\begin{align*}
x_i\eta \lhook d &= x_i\left(d_i \shuffle \left(\eta\lhook d\right)\right).
\end{align*}

Thus, $\supp\left(x_i\eta \lhook d\right) \subseteq x_iX^{\ast}$, where $x_i \neq x_0$. Therefore,
$\supp\left(c^{\prime}\lhook d\right) \not \subseteq x_0X^{\ast}$.

The following theorem quantifies that class is invariant under the multiplicative mixed composition product
Let $c,d \in \allseries{}$, then $\mathscr{C}\left(c\lhook d\right) = \mathscr{C}\left(c\right)$.
Suppose the series $c \in \allseries{}$ is of $r$-class, then the series $c$ can be written as:

\begin{align*}
c = c_N +  x_0^{r-1}c^{\prime},
\end{align*}

where $c^{\prime}$ is a proper series such that $\supp\left(c^{\prime}\right) \not \subseteq x_0X^{\ast}$. Hence by Theorem <ref>,

\begin{align*}
c \lhook d &= \left(c_N \lhook d\right) + \left(x_0^{r-1}c^{\prime} \lhook d\right).
\end{align*}

Using eqn:mixprocomp_def,

\begin{align*}
c \lhook d &= \left(c_N \lhook d\right) + x_0^{r-1}\left(c^{\prime}\lhook d\right).
\end{align*}

Since $\supp\left(c^{\prime}\right) \not \subseteq x_0\allwords$, then by applying Lemma <ref>,

\begin{align*}
c \lhook d &= c_N + x_0^{r-1}\left(c^{\prime}\lhook d\right),
\end{align*}

with $\supp\left(c^{\prime}\lhook d\right) \not\subseteq x_0\allwords$. Given that $c^{\prime} \in \allproperseries{}$, whence $\supp\left(c\lhook d\right)_F \subseteq x_0^{r-1}X^{+}$ and $\supp\left(c\lhook d \right)_F \not \subseteq x_0^{r}X^{+}$. Therefore, $\mathscr{C}\left(c\lhook d\right) = r = \mathscr{C}\left(c\right)$.

Consider the series $c$ in Example <ref>, given by $c = 1+x_0^2x_1+x_0x_1^2$ and $d = 1+x_1 \in \allseries{}$. Using eqn:mixprocomp_def, the multiplicative mixed composition product of $c$ and $d$ is computed as:

\begin{align*}
c \lhook d = 1 +x_0x_1^2 + 3x_0x_1^3 + 3x_0x_1^4 + x_0^2x_1 +x_0^2x_1^2.
\end{align*}

Observe that $\mathscr{C}\left(c\lhook d\right) = 2 = \mathscr{C}\left(c\right)$, as in Example <ref>.

The following theorem asserts that class of a series is preserved under the multiplicative dynamic feedback product which is one of the prime goal of this subsection.

If $c \in \allseries{}$ with $\mathscr{C}\left(c\right) = r$, and $d \in \allpiseries{}$, then $\mathscr{C}\left(c\check{@}d\right) = r = \mathscr{C}\left(c\right)$.

From Theorem <ref>,

\begin{align*}
c \check{@}d = c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1}.
\end{align*}

Since $\mathscr{C}\left(c\right) = r$, whence applying Theorem <ref>,

\begin{align*}
\mathscr{C}\left(c\check{@}d\right) &= \mathscr{C}\left( c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star-1} \right) \\
&= r = \mathscr{C}\left(c\right).
\end{align*}

The preservation of class under the multiplicative dynamic feedback connections as asserted in Theorem <ref> is further illustrated in the following example.
Let $c,d \in \allseries{}$ $c = x_1$ and $d = 1 + \sum_{k \in \nat} k!x_1^k$. Note that the class of series $\mathscr{C}\left(c\right) = 1$. Using Theorem <ref> the multiplicative feedback product is computed as:

\begin{align*}
c\check{@}d = x_1 + x_1x_0x_1 + 3x_1x_0x_1x_0x_1 + 4x_1x_0^2x_1^2 + \cdots.
\end{align*}

Infer from Definition <ref> that $\mathscr{C}\left(c\check{@}d\right) = \mathscr{C}\left(c\right) = 1$.

Finally, the main definition of the section details the concept of relative degree in the context of Chen–Fliess series which is characterized on its generating series.

[Gray & Venkatesh, 2019]
A series $c \in \allseries{}$ has relative degree $r$ if $\mathscr{C}(c) = r$ and the word $x_{0}^{r-1}x_{1} \in \supp(c_F)$. Otherwise, $c$ does not have relative degree.

The following theorem asserts the quantification of relative degree under multiplicative mixed composition product.

If $c \in \allseries{}$ with relative degree $r_c$ and $d \in \allseries{}$ be non-proper, then $c \lhook d$ has relative degree $r_c$.

From Theorem <ref>, $\mathscr{C}\left(c\lhook d\right) = r_c$. It remains to prove that $x_0^{r_c-1}x_1 \in \supp\left(c\lhook d\right)$.

Given that $c \in \allseries{}$ has relative degree $r_c$, then $c$ can be decomposed as:

\begin{align*}
c &= c_N + \lambda x_0^{r_c-1}x_1 + x_0^{r_c-1}c^{\prime},
\end{align*}

where $\lambda \neq 0$ and $c^{\prime}$ is a proper series such that $x_1 \not \in \supp\left(c^{\prime}\right)$. Then,

\begin{align*}
c \lhook d = \left(c_N + \lambda x_0^{r_c-1}x_1 + x_0^{r_c-1}c^{\prime}\right) \lhook d.
\end{align*}

Applying Theorem <ref>,

\begin{align*}
c \lhook d = \left(c_N \lhook d\right) + \lambda \left(x_0^{r_c-1}x_1 \lhook d\right) + \left(x_0^{r_c-1}c^{\prime}\lhook d\right).
\end{align*}

Using eqn:mixprocomp_def and Lemma <ref>,

\begin{align*}
c \lhook d = c_N + \lambda x_0^{r_c-1}x_1d + x_0^{r_c-1}\left(c^{\prime} \lhook d\right).
\end{align*}

Since $d \in \allpiseries{} \longrightarrow d = \alpha + d^{\prime}$, where $\alpha \neq 0$ and $d^{\prime}$ is a proper series. Hence,

\begin{align*}
c \lhook d = c_N + \lambda\alpha x_0^{r_c-1}x_1 + x_0^{r_c-1}x_1d^{\prime} + x_0^{r_c-1}\left(c^{\prime}\lhook d\right).
\end{align*}

Observe from eqn:mixprocomp_def, $x_1 \not \in \supp\left(c^{\prime}\right) \Longrightarrow x_1 \not \in \supp\left(c^{\prime}\lhook d\right)$ and also that $\alpha\lambda \neq 0$.

Therefore $x_0^{r_c-1}x_1 \in \supp\left(c\lhook d\right)$, whence the relative degree of $c \lhook d$ is $r_c$, when $d$ is a non-proper series.

The following example illustrates the statement from Theorem <ref>.
Let $c = 1 + x_0^2 + x_0x_1 + x_0^2x_1$ and $d = 1+x_1$. Observe that by Definition <ref>, the relative degree of $c$ is $r_c = 2$ and also that $d$ is non-proper. The multiplicative mixed composition product of $c$ and $d$ to computed as:

\begin{align*}
c \lhook d = 1 + x_0^2 + x_0x_1 + x_0x_1^2 + x_0^2x_1 + x_0^2x_1^2.
\end{align*}

Using Definition <ref>, note that the relative degree of $c \lhook d$ is $2 = r_c$.

The following theorem is the prime objective of this section stating that the relative degree of a series remains invariant under multiplicative dynamic feedback product.

If $c \in \allseries{}$ with relative degree $r_c$ and $d \in \allpiseries{}$, then the relative degree of $\left(c\check{@}d\right)$ is $r_c$.

Since $c \in \allseries{}$ and $d \in \allpiseries{}$, then by Theorem <ref>,

\begin{align*}
c \check{@} d = c\lhook \left(d^{\shuffle-1}\circ c\right)^{\star -1}.
\end{align*}

Observe that $d \in \allpiseries{} \Leftrightarrow d^{\shuffle -1} \in \allpiseries{}$. Then by Theorem <ref> $\left(d^{\shuffle-1}\circ c\right) \in \allpiseries{}$. As per Theorem <ref>, the group inverse

\begin{align*}
\left(d^{\shuffle -1}\circ c\right)^{\star -1} \in \allpiseries{}.
\end{align*}

Hence by Theorem <ref>,

\begin{align*}
c \check{@} d = c \lhook \left(d^{\shuffle -1}\circ c\right)^{\star -1}.
\end{align*}

has relative degree $r_c$.

The invariance of the relative degree of a Chen–Fliess series under  multiplicative dynamic feedback connections as stated in Theorem <ref> is illustrated through the following example.

Consider the Example <ref> again where $c = x_1$ and $d = 1 + \sum_{k \in \nat}k!x_1^k$. Observe that by Definition <ref>, the relative degree of $c$ is $r_c = 1$. The multiplicative feedback product is computed as:

\begin{align*}
c \check{@} d = x_1 + x_1x_0x_1 + 3x_1x_0x_1x_0x_1 + 4x_1x_0^2x_1^2 + \cdots
\end{align*}

Infer that the relative degree of $c\check{@}d = 1 = r_c$ as stated in Theorem <ref>.

§ COMPUTATIONAL FRAMEWORK FOR MULTIPLICATIVE MIXED COMPOSITION & DYNAMIC FEEDBACK PRODUCT

The goal of this section is to describe the computational framework for multiplicative dynamic feedback product as explained in Section <ref>. The section further illustrates the framework with examples but prior to that it is imperative to understand the dual bialgebra and Hopf algebra constructions corresponding to the multiplicative dynamic output feedback group.

§.§ Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Subgroup

The goal of the subsection is to construct a dual Hopf algebra reflecting the group structure of the multiplicative dynamic feedback subgroup $M$ as asserted in Theorem <ref>. The group inverse is computed the antipode of the constructed Hopf algebra and thus provides a computational framework to compute the multiplicative dynamic feedback group inverse. As a recall, the group $M$ is defined as

\begin{align*}
M &= \{ \doubleone + d \:: d \in \allproperseries{m}\},
\end{align*}

where $\doubleone = \left[ 1 \cdots 1 \: 1 \right]^{T}\in \re^m$. In light of Theorem <ref>, $(M, \star)$ forms a subgroup of the multiplicative dynamic feedback group. The algebra structure is same as the algebra of $H_{\shuffle}$ in Section <ref>.
Let the set $W_b \subset \allseries{m}^{\ast}$ (dual module of $\allseries{m}$) be defined as the collection of coordinate maps defined as:

\begin{align*}
W_b &= \{ a_{\eta} \: : a_{\eta}(c) = (c,\eta) \: : \: \eta \in \allwords\},
\end{align*}

where $c \in \allseries{m}$. Define $W$ to be the free $\re^m$-module spanned by the set $W_b$. Let $H$ denote the reduced symmetric algebra generated by the module $W$. The unit map $\xi : \re^m \longrightarrow W$ is defined by $\xi(\doubleone) = a_{\emptyset}$. Note that $a_{\emptyset}\left(c\right) = \doubleone \, \forall c \in M$. By construction $H$ is an $\re^m$-associative, commutative and unital algebra with addition, scalar multiplication and product defined, respectively, as

\begin{align*}
(a_{\eta} + a_{\zeta})(c) &= a_{\eta}(c) + a_{\zeta}(c)\\
(k a_{\eta})(c) &= k (a_{\eta}^i(c)) \\
\HAprod(a_{\eta},a_{\zeta})(c) &= a_{\eta}(c)a_{\zeta}(c),
\end{align*}

where $c \in \allseries{m}$. Then $H$ is given a coproduct $\Delta_H : H \longrightarrow H \bigotimes H$ such that for all $c,d \in M$: $\Delta_H a_{\eta}^i(c,d) = a_{\eta}^i(c\star d)  = \left(\left(c\star d\right)_i, \eta\right) \, \forall \eta \in X^{+}$. The counit map $\epsilon : H \longrightarrow \re$ is defined as

\begin{align*}
\epsilon(h) = \begin{cases}
\doubleone \; : & \!\!\! h = a_{\emptyset} \\
0\; : & \!\!\!\text{otherwise}.
\end{cases}
\end{align*}

Since $\circ$ is associative (from Theorem <ref>), thus by the dual the coproduct $\Delta_H$ is coassociative. Therefore, $\left(H,\HAprod,\xi,\Delta_H,\epsilon\right)$ forms a $\re^m$-bialgebra. Owing to the group structure of $\left(M, \circ\right)$, the bialgebra $H$ is equipped with antipode $S$ defined as:

\begin{align*}
Sa_{\eta} \left(c\right) = a_{\eta} \left(c^{\star -1}\right) = \left(c^{\star -1}, \eta\right),
\end{align*}

for all $i = 1,2,\ldots,m$ and $\eta\in X^{+}$. Hence $H$ is a $\re^m$-Hopf algebra. The computation of coproduct $\Delta_H$ is well-understood through the right coaction of Hopf algebra $H$ on the Hopf algebra $H_{\shuffle}$. Prior to that, it is imperative to understand the right coaction of Hopf algebra $H$ on the non-unital algebra of coordinate functions.

§.§ Coaction of Hopf algebra $H$ on Algebra of Coordinate Map

The subsection explains the coaction of the Hopf algebra $H$ defined in Section <ref> on the algebra of coordinate functions. The results in this subsection are utilized subsequently to explain the coaction of $H$ on the bialgebra $H_{\shuffle}$, particularly in proofs in Section <ref>. The right coaction of the Hopf algebra $H$ is on $\re^m$-algebra of coordinate maps $S^{+}\left(W\right)$ constructed in Section <ref>.

The right coaction map $\tilde{\Delta} : S^{+}\left(W\right) \longrightarrow S^{+}\left(W\right) \bigotimes H$ is defined such that for all $c \in \allseries{m}$, $d \in M$ and $\eta \in \allwords$,

\begin{align}\label{eqn:red_coprod}
\tilde{\Delta}a_{\eta}\left(c,d\right) = \left(c\lhook d,\eta\right).
\end{align}

The map $\tilde{\Delta}$ being a right coaction map is a reflection of Theorem <ref>. It remains to show how the coaction map $\tilde{\Delta}$ is computed on $S^{+}(W)$, for which it is sufficient to define its computation on the module $W$.  Observe that for all $a_{\eta} \in W$,

\begin{align*}
\tilde{\Delta}a_{\eta} = \left[\tilde{\Delta}\circ \pi^1 \;\; \tilde{\Delta} \circ \pi^2 \;\; \cdots \; \tilde{\Delta} \circ \pi^m\right]^ta_{\eta}.
\end{align*}

On the dual side, the above statement infers that for all $c \in \allseries{m}$, $d \in M$ and $\eta \in \allwords$,

\begin{align*}
(c\lhook d, \eta) = \left[\left(\left(c \lhook d\right)_1, \eta\right) \, \cdots \, \left(\left(c\lhook d\right)_m, \eta\right) \right]^t.
\end{align*}

Hence, the notation $\tilde{\Delta}a_{\eta}^i := \tilde{\Delta} \circ \pi^i a_{\eta}$ for all $\eta \in \allwords$ and $i = 1,2,\ldots,m$. The following proposition provides a recursive definition to compute $\tilde{\Delta}$ on the module $V$ viz to compute the $\tilde{\Delta}\left(a_{\eta}^j\right) \, \forall \eta \in \allwords$ and $j = 1,2,\dots,m$.

For all $i = 1,\ldots,m$:

* $\tilde{\Delta}a_{\emptyset}^i = a_{\emptyset}^i\otimes a_{\emptyset}^i$.

* $\tilde{\Delta} \circ \theta_0 = \left(\theta_0\otimes \id \right) \circ \tilde{\Delta}$.

* $\tilde{\Delta}\circ \theta_i = \left(\theta_i\otimes\HAprod\right)\circ \left(\tilde{\Delta}\otimes \id \right) \circ \rho_{\shuffle}^i$,
where $\rho_{\shuffle}$ is the coaction map of Hopf algebra $H_{\shuffle}$ on $S^{+}\left(W\right)$ as defined in Section <ref>.

Observe that $\forall c \in \allseries{m}$ and $d \in M$,

\begin{align*}
c &= \left(c,\emptyset\right) + \sum_{j=0}^m x_j\left(x_j^{-1}\left(c\right)\right).
\end{align*}

Hence by Theorem <ref>,

\begin{align}\label{eqn:mixprocomp_det}
c \lhook d &= \left(c,\emptyset\right) + x_0\left(x_0^{-1}\left(c\right)\lhook d\right) + \sum_{j=1}^m x_j\left(d_j \shuffle \left(x_j^{-1}\left(c\right)\lhook d\right)\right).
\end{align}

The proof for each of the statement as follows:

* Let $c,d \in \allseries{m}$. From eqn:red_coprod and eqn:mixprocomp_det,

\begin{align*}
\tilde{\Delta}a_{\emptyset}^i\left(c,d\right) &= \left(\left(c\lhook d\right)_i,\emptyset\right) \\
&= \left(c_i \lhook d,\emptyset\right) =\left(c_i,\emptyset\right).1 = a_{\emptyset}^i\otimes a_{\emptyset}^i\left(c,d\right).
\end{align*}

Therefore, $\tilde{\Delta}a_{\emptyset}^i = a_{\emptyset}^i\otimes a_{\emptyset}^i$.

* Let $c,d \in \allseries{m}$, $\eta \in \allwords$ and $\forall\, j = 1,2,\ldots,m$. Then,

\begin{align*}
\left(\tilde{\Delta}\circ \theta_0\right)a_{\eta}^j\left(c,d\right) &= \left(\left(c\lhook d\right)_j, x_0\eta\right)\\
&=\left(x_0^{-1}\left(c\lhook d\right)_j,\eta\right) \\
\end{align*}

From eqn:mixprocomp_det,

\begin{align*}
\left(\tilde{\Delta}\circ \theta_0\right)a_{\eta}^j\left(c,d\right) &= \left(x_0^{-1}\left(c_j\right)\lhook d,\eta\right) \\
&= \tilde{\Delta}a_{\eta}^j\left(x_0^{-1}\left(c\right),d\right) \\
&= \left(\theta_0 \otimes \id \right) \circ \tilde{\Delta}a_{\eta}\left(c,d\right).
\end{align*}

Therefore, $\tilde{\Delta} \circ \theta_0 = \left(\theta_0 \otimes \id \right) \circ \tilde{\Delta}$.

* Let $c,d \in \allseries{m}$ and $\eta \in \allwords$. Then $\, \forall \, i,j = 1,2,\ldots,m$,

\begin{align*}
\left(\tilde{\Delta}\circ\theta_i\right)a_{\eta}^j\left(c,d\right) &= \left(\left(c\lhook d\right)_j,x_i\eta\right)\\
&=\left(x_i^{-1}\left(c\lhook d\right)_j,\eta\right)\\
\end{align*}

From eqn:mixprocomp_det,

\begin{align*}
\left(\tilde{\Delta}\circ\theta_i\right)a_{\eta}^j\left(c,d\right) &= \left(d_i \shuffle x_i^{-1}\left(c_j\right)\lhook d, \eta\right)\\
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d,d\right) \\
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d\right)\\
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d,d\right)\\
&= \left(\tilde{\Delta}\otimes \id\right)\circ \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right),d,d\right) \\
&= \left(\theta_i \otimes \HAprod\right)\circ \left(\tilde{\Delta}\otimes \id\right) \circ \rho_{\shuffle}^ia_{\eta}^j\left(c,d\right).
\end{align*}

Therefore, $\tilde{\Delta}\circ \theta_i = \left(\theta_i\otimes\HAprod\right)\circ \left(\tilde{\Delta}\otimes\id \right) \circ \rho_{\shuffle}^i \,\, \forall i = 1,2,\ldots,m$.

A few examples of the computation of $\tilde{\Delta}$ on $V$ using Proposition <ref> are given as follows(indices $i,j,k = 1,2,\ldots,m$.):

\begin{align*}
\tilde{\Delta}a_{\emptyset}^i &= a_{\emptyset}^i \otimes a_{\emptyset}^i. \\
\tilde{\Delta}a_{x_0}^i &= a_{x_0}^i \otimes a_{\emptyset}^i. \\
\tilde{\Delta}a_{x_i}^j &= a_{x_i}^j \otimes a_{\emptyset}^i. \\
\tilde{\Delta}a_{x_0^2}^i &= a_{x_0^2}^i \otimes a_{\emptyset}^i. \\
\tilde{\Delta}a_{x_0x_i}^j &= a_{x_0x_i}^j \otimes a_{\emptyset}^i.  \\
\tilde{\Delta}a_{x_ix_0}^j &= \left(a_{x_ix_0}^j \otimes a_{\emptyset}^i\right) + \left(a_{x_i}^j\otimes a_{x_0}^i\right). \\
\tilde{\Delta}a_{x_ix_j}^k &= \left(a_{x_ix_j}^k \otimes a_{\emptyset}^ja_{\emptyset}^i\right) + \left(a_{x_i}^k \otimes a_{x_j}^i\right).
\end{align*}

The coaction map $\tilde{\Delta}$ thus provides a framework to compute the multiplicative mixed composition product and multiplicative dynamic feedback group product whenever $c\in \allseries{m}$ and $d \in M \subsetneq \allseries{m}$. For computing the multiplicative mixed composition product for $c \in \allseries{p}$ and $d \in M \subsetneq \allseries{m}$ where $p =m$,

* If $p < m$, then define $\check{c} \in \allseries{m} $ such that $\check{c}_i = c_i \; \forall\, i = 1,2,\ldots,p$ and $\check{c}_i = 0 \; \forall\, i = p+1,p+2,\ldots,m$. Then for all $\eta \in \allwords$,

\begin{align*}
\left(\left(c \lhook d\right)_i, \eta\right) = \tilde{\Delta}a_{\eta}^i \left(\check{c},d\right) \quad \forall i=1,2,\ldots,p.
\end{align*}

Note that $\left(\check{c}\lhook d\right)_j = 0 \; \forall j = p+1,p+2,\ldots,m$.

* If $p > m$, then this can be reduced to Case $1$ by performing computations component wise viz computing $c_i \lhook d$ for all $i = 1,2,\ldots,p$.

Thus the computational framework to compute the multiplicative mixed composition product of $c \in \allseries{p}$  and $d \in M$, denoted by $c \lhook d$ for arbitrary $p$ and $m$ is well-defined via the coaction map $\tilde{\Delta}$. The computations of the coproduct $\Delta_H$ and antipode $S$ (defined in Section <ref>) are well-understood once the right coaction of Hopf algebra $H$ on Hopf algebra $H_{\shuffle}$.

§.§ Coaction of Hopf algbera $H$ on the Hopf algebra $H_{\shuffle}$

The objective of the subsection is to define the right coaction map  of Hopf algebra $H$ on the unshuffle Hopf algebra $H_{\shuffle}$ defined in Section <ref>. The right coaction is pivotal in computation of the coproduct and antipode of Hopf algebra $H$ which in turn are essential to compute the multiplicative dynamic feedback product.

The right coaction map of $H$ on $H_{\shuffle}$ is defined to be $\tilde{\Delta}_H : H_{\shuffle} \longrightarrow H_{\shuffle} \bigotimes H$ such that for all $c,d \in M$ (the underlying sets of $M$ and $M_{\shuffle}$ are identical) and $\eta \in \allwords$,

\begin{align}\label{eqn:red_coprod_hopf_sec}
\tilde{\Delta}_H a_{\eta}\left(c,d\right) = \left(c\lhook d, \eta\right).
\end{align}

Observe that the algebra of coordinate functions $S^{+}(W)$ and $H_{\shuffle}$ are isomorphic as $\re^m$-modules. Thus it is vital to understand the relationship between the operator $\tilde{\Delta}$ operating on the module $S^{+}(W)$ and operator $\tilde{\Delta}_H$ operating on $H_{\shuffle}$, which is stated in the following lemma.

If $c,d \in M$, then for all $\eta \in \allwords$

\begin{align*}
\tilde{\Delta}_Ha_{\eta}\left(c,d\right) &= \tilde{\Delta}a_{\eta}\left(c,d\right).
\end{align*}

If $c,d \in M$ and $\eta \in X^{+}$,

\begin{align*}
\tilde{\Delta}_Ha_{\eta}\left(c,d\right) = \left(c\lhook d,\eta\right) = \tilde{\Delta}a_{\eta}\left(c,d\right).
\end{align*}

Despite the statement of Lemma <ref>, it is vital to understand the difference between the coaction maps $\tilde{\Delta}$ and $\tilde{\Delta}_H$. The coaction map $\tilde{\Delta}_H$ is compatible with the Hopf algebra structure of $H_{\shuffle}$ viz.

\begin{align*}
\HAprod_{1,3,24} \circ \left(\tilde{\Delta}_H\otimes \tilde{\Delta}_H\right)\circ \Delta_{\shuffle} &= \left(\Delta_{\shuffle} \otimes \id \right)\circ \tilde{\Delta}_H, \\
\tilde{\Delta}_H \circ S &= \left(S_{\shuffle} \otimes \id\right) \circ \tilde{\Delta}_H,
\end{align*}

where $\HAprod_{1,3,24} = \left(\HAprod \otimes \HAprod\right) \circ \left(\id \otimes \tau \otimes \id\right)$.

Thus the coaction map $\tilde{\Delta}_H$ makes $H_{\shuffle}$ a comodule-Hopf algebra over $H$. Equivalently, the coaction map $\tilde{\Delta}_H$ is a corepresentation of Hopf algebra $H$ over unshuffle Hopf algebra $H_{\shuffle}$. Similar to Section <ref>, for all $a_{\eta} \in W$,

\begin{align*}
\tilde{\Delta}_Ha_{\eta} = \left[\tilde{\Delta}_H\circ \pi^1 \; \tilde{\Delta}_H \circ \pi^2 \; \cdots \; \tilde{\Delta}_H \circ \pi^m\right]a_{\eta}.
\end{align*}

The map  to compute the $\tilde{\Delta}_H\left(a_{\eta}^j\right) \, \forall \eta \in \allwords$ and $j = 1,2,\dots,m$ is g module $W$ is stated in the following proposition.

For all $i,j = 1,2\ldots,m$ and $\eta \in \allwords$:

* $\tilde{\Delta}_Ha_{\emptyset}^i = a_{\emptyset}^i\otimes a_{\emptyset}^i$.

* $\tilde{\Delta}_H \circ \theta_0 a_{\eta}^j = \left(\theta_0\otimes \id \right) \circ \tilde{\Delta}_H a_{\eta}^j$.

* $\left(\tilde{\Delta}_H\circ\theta_i\right)a_{\eta}^j = \left(\theta_i\otimes\HAprod\right)\circ \left(\tilde{\Delta}_H\otimes \id \right) \circ \Delta_{\shuffle}^i a_{\eta}^j$,
where $\Delta_{\shuffle}$ is the unshuffle coproduct defined in Section <ref>.

Observe that $\forall c \in M$,

\begin{align*}
c &= \doubleone + \sum_{j=0}^m x_j\left(x_j^{-1}\left(c\right)\right).
\end{align*}

Hence by Theorem <ref>,

\begin{align}\label{eqn:mixprocomp_det_M}
c \lhook d &= \doubleone + x_0\left(x_0^{-1}\left(c\right)\lhook d\right) + \sum_{j=1}^m x_j\left(d_j \shuffle \left(x_j^{-1}\left(c\right)\lhook d\right)\right).
\end{align}

The proof for each of the statement as follows:

* Let $c,d \in M$. From eqn:red_coprod_hopf_sec and eqn:mixprocomp_det_M,

\begin{align*}
\tilde{\Delta}_Ha_{\emptyset}^i\left(c,d\right) &= \left(\left(c\lhook d\right)_i,\emptyset\right) \\
&= \left(c_i \lhook d, \emptyset\right) = 1 = (c_i, \emptyset)(d_i,\emptyset) \\
&= a_{\emptyset}^i \otimes a_{\emptyset}^i(c,d).
\end{align*}

Therefore, $\tilde{\Delta}_Ha_{\emptyset}^i = a_{\emptyset}^i\otimes a_{\emptyset}^i$.

* Let $c,d \in M$, $\eta \in \allwords$ and $\forall\, j = 1,2,\ldots,m$. Then,

\begin{align*}
\left(\tilde{\Delta}_H\circ \theta_0\right)a_{\eta}^j\left(c,d\right) &= \left(\left(c\lhook d\right)_j, x_0\eta\right)\\
&=\left(x_0^{-1}\left(c\lhook d\right)_j,\eta\right) \\
\end{align*}

Observe that $x_0^{-1}\left(c\right)$ may not belong to $M$ and from eqn:mixprocomp_det_M,

\begin{align*}
\left(\tilde{\Delta}_H\circ \theta_0\right)a_{\eta}^j\left(c,d\right) &= \left(x_0^{-1}\left(c_j\right)\lhook d,\eta\right) \\
&= \tilde{\Delta}a_{\eta}^j\left(x_0^{-1}\left(c\right),d\right) \\
&= \left(\theta_0 \otimes \id \right) \circ \tilde{\Delta}a_{\eta}\left(c,d\right).
\end{align*}

Since $\eta \in X^{+}$ and $c,d \in M$, then by Lemma <ref>

\begin{align*}
\left(\tilde{\Delta}_H\circ \theta_0\right)a_{\eta}^j\left(c,d\right) &= \left(\theta_0 \otimes \id \right) \circ \tilde{\Delta}_Ha_{\eta}\left(c,d\right).
\end{align*}

Therefore, $\tilde{\Delta}_H \circ \theta_0 = \left(\theta_0 \otimes \id \right) \circ \tilde{\Delta}_H$.

* Let $c,d \in M$ and $\eta \in \allwords$. Then $\, \forall \, i,j = 1,2,\ldots,m$,

\begin{align*}
\left(\tilde{\Delta}_H\circ\theta_i\right)a_{\eta}^j\left(c,d\right) &= \left(\left(c\lhook d\right)_j,x_i\eta\right)\\
&=\left(x_i^{-1}\left(c\lhook d\right)_j,\eta\right)\\
\end{align*}

From eqn:mixprocomp_det_M,

\begin{align*}
\left(\tilde{\Delta}_H\circ\theta_i\right)a_{\eta}^j\left(c,d\right) &= \left(d_i \shuffle x_i^{-1}\left(c_j\right)\lhook d, \eta\right).
\end{align*}

Since $x_i^{-1}\left(c\right)$ may not belong to group $M$ (also $M_{\shuffle}$),

\begin{align*}
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d,d\right) \\
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d\right)\\
&= \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right)\lhook d,d\right)\\
&= \left(\tilde{\Delta}\otimes \id\right)\circ \rho_{\shuffle}^ia_{\eta}^j\left(x_i^{-1}\left(c\right),d,d\right) \\
&= \left(\theta_i \otimes \HAprod\right)\circ \left(\tilde{\Delta}\otimes \id\right) \circ \rho_{\shuffle}^ia_{\eta}^j\left(c,d\right).
\end{align*}

Since $\eta \in X^{+}$ and $c,d \in M$, then by Lemma <ref> and Lemma <ref>,

\begin{align*}
\left(\tilde{\Delta}_H\circ\theta_i\right)a_{\eta}^j\left(c,d\right) = \left(\theta_i\otimes\HAprod\right)\circ \left(\tilde{\Delta}_H\otimes \id \right) \circ \Delta_{\shuffle}^ia_{\eta}^j.
\end{align*}

\begin{align*}
\left(\tilde{\Delta}_H\circ\theta_i\right) = \left(\theta_i\otimes\HAprod\right)\circ \left(\tilde{\Delta}_H\otimes \id \right) \circ \Delta_{\shuffle}^i
\end{align*}

for all $i = 1,2,\ldots,m$.

§.§ Coproduct, Antipode Computations and Grading of Hopf algebra $H$

The objective of this subsection is to define and illustrate the computation of coproduct $\Delta_H$ of the bialgebra $H$. Further, a graded and connected structure is endowed with the bialgebra owing to which the antipode computation is possible owing to Theorem <ref>. The following proposition asserts the essential reason behind the definition of $\tilde{\Delta}_H$.

For all $\eta \in \allwords$ and $i = 1,2,\ldots,m$,

\begin{align*}
\Delta_H a_{\eta}^i &= \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta}_H\otimes \id \right) \circ \hat{\Delta}_{\shuffle}^{i}a_{\eta}^i.
\end{align*}

Observe that for all $c,d \in M$ and $\eta \in \allwords$,

\begin{align*}
\Delta a_{\eta}^i \left(c,d\right) &= \left(\left(c\star d\right)_i, \eta\right) \quad \forall\, i = 1,2,\ldots,m.
\end{align*}

Using eqn:grp_prod,

\begin{align*}
\Delta_H a_{\eta}^i \left(c,d\right) &= \left(d_i \shuffle c_i\lhook d, \eta\right) \\
&= \hat{\Delta}_{\shuffle}^ia_{\eta}^i(c \lhook d, d)\\
&=\left(\tilde{\Delta}_H\otimes \id \right)\circ \hat{\Delta}_{\shuffle}^ia_{\eta}^i\left(c,d,d\right) \\
&= \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta}_H \otimes \id\right) \circ \hat{\Delta}_{\shuffle}^ia_{\eta}^i\left(c,d\right).
\end{align*}

Proposition <ref> asserts that the computation of coproduct $\Delta_H$ on the module $W$ (subsequently on the algebra $H$) can be carried out post the computation of the operator $\tilde{\Delta}_H$ on $W$. The computation of the coproduct $\Delta_H$ for the some coordinate maps are given as follows:

\begin{align*}
\Delta_H a_{\emptyset}^i &= a_{\emptyset}^i \otimes a_{\emptyset}^i. \\
\Delta_H a_{x_0}^i &= \left(a_{x_0}^i \otimes a_{\emptyset}^i\right) + \left(a_{\emptyset}^i \otimes a_{x_0}^i\right).\\
\Delta_H a_{x_i}^j &= \left(a_{x_i}^j \otimes a_{\emptyset}^i a_{\emptyset}^j\right) + \left(a_{\emptyset}^j \otimes a_{x_i}^j\right). \\
\Delta_H a_{x_0^2}^i &= \left(a_{x_0^2}^i \otimes a_{\emptyset}^i\right) + 2\left(a_{x_0}^i \otimes a_{x_0}^i\right) + \left(a_{\emptyset}^i \otimes a_{x_0^2}^i\right). \\
\Delta_H a_{x_0x_i}^j &= \left(a_{x_0x_i}^j \otimes a_{\emptyset}^j\right) + \left(a_{x_0}^j \otimes a_{x_i}^j\right) + \left(a_{x_i}^j \otimes a_{\emptyset}^ia_{x_0}^j\right) + \left(a_{\emptyset}^j \otimes a_{x_0x_i}^j\right).\\
\Delta_H a_{x_ix_0}^j &= \left(a_{x_ix_0}^j \otimes a_{\emptyset}^ia_{\emptyset}^j\right) + \left(a_{x_i}^j \otimes a_{x_0}^ia_{\emptyset}^j\right) + \left(a_{x_i}^j \otimes a_{\emptyset}^ia_{x_0}^j\right) + \left(a_{x_0}^j \otimes a_{x_i}^j\right) + \left(a_{\emptyset}^j \otimes a_{x_ix_0}^j\right).\\
\Delta_H a_{x_ix_j}^k &= \left(a_{x_ix_j}^k \otimes a_{\emptyset}^ja_{\emptyset}^ia_{\emptyset}^k\right) + \left(a_{x_i}^k \otimes a_{x_j}^ia_{\emptyset}^k\right) + \left(a_{x_i}^k\otimes a_{\emptyset}^ia_{x_j}^k\right) + \left(a_{x_j}^k \otimes a_{\emptyset}^ja_{x_i}^k\right) + \left(a_{\emptyset}^k \otimes a_{x_ix_j}^k\right).
\end{align*}

If $m = 2$ (two input-two output MIMO case) viz. $X = \{x_0,x_1,x_2\}$, then from above computations

\begin{align*}
\Delta_H a_{x_1x_2} &= \begin{bmatrix}
\left(a_{x_1x_2}^1 \otimes \left(a_{\emptyset}^1\right)^2a_{\emptyset}^2\right) + \left(a_{x_1}^1 \otimes a_{x_2}^1a_{\emptyset}^1\right) + \left(a_{x_1}^1 \otimes a_{\emptyset}^1a_{x_2}^1\right) + \left(a_{x_2}^1 \otimes a_{\emptyset}^2a_{x_1}^1\right) + \left(a_{\emptyset}^1 \otimes a_{x_1x_2}^1\right) \\
\\
\left(a_{x_1x_2}^2 \otimes a_{\emptyset}^1\left(a_{\emptyset}^2\right)^2\right) + \left(a_{x_1}^2 \otimes a_{x_2}^1a_{\emptyset}^2\right) + \left(a_{x_1}^2 \otimes a_{\emptyset}^1a_{x_2}^2\right) + \left(a_{x_2}^2 \otimes a_{\emptyset}^2a_{x_1}^2\right) + \left(a_{\emptyset}^2 \otimes a_{x_1x_2}^2\right)
\end{bmatrix}
\end{align*}

which can be rewritten as

\begin{align*}
\Delta_H a_{x_1x_2} &= \left(a_{x_1x_2} \otimes (a_{\emptyset}^1a_{\emptyset}^2\doubleone) a_{\emptyset}\right) + \left(a_{x_1} \otimes (a_{x_2}^1 \doubleone) a_{\emptyset}\right) + \left(a_{x_1} \otimes (a_{\emptyset}^1 \doubleone) a_{x_2}\right) + \\
& \quad \quad \quad  \left(a_{x_2} \otimes (a_{\emptyset}^2 \doubleone)a_{x_1}\right) + \left(a_{\emptyset} \otimes a_{x_1x_2}\right),
\end{align*}

where $\doubleone = [1\; 1]^t$. It is vital to observe that the term $\left(a_{x_1x_2} \otimes (a_{\emptyset}^1a_{\emptyset}^2 \doubleone)a_{\emptyset}\right)$ is a primitive term of the coproduct as
$a_{\emptyset}^1a_{\emptyset}^2\doubleone \cong \doubleone$ since $a_{\emptyset}$ is the unit of $H$.

The following corollary is resultant of the Proposition <ref> to the words of the form $x_0^n$ for all $n \geq 0$.

If $n \in \nat_0$, then for all $i = 1,2,\ldots,m$ (defining $x_0^0 := \emptyset$):

\begin{align*}
\tilde{\Delta}_Ha_{x_0^n}^i &= a_{x_0^n}^i \otimes a_{\emptyset}^i. \\
\Delta_H a_{x_0^n}^i &= \sum_{k = 0}^n \binom{n}{k} a_{x_0^k}^i \otimes a_{\emptyset}^ia_{x_0^{n-k}}^i.
\end{align*}

The proof is by induction on $n \in \nat_0$. The base case $(n= 0)$ :

\begin{align*}
\tilde{\Delta}_Ha_{\emptyset}^i &= a_{\emptyset}^i \otimes a_{\emptyset}^i,
\end{align*}

is proved in Proposition <ref>. Assume the statement is true for $n = k$, then

\begin{align*}
\tilde{\Delta}a_{x_0^{k+1}}^i &= \left(\tilde{\Delta}\circ\theta_0\right)a_{x_0^k}^i.
\end{align*}

Using Proposition <ref>,

\begin{align*}
\tilde{\Delta}a_{x_0^{k+1}}^i &= \left(\theta_0 \otimes \id\right) \circ \tilde{\Delta}a_{x_0^k}^i \\
&= \left(\theta_0 \otimes \id\right)\{a_{x_0^k}^i \otimes a_{\emptyset}^i\} \\
&= a_{x_0^{k+1}}^i \otimes a_{\emptyset}^i.
\end{align*}

Hence proved by induction on $n \in \nat_0$ that: $\tilde{\Delta}a_{x_0^{n}}^i = a_{x_0^n}^i \otimes 1$. Observe that from Proposition <ref>

\begin{align*}
\Delta a_{x_0^n}^i &= \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta}\otimes \id \right) \circ \Delta_{\shuffle}^{i}a_{x_0^n}^i.
\end{align*}

Using Corollary <ref>,

\begin{align*}
\Delta a_{x_0^n}^i &= \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta}\otimes \id \right) \left(\sum_{k = 0}^n \binom{n}{k}a_{x_0^k}^i \otimes a_{x_0^{n-k}}^{i}\right) \\
&= \left(\id \otimes \HAprod\right) \left(\sum_{k = 0}^n \binom{n}{k}\tilde{\Delta}a_{x_0^k}^i \otimes a_{x_0^{n-k}}^{i}\right) \\
&=  \left(\id \otimes \HAprod\right) \left(\sum_{k = 0}^n \binom{n}{k} a_{x_0^k}^i \otimes a_{\emptyset}^i \otimes a_{x_0^{n-k}}^{i}\right) \\
&= \sum_{k = 0}^n \binom{n}{k} a_{x_0^k}^i \otimes a_{\emptyset}^ia_{x_0^{n-k}}^{i}.
\end{align*}

Proposition <ref> asserted that the calculation of coproduct $\Delta_H$ is carried out post the computation of $\tilde{\Delta}_H$. However the converse is also true viz. the computation of $\tilde{\Delta}_H$ can be carried out if the evaluation fo the coproduct $\Delta_H$ is known a priori which is well-asserted in the following proposition.

For all $\eta \in X^{+}$ and for all $i=1,2,\ldots,m$,

\begin{align*}
\tilde{\Delta}_H a_{\eta}^i\left(c,d\right) = \left(\id \otimes \HAprod \right) \circ \left(\Delta_H \otimes S_{\shuffle}\right) \circ \hat{\Delta}_{\shuffle}^i a_{\eta}^i.
\end{align*}

Given $c,d \in M$, by Theorem eqn:grp_prod

\begin{align*}
\left(c \star d\right) = \left(d \shuffle \left(c \lhook d\right)\right).
\end{align*}

Observe that $d \in M$ implies that $d$ is shuffle invertible. Thus for any $\eta \in X^{+}$,

\begin{align*}
\left(\left(c \lhook d\right)_i,\eta\right) = \left(d_i^{\shuffle -1} \shuffle \left(c \star d\right)_i, \eta \right),
\end{align*}

for all $i = 1,2,\ldots,m$. Hence,

\begin{align*}
\left(\left(c \lhook d\right)_i, \eta\right) &= \tilde{\Delta}_H a_{\eta}^i\left(c,d\right)\\
&=\hat{\Delta}_{\shuffle}^ia_{\eta}^i\left(c \star d, d^{\shuffle -1}\right). \\
&= \left(\Delta_H \otimes S_{\shuffle}\right)\circ \hat{\Delta}_{\shuffle}^ia_{\eta}^i\left(c,d,d\right). \\
&= \left(\id \otimes \HAprod\right) \circ \left(\Delta_H \otimes S_{\shuffle}\right) \circ \hat{\Delta}_{\shuffle}^ia_{\eta}^i \left(c,d\right).
\end{align*}

The key point of the Proposition <ref> is the shuffle-invertibility of a series $c \in M$

The goal of this subsection is to provide a graded structure on the $\re$-module $W$ and consequently on the underlying $\re$-module structure of the Hopf algebra $H$ such that $H$ is connected and the homogeneous components of $H$ are finite-dimensional.

Given a word $\eta \in X^{+}$, denote the degree of the word as $\deg\left(\eta\right)$ and define $\deg\left(\eta\right) = \abs{\eta}$ and for all $k \geq 1$:

\begin{align*}
X_k := \{a_{\eta} : \deg\left(\eta\right) = k\}.
\end{align*}

* Define gradation on the $\re$-module $W$ viz.

\begin{align*}
W = \bigoplus_{k \geq 1} W_k,
\end{align*}

where $W_k$ is the free $\re$-module spanned by $X_k$.

* The gradation on the module $W$ induces a graded structure on the algebra $H$ as

\begin{align*}
H = \bigoplus_{n \in \nat_0}H_n,
\end{align*}

with $H_0 \cong \re$ in the category of $\re$-modules.

The following lemma aids in proving that the gradation in Definition <ref> makes the Hopf algebra $H$ is well-defined.

If $\eta \in \allwords$ such that $\deg\left(\eta\right) = n$ then

\begin{align*}
\tilde{\Delta}_H\left(a_{\eta}\right) \in \bigoplus_{i+j = n} W_i \otimes H_j,
\end{align*}

for all $k = 1,2,\ldots,m$.

The following observations will help in proving the lemma.

* The map $\{\theta_i\}_{i = 0}^m$  is a homogeneous operator of degree $1$ on the module $W$. If $\deg\left(\eta\right) = \abs{\eta} = n$ for some $\eta \in \allwords$, then $\abs{x_i\eta} = n+1$ for all $i = 0,1,\ldots,m$. Hence,

\begin{align*}
\theta_i : W_n \longrightarrow W_{n+1}
\end{align*}

for all $i = 0,1,\ldots,m$ and $n \geq 1$.

* Observe that if $\eta,\zeta,\gamma \in \allwords$ such that $\abs{\gamma} = n$ and $\gamma \in \supp\left(\eta \shuffle \zeta\right)$ then $\abs{\gamma} = n = \abs{\zeta} + \abs{\eta}$. Thus, the reduced coproduct $\hat{\Delta}_{\shuffle} : W \longrightarrow W \otimes W $ is homogeneous operator of degree $0$ viz.

\begin{align*}
\hat{\Delta}_{\shuffle} : W_n \longrightarrow \left(W \otimes W\right)_n.
\end{align*}

Let us prove the statement ot the lemma by induction on degree (equivalently length) $n$ of the word $\eta \in \allwords$. The base case is $n = 0 \Leftrightarrow \eta = \emptyset$. From Proposition <ref>,

\begin{align*}
\tilde{\Delta}_H a_{\emptyset} = a_{\emptyset} \otimes a_{\emptyset} \quad \in W_0 \otimes H_0,
\end{align*}

Thus the statement holds true for the base case. Assume that the statement of theorem holds true for all $\eta \in \allwords$ such that $\deg\left(\eta\right) \leq k$. Let $\eta^{\prime}$ such that $\deg\left(\eta^{\prime}\right) = k+1$. Then two cases can occur.

* Let $\eta^{\prime} = x_0\eta$ where $\abs{\eta} = k$. Then

\begin{align*}
\tilde{\Delta}_{H}a_{\eta^{\prime}} = \tilde{\Delta}_H \circ \theta_0 a_{\eta}^j.
\end{align*}

By Proposition <ref>,

\begin{align*}
\tilde{\Delta}_Ha_{\eta^{\prime}} = \left(\theta_0 \otimes \id\right) \circ \tilde{\Delta}_H a_{\eta}.
\end{align*}

Since $a_{\eta} \in W_k$, then by the induction hypothesis $\tilde{\Delta}_{H}\left(a_{\eta}\right) \subseteq \bigoplus_{i + j = k} W_i \otimes H_j$. Then,

\begin{align*}
\left(\theta_0 \otimes \id\right) \left(\bigoplus_{i + j =k} W_i \otimes H_j\right) &\subseteq \bigoplus_{i + j = k} W_{i+1} \otimes H_j\\
&\subseteq \bigoplus_{i + j = k+1} W_{i} \otimes H_j.
\end{align*}

Thus, $\tilde{\Delta}_Ha_{{\eta}^{\prime}} \in \bigoplus_{i+j = k+1} W_i \otimes H_j$ where $\abs{\eta^{\prime}} = k+1$.

* Let $\eta^{\prime} = x_i\eta$ where $\abs{\eta} = k$ and $x_i \neq x_0$. Then from Proposition <ref>,

\begin{align*}
\left(\tilde{\Delta}_H \circ \pi_j\right) a_{\eta^{\prime}} &= \left(\theta_i \otimes \HAprod\right) \circ \left(\tilde{\Delta}_H \otimes \id \right) \circ \left(\pi_j \otimes \pi_i\right) \circ \tilde{\Delta}_{\shuffle} a_{\eta}. \\
&= \left(\theta_i \otimes \HAprod\right) \circ \left((\tilde{\Delta}_H \circ \pi_j) \otimes \pi_i \right) \circ \tilde{\Delta}_{\shuffle} a_{\eta}
\end{align*}

\begin{align*}
\tilde{\Delta}_H a_{\eta^{\prime}} &= \left(\theta_i \otimes \HAprod\right) \circ \left(\tilde{\Delta}_H \otimes \doubleone.\pi_i\right) \tilde{\Delta}_{\shuffle}a_{\eta},
\end{align*}

where $\doubleone. \pi_i = [\pi_i \; \pi_i \cdots \pi_i]^t$. Since $\deg(\eta) = k$,

\begin{align*}
\tilde{\Delta}_{\shuffle} a_{\eta} \subseteq (W \otimes W)_k.
\end{align*}

Note that $\doubleone.\pi_i (a_{\eta})(c) = [a_{\eta}^i \; a_{\eta}^i \; \cdots a_{\eta}^i ](c) = a_{\eta}[c_i\; c_i \cdots c_i]$. Thus $\doubleone.\pi_i a_{\eta} \in W$ and then applying the induction hypothesis $\tilde{\Delta}_H W_n \subseteq (W \otimes H)_n$ for $n \leq k$,

\begin{align*}
\left(\tilde{\Delta}_H \otimes \doubleone.\pi_i\right) \left(W \otimes W\right)_k \subseteq \left(W \otimes H \otimes W\right)_k.
\end{align*}

\begin{align*}
\left(\theta_i \otimes \HAprod\right) \left(W \otimes H \otimes W\right)_k \subseteq (W \otimes H)_{k+1},
\end{align*}

as $\theta_i$ is homogeneous operator of degree $1$.
Thus, $\tilde{\Delta}_Ha_{{\eta}^{\prime}} \in \bigoplus_{i+j = k+1} W_i \otimes H_j$ where $\abs{\eta^{\prime}} = k+1$.

Hence proved by induction that for all $n \geq 0$: $\tilde{\Delta}_H\left(a_{\eta}\right) \in \bigoplus_{i+j = n} W_i \otimes H_j$ where $\abs{\eta} = n$.

The following proposition asserts that the grading on $H$ in Definition
<ref> is compatible with bialgebraic structure of $H$.

With the grading on the Hopf algebra $H$ as in Definition <ref>,

\begin{align*}
\Delta_{H}\left(H_n\right) \subseteq \bigoplus_{i + j =n} H_i \otimes H_j
\end{align*}

for all $n \geq 0$.

Observe that the statement is true for $n=0$. Prior to the proving the statement for $n \geq 1$, the following statement needs to be proved:

\begin{align*}
\Delta_{H}\left(W_n\right) \subseteq \bigoplus_{i+j = n} W_i \otimes H_j \quad \forall\, n \geq 0.
\end{align*}

Observe that from Proposition <ref>,

\begin{align*}
\Delta_H \circ \pi_i = \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta}_H \otimes \id\right) \circ \left(\pi_i \otimes \pi_i\right) \circ \tilde{\Delta}_{\shuffle}.
\end{align*}

Thus (by grouping them along the coordinate $i$),

\begin{align*}
\Delta_H = \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta} \circ \id \right) \circ \tilde{\Delta}_{\shuffle}.
\end{align*}

\begin{align*}
\Delta_H (W_n) = \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta} \circ \id \right) \circ \tilde{\Delta}_{\shuffle}(W_n) \\
\subseteq \left(\id \otimes \HAprod\right) \circ \left(\tilde{\Delta} \circ \id \right) (W \otimes W)_n.
\end{align*}

Using Proposition <ref>,

\begin{align*}
\Delta_H(W_n) \subseteq \left(\id \otimes \HAprod\right) (W \otimes H \otimes W)_n \\
\subseteq (W \otimes H)_n.
\end{align*}

Therefore, the intermediate statement holds true viz.

\begin{align*}
\Delta_{H}\left(W_n\right) \subseteq \bigoplus_{i+ j =n} W_i \otimes H_j \quad \forall\, n \geq 0.
\end{align*}

The statement of the theorem then holds true as $\Delta$ is an $\re^n$-algebra morphism  from $H$ to $H \otimes H$.

Thus Proposition <ref> asserts that the grading defined on the Hopf algebra $H$ in Definition <ref> is well-defined and connected. The homogeneous components are finite-dimensional and dimensions respect the Proposition <ref> since the bialgebras $H$ and $H_{\shuffle}$ are isomorphic with respect to the underlying graded algebraic structures.

The following example is rework of the Example $4.10$ in <cit.> acting as a check for the computation of feedback group inverse in one-dimensional case.
Let $c = 1 - x_1 \, \in \allseries{}$. The series $c^{\circ -1} = 1+\cdots+\cdots$. Using the recursive computation formula for antipode as in Theorem <ref>

\begin{align*}
a_{x_1}(c^{\circ -1}) = Sa_{x_1}\left(c\right) = -a_{x_1}(c) = 1.
\end{align*}

Observe that

\begin{align*}
\Delta_{H}^{\prime}a_{x_1^2} = 3 a_{x_1} \otimes a_{x_1}.
\end{align*}

\begin{align*}
a_{x_1^2}\left(c^{\circ -1}\right) &= Sa_{x_1^2}\left(c\right) \\
&= -a_{x_1^2} - 3a_{x_1}.Sa_{x_1} = -a_{x_1^2} + 3a_{x_1}^2.
\end{align*}

Therefore, $a_{x_1^2}\left(c^{\circ -1}\right) = 0 + 3(1)^2 = 3$. In similar fashion the reduced coproduct of $a_{x_1}^3$ is

\begin{align*}
\Delta_{H}^{\prime}a_{x_1^3} = 4a_{x_1} \otimes a_{x_1^2} + 6a_{x_1^2} \otimes a_{x_1} + 3 a_{x_1} \otimes a_{x_1}^2.
\end{align*}

\begin{align*}
a_{x_1^3}\left(c^{\circ -1}\right) &= \left[-a_{x_1^3} - 4a_{x_1}.Sa_{x_1^2} -6a_{x_1^2}.Sa_{x_1} - 3a_{x_1}.\left(Sa_{x_1}\right)^2\right](c)\\
&= 0 -4(-1)(3) -6(0)(-1) -3 (-1)(1)^2 = 15.
\end{align*}

Therefore $c^{\circ -1} = 1 + x_1 + 3x_1^2 + 15x_1^3 + 105 x_1^4 + \cdots$.

The result matches exactly with that of Example $4.10$ in <cit.>.

§ CONCLUSIONS AND FUTURE WORK

It was shown that the closed-loop system of a plant in Chen–Fliess series description in multiplicative output feedback with another system, given by Chen–Fliess series, has a Chen–Fliess series representation. An explicit expression of the closed-loop generating series was derived and the multiplicative dynamic feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework has been devised utilizing the dual Hopf algebras corresponding to the shuffle group and multiplicative output dynamic feedback group. Future work will be to address the solemn problem regarding the local convergence of the both multiplicative dynamic and static output feedback connections and to identify both the multiplicative dynamic and static feedback invariants.

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