The realization of input-output maps using bialgebras

TL;DR

This paper employs bialgebra theory to unify classical and modern state space realization results for nonlinear control systems, providing an algebraic framework for input-output maps.

Contribution

It introduces a novel algebraic approach using bialgebras to analyze and characterize state space realizations of nonlinear systems.

Findings

Unified classical and modern realization results

Characterization of differentially produced functionals

Algebraic framework for input-output map analysis

Abstract

We use the theory of bialgebras to provide the algebraic background for state space realization theorems for input-output maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent results involving analysis related to families of trees. If $H$ is a bialgebra, we say that $p \in H^*$ is differentially produced by the algebra $R$ with the augmentation $\epsilon$ if there is right $H$-module algebra structure on $R$ and there exists $f \in R$ satisfying $p(h) = \epsilon(f \cdot h)$. We characterize those $p \in H^*$ which are differentially produced.