Lyndon Word Transduction to Improve the Efficiency of Computing Chen-Fliess Series

Abstract

The class of input-output systems representable as Chen-Fliess series arises often in control theory. One well known drawback of this representation, however, is that the iterated integrals which appear in these series are algebraically related by the shuffle product. This becomes relevant when one wants to numerically evaluate these series in applications, as this redundancy leads to unnecessary computational expense. The general goal of this paper is to present a computationally efficient way to evaluate these series by introducing what amounts to a change of basis for the computation. The key idea is to use the fact that the shuffle algebra on (proper) polynomials over a finite alphabet is isomorphic to the polynomial algebra generated by the Lyndon words over this alphabet. The iterated integrals indexed by Lyndon words contain all the input information needed to compute the output. The change of basis is accomplished by applying a transduction, that is, a linear map between formal power series in different alphabets, to re-index the Chen-Fliess series in terms of Lyndon monomials. The method is illustrated using a simulation of a continuously stirred-tank reactor system under a cyber-physical attack.