arXiv subjects

W. Steven Gray

Publications and source records attributed to W. Steven Gray.

At least 19 recordsLinked to original sources

arXiv2025

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

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.

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arXiv2025

On Symmetries in Analytic Input-Output Systems

There are many notions of symmetry for state space models. They play a role in understanding when systems are time reversible, provide a system theoretic interpretation of thermodynamics, and have applications in certain stabilization and optimal control problems. The earliest form of symmetry for analytic input-output systems is due to Fliess who introduced systems described by an exchangeable generating series. In this case, one is able to write the output as a memoryless analytic function of the integral of each input. The first goal of this paper is to describe two new types of symmetry for such Chen--Fliess input-output systems, namely, coefficient reversible symmetry and palindromic symmetry. Each concept is then related to the notion of an exchangeable series. The second goal of the paper is to provide an in-depth analysis of Chen--Fliess input-output systems whose generating series are linear time-varying, palindromic, and have generating series coefficients growing at a maximal rate while ensuring some type of convergence. It is shown that such series have an infinite Hankel rank and Lie rank, have a certain infinite dimensional state space realization, and a description of their relative degree and zero dynamics is given.

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arXiv2024

Decompositions of Nonlinear Input-Output Systems to Zero the Output

Consider an input-output system where the output is the tracking error given some desired reference signal. It is natural to consider under what conditions the problem has an exact solution, that is, the tracking error is exactly the zero function. If the system has a well defined relative degree and the zero function is in the range of the input-output map, then it is well known that the system is locally left invertible, and thus, the problem has a unique exact solution. A system will fail to have relative degree when more than one exact solution exists. The general goal of this paper is to describe a decomposition of an input-output system having a Chen-Fliess series representation into a parallel product of subsystems in order to identify possible solutions to the problem of zeroing the output. For computational purposes, the focus is on systems whose generating series are polynomials. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying the subsystems in a parallel product decomposition. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words.

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arXiv2024

Parameter Dependent Chen--Fliess Series and Their Nonrecursive Interconnections

A class of parameter dependent Chen--Fliess series is introduced where the series coefficients are taken from a noncommutative ring of multivariable differential operators. Such series are shown in the linear case to represent formal solutions to Cauchy initial value problems for nonhomogeneous PDEs and thus are useful for characterizing the input-output maps of distributed control systems. It is also shown that this class of functional series is almost closed under the set of nonrecursive interconnections, that is, any finite combination of parallel and series interconnections without a closed-loop. Some sufficient conditions are needed for the series interconnection. Specific examples are given involving the transport equation and the wave equation.

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arXiv2024

On the Post-Lie Structure in SISO Affine Feedback Control Systems

The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.

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arXiv2022

Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions.

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arXiv2021

Formal Power Series Approach to Nonlinear Systems with Additive Static Feedback

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and an additive static output feedback is applied. The first step is to consider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output feedback, and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener-Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener-Fliess composition product and the additive static feedback product are completely characterized.

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arXiv2021

Additive Networks of Chen-Fliess Series: Local Convergence and Relative Degree

Given an additive network of input-output systems where each node of the network is modeled by a locally convergent Chen-Fliess series, two basic properties of the network are established. First, it is shown that every input-output map between a given pair of nodes has a locally convergent Chen-Fliess series representation. Second, sufficient conditions are given under which the input-output map between a pair of nodes has a well defined relative degree as defined by its generating series. This analysis leads to the conclusion that this relative degree property is generic in a certain sense.

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arXiv2021

Continuity of Formal Power Series Products in Nonlinear Control Theory

Formal power series products appear in nonlinear control theory when systems modeled by Chen-Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal of the present article is to prove the continuity and analyticity of such products with respect to several natural (locally convex) topologies on spaces of locally convergent formal power series in order to establish foundational properties behind these technologies. In addition, it is shown that a transformation group central to describing the output feedback connection is in fact an analytic Lie group in this setting with certain regularity properties.

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arXiv2020

Generating Series for Networks of Chen-Fliess Series

Consider a set of single-input, single-output nonlinear systems whose input-output maps are described only in terms of convergent Chen-Fliess series without any assumption that finite dimensional state space models are available. It is shown that any additive or multiplicative interconnection of such systems always has a Chen-Fliess series representation that can be computed explicitly in terms of iterated formal Lie derivatives.

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arXiv2020

Combining Learning and Model Based Control via Discrete-Time Chen-Fliess Series

A learning control system is presented suitable for control affine nonlinear plants based on discrete-time Chen-Fliess series and capable of incorporating knowledge of a given physical model. The underlying noncommutative algebraic and combinatorial structures needed to realize the multivariable case are also described. The method is demonstrated using a two-input, two-output Lotka-Volterra system.

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arXiv2020

Continuity of Chen-Fliess Series for Applications in System Identification and Machine Learning

Model continuity plays an important role in applications like system identification, adaptive control, and machine learning. This paper provides sufficient conditions under which input-output systems represented by locally convergent Chen-Fliess series are jointly continuous with respect to their generating series and as operators mapping a ball in an $L_p$-space to a ball in an $L_q$-space, where $p$ and $q$ are conjugate exponents. The starting point is to introduce a class of topological vector spaces known as Silva spaces to frame the problem and then to employ the concept of a direct limit to describe convergence. The proof of the main continuity result combines elements of proofs for other forms of continuity appearing in the literature to produce the desired conclusion.

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arXiv2017

The Faa di Bruno Hopf algebra for multivariable feedback recursions in the center problem for higher order Abel equations

Poincare's center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a convenient coordinate system. In 1989, Devlin described an algebraic approach for constructing sufficient conditions for a center using a linear recursion for the generating series of the solution to the Abel equation. Subsequent work by the authors linked this recursion to feedback structures in control theory and combinatorial Hopf algebras, but only for the lowest degree case. The present work introduces what turns out to be the nontrivial multivariable generalization of this connection between the center problem, feedback control, and combinatorial Hopf algebras. Once the picture is completed, it is possible to provide generalizations of some known identities involving the Abel generating series. A linear recursion for the antipode of this new Hopf algebra is also developed using coderivations. Finally, the results are used to further explore what is called the composition condition for the center problem.

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arXiv2017

Post-Lie Algebras, Factorization Theorems and Isospectral-Flows

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.

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arXiv2017

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

The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen-Fliess functional expansions or Fliess operators and are independent of the existence of any state space models. This interconnection, called an output affine feedback connection, is distinguished from conventional output feedback by the presence of a multiplier in an outer loop. Once this transformation group is established, three basic questions are addressed. How can this transformation group be used to provide an explicit Fliess operator representation of such a closed-loop system? Is it possible to use this feedback scheme to do system inversion purely in an input-output setting? In particular, can feedback input-output linearization be posed and solved entirely in this framework, i.e., without the need for any state space realization? Lastly, what can be said about feedback invariants under this transformation group? A final objective of the paper is to describe the Lie algebra of infinitesimal characters associated with the group in terms of a pre-Lie product.

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arXiv2016

Collision Avoidance for Bi-Steerable Car Using Analytic Left Inversion

A case study is presented of a collision avoidance system that directly integrates the kinematics of a bi-steerable car with a suitable path planning algorithm. The first step is to identify a path using the method of rapidly exploring random trees, and then a spline approximation is computed. The second step is to solve the output tracking problem by explicitly computing the left inverse of the kinematics of the system to render the Taylor series of the desired input for each polynomial section of the spline approximation. The method is demonstrated by numerical simulation.

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arXiv2016

Evaluating Generating Functions for Periodic Multiple Polylogarithms

The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.

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arXiv2015

Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback

A combinatorial interpretation is given of Devlin's word problem underlying the classical center-focus problem of Poincare for non-autonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras intimately related to the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by passing through control theory since the Abel equation, which describes a center, is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived from the composition of iterated integrals rather than just the products of iterated integrals, which yields the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faa di Bruno type Hopf algebra, which is defined in terms of the shuffle product but is a distinct algebraic structure.

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