Fliess Operators Research Articles

Discrete-time approximations of Fliess operators

A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are achievable in the sense that worst case inputs can be identified which hit the error bound. The paper then focuses on the special case where the operators are rational, i.e., they have rational generating series. It is shown in this situation that the iterated sum approximation can be realized by a discrete-time state space model which is a rational function of the input and state affine. In addition, this model comes from a specific discretization of the bilinear realization of the rational Fliess operator.

SISO Output Affine Feedback Transformation Group and Its Faà 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.

Dendriform–Tree Setting for Fliess Operators

This paper provides a dendriform-tree setting for Fliess operators with matrix-valued inputs. This class of analytic nonlinear input-output systems is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence of the defining series are also given.

A combinatorial Hopf algebra for nonlinear output feedback control systems

In this work a combinatorial description is provided of a Faà di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given.

The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finite-dimensional, connected grading. Dually, the vector space ℝ ⟨ x 0, x 1 ⟩ is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibility which makes ℝ ⟨ x 0, x 1 ⟩ a Com-Pre-Lie algebra. We give a presentation of this object as a pre-Lie algebra.

A pre-Lie algebra associated to a linear endomorphism and related algebraic structures

We construct a functor $$T$$ from the category of endomorphisms of vector spaces to the category of Com-PreLie algebras. For any endomorphism $$f$$ of a vector space $$V$$ , we describe the enveloping algebra of the pre-Lie algebra $$T(V,f)$$ , the dual Hopf algebra and the associated group of characters. For $$f=\mathrm{Id}_V$$ , we find the algebra of formal diffeomorphisms, seen as a subalgebra of the Connes–Kreimer Hopf algebra of rooted trees in the context of QFT; for other well-chosen nilpotent $$f$$ , we obtain the groups of Fliess operators in Control Theory. An algebraic study of these Com-PreLie Hopf algebras is carried out: gradations, generation, subobject generated by $$V$$ , etc.

Functional Series Expansions for Continuous-Time Switched Systems

The main objective of this paper is to describe a class of functional series expansions, known as Fliess operators, which admit inputs from a ball in an L p space as well as Poisson random processes. Conditions are given under which these functional series expansions converge absolutely in the mean. Then, it is shown that a continuous-time switched input-affine nonlinear system with a Poisson switching signal can be represented as a Fliess operator and that for certain cases a closed form solution can be obtained in terms of Poisson integrals.

Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators

Given two nonlinear input–output systems written in terms of Chen–Fliess functional expansions, i.e., Fliess operators, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed.

A unified approach to generating series for mixed cascades of analytic nonlinear input–output systems

The primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input–output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators (Fliess operators) and analytic functions provided that the composite system is well defined and resides in one of these two classes. In each case, the radius of convergence of the interconnection is computed.

On Fliess operators driven by L 2-Itô processes

Fliess operators, which are a type of functional series expansion, have been used to describe a broad class of nonlinear input–output maps driven by deterministic inputs. But in most applications, a system's inputs have noise components. This paper has three objectives. The first objective is to show that the notion of a Fliess operator can be generalized to admit a class of L 2-Itô stochastic input processes. The next objective is to show that they converge absolutely over an arbitrarily large but finite time interval when a certain coefficient growth condition is met. However, a significant number of systems fail to meet this condition. Thus, the final objective is to consider an interval of convergence having a random length so that a Fliess operator might converge under less restrictive growth conditions.

On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently.

A Faà di Bruno Hopf algebra for a group of Fliess operators with applications to feedback

A Faà di Bruno type Hopf algebra is developed for a group of integral operators known as Fliess operators, where operator composition is the group product. Such operators are normally written in terms of generating series over a noncommutative alphabet. Using a general series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. The result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product, that is, the generating series for the Fliess operator representation of the closed-loop system written in terms of the generating series of the Fliess operator component systems. This formula is employed to provide a proof that local convergence is preserved under feedback.

Non-causal Fliess operators and their shuffle algebra

Fliess operators as a class of non-linear operators have been well studied in several respects. They have a well developed realization theory and convenient representations in terms of directed infinite products of exponential Lie series. Their interconnection as subsystems has been studied, as has their relationship to rational systems. They find applications in such diverse areas as discretization methods for controls systems, optimal control, neural network analysis, and the numerical solution of stochastic differential equations. One issue concerning Fliess operators, however, that has received little attention is their possible generalization to the non-causal case. Examples of such operators appear implicitly in the literature addressing Hilbert adjoints of causal non-linear operators and system inversion for the purpose of output tracking. But a general, systematic treatment of the subject has not appeared. In this paper, a non-causal extension of a Fliess operator is developed with the primary focus being on local convergence, continuity, the associated shuffle algebra, and computing adjoint operators.

The formal Laplace‐Borel transform of Fliess operatorsand the composition product

The formal Laplace‐Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace‐Borel transform is shown to provide an isomorphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition product. Finally, the formal Laplace‐Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the operational calculus of Fliess for computing the output response of an analytic nonlinear system.

Generating Series for Interconnected Analytic Nonlinear Systems

Given two analytic nonlinear input-output systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection, and feedback connection. In each case, the corresponding generating series is produced and conditions for the convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series has its set of known properties significantly expanded. In addition, the notion of a feedback product for formal power series is shown to be well defined in a broad context, and its basic properties are characterized.

Fliess operators on Lp spaces: convergence and continuity

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, L p function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in L p [ t 0, t 0+ T], where T is bounded and p⩾1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on L p [ t 0, t 0+ T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition.