Documentation

Fast and Numerical Chen–Fliess Series in Julia

A lightweight, high‑performance Julia package for computing iterated integrals, Lie derivatives, and truncated Chen–Fliess expansions for nonlinear control‑affine systems.

Features at a glance

Iterated integrals

Compute all iterated integrals up to a chosen word length using recursive schemes based on Chen’s identity.

Lie derivatives

Generate Lie derivatives indexed by words in the free monoid, symbolically or numerically.

Truncated Chen–Fliess series

Assemble truncated series and compare them against ODE simulations for validation and benchmarking.

Minimal dependencies

Built on top of Symbolics.jl and LinearAlgebra for a lean, composable stack.

What are Chen–Fliess series?

Chen–Fliess series provide an input–output representation of analytic nonlinear control‑affine systems. They express the output as a formal power series whose coefficients are Lie derivatives of the output function and whose terms are iterated integrals of the input.

Formally:

\[F_c[u](t) = \sum_{\eta \in X^*} (c,\eta)\, E_\eta[u](t),\]

where:

• \(\eta\) is a word over the alphabet \(X = \{x_0,\ldots,x_m\}\),

• \((c,\eta)\) is the coefficient given by a Lie derivative,

• \(E_\eta[u](t)\) is the iterated integral associated with \(\eta\).

These series play a central role in nonlinear control theory, system approximation, and operator‑theoretic analysis.

Overview

ChenFliessSeries.jl implements efficient numerical and symbolic tools for:

• generating all iterated integrals indexed by words up to a chosen length,

• computing Lie derivatives associated with words in the free monoid,

• assembling truncated Chen–Fliess series,

• validating approximations against ODE simulations.

The implementation is based on Chen’s identity, which provides recursive relations between iterated integrals and Lie derivatives, significantly reducing redundant computation.

This package is the first publicly available open‑source implementation of these tools in the Julia ecosystem, relying only on Symbolics.jl and LinearAlgebra.jl for minimal and efficient dependencies.

Why Julia?

ChenFliessSeries.jl is written in Julia to leverage:

• High performance: JIT compilation and type specialization enable tight inner loops for iterated integrals and Lie derivatives.

• Multiple dispatch: Natural expression of different system types, vector field representations, and numeric backends.

• Symbolic–numeric synergy: Seamless integration with Symbolics.jl for symbolic Lie derivatives and LinearAlgebra for efficient evaluation.

• Interactive workflows: Notebooks and REPL‑driven exploration make it easy to prototype, visualize, and benchmark Chen–Fliess expansions.

Citing this package

If you use ChenFliessSeries.jl in academic work, please cite it as:

@misc{ChenFliessSeriesJL,
  title        = {ChenFliessSeries.jl: A Julia package for computing Chen-Fliess series},
  author       = {Perez Avellaneda, Ivan},
  year         = {2026},
  url          = {https://github.com/iperezav/ChenFliessSeries.jl},
  note         = {Julia package version 1.0.0}
}

A CITATION.bib file is also included in the repository.

Example gallery

Nonlinear pendulum
Chen–Fliess approximation vs ODE solution.

2D Planar quadrotor
Control‑affine model and truncated series behavior.

Contents

• Introduction
  • Why Chen–Fliess series?
  • Understanding the algebraic structure
  • Definition of iterated integrals
  • Definition of Lie derivatives
  • A concrete example
  • Practical use and truncation
  • What the package does
  • Summary of the Chen–Fliess pipeline
• Getting Started
  • Contents
  • Installation
  • Basic Usage (High-Level Overview)
  • Computing iterated integrals
  • Defining vector fields and outputs
  • Computing Lie derivatives
  • Computing truncated Chen–Fliess series
  • Comparing with ODE simulation
  • Common pitfalls
  • More examples
• Examples
  • 2D Planar Quadrotor
    • System dynamics
    • Defining the system in Julia
    • Input signal
    • Computing iterated integrals
    • Computing Lie derivatives
    • Assembling the truncated Chen-Fliess series
    • Simulating the true 2D planar quadrotor dynamics
    • Comparison plot
  • Pendulum Example
    • System dynamics
    • Defining the system in Julia
    • Input signal
    • Computing iterated integrals
    • Computing Lie derivatives
    • Assembling the truncated Chen–Fliess series
    • Simulating the true pendulum dynamics
    • Comparison plot
    • Error analysis
    • Interpretation
• Contributing
  • How to Contribute
  • Guidelines