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Open-loop control design for contraction in affine nonlinear systems

Mohamed Yassine Arkhis, Denis Efimov

TL;DR

The paper shows that contractive nonlinear time-varying systems are incrementally exponentially stable, and uses this to develop a constructive open-loop feedforward input design for affine-in-control systems to enforce contraction/IES under mild assumptions. The core result connects contraction to IES via displacements along curves between trajectories, producing an exponential decay bound. It then provides a systematic input-design procedure that leverages a piecewise time partition where the input modulates the Jacobian to make the linearized system uniformly negative definite, and extends to periodic settings. The work is illustrated with scalar examples and discusses the conservativeness and limitations of the assumptions, offering directions for future research on direct nonlinear-system formulations and broader applicability of incremental stability-based control design.

Abstract

In this paper, first, it is shown that if a nonlinear time-varying system is contractive, then it is incrementally exponentially stable. Second, leveraging this result, under mild restrictions, an approach is proposed to design feedforward inputs for affine in control systems providing contraction/incremental exponential stability. Unlike standard stability notions, which have well-established control design techniques, this note can be considered among the first ones to provide such a tool for a kind of incremental stability. The theoretical findings are illustrated by examples.

Open-loop control design for contraction in affine nonlinear systems

TL;DR

The paper shows that contractive nonlinear time-varying systems are incrementally exponentially stable, and uses this to develop a constructive open-loop feedforward input design for affine-in-control systems to enforce contraction/IES under mild assumptions. The core result connects contraction to IES via displacements along curves between trajectories, producing an exponential decay bound. It then provides a systematic input-design procedure that leverages a piecewise time partition where the input modulates the Jacobian to make the linearized system uniformly negative definite, and extends to periodic settings. The work is illustrated with scalar examples and discusses the conservativeness and limitations of the assumptions, offering directions for future research on direct nonlinear-system formulations and broader applicability of incremental stability-based control design.

Abstract

In this paper, first, it is shown that if a nonlinear time-varying system is contractive, then it is incrementally exponentially stable. Second, leveraging this result, under mild restrictions, an approach is proposed to design feedforward inputs for affine in control systems providing contraction/incremental exponential stability. Unlike standard stability notions, which have well-established control design techniques, this note can be considered among the first ones to provide such a tool for a kind of incremental stability. The theoretical findings are illustrated by examples.

Paper Structure

This paper contains 10 sections, 3 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement
  4. Main results
  5. Connection between definitions \ref{['def 2']} and \ref{['def 3']}
  6. Control design for contraction/incremental exponential stability
  7. Examples
  8. Illustration for Corollary \ref{['cor 1']}
  9. Conservativeness of conditions
  10. Conclusion

Key Result

Theorem 1

If the system 8 is contractive, then it is IES.

Figures (4)

  • Figure 1: Trajectories of the system \ref{['47']} with $u(t):=-3\sin(t)$ and $g(t)\equiv 0$.
  • Figure 2: Trajectories of the system \ref{['47']} with $u(t):=-3\sin(t)$ and $g(t)=t \cos (t)$.
  • Figure 3: Trajectories of the system \ref{['48']} with $u(t):=-2$.
  • Figure 4: Trajectories of the system \ref{['49']} with $u(t):=3$.

Theorems & Definitions (13)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Theorem 2
  • proof
  • ...and 3 more