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Contraction 2.0 Natural Metrics in Contraction Analysis

Winfried Lohmiller, Jean-Jacques Slotine

TL;DR

This work introduces a systematic, coordinate-invariant method to derive contraction metrics for general nonlinear time-varying systems by recasting dynamics as a covariant natural gradient with a symmetric invertible metric $M_{jk}$. Through a local geodesic-coordinate decomposition, it yields exact, decoupled contraction rates as solutions to a generalized eigenvalue problem $\theta_a^k (U_{| jk} - \lambda_a M_{jk}) = 0$, applicable to unconstrained, inequality-constrained, and Hamiltonian dynamics. The approach unifies contraction analysis with geometric concepts, enabling precise computation of incremental stability across diverse systems and providing a foundation for robust control and estimation in complex settings. The results are coordinate-invariant and supported by tensor-based derivations, with practical demonstrations on classical mechanical and robotic systems and outlines for extending to nonlinear observers, discrete-time frameworks, and machine learning applications.

Abstract

Contraction analysis establishes exponential incremental convergence of a nonlinear system by solving a linear matrix inequality for a contraction metric, and has become a standard resource for solving problems in nonlinear control and estimation. This paper shows that, for a general nonlinear time-varying system, a contraction metric can be systematically derived by rewriting the system dynamics as a complex natural gradient dynamics. In this form, the variational dynamics can be modally decomposed with geodesic coordinates, and exact exponential convergence rates can be computed. The results are extended to systems with nonlinear inequality constraints. All derivations are tensor-based, and the computed eigenvalues themselves are coordinate-invariant, i.e., the contraction rates are independent of the chosen coordinate system. Simple examples including a gravity pendulum, gradient descent with non-convex cost, Schuler dynamics, and a two-link manipulator, illustrate that the computation of the decomposed convergence rates is straightforward. The role of inequality constraints is illustrated for a controller confined to an operational envelope.

Contraction 2.0 Natural Metrics in Contraction Analysis

TL;DR

This work introduces a systematic, coordinate-invariant method to derive contraction metrics for general nonlinear time-varying systems by recasting dynamics as a covariant natural gradient with a symmetric invertible metric . Through a local geodesic-coordinate decomposition, it yields exact, decoupled contraction rates as solutions to a generalized eigenvalue problem , applicable to unconstrained, inequality-constrained, and Hamiltonian dynamics. The approach unifies contraction analysis with geometric concepts, enabling precise computation of incremental stability across diverse systems and providing a foundation for robust control and estimation in complex settings. The results are coordinate-invariant and supported by tensor-based derivations, with practical demonstrations on classical mechanical and robotic systems and outlines for extending to nonlinear observers, discrete-time frameworks, and machine learning applications.

Abstract

Contraction analysis establishes exponential incremental convergence of a nonlinear system by solving a linear matrix inequality for a contraction metric, and has become a standard resource for solving problems in nonlinear control and estimation. This paper shows that, for a general nonlinear time-varying system, a contraction metric can be systematically derived by rewriting the system dynamics as a complex natural gradient dynamics. In this form, the variational dynamics can be modally decomposed with geodesic coordinates, and exact exponential convergence rates can be computed. The results are extended to systems with nonlinear inequality constraints. All derivations are tensor-based, and the computed eigenvalues themselves are coordinate-invariant, i.e., the contraction rates are independent of the chosen coordinate system. Simple examples including a gravity pendulum, gradient descent with non-convex cost, Schuler dynamics, and a two-link manipulator, illustrate that the computation of the decomposed convergence rates is straightforward. The role of inequality constraints is illustrated for a controller confined to an operational envelope.
Paper Structure (6 sections, 9 theorems, 41 equations)

This paper contains 6 sections, 9 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Natural gradient dynamics and its exact decomposition
  3. Exact decomposition of constrained dynamics
  4. Exact decomposition of Hamiltonian dynamics
  5. Summary
  6. Appendix

Key Result

Lemma 1

If $A_{jk}(x^h)$ is a complex symmetric matrix w.r.t. complex symmetric metric $M_{lm}(x^h)$ then hold with the Kronecker delta $\delta_{lm}$kronecker1868, the complex orthonormal eigenvector matrix $\Theta_l^k(x^n) = (\theta_1^k, ..., \theta_n^k)$ and the diagonal eigenvalue matrix $\Lambda_{lm}(x^n) = diag(\lambda_1, ..., \lambda_n)$ from Definition def:eigenval.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition 3
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • ...and 3 more