Natural Metrics in Contraction Analysis

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 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 quadratic geodesic coordinates, and exact exponential convergence rates can be computed analytically. Specializing the results above to Hamiltonian systems shows that differential lengths of general Hamiltonian dynamics correspond to exact complex analytic exponential functions, whose eigenvalues can be analytically computed from the metric, damping, curvature, second covariant derivative of the potential energy, and first covariant derivative of the vector potential, a result which applies to both classical and relativistic systems. Incorporating nonlinear inequality constraints is also discussed. 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.