Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design

TL;DR

This paper introduces control contraction metrics, providing convex, coordinate-invariant conditions for nonlinear feedback stabilization, including new criteria for stabilizing nonlinear submanifolds, with practical controller design via sum-of-squares programming.

Contribution

It extends contraction analysis to nonlinear control design through convex, invariant conditions and introduces novel criteria for stabilizing nonlinear submanifolds.

Findings

Conditions are necessary and sufficient for feedback linearizable systems.

Convex criteria enable controller synthesis using sum-of-squares programming.

Demonstrated stabilization of an unstable polynomial system with combined local and global stability.

Abstract

We introduce the concept of a control contraction metric, extending contraction analysis to constructive nonlinear control design. We derive sufficient conditions for exponential stabilizability of all trajectories of a nonlinear control system. The conditions have a simple geometrical interpretation, can be written as a convex feasibility problem, and are invariant under coordinate changes. We show that these conditions are necessary and sufficient for feedback linearizable systems, and also derive novel convex criteria for exponential stabilization of a nonlinear submanifold of state space. We illustrate the benefits of convexity by constructing a controller for an unstable polynomial system that combines local optimality and global stability, using a metric found via sum-of-squares programming.