Concave Comparison Functions for Accelerating Constrained Lyapunov Decay

TL;DR

This paper demonstrates that using concave comparison functions in Lyapunov analysis can strictly improve decay guarantees and actuation efficiency for control systems, surpassing linear and convex approaches.

Contribution

It introduces the superiority of concave comparison functions for Lyapunov decay acceleration and provides a constructive design method compatible with CLF QP.

Findings

Concave functions outperform linear and convex ones in decay guarantees.

Strict concavity reduces required actuation levels for target rates.

A tunable rational concave factor enables practical Lyapunov decay acceleration.

Abstract

What limits how fast a Lyapunov function can decay under input bounds? We address this question by showing how the shape of Lyapunov comparison functions governs guaranteed decay for control affine systems. Using a windowed nominal exponential rate together with the endpoint cap induced by actuator limits, we establish a strict ordering: concave comparison functions strictly outperform linear and convex ones, and strict concavity is necessary to improve the best achievable global exponential rate under a fixed endpoint cap. We derive a computable lower bound on the required actuation level for a target nominal rate and show that only concave shaping can reduce this level under the endpoint cap. We then establish a feasibility-preserving acceleration result: whenever a margin exists on a sublevel set, a feasible linear comparison can be replaced by a concave one that preserves feasibility while strictly increasing the guaranteed windowed decay. Finally, we give a tunable rational concave factor with controlled slope that yields a constructive design and integrates with CLF QP, as illustrated by examples.