Lyapunov Functions can Exactly Quantify Rate Performance of Nonlinear Differential Equations

TL;DR

This paper demonstrates that Lyapunov functions can precisely quantify the rate performance of nonlinear ODEs, providing necessary and sufficient conditions for various decay rates and enabling tight bounds via SOS programming.

Contribution

It introduces a generalized rate performance notion for ODEs, establishing Lyapunov conditions that are both necessary and sufficient, and applies SOS programming for accurate bounds.

Findings

SOS-based tests achieve tight bounds on decay rates.

Lyapunov conditions are necessary and sufficient for rate performance.

Numerical examples confirm the accuracy of the bounds.

Abstract

Pointwise-in-time stability notions for Ordinary Differential Equations (ODEs) provide quantitative metrics for system performance by establishing bounds on the rate of decay of the system state in terms of initial condition -- allowing stability to be quantified by e.g. the maximum provable decay rate. Such bounds may be obtained by finding suitable Lyapunov functions using, e.g. Sum-of-Squares (SOS) optimization. While Lyapunov tests have been proposed for numerous pointwise-in-time stability notions, including exponential, rational, and finite-time stability, it is unclear whether these characterizations are able to provide accurate bounds on system performance. In this paper, we start by proposing a generalized notion of rate performance -- with exponential, rational, and finite-time decay rates being special cases. Then, for any such notion and rate, we associate a Lyapunov condition which is shown to be necessary and sufficient for a system to achieve that rate. Finally, we show how the proposed conditions can be enforced using SOS programming in the case of exponential, rational, and finite-time stability. Numerical examples in each case demonstrate that the corresponding SOS test can achieve tight bounds on the rate performance with accurate inner bounds on the associated regions of performance.