有理稳定性的区域速率性能的精确 Lyapunov 刻画
An Exact Lyapunov Characterization of Regional Rate Performance for Rational Stability
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中文总结 AI 辅助
本文针对非线性常微分方程的有理稳定性，提出精确的逆 Lyapunov 刻画以规定速率性能，并给出分层凸松弛与 SOS 规划验证，优于经典方法。
中文摘要 AI 辅助
有理稳定性是非线性常微分方程（ODEs）的一种定量稳定性概念，在该概念下，收敛性通过解的有理时间衰减界来刻画。尽管已有有理稳定性的 Lyapunov 刻画被提出，但尚无现有条件被证明能精确刻画有理速率性能。本文通过提供具有规定速率性能的区域有理稳定性的精确逆 Lyapunov 刻画来解决这一问题。为了数值检验由此产生的逆 Lyapunov 条件，提出了一组分层的凸松弛，可通过平方和（SOS）规划来实施。每个层级关联一个推测的速率性能保守性缩放因子，并通过数值验证。所提出的第一个层级似乎是新颖的，并被证明优于经典的基于 SOS 的方法。数值算例用于验证结果，并计算状态空间上可保证规定性能水平的嵌套区域。
英文摘要
Rational stability is a quantitative stability notion for nonlinear ordinary differential equations (ODEs), under which convergence is characterized by a rationally-in-time decaying bound on solutions. Although Lyapunov characterizations of rational stability have been proposed, no existing condition has been shown to exactly characterize rational rate performance. The paper resolves this issue by providing an exact converse Lyapunov characterization of regional rational stability with prescribed rate performance. To test the resulting converse Lyapunov conditions numerically, a tiered set of convex relaxations is proposed which can be enforced using Sum-of-Squares (SOS) programming. A conjectured conservatism scaling factor in rate performance is associated with each tier and validated numerically. The first proposed tier appears to be novel and is shown to outperform classical SOS-based approaches. Numerical examples are used to validate the results and compute nested regions of state-space on which prescribed levels of performance can be guaranteed.
