用正性约束分析神经网络反馈系统的局部稳定性与吸引域。
Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints
- 基于局部Aizerman猜想，结合正性系统约束设计稳定性判据。
- 提出两种方法：基于LMI的李雅普诺夫法与逐层线性松弛的扇区界计算。
- 相比传统方法，吸引域更大且可扩展，适合高维神经网络系统分析。
研究了以前馈神经网络（FFNN）实现静态非线性反馈的Lur'e型非线性系统的局部稳定性。通过利用正性系统约束，采用局部化Aizerman猜想，给出了轨迹被限制在紧集内时指数稳定的充分条件。在此基础上，发展了两种吸引域（ROA）估计方法：(i) 基于李雅普诺夫的非保守方法，通过构造满足线性矩阵不等式（LMI）的二次函数不变子水平集；(ii) 一种新颖的逐层传播线性松弛技术，用于计算FFNN的紧致局部扇区界。这些扇区界被整合进局部Aizerman框架以认证局部指数稳定性。数值结果表明，该方法在吸引域大小和可扩展性方面显著优于现有的积分二次约束方法。
原文摘要 · Abstract (English)
We study the local stability of nonlinear systems in the Lur'e form with static nonlinear feedback realized by feedforward neural networks (FFNNs). By leveraging positivity system constraints, we employ a localized variant of the Aizerman conjecture, which provides sufficient conditions for exponential stability of trajectories confined to a compact set. Using this foundation, we develop two distinct methods for estimating the Region of Attraction (ROA): (i) a less conservative Lyapunov-based approach that constructs invariant sublevel sets of a quadratic function satisfying a linear matrix inequality (LMI), and (ii) a novel technique for computing tight local sector bounds for FFNNs via layer-wise propagation of linear relaxations. These bounds are integrated into the localized Aizerman framework to certify local exponential stability. Numerical results demonstrate substantial improvements over existing integral quadratic constraint-based approaches in both ROA size and scalability.
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