Stability Analysis of Feedback Systems with ReLU Nonlinearities via Semialgebraic Set Representation

Q3 Engineering

IFAC-PapersOnLine Pub Date : 2024-01-01 DOI:10.1016/j.ifacol.2024.10.160

Tsuyoshi Yuno , Shingo Nishinaka , Rin Saeki , Yoshio Ebihara , Victor Magron , Dimitri Peaucelle , Samuele Zoboli , Sophie Tarbouriech

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引用次数: 0

Abstract

This paper is concerned with the stability/instability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a new linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is unstable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.

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通过半代数集合表示法分析具有 ReLU 非线性的反馈系统的稳定性

本文关注具有整流线性单元（ReLU）非线性的反馈系统的稳定性/不稳定性分析问题。当我们对动态（递归）神经网络（NN）和 NN 驱动控制系统建模时，就会出现这种反馈系统，其中 NN 的所有激活函数都是 ReLU。在本研究中，我们将重点放在表征 ReLUs 输入输出特性的半代数集合表示法上。这使我们能够在积分二次约束的框架内使用一种新的共正乘法器，从而为反馈系统的稳定性分析推导出一种新的线性矩阵不等式（LMI）条件。然而，由于线性矩阵不等式的保守性，我们无法得出系统稳定性的结论。这促使我们考虑其对偶 LMI。通过研究对偶解的结构，我们得出了对偶变量的秩条件，证明当前系统是不稳定的。此外，我们还构建了对偶 LMI 的层次结构，从而改进了不稳定性检测。我们通过几个数值示例来说明所提方法的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

IFAC-PapersOnLine Engineering-Control and Systems Engineering

CiteScore

1.70

自引率

0.00%

发文量

1122

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