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Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints

Hamidreza Montazeri Hedesh, Moh Kamalul Wafi, Milad Siami

TL;DR

The paper addresses the challenge of certifying local stability for neural-network-in-the-loop control under positivity constraints by formulating a localized positive Lur'e framework and a local positive Aizerman conjecture. It introduces two scalable ROA estimation approaches: (i) a Lyapunov-based method using LMIs to certify local stability for general sector-bounded nonlinearities, and (ii) a novel local sector bound for FFNNs obtained via layer-wise linear relaxations that integrates into the Aizerman framework. The methods yield larger ROAs and improved computational efficiency compared to IQC-based approaches, demonstrating practical applicability to NN controllers in safety-critical settings. Overall, the work blends positive system theory with neural-network verification to enable certifiable, local stability for NN feedback loops, with potential extensions to biased/recurrent nets and broader stability frameworks.

Abstract

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.

Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints

TL;DR

The paper addresses the challenge of certifying local stability for neural-network-in-the-loop control under positivity constraints by formulating a localized positive Lur'e framework and a local positive Aizerman conjecture. It introduces two scalable ROA estimation approaches: (i) a Lyapunov-based method using LMIs to certify local stability for general sector-bounded nonlinearities, and (ii) a novel local sector bound for FFNNs obtained via layer-wise linear relaxations that integrates into the Aizerman framework. The methods yield larger ROAs and improved computational efficiency compared to IQC-based approaches, demonstrating practical applicability to NN controllers in safety-critical settings. Overall, the work blends positive system theory with neural-network verification to enable certifiable, local stability for NN feedback loops, with potential extensions to biased/recurrent nets and broader stability frameworks.

Abstract

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.

Paper Structure

This paper contains 20 sections, 8 theorems, 38 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Proposed Theoretical Framework
  4. Positive Lur’e Systems
  5. Positive Aizerman Conjecture
  6. Problem Formulation
  7. Local Stability via Positive Aizerman Conjecture
  8. Lyapunov Based Method
  9. Local Sector Bounds for FFNN
  10. Propagation of Sector Bounds through Activation Functions
  11. Computation of pre-Activation logits
  12. Sector relaxation of activation functions
  13. Propagation of Sector Bounds through Weight Matrices
  14. Sector Bound for the Entire Neural Network
  15. Numerical Example
  16. ...and 5 more sections

Key Result

Lemma 1

The system in eq:generallti is positive if and only if $A$ is a Metzler matrix, $B \in \mathbb R_+^{n\times m}$, and $C \in \mathbb R_+^{p\times n}$.

Figures (6)

  • Figure 1: Lur'e system with plant $G$ and nonlinear controller $\Phi$.
  • Figure 2: Illustration of linear relaxation of $\phi = \tanh(\nu)$ under different intervals for $\nu$.
  • Figure 3: Local sector bounds calculated for given $y\in\Gamma$ using equation \ref{['eq:localsectorbound']}. The title of each subplot defines the $\Gamma$ set and the corresponding $[\gamma_1,\gamma_2]$.
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition 2: $\Gamma-$Sector Bounded Function
  • Theorem 2
  • Lemma 3
  • proof
  • Remark 1
  • Theorem 3: Quadratic certificate
  • ...and 4 more