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Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers

arXiv 2103.17106 ↗First posted 31 Mar 2021eess.SY cs.LG cs.SYcited by 6

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Abstract (arXiv)

In this paper, we analyze the stability of feedback interconnections of a linear time-invariant system with a neural network nonlinearity in discrete time. Our analysis is based on abstracting neural networks using integral quadratic constraints (IQCs), exploiting the sector-bounded and slope-restricted structure of the underlying activation functions. In contrast to existing approaches, we leverage the full potential of dynamic IQCs to describe the nonlinear activation functions in a less conservative fashion. To be precise, we consider multipliers based on the full-block Yakubovich / circle criterion in combination with acausal Zames-Falb multipliers, leading to linear matrix inequality based stability certificates. Our approach provides a flexible and versatile framework for stability analysis of feedback interconnections with neural network nonlinearities, allowing to trade off computational efficiency and conservatism. Finally, we provide numerical examples that demonstrate the applicability of the proposed framework and the achievable improvements over previous approaches.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 9 of its 40 references that are in the map. First posted on arXiv 31 Mar 2021.

Builds on

9 of 40 references.

References · 9

  1. 2018On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Modelscited by 322
  2. 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 290
  3. 2014Analysis and Design of Optimization Algorithms via Integral Quadratic Constraintscited by 105
  4. 2019Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programmingcited by 100
  5. 2020Training robust neural networks using Lipschitz boundscited by 98
  6. 2020Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllerscited by 42
  7. 2020Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimizationcited by 17
  8. 2020Offset-free setpoint tracking using neural network controllerscited by 10
  9. 2018Convex searches for discrete-time Zames-Falb multiplierscited by 4

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Cited by year

Citing papers per year: 2021: 1; 2022: 1; 2023: 2; 2024: 2.

6 papers.

  1. 2021Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systemscited by 15
  2. 2022Neural network training under semidefinite constraintscited by 11
  3. 2023Convolutional Neural Networks as 2-D systemscited by 5
  4. 2024Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guaranteescited by 4
  5. 2023Rational Neural Network Controllerscited by 0
  6. 2024Convex neural network synthesis for robustness in the 1-normcited by 0