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

In this paper, we analyze the stability of feedback interconnections of a\nlinear time-invariant system with a neural network nonlinearity in discrete\ntime. Our analysis is based on abstracting neural networks using integral\nquadratic constraints (IQCs), exploiting the sector-bounded and\nslope-restricted structure of the underlying activation functions. In contrast\nto existing approaches, we leverage the full potential of dynamic IQCs to\ndescribe the nonlinear activation functions in a less conservative fashion. To\nbe precise, we consider multipliers based on the full-block Yakubovich / circle\ncriterion in combination with acausal Zames-Falb multipliers, leading to linear\nmatrix inequality based stability certificates. Our approach provides a\nflexible and versatile framework for stability analysis of feedback\ninterconnections with neural network nonlinearities, allowing to trade off\ncomputational efficiency and conservatism. Finally, we provide numerical\nexamples that demonstrate the applicability of the proposed framework and the\nachievable improvements over previous approaches.\n