Strengthened Circle and Popov Criteria for the Stability Analysis of Feedback Systems With ReLU Neural Networks

Abstract

This paper considers the stability analysis of a Lurie system with a static repeated ReLU (rectified linear unit) nonlinearity. Properties of the ReLU function are leveraged to derive new tailored quadratic constraints (QCs) which are satisfied by the repeated ReLU. These QCs are used to strengthen the Circle and Popov Criteria for this specialised Lurie system. It is shown that the criteria can be cast as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. Many systems involving a neural network (NN) with ReLU activations are important instances of this specialised Lurie system; for example, a continuous time recurrent neural network (RNN) or the interconnection of a linear system with a feedforward NN. Numerical examples show the strengthened criteria strike an appealing balance between reduced conservatism and complexity, compared to existing criteria.