Analysis of Lurie Systems With Magnitude Nonlinearities and Connections to Neural Network Stability Analysis

This paper considers the interconnection of a continuous time linear time-invariant system and a multivariable magnitude nonlinearity. A number of different quadratic constraints are established for the magnitude nonlinearity and then used to derive stability criteria based on quadratic and Lurie-type Lyapunov functions. The new stability criteria are cast as matrix inequalities and in some cases solved using semi-definite programming. Connections are made between the magnitude nonlinearity and neural network activation functions, such as the rectified linear unit (ReLU) and leaky ReLU, effectively allowing the stability criteria derived here to be used to analyse interconnections of dynamical systems and neural networks. Using the positive homogeneity property, shared by the (leaky) ReLU and magnitude functions, mild conditions are also established to show that the existence of a unique equilibrium point is sufficient for local and global stability to be equivalent. Finally, the new global stability criteria are tested on several numerical examples, including a Hopfield network with 100 states and neurons, and compare favourably with competing criteria from the literature.