Analysis of Lurie systems for learning and control

Abstract

Due to the remarkable success of deep learning in a wide variety of applications, neural networks (NNs) are increasingly being adopted to model and control dynamical systems. When controlling dynamical systems, it is key to provide robust guarantees of the system's behaviour, especially in safety-critical applications. However, efficiently verifying such safety and robustness properties for systems involving NNs is challenging, due to the presence and large number of activation functions inside the NN. Furthermore, it is vital to learn robust and generalisable models of dynamical systems, to deal with noise and the, potentially, large space of initial conditions. This is particularly difficult since learning generic, high-dimensional functions is a cursed estimation problem.

This thesis focuses on the forced Lurie system as a general modelling framework which, amongst others, captures many systems involving NNs as special cases. Examples include recurrent neural networks, neural oscillators, and the interconnection of a linear time invariant system with a feed-forward NN. The analysis of Lurie systems has been well-studied in the control theory literature, with the absolute stability problem being a pertinent example. This thesis builds upon this rich literature to develop less conservative stability criteria for Lurie systems involving NNs and to exploit known properties for constructing robust and generalisable models of convergent dynamical systems.

The first contribution develops less conservative stability criteria for Lurie systems, involving NNs with only ReLU activations. Properties of the ReLU function are leveraged to construct tailored quadratic constraints which are incorporated in the stability criteria, posed as semidefinite programs, via the S-procedure. Both continuous and discrete-time cases are examined. The second contribution repeats the same steps for Lurie systems with magnitude nonlinearities. This is a generalisation of the first contribution since a loop transformation between the magnitude and (leaky) ReLU is shown to exist, meaning the stability of Lurie systems involving ReLU or leaky ReLU activations can be verified. Numerical examples highlight the reduced conservatism, particularly for high-dimensional systems.

The final contribution proposes the k-contracting Lurie network (LN) as a robust and generalisable model of convergent dynamical systems. The absolute stability framework is used to establish conditions which guarantee the LN is k-contracting for all slope-restricted nonlinearities. Unconstrained parametrisations of these conditions are then established to restrict training to only search over LNs satisfying the k-contraction property. When tested on dynamical systems datasets involving multiple equilibrium points and limit cycles, the k-contracting LN is an order of magnitude more accurate than existing models, when initial conditions are sampled outside the training distribution and subjected to additive noise.

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Identifiers

ORCID for Carl Robert Richardson: orcid.org/0000-0001-9799-896X

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