Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness

This paper introduces recurrent equilibrium networks (RENs), a new class of nonlinear dynamical models for applications in machine learning, system identification and control. The new model class has ``built in'' guarantees of stability and robustness: all models in the class are contracting - a strong form of nonlinear stability - and models can satisfy prescribed incremental integral quadratic constraints (IQC), including Lipschitz bounds and incremental passivity. RENs are otherwise very flexible: they can represent all stable linear systems, all previously-known sets of contracting recurrent neural networks and echo state networks, all deep feedforward neural networks, and all stable Wiener/Hammerstein models. RENs are parameterized directly by a vector in R^N, i.e. stability and robustness are ensured without parameter constraints, which simplifies learning since generic methods for unconstrained optimization can be used. The performance and robustness of the new model set is evaluated on benchmark nonlinear system identification problems, and the paper also presents applications in data-driven nonlinear observer design and control with stability guarantees.Comment: Journal submission, extended version of conference paper (v1 of this arxiv preprint

A Behavioral Approach to Robust Machine Learning

Machine learning is revolutionizing almost all fields of science and technology and has been proposed as a pathway to solving many previously intractable problems such as autonomous driving and other complex robotics tasks. While the field has demonstrated impressive results on certain problems, many of these results have not translated to applications in physical systems, partly due to the cost of system fail- ure and partly due to the difficulty of ensuring reliable and robust model behavior. Deep neural networks, for instance, have simultaneously demonstrated both incredible performance in game playing and image processing, and remarkable fragility. This combination of high average performance and a catastrophically bad worst case performance presents a serious danger as deep neural networks are currently being used in safety critical tasks such as assisted driving. In this thesis, we propose a new approach to training models that have built in robustness guarantees. Our approach to ensuring stability and robustness of the models trained is distinct from prior methods; where prior methods learn a model and then attempt to verify robustness/stability, we directly optimize over sets of models where the necessary properties are known to hold. Specifically, we apply methods from robust and nonlinear control to the analysis and synthesis of recurrent neural networks, equilibrium neural networks, and recurrent equilibrium neural networks. The techniques developed allow us to enforce properties such as incremental stability, incremental passivity, and incremental l2 gain bounds / Lipschitz bounds. A central consideration in the development of our model sets is the difficulty of fitting models. All models can be placed in the image of a convex set, or even R^N , allowing useful properties to be easily imposed during the training procedure via simple interior point methods, penalty methods, or unconstrained optimization. In the final chapter, we study the problem of learning networks of interacting models with guarantees that the resulting networked system is stable and/or monotone, i.e., the order relations between states are preserved. While our approach to learning in this chapter is similar to the previous chapters, the model set that we propose has a separable structure that allows for the scalable and distributed identification of large-scale systems via the alternating directions method of multipliers (ADMM)

Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System

This paper proposes a nonlinear policy architecture for control of partially-observed linear dynamical systems providing built-in closed-loop stability guarantees. The policy is based on a nonlinear version of the Youla parameterization, and augments a known stabilizing linear controller with a nonlinear operator from a recently developed class of dynamic neural network models called the recurrent equilibrium network (REN). We prove that RENs are universal approximators of contracting and Lipschitz nonlinear systems, and subsequently show that the the proposed Youla-REN architecture is a universal approximator of stabilizing nonlinear controllers. The REN architecture simplifies learning since unconstrained optimization can be applied, and we consider both a model-based case where exact gradients are available and reinforcement learning using random search with zeroth-order oracles. In simulation examples our method converges faster to better controllers and is more scalable than existing methods, while guaranteeing stability during learning transients

RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness

Neural networks are typically sensitive to small input perturbations, leading to unexpected or brittle behaviour. We present RobustNeuralNetworks.jl: a Julia package for neural network models that are constructed to naturally satisfy a set of user-defined robustness constraints. The package is based on the recently proposed Recurrent Equilibrium Network (REN) and Lipschitz-Bounded Deep Network (LBDN) model classes, and is designed to interface directly with Julia's most widely-used machine learning package, Flux.jl. We discuss the theory behind our model parameterization, give an overview of the package, and provide a tutorial demonstrating its use in image classification, reinforcement learning, and nonlinear state-observer design