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

This article 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 admits “built-in” behavioral guarantees of stability and robustness. All models in the proposed class are contracting—a strong form of nonlinear stability—and can satisfy prescribed incremental integral quadratic constraints, 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, and can approximate all fading memory and contracting nonlinear systems. RENs are parameterized directly by a vector in
$\mathbb {R}^{N}$
, i.e., stability and robustness are ensured without parameter constraints, which simplifies learning since generic methods for unconstrained optimization such as stochastic gradient descent and its variants can be used. The performance and robustness of the new model set are evaluated on benchmark nonlinear system identification problems. This article also presents applications in data-driven nonlinear observer design and control with stability guarantees.