Learning dynamical systems with regional stability certificates

Regionally stable systems produce bounded outputs only on part of their state and input space, outside it, their outputs grow unbounded even for bounded inputs. Unconstrained models can reproduce such behavior but provide no guarantees, whereas globally stable models provide guarantees but cannot predict unbounded outputs. We propose a dynamic recurrent neural network in Lur'e form with deadzone nonlinearity. Exploiting the deadzone, we derive sufficient conditions on the learnable parameters that yield a regional stability certificate, which consists of an invariant state set, a state-dependent input map, and an output set that contains the model predictions whenever the initial state and inputs are admissible. During training, we enforce these conditions with logarithmic barrier functions and enlarge the admissible input map with a semidefinite program. On two synthetic systems, the learned model recovers the regional stability properties of the data-generating system and reproduces its unbounded response. On datasets from a nonlinear system identification benchmark that do not contain unbounded outputs, they match the accuracy of globally stable models, while unconstrained models are more accurate but uncertified.