Papers by N. Ozay and E.D. Sontag

Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the non-existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it remained uncertain whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous one-to-one immersion to a class of systems including linear systems cannot distinguish different omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results.

Data-driven control (DDC), that is the design of controllers directly from observed data, has attracted substantial attention in recent years due to its advantages over model-based control. DDC avoids a computationally expensive, potentially conservative model identification step and bypasses practically difficult questions such as model order/class selection. This tutorial paper seeks to offer a sampling of the different approaches that have been recently used to synthesize data driven controllers and filters, covering both analytic approaches and learning enabled ones, indicating the relative strengths of each. A second objective is to provide a key to the rapidly expanding literature in the subject, to help researchers newly interested in this field to quickly come up to speed.

Linear immersions (or Koopman eigenmappings) of a nonlinear system have wide applications in prediction and control. In this work, we study the existence of one-to-one linear immersions for nonlinear systems with multiple omega-limit sets. For this class of systems, existing work shows that a discontinuous one-to-one linear immersion may exist, but it is unclear if a continuous one-to-one linear immersion exists. Under mild conditions, we prove that systems with multiple omega-limit sets cannot admit a continuous one-to-one immersion to a class of systems including linear systems.