Search for: All records

Transfer operators provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements, motivating dataset sparsification as a precursory step to computation. Further, in practice, data is available in the form of trajectories, introducing correlation between samples. In this work, we present a method for sparse learning of transfer operators from $$\beta$$-mixing stochastic processes, in both discrete and continuous time, and provide sample complexity analysis extending existing theoretical guarantees for learning from non-sparse, i.i.d. data. In addressing continuous-time settings, we develop precise descriptions using covariance-type operators for the infinitesimal generator that aids in the sample complexity analysis. We empirically illustrate the efficacy of our sparse embedding approach through deterministic and stochastic nonlinear system examples. 

This paper presents a data-driven framework to discover underlying dynamics on a scaled F1TENTH vehicle using the Koopman operator linear predictor. Traditionally, a range of white, gray, or black-box models are used to develop controllers for vehicle path tracking. However, these models are constrained to either linearized operational domains, unable to handle significant variability or lose explainability through end-2-end operational settings. The Koopman Extended Dynamic Mode Decomposition (EDMD) linear predictor seeks to utilize data-driven model learning whilst providing benefits like explainability, model analysis and the ability to utilize linear model-based control techniques. Consider a trajectory-tracking problem for our scaled vehicle platform. We collect pose measurements of our F1TENTH car undergoing standard vehicle dynamics benchmark maneuvers with an OptiTrack indoor localization system. Utilizing these uniformly spaced temporal snapshots of the states and control inputs, a data-driven Koopman EDMD model is identified. This model serves as a linear predictor for state propagation, upon which an MPC feedback law is designed to enable trajectory tracking. The prediction and control capabilities of our framework are highlighted through real-time deployment on our scaled vehicle. 

The path-tracking control performance of an autonomous vehicle (AV) is crucially dependent upon modeling choices and subsequent system-identification updates. Traditionally, automotive engineering has built upon increasing fidelity of white- and gray-box models coupled with system identification. While these models offer explainability, they suffer from modeling inaccuracies, non-linearities, and parameter variation. On the other end, end-to-end black-box methods like behavior cloning and reinforcement learning provide increased adaptability but at the expense of explainability, generalizability, and the sim2real gap. In this regard, hybrid data-driven techniques like Koopman Extended Dynamic Mode Decomposition (KEDMD) can achieve linear embedding of non-linear dynamics through a selection of “lifting functions”. However, the success of this method is primarily predicated on the choice of lifting function(s) and optimization parameters. In this study, we present an analytical approach to construct these lifting functions using the iterative Lie bracket vector fields considering holonomic and non-holonomic constraints on the configuration manifold of our Ackermann-steered autonomous mobile robot. The prediction and control capabilities of the obtained linear KEDMD model are showcased using trajectory tracking of standard vehicle dynamics maneuvers and along a closed-loop racetrack.