Abstract
The design of the next-generation intelligent robotics systems with a certain degree of robustness and safety guarantee requires integrating a traditional model-based approach and learning-based, data-driven methods. Traditional system models do not incorporate the perception and sensory inputs in the design process, which is essential for capturing the system's interaction with the environment. In contrast, machine learning methods, including reinforcement learning, can directly work with sensory data but lack generalizability and predictive capabilities. This talk discusses unifying these complementary approaches under the linear operator theoretic framework involving Koopman and Perron-Frobenius operators. The linear operators provide a linear representation of the dynamical system. They are ideally suited for integrating traditional physics-based models and sensory data, providing a unified representation of the system interacting with the environment.
In the first part of this talk, we will present the results of the Koopman theory for controlling dynamic systems. We discover the connection between the spectral properties of the Koopman operator and the Hamilton Jacobi (HJ) solution. The HJ equation is the cornerstone of various problems in system theory, including optimal control, robust control, input-output analysis, dissipativity theory, and reachability analysis. The connection discovered between the Koopman spectrum and the HJ solution opens the possibility of exploiting the spectral properties of the Koopman operator to solve various control problems in the data-driven setting. We show the spectral properties of the Koopman operator is used to transfer the curse of dimensionality problem associated with solving the HJ equation to the curse of complexity problem. In the second part of this talk, we present results on safe control design using the Perron-Frobenius operator. One of the main contributions of this work is the analytical construction of the navigation density function used for solving the safe navigation problem in static and dynamic environments. This is the first systematic result providing a closed-form expression of a feedback controller for safe navigation. We will also discuss the result of the convex formulation of the optimal control problem with safety constraints using the Perron-Frobenius operator in the dual space of density. Finally, the operator theoretic framework will be applied to the control of autonomous vehicles in the off-road environment.
Bio
Umesh Vaidya received a Ph.D. in Mechanical Engineering from the University of California at Santa Barbara, Santa Barbara, CA, in 2004. He was a research engineer at the United Technologies Research Center (UTRC), East Hartford, CT. Umesh Vaidya is a Professor of Mechanical Engineering at Clemson University, SC. Before joining Clemson University in 2019, and since 2006, he was a faculty member with the Department of Electrical and Computer Engineering at Iowa State University, Ames, IA. He is the recipient of the 2012 National Science Foundation CAREER award. His current research interests include dynamical systems and control theory with application to power systems, robotic systems, and vehicle autonomy.
