This talk will present results on applying linear transfer operator theory involving Perron-Frobenius and Koopman operators for data-driven control problems. In the first part of this talk, I will show the results of developing Koopman theory for control dynamical systems. One of the main challenges in using Koopman theory for control problems arises due to the bilinear lifting of the control dynamical system. We circumvent the bilinear lifting problem by establishing a connection between the spectrum of the Koopman operator and the Hamilton Jacobi (HJ) equation. We show that the solution to the HJ equation can be extracted from the spectrum of the Koopman operator. The HJ equation is the cornerstone of various problems in control theory, including optimal control, robust control, input-output analysis, dissipativity theory, and reachability/safety analysis. The connection between the Koopman spectrum and the HJ solution opens the possibility of exploiting the Koopman spectrum for various control problems. One of the main advantages of using the Koopman theory is that the Koopman operator and its spectrum can be approximated using data. We present novel approaches for computing the Koopman spectrum from data, thereby leading to systematic convex optimization-based methods for solving the HJ equation with application to optimal control design and input-output analysis of a nonlinear system.
In the second part of this talk, we will present results involving the Perron-Frobenius operator for the convex formulation of the optimal control problem with safety constraints. The convex problem is formulated over the space of densities defined only over the state space. The convex formulation is attractive for multiple reasons. First, the convex optimization problem can be constructed based on the data-driven approximation of the Koopman operator, dual to the Perron-Frobenius operator. Second, the convex incorporation of safety constraints allows us to provide a novel approach for the analytical construction of density functions for navigation. The proposed density function is used for navigation in a complex environment and high dimensional configuration space. The proposed construction overcomes the problem associated with navigation based on navigation functions, which are known to exist but challenging to construct, and potential functions that suffer from the existence of local minima. Finally, we demonstrate the application of the developed results for controlling the robotic system and vehicle autonomy.
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. Dr. 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.