Exploring Interplay between Dynamical Systems and Function Spaces: A Unifying Presentation of Dynamics Mode Decomposition and Occupation Measures

Overview

Two different perspectives of lifting finite dimensional nonlinear problems into infinite dimensional linear problems have been gaining significant traction over the past decade: Dynamic Mode Decomposition (DMD), which aims to establish "equation-free" models from snapshots of a dynamical system by exploiting properties of the Koopman operators over Hilbert function spaces, and methods inspired by Liouville operators that utilize occupation Measures and occupation kernels to reformulate nonlinear optimal control problems as infinite dimensional linear programs.

The purpose of this workshop is to bring together practitioners of both fields together to enable a unifying discourse concerning nonlinear dynamical systems and their connections to infinite dimensional spaces. The presentations will include topics such as DMD, moment problems, reproducing Kernel Hilbert spaces, and Lyapunov measures. The workshop will conclude with several talks connecting DMD with Liouville operators using newly introduced occupation kernels.

This workshop aims to provide a comprehensive treatment of Dynamic Mode Decomositions and moment problems using Occupation Measures. The attendees will leave with a thorough understanding of how to cast finite dimensional nonlinear problems into infinite dimensional linear problems, and will understand this approach from multiple perspectives. Attendees who are already familiar with both methods will be introduced to occupation kernels and Liouville operators which can be leveraged to blend DMD with the theory of occupation measures via a RKHS framework.

The workshop will be beneficial to a wide audience. These include researchers working on nonlinear controls problems, operator theory, data science, optimal control, and function spaces. Dynamic mode decomposition give data driven approach to feature extraction and prediction for nonlinear dynamical systems, which has far reaching applications in fluid dynamics, computer vision, neuroscience, etc.

For more information on ACC2020 workshops, see http://acc2020.a2c2.org/technical-program/workshops/.

Organizers

Dr. Joel Rosenfeld

Assistant Professor

Department of Mathematics and Statistics

University of South Florida Dr. Rushikesh Kamalapurkar

Assistant Professor

School of Mechanical and Aerospace Engineering

Oklahoma State University

Agenda (Timezone: Mountain Daylight Time)

Speakers and Abstracts

Moment-SOS hierarchy and its dynamical systems applications Dr. Jean B. Lasserre Abstract: We first provide a brief description of the moment-SOS hierarchy. Initially designed for solving polynomial optimization problems, it is based on powerful positivity certificates from real algebraic geometry combined with semidefinite programming (an efficient technique from convex optimization). It consists of a hierarchy of convex (semidefinite) relaxations whose size increases and whose associated monotone sequence of optimal values converges to the global minimum. Finite convergence is generic and fast in practice. In fact this methodology also applies to solve the Generalized Problem of Moments (GPM) (of which global optimization is only a particular instance, and even the simplest). Then we briefly describe its application to several of many other applications outside optimization, notably in dynamical systems (control, optimal control, and analysis of some non-linear hyperbolic PDEs)

Biography: Graduated from "Ecole Nationale Supérieure d'Informatique et Mathematiques Appliquees" (ENSIMAG) in Grenoble, then got my PhD (1978) and "Doctorat d'Etat" (1984) degrees both from Paul Sabatier University in Toulouse (France). Dr. Lasserre has been at LAAS-CNRS in Toulouse since 1980, where Dr. Lasserre is currently Directeur de Recherche. Dr. Lasserre is also a member of IMT, the Institute of Mathematics of Toulouse. Twice a one-year visitor (1978-79 and 1985-86) at the Electrical Engineering Department of the University of California at Berkeley with a fellowship from INRIA and NSF. Dr. Lasserre has done several one-month visits to Stanford University (Stanford, California), the Massachusetts Institute of Technology (MIT, Cambridge), the Mathematical Sciences Research Institute (MSRI, Berkeley), the Fields Institute (Fields, Toronto), the Institute for Mathematics and its Applications (IMA, Minneapolis), the Institute for Pure and Applied Mathematics (IPAM, UCLA), Cinvestav-IPN (Cinvestav, Mexico), Leiden University (Leiden, The Netherlands), the Tinbergen Institute (Tinbergen, Amsterdam, The Netherlands), the University of Adelaide (Adelaide, Australia), the University of South Australia (UniSA, Adelaide), the University of New South wales (UNSW, Sydney), the University of British Columbia (UBC, Vancouver). Dr. Lasserre has been awarded several distinctions, such as the ISSAC' 2019 Distinguished Paper Award (Beijing, July 2019) with Florent Bréhard and Mioara Joldès, a Simons CRM Professor at the CRM in Montreal (October 2019), an Invited Speaker at the International Congress of Mathematicians (ICM 2018) at Rio de Janeiro in August 2018: Section 16: Control Theory & Optimization, the 2015 John von Neumann Theory prize of the INFORMS society, the 2015 Khachiyan prize of the Optimization Society of INFORMS, the 2009 Lagrange prize in Continuous optimization (awarded jointly every 3 years by SIAM and the Mathematical Optimization Society), a SIAM Fellow (class 2014), and a 2014 Laureate of an ERC-Advanced Grant from the European Research Council (ERC) for the TAMING project. Dr. Lasserre is also the author of about 190 papers and eight books including: "An Integrated Approach in Production Planning and Scheduling", Lecture Notes in Economics and Mathematical Systems" (Springer, Berlin 1994), "Discrete-Time Markov Control Processes: Basic Optimality Criteria" (Springer, New York, 1996) "Further Topics in Discrete-Time Markov Control Processes" (Springer, New York, 1999), "Markov Chains and Invariant Probabilities" (Birkhauser, 2003), "Linear \& Integer Programming vs Linear Integration and Counting", (Springer, New York, 2009), "Moments, Positive Polynomials and Their Applications" (Imperial College Press, London, 2009), "Modern Optimization Modelling Techniques" (Springer, New York, 2012), and "An Introduction to Polynomial and Semi-Algebraic Optimization" (Cambridge University Press, Cambridge, UK, 2015) Back to Agenda

How I Learned to Stop Worrying and Start Loving Lifting to Infinite Dimensions While Controlling Robotic Systems Dr. Ram Vasudevan Abstract: Dynamic mode decomposition and occupation measures present attractive alternatives to typical control synthesis techniques due to their ability to transform control design problems into convex optimization programs. However, the most appropriate choice between these approaches can depend a great deal on the specific application in question. This talk will describe the application of these methods to the control of robotic systems. In particular, this talk will try to highlight some of the pitfalls and lessons learned while applying these methods to control of autonomous vehicles, soft robots, and walking robots.

Biography: Ram Vasudevan received the B.S. degree in electrical engineering and computer sciences, the M.S. degree in electrical engineering, and the Ph.D. degree in electrical engineering from the University of California at Berkeley in 2006, 2009, and 2012, respectively. He is currently an Assistant Professor in mechanical engineering with the University of Michigan, Ann Arbor, with an appointment in the University of Michigan’s Robotics Program. His research interests include the development and application of optimization and systems theory to quantify and improve human and robot interaction. Back to Agenda

Data-Driven Control of Dynamical Systems Using Linear Operators Dr. Umesh Vaidya Abstract: In this talk, I will discuss results on the duality in the stability theory of dynamical systems based on duality between linear Perron Frobenius and Koopman operators. In particular, Lyapunov measure is introduced as a dual to Lyapunov function, capturing a weaker notion of almost everywhere stability. While Lyapunov function is shown to be related to the Koopman operator, the Lyapunov measure is derived using the Perron-Frobenius operator. Using recent advances in the data-driven approximation of linear Koopman operator, we propose an approach for the approximation of the Perron-Frobenius operator preserving positivity and Markov properties of the operator. These properties of the P-F operator are exploited to formulate an optimal control for a nonlinear system using Lyapunov measure. In particular, we show that the data-driven optimal control problem for a dynamical system can be solved as a linear program.

Biography: Umesh Vaidya received the Ph.D. degree in mechanical engineering from the University of California at Santa Barbara, Santa Barbara, CA, USA, in 2004. He was a Research Engineer at the United Technologies Research Center (UTRC), East Hartford, CT, USA. He is currently an Associate Professor in the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, USA. He is the recipient of 2012, National Science Foundation CAREER award. His current research interests include dynamical systems and control theory. Back to Agenda

From Fourier to Koopman: Long-term forecasting of non-linear high dimensional systems Dr. Henning Lange Abstract: In this talk, generic algorithms for long-term forecasting of high-dimensional data streams will be discussed. Specifically, by combining insights from operator theory, signal processing and machine learning, data driven algorithms with similarities to Dynamic Mode Decomposition are introduced. These algorithms are highly scalable because solutions can partially be obtained using the Fast Fourier Transform and because future state predictions are independent given model parameters, i.e. errors do not compound over time, we can show that these algorithms exhibit linear-in-time error bounds. The performance of these algorithms is showcased by predicting signals from various domains such as computational chemistry, video frame prediction and fluid flows.

Biography: Dr. Lange received a BSc in Cognitive Science from the University of Osnabrueck, Germany in 2012, an MSc degree in Machine Learning & Data Mining from Aalto University, Finland, in 2016 and a PhD in Advanced Infrastructure Systems from Carnegie Mellon University in 2019 under supervision of Mario Berges. He currently serves as a Research Associate in Applied Mathematics under supervision of Nathan Kutz at the University of Washington. Back to Agenda

Applications of Liouville operators in system identification and control Dr. Rushikesh Kamalapurkar Abstract: The relationship between Liouville operators and occupation kernels can be utilized to formulate system identification and optimal control problems as infinite dimensional linear programs that can be solved efficiently using finite-dimensional numerical approximation. In this talk, the occupation kernel approach to lift identification and control problems to infinite dimensional linear programs is discussed along with associated technical challenges and open problems. The utility of the approach is also demonstrated using several system identification and optimal control examples.

Biography: Rushikesh Kamalapurkar received his M.S. and his Ph.D. degree in 2011 and 2014, respectively, from the Department of Mechanical and Aerospace Engineering at the University of Florida. After working for a year as a postdoctoral researcher with Dr. Warren E. Dixon, he was appointed as the 2015-16 MAE postdoctoral teaching fellow. In 2016 he joined the School of Mechanical and Aerospace Engineering at the Oklahoma State University as an Assistant professor. His primary research interest is intelligent, learning-based optimal control of uncertain nonlinear dynamical systems. He has published a book, multiple book chapters, over 25 peer reviewed journal papers, and over 25 peer reviewed conference papers. Back to Agenda

Interfacing Occupation Kernels with Dynamic Mode Decomposition for the Analysis of Continuous Time Systems. Dr. Joel A. Rosenfeld Abstract: In this presentation, we will demonstrate how to incorporate trajectories as the fundamental unit of data in a reproducing kernel Hilbert space (RKHS) framework. The principle objects of study are occupation kernels, which interact with Liouville operators over RKHSs in much the same way that Liouville operators interact with occupation measures over the Banach space of continuous functions as investigated by Lasserre and his colleagues. The objective of this presentation is to demonstrate how these occupation kernels can be integrated into the kernel-based extended DMD framework initiated by Williams and Rowley to enable the direct DMD analysis of continuous time systems.

Biography: Dr. Joel A. Rosenfeld received his Ph.D. from the University of Florida in Mathematics studying densely defined operators over reproducing kernel Hilbert spaces under Dr. Michael T. Jury. After graduating in 2013, Dr. Rosenfeld was a postdoctoral researcher in the Department of Mechanical and Aerospace Engineering at the University of Florida under the direction of Dr. Warren E. Dixon, and then was a postdoctoral researcher under Dr. Taylor T. Johnson at Vanderbilt University. His research is focused on data driven problems in control theory, such as system identification and dynamic mode decomposition. His methods intertwine his experience in operator theory and control theory to enable novel approaches using kernel methods. Back to Agenda