CDC_2024 Workshop

This talk will cover the basics in Koopman operator theory which are needed for further application to nonlinear systems analysis and control. We will first review the definitions and main properties of the Koopman operator, with a specific focus on spectral properties. Next, we will provide a broad overview of finite-dimensional approximation methods that are used in numerical implementations. The concepts and methods presented in this talk will be illustrated in the context of stability analysis.

Author: Alexandre Mauroy (alexandre.mauroy@unamur.be), University of Namur, Belgium

Alexandre Mauroy is an Associate Professor at the University of Namur, Belgium (Department of Mathematics and Namur Institute for Complex Systems). He received the M.S. degree in aerospace engineering and the Ph.D. degree in applied mathematics from the University of Liége, Belgium, in 2007 and 2011, respectively. Prior to joining UNamur, he was a postdoctoral researcher at the University of California Santa Barbara from 2011 to 2013, at the University of Li`ege from 2013 to 2015, and at the University of Luxembourg in 2016. He is also the main co-editor of the multi-contributed book ”Koopman Operator in Systems and Control”. His research interests include synchronization in complex networks, network identification, and applications of (Koopman) operator-theoretic methods to dynamical systems and control.

We present two themes from a recent development of the Dynamic Mode Decomposition (DMD). First, the DMD is presented as a data driven Rayleigh-Ritz extraction of spectral information. This (unlike the mere regression aspect) allows for a better connection with the Koopman operator, and provides better understanding of the dynamics under study. Computable residuals can be used to select physically meaningful eigenvalues and modes, and to guide sparse representation of the snapshot in the KMD (Koopman Mode Decomposition). We believe that this is the proper approach to the DMD as a numerical toolbox for Koopman operator based data driven analysis of nonlinear dynamics. Then, we discuss the problem of ill-conditioned eigenvectors in the non-normal case and show how the recently proposed Koopman-Schur decomposition can be used both in the operator setting and in numerical computation.

Author: Zlatko Drmač (zlatko.drmac@gmail.com), University of Zagreb, Croatia

Zlatko Drmač received the B.S. and M.S. degrees in mathematics from the University of Zagreb, Croatia, in 1989 and 1991, respectively. He holds a Ph.D. in mathematics (1994) from the University of Hagen, Germany. He is a Full Professor at the Department of Mathematics at the University of Zagreb. His area of expertise include numerical mathematics (SVD, GSVD, linear and nonlinear eigenvalue problems and applications), model reduction, data driven analysis of nonlinear dynamical systems, matrix theory and numerical software development with software contributions to the LAPACK library. Dr. Drmač received the SIAM/SIAG Linear Algebra prize in 2009 for most outstanding paper (coauthored with K. Veselić) on a topic in applicable linear algebra published in English in a peer reviewed journal in the three year award period, and in 2022 he was elected Fellow of the Society of Industrial and Applied Mathematics (SIAM).

The key to data-driven methods for learning dynamical systems lies in representing system trajectories through suitable feature representations. While Koopman operator theory provides a theoretical justification for the existence of such features, provable and practical ways for obtaining these representations are not well-explored.

In this talk, we present a universal framework for Koopman-based representation learning using reproducing kernels. This framework enables learning in invariant subspaces of the Koopman operator by exploiting the direct connection between a kernel's hypothesis space and its feature map, such that suitable constructions of the kernel allow for learning Koopman operator models. We demonstrate the effectiveness of this approach with two representation learning formulations: one for kernelized regression with invariance guarantees and the other for Gaussian process regression ensuring Koopman-based equivariance certificates. The benefits of the representation learning perspective, e.g., convergence rates and automated model selection, are theoretically demonstrated and illustrated in numerical experiments.

Author:

Armin Lederer (armin.lederer@inf.ethz.ch), ETH Zurich, Switzerland
Armin Lederer received the B.Sc., M.Sc., and Dr. Ing. degree in electrical and computer engineering from the Technical University of Munich, Germany, in 2015, 2018, and 2023, respectively. Since 2023, he is a postdoctoral researcher at the Learning and Adaptive Systems Group in the Department of Computer Science at ETH Zurich, Switzerland. He was recipient of the Rohde & Schwarz PhD thesis award in 2023, and a scholarship of the German National Scholarship Foundation between 2019 and 2022. His current research interests include the stability and safety of data-driven control systems, control with stochastic models, and probabilistic machine learning in closed-loop systems.

Max Beier, TU Munich, Germany
Max Beier obtained his Bachelor of Mechanical Engineering in 2020 at Baden-Wuerttemberg Cooperative State University Stuttgart with Bosch. His interest in learning from data where models from first principles fail drove him to enroll in the Robotics, Cognition, Intelligence Master at TUM, where he graduated in 2023. During his studies, he came to love systems and control as a framework for building models in the face of complex interactions. Since fall 2023, he has been pursuing his PhD with Prof. Hirche. He is particularly interested in learning practical dynamical systems representations in terms of utilizing the inherent structure in dynamical systems data, provable learning theoretic guarantees and uncertainty quantification.

Modern data-driven techniques have rapidly progressed beyond modelling and systems identification, with a growing interest in learning high-level dynamical properties of a system, such as safe-set invariance, reachability, input-to-state stability etc. In particular, developing Lyapunov-based formal guarantees for safety and performance of dynamical systems is an essential step in the design and control of cyber-physical systems. In the absence of a well-known dynamical model, leveraging trajectory data for control and analysis is particularly well-suited. Even for the cases where we have a model, it often takes expert intuition and knowledge to handcraft Lyapunov functions.

In this talk, we are interested in exploring how one can leverage Koopman Operator theory towards learning Lyapunov-based guarantees by efficient utilization of trajectory data, in a manner that is highly interpretable and scalable thanks to linearity of the operator. We explore how the Koopman paradigm can render the computed functions amenable to formal verification via tools such as polynomial sum-of-squares programming, SMT solvers or off-the-shelf neural-net verification tools. 

Author: Shankar A. Deka (shankar.deka@aalto.fi), Aalto University, Finland

Shankar A. Deka is an Assistant Professor of Automatic Control in the Department of Electrical Engineering and Automation at Aalto University. His research focus is on nonlinear stability theory, robust and optimal control, and learning for dynamics and controls, in the application areas of medical robotics and precision agriculture. He previously held postdoc positions at KTH Royal Institute of Technology, Stockholm and University of California, Berkeley between 2019 and 2023. He is currently a guest editor for the Journal of Marine Science and Engineering, Drones, Electronics, Energies, and Mathematics.

Data-driven and learning-based methods have attracted considerable attention in recent years both for the analysis of dynamical systems and for control design. While there are many interesting and exciting results in this direction, our understanding of fundamental limitations of learning for dynamics and control is lagging. In this talk, I will discuss a new observation on immersions for systems with multiple limit sets and how it reveals some potential limitations in learning Koopman embeddings. While this result might seem negative, I will conclude the talk with some thoughts on how it can inspire interesting future directions. Based on joint work with Zexiang Liu and Eduardo Sontag.

Author: Necmiye Ozay (necmiye@umich.edu), University of Michigan, Ann Arbor, USA

Necmiye Ozay is the Chen-Luan Family Faculty Development Professor of Electrical and Computer Engineering, and an associate professor of Electrical Engineering and Computer Science, and Robotics at the University of Michigan, Ann Arbor. She received her PhD in Electrical Engineering from Northeastern University in 2010. After a postdoctoral position at Caltech in Computing and Mathematical Sciences, she joined Michigan in 2013. Her research interests include dynamical systems, control, optimization, and formal methods with applications in learning-enabled cyber-physical systems, system identification, verification and validation, and safe autonomy. She received the 1938E. 
Award and a Henry Russel Award from the University of Michigan for her contributions to teaching and research. She received five young investigator awards, including NSF CAREER Award, NASA Early Career Faculty Award, DARPA Young Faculty Award, and ONR Young Investigator Award. She is also the recipient of the 2021 Antonio Ruberti Young Researcher Prize from the IEEE Control Systems Society for her fundamental contributions to the control and identification of hybrid and cyber-physical systems.

Extended dynamic mode decomposition (EDMD), embedded in the Koopman framework, is a widely-applied technique for prediction and control of dynamical control systems. However, despite its popularity, the error analysis is still fragmentary. We provide a complete and rigorous error analysis [1] for dynamical systems and its extension to control-affine systems [2]. To this end, we show Koopman invariance of suitably-constructed reproducing kernel Hilbert spaces (RKHS). Then, we demonstrate the applicability of the data-driven surrogate for model predictive control [3] as well as robust controller design of nonlinear systems [4].

Author: Karl Worthmann (karl.worthmann@tu-ilmenau.de), TU Ilmenau, Germany
Karl Worthmann received his Ph.D. degree in mathematics from the University of Bayreuth, Germany, in 2012. 2014 he became assistant professor for ”Differential Equations” at Technische Universität Ilmenau (TU Ilmenau), Germany. 2019 he was promoted to full professor after receiving the Heisenberg-professorship ”Optimization-based Control” by the German Research Foundation. He was recipient of the Ph.D. Award from the City of Bayreuth, Germany, and stipend of the German National Academic Foundation. 2013 he has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM), where he served as speaker in 2014 and 2015. Karl Worthmann’s current research interests include systems and control theory with a particular focus on nonlinear model predictive control, stability analysis, and data-driven control.

The linear structure of the Koopman representation provides tremendous opportunities for control and permits very efficient solutions for optimal control when the Koopman dynamics are linear on the control input. However, linearity of the Koopman representation in the control input severely limits the class of applicable nonlinear dynamics. While bilinear representations are more widely applicable, they are also much harder to control. 

When the set of admissible values for the control input is finite, the Koopman representation can be viewed as a switched linear system, where the optimal control problem becomes the optimal selection, at each time step, of one out of several admissible dynamics. 

While the optimal control of switched system it typically computationally difficult, when the optimization criterion has a linear representation in the lifted state, the dynamic programming cost-to-go is concave and piecewise linear, with a simple representation in terms of the minimum over a finite set of linear functions. This observation enabled the design of efficient algorithms that combine dynamic programming with dynamic pruning.

This same structure of the Koopman representation also enables the development of very efficient Markov Chain Monte Carlo (MCMC) sampling methods for optimal control, based on Gibbs sampling with parallel tempering.

This talk outlines these approaches to nonlinear optimal control and discusses issues that arise when applied to representative problems.

Author: João Hespanha (hespanha@ece.ucsb.edu), University of California, Santa Barbara, USA

João P. Hespanha received his Ph.D. degree in electrical engineering and applied science from Yale University, New Haven, Connecticut in 1998. From 1999 to 2001, he was Assistant Professor at the University of Southern California, Los Angeles. He moved to the University of California, Santa Barbara in 2002, where he currently holds a Professor position with the Department of Electrical and Computer Engineering. Dr. Hespanha is the recipient of the Yale University’s Henry Prentiss Becton Graduate Prize for exceptional achievement in research in Engineering and Applied Science, a National Science Foundation CAREER Award, the 2005 best paper award at the 2nd Int. Conf. on Intelligent Sensing and Information Processing, the 2005 Automatica Theory/Methodology best paper prize, the 2006 George S. Axelby Outstanding Paper Award, and the 2009 Ruberti Young Researcher Prize. Dr. Hespanha is a Fellow of the International Federation of Automatic Control (IFAC) and of the IEEE. He was an IEEE distinguished lecturer from 2007 to 2013. His current research interests include hybrid and switched systems; multi-agent control systems; game theory; optimization; distributed control over communication networks (also known as networked control systems); and network security.

The Koopman framework has been introduced to provide linear surrogate representations of finite-dimensional nonlinear systems through a generally infinite-dimensional global linear embedding. Powerful results of the framework have been successfully applied in a wide range of applications, showing their tremendous potential and practical usefulness. However, while the Koopman theory has become well established for autonomous systems, the general theory for systems with inputs, which has primary importance for control, has been lagging behind with results restricted to special cases and assumptions taken based on tradition and mathematical convenience. In this talk, we systematically investigate and analytically derive exact Koopman lifted forms under inputs for a rather wide class of nonlinear systems in both continuous and discrete time by first embedding the autonomous part of the system and then deriving the resulting controlled lifted forms. Furthermore, we discuss a universal framework via control-affine reproducing kernels that enables direct and efficient estimation of a single operator for control systems without suffering from the curse of control input dimensionality. Finally, we show some recent results on when finite dimensional embedding of nonlinear systems is possible and provide algorithms for computing them. The validity and efficiency of our results are demonstrated on a wide range of simulation studies and benchmark examples.

Authors: Roland Tóth, Lucian Cristian Iacob, and Maarten Schoukens

Roland Tóth (R.Toth@tue.nl), Eindhoven University of Technology, the Netherlands
Roland Tóth received his Ph.D. degree with Cum Laude distinction at the Delft University of Technology (TUDelft) in 2008. He was a post-doctoral researcher at TUDelft in 2009 and at the Berkeley Center for Control and Identification, University of California in 2010. He held a position at TUDelft in 2011-12, then he joined to the Control Systems (CS) Group at the Eindhoven University of Technology (TU/e). Currently, he is a Full Professor at the CS Group, TU/e and a Senior Researcher at the Systems and Control Laboratory, HUN-REN Institute for Computer Science and Control (SZTAKI) in Budapest, Hungary. He is Senior Editor of the IEEE Transactions on Control Systems Technology. His research interests are in identification and control of linear parameter-varying (LPV) and nonlinear systems, developing data-driven and machine learning methods with performance and stability guarantees for modeling and control, model predictive control and behavioral system theory. On the application side, his research focuses on advancing reliability and performance of precision mechatronics and autonomous robots/vehicles with nonlinear, LPV and learning-based motion control. He has received the TUDelft Young Researcher Fellowship Award in 2010, the VENI award of The Netherlands Organization for Scientific Research in 2011, the Starting Grant of the European Research Council in 2016 and the DCRG Fellowship of Mathworks in 2022.

There is a fundamental limitation to the current Koopman Operator theory when applying it to dynamical systems with control. Prior works assume that control inputs are linearly involved in state equations and apply least squares estimate for approximating the input matrix. Bilinear approximation provides better accuracy, but the resultant Koopman model is no longer linear in control.

In this talk, a new approach based on physical system modeling is presented to extend the Koopman method to a broad class of non-autonomous systems with control. Control Coherent Koopman modeling allows us to represent nonlinear controlled systems as linear time-invariant systems without control approximation. Dynamics of actuators are modeled as subsystems possessing independent state variables, which are driven by a source control input linearly. It is shown that the Koopman modeling problem is reduced to that of standard autonomous systems and that this actuator subsystem modeling applies to broad engineered dynamical systems.

As an example, the proposed method is applied to multi degree-of-freedom robotic arms, which are controlled with Model Predictive Control (MPC). It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate input matrix does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct input matrix, even performing better than the bilinear formulation of the Koopman operator.

Next, the Control Coherent Koopman modeling is applied to diverse control problems, including multi-cable manipulation, rimless wheel dynamic walking, and pushing manipulation, all of them are switched dynamical systems. Although governing equations must be switched as the robot makes and breaks contact with an object or a ground, a unified, global Koopman model allows us to apply linear Model Predictive Control to these otherwise complex nonlinear systems.

Author: Haruhiko Harry Asada (asada@mit.edu), Massachusetts Institute of Technology, USA
H. Harry Asada is Ford Professor of Mechanical Engineering and Director of the Brit and Alex d’Arbeloff Laboratory for Information Systems and Technology in the Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA. He specializes in robotics, system dynamics and control, and biomedical engineering. His current research includes Koopman operator theory, assistive robotics for eldercare, supernumerary robotic limbs, and multi-cable manipulation. He received Best Paper Awards at the IEEE International Conference on Robotics and Automation (ICRA) in 1993, 1997, 1999, and 2010, the best application paper award at the 2017 IEEE/RSJ International Conference on Intelligent Robotics and Systems (IROS), the O. Hugo Schuck Best Paper Award from the American Control Council in 1985, and other 6 best paper awards of major journals and conferences. He was the recipient of the Henry Paynter Outstanding Researcher Award from ASME Dynamic Systems and Control in 1998, the 2011 Rufus Oldenburger Medal from ASME, and the 2012 Ruth and Joel Spira Award for Distinguished Teaching from School of Engineering, MIT. More recently he received the 2023 IEEE Robotics and Automation Pioneer Award. He is a fellow of IEEE and ASME.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to Koopman-based data-driven modelling, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. Starting from the review of recent results in statistical theory of learning transfer operators and their smeigorup generators,, in particular highlighting the role of approximation error and metric distortion, we show how the search for a good representation can be cast as an optimization problem over neural networks. In this talk, we first present results for fully-data driven approach, and then introduce physics-informed representation learning. Finally, we illustrate how learning representations improves the ability to model, interpret and forecast flows of probability distributions in several important applications.

Author: Vladimir R. Kostić (vladimir.kostic@iit.it)

Vladimir R. Kostić received the B.Sc. and M.Sc. degrees in Applied Mathematics from the Faculty of Science University of Novi Sad Serbia in 2003 and 2009 respectively. He obtained his Ph.D. in Numerical Mathematics (2010) from the same university, where he currently holds an Associated Professorship, and received BMS Einstein Foundation Fellowship for post-doctoral research at the Technical University of Berlin in 2013. He is currently a Researcher in Machine Learning and Computational Statistics at the Italian Institute of Technology in Genova Italy, and a visiting scientist at École Polytechnique in Paris. His contributions span multiple domains: advancing fundamental research in matrix analysis and numerical algorithms, developing advanced optimization algorithms, creating novel machine learning methodologies for dynamical systems, modeling and establishing theoretical foundations for learning complex stochastic processes. 

This presentation provides an overview of recent advancements in solving nonlinear stochastic optimal control problems by leveraging on Koopman and Fokker-Planck-Kolmogorov operators. The central concept of these operator-theoretic methods involves transforming finite-dimensional nonlinear closed-loop control systems into equivalent infinite-dimensional open-loop controlled linear systems. This transformation allows for the application of modern convex PDE-constrained optimization and control techniques to solve nonlinear optimal control problems to global optimality.

The talk discusses the methodologies used to solve specific classes of nonlinear partial differential equations (PDEs), such as Hamilton-Jacobi-Bellman (HJB) equations, using approaches originally designed for linear parabolic and elliptic PDEs. Additionally, we introduce a novel operator-theoretic dynamic programming recursion, which finds applications in data-driven optimal control. In these applications, the operators are directly learned from data using techniques such as Kernel Ridge Regression.

The presentation concludes by summarizing the significant implications of these findings for researchers in nonlinear control theory and developers of data-driven optimal control algorithms. This work builds on my recent articles on "Convex Operator-Theoretic Methods in Stochastic Control" and "Data-Driven Optimal Feedback Laws via Kernel Mean Embeddings," co-authored with Petar Bevanda, Nicolas Hoischen, Stefan Sosnowski, and Sandra Hirche.

Author: Boris Houska (borish@shanghaitech.edu.cn), ShanghaiTech University, China
Dr. Houska is an associate professor at the School of Informations Science and Technology at ShanghaiTech University. He received a diploma in mathematics and physics from the University of Heidelberg in 2007, and a Ph.D. in Electrical Engineering from KU Leuven in 2011. From 2012 to 2013 he was a postdoctoral researcher at the Centre for Process Systems Engineering at Imperial College London. Subsequently, from 2013-14, Dr. Houska has worked as an associate professor at Shanghai Jiao Tong University. Moreover, he has held visiting professor positions at the Freiburg Institute for Advanced Studies as well as at the Institute for Microsystems Engineering at the University of Freiburg (both in 2014) and various shorter academic visiting appointments, for example, at UC Berkeley during Winter 2017, and Imperial College London during Summer 2018. Since 2024, he works as a visiting professor at IMT Lucca. Dr. Houska has been recipient of awards including ICCOPT Best Paper Prize for a Young Researcher in Continuous Optimization (Finalist, Top 3), a Marie-Curie Fellowship for the project Next Generation Algorithms for Robust and Global Optimization of Dynamic Systems, as well as an ShanghaiTech Excellent Professor Award from ShanghaiTech University. His paper on ”ACADO Toolkit—An open source framework for automatic control and dynamic optimization” has been listed as highly cited paper by Web of Science. Moreover, Dr. Houska is a member of the scientific advisory board of the EU project ELO-X, and serves as program committee member and associate editor for various international journals and conferences. Boris Houska’s research interests include numerical optimization and optimal control, robust and global optimization, as well as fast model predictive control algorithms.

Learning controllers from data depends on successfully representing the dynamics and effectively utilizing this representation for control design. While state-space models (SSM) are the de-facto standard modeling approach, data-driven SSMs can be difficult to exploit for prediction and control, especially in data-driven and higher-dimensional settings. In contrast, operator-theoretic models that describe the dynamics of parameters instead of states, hold promise for simplified prediction and convexification of nonlinear optimal control.
This talk presents nonparametric system identification for operator learning using reproducing kernel Hilbert spaces (RKHS), covering system identification in both discrete and continuous time. The nonparametric nature of the approach allows for working in expressive infinite-dimensional spaces while providing tractable forecasting and control through the use of the "kernel-trick". We demonstrate the effectiveness of the approaches on examples ranging from synthetic ODEs, fluid flow datasets and robotic simulators. The proposed data-driven methodology offers superior prediction performance to classical extended dynamic mode decomposition (DMD) and shows advantages over classical gradient based numerical solvers for model-based optimal control.

Author: Petar Bevanda (petar.bevanda@tum.de), Technical University of Munich, Germany
Petar Bevanda received his B.Sc. and M.Sc. degrees in electrical engineering and information technology from the University of Zagreb, Croatia and the Technical University of Munich, Germany, respectively in 2017 and 2019. Since 2020, he has been working towards a Ph.D. degree with the Chair of Information-oriented Control, TUM School of Computation, Information and Technology at the Technical University of Munich. His current research interests include operator-theoretic machine learning for dynamical systems, probabilistic forecasting and (stochastic) optimal control.