Data-driven control: theory and applications

Abstract: Using the behavioral approach to system theory, we derive a nonparametric representation of linear time-invariant systems based on Hankel matrices constructed from data. The data-driven representation leads to new system identification, signal processing, and control methods. In this talk, we show how the data-driven representation can be used for solving missing data estimation problems. The theory leads to algorithms that are general---can deal simultaneously with missing, exact, and noisy data of multivariable systems---and simple---require basic linear algebra operations only. The results open a practical computational way of doing system theory and signal processing directly from data without identification of a transfer function or a state space representation and doing model-based design. 

This is joint work with Florian Dörfler (ETH Zürich).

Abstract: The “fundamental lemma” introduced by Willems et al. in 2005 provides a parametrization of all finite-length trajectories produced by a linear time-invariant discrete-time system. Recently several formulations of analogous results for continuous-time systems have appeared in the literature. I have had the privilege of working on some of them in collaboration with K. Camlibel, T. Faulwasser, V. Lopez, M. Mueller, Y. Ohta, P. Schmitz, H. van Waarde and K. Worthmann. In this talk I illustrate the main features and discuss the main advantages and limitations of these approaches. 

Abstract: In this talk, we consider the following problem: given an upper bound of the state-space dimension and lag of a linear time-invariant system, design a sequence of inputs so that the system dynamics can be recovered from the resulting input-output data. As our main result we propose a new online experiment design method, meaning that the selection of the inputs is iterative and guided by past data samples. We show that this approach leads to the shortest possible experiments for linear system identication. In terms of sample complexity, the proposed method outperforms offline methods based on persistency of excitation as well as existing online experiment design methods.

This is joint work with Kanat Camlibel (University of Groningen) and Paolo Rapisarda (University of Southampton).

Abstract: Advances in computer science have spurred a large interest in developing data-driven prediction, decision making, and control methods for real-world systems. One grand challenge in data-driven control is to design data-driven decision-making algorithms that perform reliably in safety-critical and real-time environments and are tractable in terms of computation and sample efficiency. This talk proposes Data-enabled Predictive Control (DeePC) leveraging behavioral systems theory and a result known as the fundamental lemma. The main idea is to replace the parametric dynamical system model with a raw data matrix of time series measurements (trajectories) and use it as a non-parametric predictive model. We study suitable regularization techniques leading to robust performance guarantees in the presence of corrupted data. We illustrate the method using simulations and real-world experiments in robotics and power systems. 

Abstract: Data-driven predictive control (DDPC) has become a central focus in control research, promising to directly use data to achieve desired control objectives while satisfying design constraints. Despite its promise, the sensitivity of DDPC to noisy data, even within the linear time-invariant framework, poses a challenge for this paradigm to take over established model-based approaches. Indeed, while years of research in system identification have yielded effective methods to cope with noise in modeling, the issue of noise handling remains an open problem in the DDPC domain. This talk explores the evolution of DDPC, starting with its precursor, Subspace Predictive Control, and delves into recent advancements in the field. The discussion bridges various approaches proposed in the literature, shedding light on the strengths and weaknesses of their strategies to address the challenges posed by noisy data. 

Abstract: In this talk, we briefly recap the extended Dynamic Mode Decomposition (EDMD) as a very popular data-driven method to predict quantities of interest are lifted into a high-, but finite-dimensional space, on which the surrogate model evolves linearly [1]. We embed EDMD in the Koopman framework to provide a rigorous error analysis depending on the amount of data by splitting up the approximation error into its two sources: projection and estimation [2]. Then, we present recent results ensuring stability w.r.t. the closed-loop using EDMD-based surrogate models in the controller design [3,4]. Finally, we provide a glimpse into a potential extension towards kernel EDMD including uniform error bounds [5]. Hereby, we rigorously show invariance of the respective reproducing kernel Hilbert space (RKHS) under the Koopman flow. 

References: 

[1] M.O. Williams, I.G. Kevrekidis, and C. Rowley: A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25, 1307-1346, 2015 

[2] F. Nüske, S. Peitz, F.M. Philipp, M. Schaller, and K. Worthmann: Finite-data error bounds for Koopman-based prediction and control. Journal of Nonlinear Science, 33(1):14, 2023 

[3] L. Bold, L. Grüne, M. Schaller, and K. Worthmann: Practical asymptotic stability of data-driven model predictive control using extended DMD. arXiv preprint arXiv:2308.00296 

[4] R. Strässer, M. Schaller, K. Worthmann, J. Berberich, and F. Allgöwer: SafEDMD: A certified learning architecture tailored to data-driven control of nonlinear dynamical systems, arXiv preprint arXiv:2402.03145 

[5] F. Köhne, F.M. Philipp, M. Schaller, A. Schiela, and K. Worthmann: L^\infty-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition. arXiv preprint arXiv:2403.18809 

Abstract: The informativity framework for control considers the problem of designing controllers for the set of all systems compatible with some given measurements. In a similar way, we can consider data-based (sub-)optimization: Pick a class of functions, collect data, and find worst-case bounds which hold for all functions compatible with the data. In either problem the first step is to go from a set of noisy measurements to a set of systems or functions. In this talk, we will discuss the relations between different types of noise models, the resulting sets of systems or functions, and how this yields a spectrum of new problems in data-driven control of potentially nonlinear systems and data-based optimization. 

Abstract: In this talk we will cover recent results on designing safe data driven control for non-linear systems. In the first part of the talk we will show that, for continuous time systems, tractable solutions can be obtained by exploiting a combination of density functions and duality. Further, these controllers can be made robust to process noise during execution. In the second part of the talk, we will discuss the prospects to extend these results to discrete time systems. We will conclude the talk by illustrating the challenges entailed in “model agnostic” data driven control and show that completely model agnostic data driven control methods can fail to stabilize simple systems. This highlights the need for design methods that are rooted in systems theory, even if models are a-priori unavailable and must be inferred from the data. 

Abstract: Recently, we proposed stochastic extensions of the fundamental lemma by Willems et al., which rely on measured input-output-disturbance data and which allow the forward propagation of stochastic non-Gaussian uncertainties using polynomial chaos expansions which date back to Norbert Wiener. In many applications, measuring disturbance realizations, however, is cumbersome or even infeasible. Hence we explore and proof the possibilities to remove disturbance data from the fundamental lemma without loss of prediction accuracy. We also comment on the prospect of data-driven stochastic optimal control with distributional uncertainty. 

Abstract: In this contribution we will first highlight the equivalence between Data Enabled Predictive Control (DeePC) and subspace predictive control (SPC). We then develop and demonstrate the performance of SPC and the repetitive variant for challenging wind energy applications. For example, we demonstrate the performance of these data-driven algorithms on the repetitive control challenge for wind turbine load mitigation. We will show both simulation and experimental results. Moreover, we will also demonstrate the performance of these data-driven control methods for feedforward control. Here we use wind and wave preview data to control a (floating) wind turbine. 

Abstract: This tutorial session will explore the potential for enhancing the integration of renewable energy sources and variable loads, such as electric vehicles, into the electrical grid through data-driven control strategies. The session will examine the limitations of traditional models in capturing the complexities and variability inherent in modern power grids. Key mathematical concepts foundational to data-driven control will be discussed, and their potential to navigate the complexities of today's electrical grids will be discussed without depending on conventional dynamical models. Practical examples from power systems, local energy distribution, and power electronics will illustrate the potential of data-driven approaches in ensuring grid stability and reliability amidst evolving challenges. The role of advanced technologies such as distribution phasor measurement units (D-PMUs) will also be explored, which are pivotal in capturing real-time grid dynamics, thus enabling more precise management of power fluctuations and enhancing grid stability. The tutorial will highlight the ongoing shift towards data-driven control approaches, underscoring the significance of data in comprehending system requirements and achieving essential objectives such as stability through numerical methods. 

Abstract: Buildings consume a substantial share of energy produced, and therefore play a crucial role in the energy market. The growing adoption of building energy management systems enables the use of advanced control strategies to provide services such as demand response. However, commissioning buildings for such functionalities requires considerable expertise and design effort, taking into account differences in building dynamics and intended use. This presentation will explore a data-driven framework based on Willems' Fundamental Lemma, which reduces development costs for reliable building energy management. We will introduce an adaptive robust data-driven predictive controller that requires low amount of data and limited tuning effort. Its effectiveness is demonstrated through indoor climate control and demand response experiments on buildings located on the EPFL campus. Furthermore, we will present algorithms that enhance the operational reliability of the framework. These include a physically consistent data filtering method that integrates engineering knowledge and a computationally efficient algorithm for adaptive controller updates. In addition, a data-driven input reconstruction method will be presented, which is useful for estimating the occupancy of buildings.