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SPATIAL DECAY OF PERTURBATIONS IN HYPERBOLIC EQUATIONS WITH OPTIMAL BOUNDARY control

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arXiv 2025年

作者： Oppeneiger, Benedikt Schaller, Manuel Worthmann, Karl Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Germany Faculty of Mathematics Chemnitz University of Technology Germany

Recently, domain-uniform stabilizability and detectability has been the central assumption to ensure robustness in the sense of exponential decay of spatially localized perturbations in optimally controlled evolution equations. In the present paper we analyze a chain of transport equations with boundary and point controls with regard to this property. Both for Dirichlet and Neumann boundary and coupling conditions, we show a necessary and sufficient criterion on control domains which allow for the domain-uniform stabilization of this equation. We illustrate the results by means of a numerical example. © 2025, CC BY-NC-ND.

关键词： Hyperbolic functions

Offset-free Nonlinear MPC with Koopman-based Surrogate Models

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arXiv 2025年

作者： Schimperna, Irene Bold, Lea Worthmann, Karl Civil Engineering and Architecture Department University of Pavia Italy Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Germany

In this paper, we design offset-free nonlinear Model Predictive control (MPC) for surrogate models based on Extended Dynamic Mode Decomposition (EDMD). The model used for prediction in MPC is augmented with a disturbance term, that is estimated by an observer. If the full information about the equilibrium of the real system is not available, a reference calculator is introduced in the algorithm to compute the MPC state and input references. The control algorithm guarantees offset-free tracking of the controlled output under the assumption that the modeling errors are asymptotically constant. The effectiveness of the proposed approach is showcased with numerical simulations for two popular benchmark systems: the van-der-Pol oscillator and the four-tanks process. © 2025, CC BY.

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Invariance of Gaussian RKHSs Under Koopman Operators of Stochastic Differential Equations With Constant Matrix Coefficients

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Invariance of Gaussian RKHSs Under Koopman Operators of Stoc...

94th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2024

作者： Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl Peitz, Sebastian Nüske, Feliks Optimization-Based Control Group Institute of Mathematics Technische Universität Ilmenau Ilmenau Germany Department of Computer Science Paderborn University Paderborn Germany Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Department of Mathematics and Computer Science Freie Universität Berlin Berlin Germany

We consider the Koopman operator semigroup (Kt)t≥0 associated with stochastic differential equations of the form dKt = AKt dt + B dWt with constant matrices A and B and Brownian motion Wt . We prove that the reproducing kernel Hilbert space ℍc generated by a Gaussian kernel with a positive definite covariance matrix C is invariant under each Koopman operator Kt if the matrices A, B, and C satisfy the following Lyapunov-like matrix inequality: AC2 + C2 A ≤ 2BB. In this course, we prove a characterization concerning the inclusion ℍc1 ⊂ ℍc2 of Gaussian RKHSs for two positive definite matrices C1 and C2 . The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem. © 2024 The Author(s). Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.

关键词： Stochastic systems

SPATIAL EXPONENTIAL DECAY OF PERTURBATIONS IN OPTIMAL control OF GENERAL EVOLUTION EQUATIONS

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arXiv 2025年

作者： Göttlich, Simone Oppeneiger, Benedikt Schaller, Manuel Worthmann, Karl School of Business Informatics and Mathematics University of Mannheim Germany Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Germany Faculty of Mathematics Chemnitz University of Technology Germany

We analyze the robustness of optimally controlled evolution equations with respect to spatially localized perturbations. We prove that if the involved operators are domain-uniformly stabilizable and detectable, then these localized perturbations only have a local effect on the optimal solution. We characterize this domain-uniform stabilizability and detectability for the transport equation with constant transport velocity, showing that even for unitary semigroups, optimality implies exponential damping. Finally, we extend our result to the case of a space-dependent transport velocity. Numerical examples in one space dimension complement the theoretical results. © 2025, CC BY-NC-ND.

关键词： Optimal control systems

KERNEL EDMD FOR DATA-DRIVEN NONLINEAR KOOPMAN MPC WITH STABILITY GUARANTEES

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arXiv 2025年

作者： Bold, Lea Schaller, Manuel Schimperna, Irene Worthmann, Karl Optimization-Based Control Group Institute of Mathematics Technische Universität Ilmenau Germany Faculty of Mathematics Chemnitz University of Technology Germany Civil Engineering and Architecture Department University of Pavia Italy

Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of the Koopman operator, i.e., the evolution of an observable function along the flow of a dynamical system. In this paper, we leverage a recently-introduced kernel EDMD method for control systems for data-driven model predictive control. Building upon pointwise error bounds proportional in the state, we rigorously show practical asymptotic stability of the origin w.r.t. the MPC closed loop without stabilizing terminal conditions. The key novelty is that we avoid restrictive invariance conditions. Last, we verify our findings by numerical simulations. © 2025, CC BY.

关键词： Predictive control systems

Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data-driven control

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arXiv 2025年

作者： Strässer, Robin Schaller, Manuel Berberich, Julian Worthmann, Karl Allgöwer, Frank Institute for Systems Theory and Automatic Control University of Stuttgart Stuttgart70550 Germany Faculty of Mathematics Chemnitz University of Technology Chemnitz09111 Germany Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Ilmenau99693 Germany

We derive novel deterministic bounds on the approximation error of data-based bilinear surrogate models for unknown nonlinear systems. The surrogate models are constructed using kernel-based extended dynamic mode decomposition to approximate the Koopman operator in a reproducing kernel Hilbert space. Unlike previous methods that require restrictive assumptions on the invariance of the dictionary, our approach leverages kernel-based dictionaries that allow us to control the projection error via pointwise error bounds, overcoming a significant limitation of existing theoretical guarantees. The derived state- and input-dependent error bounds allow for direct integration into Koopman-based robust controller designs with closed-loop guarantees for the unknown nonlinear system. Numerical examples illustrate the effectiveness of the proposed framework. Copyright © 2025, The Authors. All rights reserved.

关键词： Hilbert spaces

Safe data-driven reference tracking with prescribed performance 27

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Safe data-driven reference tracking with prescribed performa...

27th International Conference on System Theory, control and Computing, ICSTCC 2023

作者： Schmitz, Philipp Lanza, Lukas Worthmann, Karl The Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Germany

ISBN: (纸本)9798350337983

We study output reference tracking for unknown continuous-time systems with arbitrary relative degree. The control objective is to keep the tracking error within predefined time-varying bounds while measurement data is only available at discrete sampling times. To achieve the control objective, we propose a two-component controller. One part is a recently developed sampled-data zero-order hold controller, which achieves reference tracking within prescribed error bounds. To further improve the control signal, we explore the system dynamics via input-output data, and include as the second component a data-driven MPC scheme based on Willems et al.'s fundamental lemma. This combination yields significantly improved input signals as illustrated by a numerical example. © 2023 IEEE.

关键词： controllers

Stage-cost design for optimal and model predictive control of linear port-Hamiltonian systems: Energy efficiency and robustness

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Stage-cost design for optimal and model predictive control o...

93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2023

作者： Şen, Gökçen Devlet Schaller, Manuel Worthmann, Karl Optimization-based Control Group Institute of Mathematics Technische Universität Ilmenau Ilmenau Germany Department of Control and Automation Engineering Istanbul Technical University Istanbul Turkey

We consider singular optimal control of port-Hamiltonian systems with minimal energy supply. We investigate the robustness of different stage-cost designs w.r.t. time discretization and show that alternative formulations that are equivalent in continuous time, differ strongly in view of discretization. Furthermore, we consider the impact of additional quadratic control regularization and demonstrate that this leads to a considerable increase in energy consumption. Then, we extend our results to the tracking problem within model predictive control and show that the intrinsic but singular choice of the cost functional as the supplied energy leads to a substantial improvement of the closed-loop performance. © 2023 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.

关键词： Model predictive control

On derivative-free sample-and-hold control with prescribed performance

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IFAC-PapersOnLine 2024年第17期58卷 121-126页

作者： Lukas Lanza Technische Universität Ilmenau Institute of Mathematics Optimization-based Control Group Ilmenau Germany

A feedback controller is proposed to perform output reference tracking with prescribed performance for nonlinear continuous-time systems of relative degree two. The controller is of sampled-data type, i.e., measurements are available only at sampling times – a typical situation in real systems when sensors are involved. Furthermore, only output information is available, i.e., neither the full state nor derivatives can be used for feedback. A sufficient uniform sampling rate is derived and the control consists of piecewise constant signals on the sampling intervals, i.e., zero-order hold. Feasibility of the controller and satisfaction of the control objective are rigorously proven. The controller is illustrated by a numerical example.

关键词： sampled-data control, zero-order hold, output feedback, tracking guarantees, nonlinear continuous-time systems 34H05, 93C15