arXiv subjects

Karl Worthmann

Publications and source records attributed to Karl Worthmann.

At least 19 recordsLinked to original sources

arXiv2026

Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation

The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we propose an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. We show that the EDMD regression problem admits a unique closed-form solution and coincides with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive $L^p$-error bounds, $2 \le p \le \infty$, on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation can be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators and for the chaotic ABC flow with dictionaries of up to $10^7$ modes.

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arXiv2026

Model Predictive Control: Systematic Design Tailored to Nonholonomic Systems

Nonholonomic systems are ubiquitous in applications such as transportation and robotics, yet their control remains challenging. In order to achieve asymptotic stability of a desired equilibrium, model predictive control (MPC) schemes without stabilizing terminal conditions have recently been proposed for nonholonomic kinematic systems. However, this approach still required lengthy system-specific calculations. This paper eliminates the need for these auxiliary calculations, as we propose a streamlined systematic design for MPC stage costs tailored to nonholonomic systems, utilizing the concepts of privileged coordinates and nonholonomic weights. Furthermore, the proposed design methodology is extended beyond purely kinematic models to a wider class of systems with non-negligible dynamics, and we highlight the geometric meaning of the resulting MPC stage costs. The systematic procedure is demonstrated using an articulated vehicle as an example.

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arXiv2026

Optimal energy conversion -- A port-Hamiltonian approach

This paper studies optimal energy conversion in port-Hamiltonian systems. We analyze state- and control-constrained finite-horizon optimal control problems that maximize energy conversion efficiencies. Under suitable reachability conditions, we show that the asymptotic optimal efficiencies exist and coincide with the corresponding optimal cyclic efficiencies. For a class of port-Hamiltonian systems describing coupled storage-conversion components, we further derive intrinsic limits on achievable energy conversion and establish a generalized Carnot-type efficiency bound. Our results provide an optimal-control framework connecting asymptotic optimal operation, cyclic operation, and fundamental efficiency limits for port-Hamiltonian systems.

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arXiv2026

ANOVA approximation for high-dimensional control systems

We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz--Zygmund inequality for the resulting ansatz space, we prove a probabilistic $L_2$-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.

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arXiv2026

Koopman operator theory: fundamentals, control, and applications

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

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arXiv2026

Nonlinear controlled port-Hamiltonian systems: Existence of (optimal) solutions

We investigate the existence of solutions of reversible and irreversible port-Hamilto\-nian systems. To this end, we utilize the associated exergy, a function that is composed of the system's Hamiltonian and entropy, to prove global existence in time for bounded control functions. Then, we rigorously verify our existence conditions for two examples, the gas-piston system and a network of heat exchangers. Last, we explore model predictive control tailored to irreversible port-Hamiltonian systems by means of a numerical case study with a heat exchanger network.

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arXiv2026

Maximum-likelihood reprojections for reliable Koopman predictors of parametric nonlinear dynamical systems

Koopman-based methods leverage a nonlinear lifting to enable linear regression techniques. Consequently, system analysis, learning, and prediction are considered through the lens of this lifting resulting in a nonlinear manifold that is invariant under the action of the Koopman operator. Typically, this invariance is not preserved in data-driven approximations such as Extended Dynamic Mode Decomposition (EDMD). In this work, we systematically generalize the coordinate reprojection to render Koopman predictors more reliable and versatile. To this end, we propose consistency projectors to align the dynamics of the Koopman surrogate model with the geometry of the nonlinear manifold using concepts from differential geometry. Further, we establish a link to maximum likelihood estimates, which can be leveraged as an event-triggering mechanism to only reproject if really necessary. In addition, we extend our approach to parameter-affine systems enabling sensitivity analysis. In numerical examples ranging from a supercritical pitchfork bifurcation over the Duffing oscillator to the Lorenz system, we showcase that the reprojected Koopman surrogate reproduces the bifurcation behaviour of the underlying system, i.e., the emerging equilibria, their stability and the resulting attractors, whereas predictions without reprojection either converge to spurious equilibria or diverge. This provides the basis for data-driven bifurcation analysis and control applications.

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arXiv2026

Energy-optimal predictive control of discrete-time port-Hamiltonian systems: Closed-loop practical stability

We study a dissipativity-based model predictive control (MPC) scheme for energy-optimal constrained output stabilization of a discrete-time nonlinear SISO port-Hamiltonian system described by difference and differential representation. For the optimal control problem to be solved in each MPC step, we establish a measure turnpike behavior. The turnpike set in our work is obtained from the underlying port-Hamiltonian structure without assuming existence of steady-states or periodic orbits. In particular, unlike existing results which typically establish the turnpike property w.r.t. a controlled forward invariant set such as optimal steady-states or optimal periodic orbits, we do not require the turnpike set to be forward control invariant. For MPC, we establish recursive feasibility and show practical stability to the turnpike set w.r.t. the MPC closed loop leveraging both dissipativity and the port-Hamiltonian structure. Finally, we demonstrate our results using two numerical examples.

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arXiv2026

Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

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arXiv2026

Data-driven feedback rectification of switched linear systems

In this paper, a data-driven method for the computation of stabilizing state-feedbacks is proposed that leads to a rectified eigenstructure of switched linear systems. This means that all switching subsystems have the same sets of eigenvectors and the rectification allows to compute a common quadratic Lyapunov function ensuring asymptotic stability of the closed-loop system. The method is illustrated for two examples including a model of an aerosonde.

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arXiv2026

Koopman meets input-output data: Data-driven output-feedback control of nonlinear systems with closed-loop guarantees

Data-driven control of nonlinear systems from input-output measurements remains a fundamental challenge, as existing approaches with rigorous closed-loop guarantees predominantly require access to full state measurements. In this paper, we address this gap by proposing a data-driven output-feedback controller design method for nonlinear systems that provides provable closed-loop guarantees while operating solely on measured input-output data. Our approach combines Koopman operator theory with an extended state representation of the nonlinear system constructed from input-output trajectories. This allows us to obtain a bilinear surrogate model directly from data, on which robust state-feedback design methods can be applied. By exploiting the observability of the underlying nonlinear system, we establish exponential stability of the extended state, which in turn implies exponential convergence of the original system state to the origin. Finally, we validate our theoretical findings in numerical simulations.

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arXiv2026

A model-free approach to control barrier functions for higher-order systems

Control barrier functions (CBFs) are a widely applied modular tool to ensure safe operation of nonlinear dynamical control systems. However, for their construction accurate knowledge of the system dynamics is typically needed. This requirement was recently alleviated for relative-degree-one systems using techniques from prescribed performance control (PPC) or funnel control (FC). This article extends the model-free CBF design to nonlinear systems of arbitrary relative degree. Moreover, we show with a simple example that a straightforward extension of existing results for relative-degree-one systems fails. Instead, we utilize novel techniques from funnel control to characterize a subset of the controls satisfying a CBF condition without requiring a dynamic model or state measurement. Finally, we demonstrate the applicability of our results on a seven degrees of freedom robotic manipulator with relative degree two.

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arXiv2026

Excitation of control-affine systems and Koopman error bounds

The Koopman operator and extended dynamic mode decomposition (EDMD) as a data-driven technique for its approximation have attracted considerable attention as a key tool for modeling, analysis, and control of complex dynamical systems. However, extensions towards control-affine systems resulting in bilinear surrogate models are prone to demanding data requirements rendering their applicability intricate. In this paper, we propose a framework for data-fitting of control-affine mappings to increase the robustness margin in the associated system identification problem and, thus, to provide reliable bilinear EDMD schemes. In particular, guidelines for input selection based on subspace angles are deduced such that a desired threshold with respect to the minimal singular value is ensured. Moreover, we derive necessary and sufficient conditions of optimality for maximizing the minimal singular value. Further, we demonstrate the usefulness of the proposed approach using bilinear EDMD with control for nonholonomic robots.

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arXiv2026

Limitations of LTI Koopman Modeling for Nonlinear Control Systems

Koopman operator theory yields powerful tools for modeling, analysis, and control of nonlinear dynamical systems. Prominently, linear time-invariant (LTI) Koopman representations have been proposed to enable the application of linear control techniques, such as LQR and convex MPC. In this work, we investigate the implications of exact LTI Koopman representations for continuous-time nonlinear control systems. In particular, we show that, assuming a mild controllability condition and the inclusion of the coordinate maps, the dynamics of the underlying control system must be affine linear. Furthermore, we study the modeling bias introduced by the LTI structure and analyze its dependency on the choice of observables.

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arXiv2026

Exponential stability of data-driven nonlinear MPC based on input/output models

We consider nonlinear model predictive control (MPC) schemes without stabilizing terminal conditions, where the model used in the optimization step is generated based on input-output data only. We establish exponential stability for sufficiently long prediction horizons assuming exponential stabilizability and a proportional error bound. Moreover, we verify the imposed condition on the approximation using kernel interpolation and demonstrate the practical applicability to nonlinear systems by numerical simulations.

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arXiv2026

A data-based image representation for continuous-time LTI systems

We derive a numerically stable method to compute an image representation of an unknown linear system only from data, leveraging a continuous-time version of Willems et al.'s fundamental lemma. To this end, we use derivatives approximated by algebraic differentiators. Our novel image representation avoids solving differential-algebraic equations and significantly reduces computational complexity by eliminating redundant degrees of freedom corresponding to the number of unknown quantities to be identified. Simulation results confirm the effectiveness of the proposed approach, even in the presence of severe measurement disturbances.

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arXiv2026

Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition

We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential.

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arXiv2026

Stability of data-driven Koopman MPC with terminal conditions

This paper derives conditions under which Model Predictive Control (MPC) with terminal conditions, using a data-driven surrogate model as a prediction model, asymptotically stabilizes the plant despite approximation errors. In particular, we prove recursive feasibility and asymptotic stability if a proportional error bound holds, where proportional means that the bound is linear in the norm of the state and the input. For a broad class of nonlinear systems, this condition can be satisfied using data-driven surrogate models generated by kernel Extended Dynamic Mode Decomposition (kEDMD) using the Koopman operator. Last, the applicability of the proposed framework is demonstrated in a numerical case study.

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