arXiv subjects

Karl Worthmann

Publications and source records attributed to Karl Worthmann.

At least 55 records · Page 3Linked to original sources

arXiv2025

Safe continual learning in model predictive control with prescribed bounds on the tracking error

We develop a three-component Model Predictive Control (MPC) algorithm to achieve output-reference tracking with prescribed performance for continuous-time nonlinear systems. One component is so-called funnel MPC, which achieves reference tracking with prescribed performance for the model output for suitable models. Recently, this MPC algorithm has been combined with a model-free reactive feedback controller (second component) to account for model-plant mismatches, bounded disturbances, and uncertainties. By construction, this two-component controller defines a robust funnel MPC algorithm. It achieves output-reference tracking within prescribed bounds on the tracking error for a class of unknown nonlinear systems. In this paper, we extend the robust funnel MPC by a machine learning component to adapt the underlying model to the system data and, thus, improve the contribution of MPC. We derive sufficient structural conditions to define a class of models for funnel MPC, and provide a characterization of suitable learning schemes. Since robust funnel MPC is inherently robust and the evolution of the tracking error in the prescribed performance funnel is guaranteed, the additional learning component is able to perform the learning task online -- even without an initial model or offline training.

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arXiv2025

Sampled-data funnel control and its use for safe continual learning

We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety-critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.'s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design.

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arXiv2025

Towards Data-Driven Multi-Stage OPF

The operation of large-scale power systems is usually scheduled ahead via numerical optimization. However, this requires models of grid topology, line parameters, and bus specifications. Classic approaches first identify the network topology, i.e., the graph of interconnections and the associated impedances. The power generation schedules are then computed by solving a multi-stage optimal power flow (OPF) problem built around the model. In this paper, we explore the prospect of data-driven approaches to multi-stage optimal power flow. Specifically, we leverage recent findings from systems and control to bypass the identification step and to construct the optimization problem directly from data. We illustrate the performance of our method on a 118-bus system and compare it with the classical identification-based approach.

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arXiv2024

Globally convergent homotopies for discrete-time optimal control

Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle).

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arXiv2024

Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry

Advances in machine learning and the growing trend towards effortless data generation in real-world systems has led to an increasing interest for data-inferred models and data-based control in robotics. It seems appealing to govern robots solely based on data, bypassing the traditional, more elaborate pipeline of system modeling through first-principles and subsequent controller design. One promising data-driven approach is the Extended Dynamic Mode Decomposition (EDMD) for control-affine systems, a system class which contains many vehicles and machines of immense practical importance including, e.g., typical wheeled mobile robots. EDMD can be highly data-efficient, computationally inexpensive, can deal with nonlinear dynamics as prevalent in robotics and mechanics, and has a sound theoretical foundation rooted in Koopman theory. On this background, this present paper examines how EDMD models can be integrated into predictive controllers for nonholonomic mobile robots. In addition to the conventional kinematic mobile robot, we also cover the complete data-driven control pipeline - from data acquisition to control design - when the robot is not treated in terms of first-order kinematics but in a second-order manner, allowing to account for actuator dynamics. Using only real-world measurement data, it is shown in both simulations and hardware experiments that the surrogate models enable high-precision predictive controllers in the studied cases. However, the findings raise significant concerns about purely data-centric approaches that overlook the underlying geometry of nonholonomic systems, showing that, for nonholonomic systems, some geometric insight seems necessary and cannot be easily compensated for with large amounts of data.

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arXiv2024

Equivariance and partial observations in Koopman operator theory for partial differential equations

The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.

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arXiv2024

Optimal control of port-Hamiltonian systems: energy, entropy, and exergy

We consider irreversible and coupled reversible-irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems, where the performance functional can be interpreted as a linear combination of energy supply, entropy generation, or exergy supply. Our results establish the integral turnpike property towards the set of thermodynamic equilibria providing a rigorous connection of optimal system trajectories to optimal steady states. Throughout the paper, we illustrate our findings by means of two examples: a network of heat exchangers and a gas-piston system.

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arXiv2024

Perturbations in PDE-constrained optimal control decay exponentially in space

For linear-quadratic optimal control problems (OCPs) governed by elliptic and parabolic partial differential equations (PDEs), we investigate the impact of perturbations on optimal solutions. Local perturbations may occur, e.g., due to discretization of the optimality system or {disturbed} problem data. Whereas these perturbations may exhibit global effects in the uncontrolled case, we prove that the ramifications are exponentially damped in space under stabilizability and detectability conditions. To this end, we prove a bound on the optimality condition's solution operator that is uniform in the domain size. Then, this uniformity is used in a scaling argument to show the exponential decay of perturbations in space. We numerically validate and illustrate our results by solving OCPs involving Helmholtz, Poisson, and advection-diffusion-reaction equations.

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arXiv2024

Modularized data-driven approximation of the Koopman operator and generator

Extended Dynamic Mode Decomposition (EDMD) is a widely-used data-driven approach to learn an approximation of the Koopman operator. Consequently, it provides a powerful tool for data-driven analysis, prediction, and control of nonlinear dynamical (control) systems. In this work, we propose a novel modularized EDMD scheme tailored to interconnected systems. To this end, we utilize the structure of the Koopman generator that allows to learn the dynamics of subsystems individually and thus alleviates the curse of dimensionality by considering observable functions on smaller state spaces. Moreover, our approach canonically enables transfer learning if a system encompasses multiple copies of a model as well as efficient adaption to topology changes without retraining. We provide finite-data bounds on the estimation error using tools from graph theory. The efficacy of the method is illustrated by means of various numerical examples.

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arXiv2024

Data-driven MPC with terminal conditions in the Koopman framework

We investigate nonlinear model predictive control (MPC) with terminal conditions in the Koopman framework using extended dynamic mode decomposition (EDMD) to generate a data-based surrogate model for prediction and optimization. We rigorously show recursive feasibility and prove practical asymptotic stability w.r.t. the approximation accuracy. To this end, finite-data error bounds are employed. The construction of the terminal conditions is based on recently derived proportional error bounds to ensure the required Lyapunov decrease. Finally, we illustrate the effectiveness of the proposed data-driven predictive controller including the design procedure to construct the terminal region and controller.

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arXiv2024

Reinforcement Learning for Docking Maneuvers with Prescribed Performance

We propose a two-component data-driven controller to safely perform docking maneuvers for satellites. Reinforcement Learning is used to deduce an optimal control policy based on measurement data. To safeguard the learning phase, an additional feedback law is implemented in the control unit, which guarantees the evolution of the system within predefined performance bounds. We define safe and safety-critical areas to train the feedback controller based on actual measurements. To avoid chattering, a dwell-time activation scheme is implemented. We provide numerical evidence for the performance of the proposed controller for a satellite docking maneuver with collision avoidance.

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arXiv2024

Data-driven MPC with stability guarantees using extended dynamic mode decomposition

For nonlinear (control) systems, extended dynamic mode decomposition (EDMD) is a popular method to obtain data-driven surrogate models. Its theoretical foundation is the Koopman framework, in which one propagates observable functions of the state to obtain a linear representation in an infinite-dimensional space. In this work, we prove practical asymptotic stability of a (controlled) equilibrium for EDMD-based model predictive control, in which the optimization step is conducted using the data-based surrogate model. To this end, we derive novel bounds on the estimation error that are proportional to the norm of state and control. This enables us to show that, if the underlying system is cost controllable, this stabilizablility property is preserved. We conduct numerical simulations illustrating the proven practical asymptotic stability.

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arXiv2024

A continuous-time fundamental lemma and its application in data-driven optimal control

Data-driven control of discrete-time and continuous-time systems is of tremendous research interest. In this paper, we explore data-driven optimal control of continuous-time linear systems using input-output data. Based on a density result, we rigorously derive error bounds for finite-order polynomial approximations of elements of the system behavior. To this end, we leverage a link between latent variables and flat outputs of controllable systems. Combined with a continuous-time counterpart of the fundamental lemma by Willems et al., we characterize the suboptimality resulting from polynomial approximations in data-driven linear-quadratic optimal control problems. Finally, we draw upon a numerical example to illustrate our results.

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arXiv2024

$L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition

Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by using the kernel's canonical features and, thus, data-informed observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.

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arXiv2024

MPC without Terminal Ingredients Tailored to the SEIR Compartmental Epidemic Model

We consider the SEIR compartmental epidemic model subject to state and input constraints (a cap on the proportion of infectious individuals and limits on the allowed social distancing and quarantining measures, respectively). We present a tailored model predictive control (MPC) scheme without terminal conditions. We rigorously show recursive feasibility and asymptotic convergence of the MPC closed loop to the continuum of disease-free equilibrium points for suitably designed quadratic running cost and a sufficiently long prediction horizon (forecast window). Moreover, we establish the viability kernel (a.k.a. the admissible set) as a domain of attraction of the continuum of equilibria.

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arXiv2024

Variance representations and convergence rates for data-driven approximations of Koopman operators

We rigorously derive novel error bounds for extended dynamic mode decomposition (EDMD) to approximate the Koopman operator for discrete- and continuous time (stochastic) systems; both for i.i.d. and ergodic sampling under non-restrictive assumptions. We show exponential convergence rates for i.i.d. sampling and provide the first superlinear convergence rates for ergodic sampling of deterministic systems. The proofs are based on novel exact variance representations for the empirical estimators of mass and stiffness matrix. Moreover, we verify the accuracy of the derived error bounds and convergence rates by means of numerical simulations for highly-complex dynamical systems including a nonlinear partial differential equation.

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arXiv2024

Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients

We consider the Koopman operator semigroup $(K^t)_{t\ge 0}$ associated with stochastic differential equations of the form $dX_t = AX_t\,dt + B\,dW_t$ with constant matrices $A$ and $B$ and Brownian motion $W_t$. We prove that the reproducing kernel Hilbert space $\bH_C$ generated by a Gaussian kernel with a positive definite covariance matrix $C$ is invariant under each Koopman operator $K^t$ if the matrices $A$, $B$, and $C$ satisfy the following Lyapunov-like matrix inequality: $AC^2 + C^2A^\top\le 2BB^\top$. In this course, we prove a characterization concerning the inclusion $\bH_{C_1}\subset\bH_{C_2}$ of Gaussian RKHSs for two positive definite matrices $C_1$ and $C_2$. The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem.

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arXiv2024

Koopman-based feedback design with stability guarantees

We present a method to design a state-feedback controller ensuring exponential stability for nonlinear systems using only measurement data. Our approach relies on Koopman-operator theory and uses robust control to explicitly account for approximation errors due to finitely many data samples. To simplify practical usage across various applications, we provide a tutorial-style exposition of the feedback design and its stability guarantees for single-input systems. Moreover, we extend this controller design to multi-input systems and more flexible nonlinear state-feedback controllers using gain-scheduling techniques to increase the guaranteed region of attraction. As the proposed controller design is framed as a semidefinite program, it allows for an efficient solution. Further, we enhance the geometry of the region of attraction through a heuristic algorithm that establishes a connection between the employed Koopman lifting and the dynamics of the system. Finally, we validate the proposed feedback design procedure by means of numerical examples.

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