Koopman-based feedback design with stability guarantees. - In: IEEE transactions on automatic control, ISSN 1558-2523, Bd. 0 (2024), 0, S. 1-16
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.
https://doi.org/10.1109/TAC.2024.3425770
Alleviating the curse of dimensionality in Minkowski sum approximations of storage flexibility. - In: IEEE transactions on smart grid, Bd. 0 (2024), 0, S. 1-11
Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over d time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to 2d vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.
https://doi.org/10.1109/TSG.2024.3420156
Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 0 (2024), 0, insges. 37 S.
The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359-377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.
https://doi.org/10.1007/s00498-024-00392-7
Funnel MPC for nonlinear systems with arbitrary relative degree. - In: Automatica, ISSN 0005-1098, Bd. 167 (2024), 111759, S. 1-10
The model predictive control (MPC) scheme funnel MPC enables output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems with stable internal dynamics. Earlier works achieved the control objective for system with relative degree restricted to one or incorporated additional feasibility constraints in the optimal control problem. Here we resolve these limitations by introducing a modified stage cost function relying on a weighted sum of the tracking error derivatives. The weights need to be sufficiently large and we state explicit lower bounds. Under these assumptions we are able to prove initial and recursive feasibility of the novel funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.
https://doi.org/10.1016/j.automatica.2024.111759
Generic observability for port-Hamiltonian descriptor systems. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 0 (2024), 0, insges. 43 S.
The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359-377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45-76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.
https://doi.org/10.1007/s00498-024-00388-3
Adjustable robust energy operation planning under uncertain renewable energy production. - In: Energies, ISSN 1996-1073, Bd. 17 (2024), 8, 1917, S. 1-14
In this paper, the application of the method of affinely adjustable robust optimization to a planning model of an energy system under uncertain parameters is presented, and the total scheduling costs in comparison with the deterministic model are evaluated. First, the basics of optimization under uncertain data are recapped, and it is described how these methods can be used in different applications for energy systems. This is followed by the methodology of adjustable robust optimization by defining the affinely adjustable robust counterpart. Finally, a numerical case study is conducted to compare the adjustable robust method with a rolling deterministic scheduling method. Both are implemented on a model of an energy system and compared with each other by simulation using real-world data. By calculating the total operating costs for both methods, it can be concluded that the adjustable robust optimization provides a significantly more cost-effective solution to the scheduling problem.
https://doi.org/10.3390/en17081917
The mystery of Carleson frames. - In: Applied and computational harmonic analysis, ISSN 1096-603X, Bd. 72 (2024), 101659, S. 1-5
In 2016 Aldroubi et al. constructed the first class of frames having the form {Tkφ}k=0∞ for a bounded linear operator on the underlying Hilbert space. In this paper we show that a subclass of these frames has a number of additional remarkable features that have not been identified for any other frames in the literature. Most importantly, the subfamily obtained by selecting each Nth element from the frame is itself a frame, regardless of the choice of N∈N. Furthermore, the frame property is kept upon removal of an arbitrarily finite number of elements.
https://doi.org/10.1016/j.acha.2024.101659
Error bounds for kernel-based approximations of the Koopman operator. - In: Applied and computational harmonic analysis, ISSN 1096-603X, Bd. 71 (2024), 101657, S. 1-25
We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.
https://doi.org/10.1016/j.acha.2024.101657
Optimale Steuerung und Regelung - Analyse, Algorithmen und Anwendungen :
Optimal control - analysis, algorithms and applications. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 72 (2024), 2, S. 77-79
Optimal control has been at the center of many pivotal developments in systems and control in the 20th century. This includes the twin breakthroughs of Richard E. Bellman’s Dynamic Programming and Lew S. Pontryagin’s Maximimum Principle as well as the optimality does not imply stability punchline by Rudolf E. Kalman. Likewise the dissipativity notion for open systems conceived by Jan C. Willems is deeply routed and closely linked to optimal control theory. Moreover, model predictive control can be regarded as an industrially impactful attempt to overcome the difficulties of analytic computation of feedback laws for constrained systems by numerical online computation. First formulations of receding-horizon ideas for optimal control can be traced back to the 1960s. With this pretext one might be tempted to conclude that contemporary research on optimal control is limited to applications. This special issue on optimal control with its particular focus on analysis, algorithms as well as applications falsifies any adhoc conclusion of this kind. Indeed, it combines different contributions which cover a wide array of topics – ranging from hydropower plants and bicycle dynamics to port-Hamiltonian formulations for adaptive structures, distributed predictive control, and moving horizon estimation. Hence, even without drawing upon the currently prevailing trends of data-driven and learning-based control – which also admit optimization-based research avenues – optimal control continues to be a supporting pillar of modern systems and control research with manifold prospects for fundamental analysis, performant algorithms, and challenging applications. Following the established structure of the journal the articles of this special issue are clustered into two categories – methods and applications.
https://doi.org/10.1515/auto-2023-0235
Energie-optimale Steuerung adaptiver Gebäude :
Energy-optimal control of adaptive structures. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 72 (2024), 2, S. 107-119
Adaptive structures are equipped with sensors and actuators to actively counteract external loads such as wind. This can significantly reduce resource consumption and emissions during the life cycle compared to conventional structures. A common approach for active damping is to derive a port-Hamiltonian model and to employ linear-quadratic control. However, the quadratic control penalization lacks physical interpretation and merely serves as a regularization term. Rather, we propose a controller, which achieves the goal of vibration damping while acting energy-optimal. Leveraging the port-Hamiltonian structure, we show that the optimal control is uniquely determined, even on singular arcs. Further, we prove a stable long-time behavior of optimal trajectories by means of a turnpike property. Last, the proposed controller’s efficiency is evaluated in a numerical study.
https://doi.org/10.1515/auto-2023-0090