Homogeneity for control systems in discrete time. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 56 (2023), 1, S. 385-390
Homogeneity, as a generalization of linearity to nonlinear systems, has proven to be a very powerful in systems and control. Nevertheless, only recently a notion of homogeneity was proposed for discrete-time control systems. However, this so-called D-Homogeneity directly couples the stability behaviour with the degree of homogeneity - in contrast to the continuous-time case. As an alternative, we propose the notion of S-Homogeneity, which avoids this coupling. S-Homogeneity uses a state-dependent time step that is compatible with sampling and discretization in time. We show that this concept preserves a contraction property and null-controllability for state-dependent sampling. For fixed sampling time, it yields (practical/semi-global) null controllability for sufficiently fast sampling, depending on the degree of homogeneity.
https://doi.org/10.1016/j.ifacol.2023.02.065
Stage-cost design for optimal and model predictive control of linear port-Hamiltonian systems: energy efficiency and robustness. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. n/a (2023), n/a, e202300296, S. 1-9
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.
https://doi.org/10.1002/pamm.202300296
Optimisation-based control in electrical grids and epidemiology. - [Berlin] : epubli, 2023. - xiv, 172 Seiten
Technische Universität Ilmenau, Dissertation 2023
ISBN 978-3-7575-7548-9
Diese Arbeit ist im Rahmen des KONSENS Projektes, gefördert durch das Bundesministerium für Bildung und Forschung (BMBF), entstanden. Hauptziel des KONSENS Konsortiums war die Konsistente Optimierung uNd Stabilisierung Elektrischer NetzwerkSysteme. Im Verlaufe der COVID-19 Pandemie, hat das Konsortium jedoch seine Expertise in mathematischer Modellierung, (numerischer) Analyse sowie optimaler Regelung auch dafür genutzt, um den Einfluss von Gegenmaßnahmen zu evaluieren und geeignete Strategien zur Eindämmung der Ausbreitung der Krankheit zu entwickeln. Infolgedessen ist diese Arbeit in zwei Teile untergliedert: Teil I: Verteilte Optimierung und Regelung von Microgrids, Teil II: Koordination von Gegenmaßnahmen gegen die Ausbreitung von Infektionskrankheiten. Der Schlüssel, um diese augenscheinlich grundlegend unterschiedlichen Probleme mittels verwandter Methoden zu adressieren, liegt in der Allgemeingültigkeit optimierungsbasierter Ansätze. Wir formulieren mathematische Modelle, um die jeweilige Systemdynamik zu beschreiben – in Teil I handelt es sich dabei um Batteriedynamiken, in Teil II um die Ausbreitung infektiöser Krankheiten. Dynamische Modelle in Verbindung mit beispielsweise ökonomischen Zielstellungen führen auf Optimalsteuerungsprobleme (engl. optimal control problems (OCPs)). Mit allen OCPs in dieser Arbeit gehen inhärente Unsicherheiten einher wie zum Beispiel der unbekannte zukünftige Nettoverbrauch (Teil I) und die grundsätzliche Entwicklung der pandemischen Lage (Teil II). Diese OCPs werden numerisch gelöst, um optimale Strategien zu ermitteln. Eine naive Implementierung der ermittelten Steuerung über den gesamten Optimierungshorizont würde während des Prozesses neu gewonnene Informationen nicht berücksichtigen. Eine Methode, die diese stetig neu gewonnenen Informationen in den Regelungsentwurf mit einbezieht, stellt die so genannte modellprädiktive Regelung (engl. model predictive control (MPC)) dar. MPC ist eine bewährte Methode, um OCPs mit Zustands- und Eingangsbeschränkungen in Echtzeit zu lösen und wird in beiden Teilen dieser Arbeit eingesetzt. Deshalb wird die grundlegende Idee von MPC in einem vorbereitenden Abschnitt erläutert.
Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again. - In: Annual reviews in control, ISSN 1872-9088, Bd. 55 (2023), S. 92-117
The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions - in particular Polynomial Chaos Expansions (PCE) of L2-random variables, which date back to Norbert Wiener’s seminal work - enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L2-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.
https://doi.org/10.1016/j.arcontrol.2023.03.005
State and parameter estimation for retinal laser treatment. - In: IEEE transactions on control systems technology, ISSN 1558-0865, Bd. 31 (2023), 3, S. 1366-1378
Adequate therapeutic retinal laser irradiation needs to be adapted to local absorption. This leads to time-consuming treatments as the laser power needs to be successively adjusted to avoid undertreatment and overtreatment caused by too low or too high temperatures. Closed-loop control can overcome this burden by means of temperature measurements. To allow for model predictive control schemes, the current state and the spot-dependent absorption need to be estimated. In this article, we thoroughly compare moving horizon estimator (MHE) and extended Kalman filter (EKF) designs for joint state and parameter estimation. We consider two different scenarios, the estimation of one or two unknown absorption coefficients. For one unknown parameter, both estimators perform very similarly. For two unknown parameters, we found that the MHE benefits from active parameter constraints at the beginning of the estimation, whereas after a settling time, both estimators perform again very similarly as long as the parameters are inside the considered parameter bounds.
https://doi.org/10.1109/TCST.2022.3228442
A Balian-Low type theorem for Gabor Riesz sequences of arbitrary density. - In: Mathematische Zeitschrift, ISSN 1432-1823, Bd. 303 (2023), 2, 48, S. 1-22
https://doi.org/10.1007/s00209-022-03182-6
Hierarchical power systems: optimal operation using grid flexibilities. - Cham : Springer International Publishing, 2023. - 1 Online-Ressource (viii, 55 Seiten). - (SpringerBriefs in Energy) ISBN 978-3-031-25699-8
Introduction -- Preliminary theory -- Providing flexibility via residential batteries -- Flexibility in the distribution grid -- Security and stability on the transmission grid -- Implementation in the distribution grid and the microgrids -- Numerical example -- Conclusion.
https://doi.org/10.1007/978-3-031-25699-8
Relatively bounded perturbations of J-non-negative operators. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 17 (2023), 1, 14, insges. 30 S.
We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with Lp-potentials. Known bounds on the non-real eigenvalues of such operators are improved.
https://doi.org/10.1007/s11785-022-01263-2
A note on the invertibility of the Gabor frame operator on certain modulation spaces. - In: The journal of Fourier analysis and applications, ISSN 1531-5851, Bd. 29 (2023), 1, 3, S. 1-20
We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$, the weighted $$L^2$$-space $$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$, and the space $$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $$L^2$$spaces. In all cases, we prove that the space of Bessel vectors in $$V_j$$is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
https://doi.org/10.1007/s00041-022-09980-0
Relative genericity of controllablity and stabilizability for differential-algebraic systems. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 35 (2023), 1, S. 45-76
https://doi.org/10.1007/s00498-022-00332-3