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
Reprojection methods for Koopman-based modelling and prediction. - In: IEEE Xplore digital library, ISSN 2473-2001, (2023), S. 315-321
Extended Dynamic Mode Decomposition (eDMD) is a powerful tool to generate data-driven surrogate models for the prediction and control of nonlinear dynamical systems in the Koopman framework. In eDMD a compression of the lifted system dynamics on the space spanned by finitely many observables is computed, in which the original space is embedded as a low-dimensional manifold. While this manifold is invariant for the infinite-dimensional Koopman operator, this invariance is typically not preserved for its eDMD-based approximation. Hence, an additional (re-)projection step is often tacitly incorporated to improve the prediction capability. We propose a novel framework for consistent reprojectors respecting the underlying manifold structure. Further, we present a new geometric reprojector based on maximum-likelihood arguments, which significantly enhances the approximation accuracy and preserves known finite-data error bounds.
https://doi.org/10.1109/CDC49753.2023.10383796
Funnel control of linear systems with arbitrary relative degree under output measurement losses. - In: IMA journal of mathematical control and information, ISSN 1471-6887, Bd. 40 (2023), 4, S. 691-713
We consider tracking control of linear minimum phase systems with known arbitrary relative degree which are subject to possible output measurement losses. We provide a control law which guarantees the evolution of the tracking error within a (shifted) prescribed performance funnel whenever the output signal is available. The result requires a maximal duration of measurement losses and a minimal time of measurement availability, which both strongly depend on the internal dynamics of the system, and are derived explicitly. The controller is illustrated by a simulation of a mass-on-car system.
https://doi.org/10.1093/imamci/dnad029
Safe data-driven reference tracking with prescribed performance. - In: 2023 27th International Conference on System Theory, Control and Computing (ICSTCC), (2023), S. 454-460
ISBN 979-8-3503-3798-3
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.
https://doi.org/10.1109/ICSTCC59206.2023.10308521
Path planning for concentric tube robots: a toolchain with application to stereotactic neurosurgery. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 56 (2023), 2, S. 2871-2876
We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear and highly nonconvex optimal control problem is solved by introducing a homotopy on the obstacle positions where at one extreme of the parameter the obstacles are removed from the operating space, and at the other extreme they are located at their intended positions. We present a detailed example (with more than a thousand obstacles) from stereotactic neurosurgery with real-world data obtained from labelled MRI scans.
https://doi.org/10.1016/j.ifacol.2023.10.1403
Towards reliable data-based optimal and predictive control using extended DMD. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 56 (2023), 1, S. 169-174
While Koopman-based techniques like extended Dynamic Mode Decomposition are nowadays ubiquitous in the data-driven approximation of dynamical systems, quantitative error estimates were only recently established. To this end, both sources of error resulting from a finite dictionary and only finitely-many data points in the generation of the surrogate model have to be taken into account. We generalize the rigorous analysis of the approximation error to the control setting while simultaneously reducing the impact of the curse of dimensionality by using a recently proposed bilinear approach. In particular, we establish uniform bounds on the approximation error of state-dependent quantities like constraints or a performance index enabling data-based optimal and predictive control with guarantees.
https://doi.org/10.1016/j.ifacol.2023.02.029
Port maps of Irreversible Port Hamiltonian Systems. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 56 (2023), 2, S. 6796-6800
Irreversible Port Hamiltonian Systems are a deviation from Port Hamiltonian Systems which embeds the definition of the irreversible phenomena taking place in the system. They are defined with respect to a quasi-Poisson bracket which ensures the positiveness of the entropy generation and is expressed in terms of the total entropy of the system. The port maps, however, associated with the conjugated port variables, were poorly justified and lacked any physical characterization. In this paper, we suggest a novel definition of the port maps which allows to recover not only the energy balance equation (when the Hamiltonian equals the total energy of the system) but also a entropy balance equation including the irreversible entropy creation term at the interface (the port) of the system in addition to the entropy creation term due to internal irreversible phenomena.
https://doi.org/10.1016/j.ifacol.2023.10.388
On the generating functions of irreversible port-Hamiltonian systems. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 56 (2023), 2, S. 10447-10452
We study the geometric structure of the drift dynamics of Irreversible port-Hamiltonian systems. This drift dynamics is defined with respect to a product of Poisson brackets, reflecting the interconnection structure and the constitutive relations of the irreversible phenomena occuring in the system. We characterise this product of Poisson brackets using a covariant 4-tensor and an associated function. We derive various conditions for which this 4-tensor and the associated function may be reduced to a product of almost Poisson brackets.
https://doi.org/10.1016/j.ifacol.2023.10.1061
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