Abstract
This paper presents synchronous and asynchronous formulations of a sensitivity-based algorithm for solving distributed continuous-time nonlinear optimal control problems with a neighbor-affine structure. Sensitivities are defined in an optimal control context and utilized to ensure coordination between agents. By using delayed data in the asynchronous formulation, idle times of agents are reduced and faster execution of the algorithm is achieved. The convergence behavior w.r.t. topology and coupling strength is investigated in a numerical simulation and the execution time is evaluated on distributed hardware with communication over TCP in comparison to the alternating direction method of multipliers (ADMM) algorithm.
Zusammenfassung
In diesem Beitrag werden synchrone und asynchrone Formulierungen eines sensitivitätsbasierten Algorithmus zur Lösung verteilter zeitkontinuierlicher nichtlinearer Optimalsteuerungsprobleme mit nachbaraffiner Struktur vorgestellt. Sensitivitäten werden für verteilte Optimalsteurungsprobleme definiert und genutzt, um die Koordination zwischen den Agenten sicherzustellen. Durch die Verwendung verzögerter Daten in einer asynchronen Formulierung werden die Wartezeiten der Agenten reduziert und eine schnellere Ausführung des Algorithmus erreicht. Das Konvergenzverhalten in Abhängigkeit von Topologie und Kopplungsstärke wird in einer numerischen Simulation untersucht und die Ausführungszeit auf verteilter Hardware mit Kommunikation über TCP im Vergleich zum Alternating Direction Method of Multipliers (ADMM)-Algorithmus bewertet.
Funding source: Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
Award Identifier / Grant number: 3870/6-1
About the authors
![](/document/doi/10.1515/auto-2023-0050/asset/graphic/j_auto-2023-0050_cv_001.jpg)
Maximilian Pierer von Esch is a research assistant at the Chair of Automatic Control at Friedrich-Alexander-Universität Erlangen-Nürnberg. His research focuses on distributed optimization, distributed nonlinear model predictive control, and communication aspects in distributed control.
![](/document/doi/10.1515/auto-2023-0050/asset/graphic/j_auto-2023-0050_cv_002.jpg)
Dr.-Ing. Andreas Völz is a lecturer at the Chair of Automatic Control at Friedrich-Alexander-Universität Erlangen-Nürnberg. His research focuses on local optimization methods for robotic applications, collision-free motion planning and nonlinear model predictive control.
![](/document/doi/10.1515/auto-2023-0050/asset/graphic/j_auto-2023-0050_cv_003.jpg)
Prof. Dr.-Ing. Knut Graichen is head of the Chair of Automatic Control at Friedrich-Alexander-Universität Erlangen-Nürnberg. Main research areas: model predictive and distributed control as well as machine learning methods for control with applications in mechatronics, robotics, and energy systems. Knut Graichen is Editor-in-Chief of Control Engineering Practice.
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Research ethics: Not applicable.
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Author contributions: The author(s) have (has) accepted responsibility for the entire content of this manuscript and approved itssubmission.
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Competing interests: The author(s) state(s) no conflict of interest.
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Research funding: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project grant number 3870/6-1.
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Data availability: The raw data can be obtained upon reasonable request from the corresponding author.
Algorithm 1 must be adapted if the states and controls in OCP (1) are subject to various constraints. At first, the case of pure box input constraints
which often occurs in MPC, is examined. While this constraint must be considered in the local solution of OCP (14), for example via the projected gradient method [31], the sensitivity-based algorithm does not need to be modified since the constraint does not depend on states and controls of any neighbors. However, the more general case in which OCP (1) is subject to local and coupled equality constraints
or local and coupled inequality constraints
requires further attention. As the local OCP (14) is now subject to coupled constraints (42b) and (43b), the calculation of the sensitivities must accommodate for the additional constraints as well.
To this end, equality and inequality constraints are adjoined to the Hamiltonian via the Lagrangian multipliers, μj,g∈Rngj , μj,h∈Rnhj , μjs,g∈Rngjs , and μjs,h∈Rnhjs , to form the extended Hamiltonian (compare [35], Section 11])
to calculate the extended partial derivatives
which replace the respective partial derivatives in (12a) and (12b). Note that again only coupled constraints similar to coupled dynamics (1b) or costs (2) need to be regarded in the calculation of the sensitivities. For this purpose, the constraint functions must be sufficiently continuously differentiable w.r.t. their arguments on t ∈ [0, T]. The required Lagrangian multipliers in each algorithm iteration can be usually obtained from the solution of the local OCP (14) in which the constraints (42a) and (43b) are for example considered by an augmented Lagrangian framework and the multipliers are calculated via steepest ascent [31]. Note that Theorem 1 is not applicable in the same manner as the optimality conditions (16) have changed and it is not straightforward to show equality of the central and distributed optimality conditions under general constraints without any further assumptions.
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