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Multiple Shooting in a Microsecond

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Multiple Shooting and Time Domain Decomposition Methods

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 9))

Abstract

Nonlinear Model Predictive Control (NMPC) is a feedback control technique that uses the most current state estimate of a nonlinear system to compute an optimal plan for the future system behavior. This plan is recomputed at every sampling time, creating feedback. Thus, NMPC needs to repeatedly solve a nonlinear optimal control problem (OCP). Direct multiple shooting is since long known as a reliable approach for discretization of OCPs. This is mainly due to the fact that the approach shows good contraction properties within the NMPC framework. Moreover, the procedure is easy to initialize and parallelize. In the context of real-time NMPC, the multiple shooting method was tailored to the Real-Time Iteration (RTI) scheme. This scheme uses a strategy known as Initial Value Embedding to deal efficiently with the transition from one optimization problem to the next. It performs two algorithmic steps in each sampling time, a long preparation phase and a short feedback phase to minimize the feedback time to the system. The two phases respectively prepare and solve a convex Quadratic Program (QP) that depends parametrically on the estimated system state. The solution of this QP delivers quickly a generalized tangential predictor to the solution of the nonlinear problem. Recent algorithmic progress makes the solution of NMPC optimization problems possible at sampling times in the milli- or even microsecond range on modern computational hardware. An essential part is the simulation of the nonlinear model together with the propagation of its derivative information. This article describes the developments and their efficient software implementations that made it possible to solve a classical NMPC benchmark problem within 1 μs sampling time.

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Acknowledgements

The authors would like to thank all reviewers for their valuable comments, which helped to improve this article. This research was supported by Research Council KUL: PFV/10/002 Optimization in Engineering Center OPTEC, GOA/10/09 MaNet and GOA/10/11 Global real-time optimal control of autonomous robots and mechatronic systems, KUL-BOF OT/10/035. Flemish Government: FWO: PhD/postdoc grants, FWO KAN2013 1.5.189.13, FWO-G.0930.13; IWT: PhD Grants, projects: Eurostars SMART; Belgian Federal Science Policy Office: IUAP P7 (DYSCO, Dynamical systems, control and optimization, 2012–2017); EU: FP7-SADCO (MC ITN-264735), FP7-TEMPO (MC ITN-607957), H2020-AWESCO (MC ITN-642682), ERC HIGHWIND (259 166). R. Quirynen holds a PhD fellowship of the Research Foundation– Flanders (FWO).

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Authors and Affiliations

  1. KU Leuven, Kasteelpark Arenberg 10, 3001, Leuven, Belgium

    Rien Quirynen, Milan Vukov & Moritz Diehl

  2. University of Freiburg, Georges-Koehler-Allee 102, 79110, Freiburg, Germany

    Moritz Diehl

Authors
  1. Rien Quirynen
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  2. Milan Vukov
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  3. Moritz Diehl
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Corresponding author

Correspondence to Rien Quirynen .

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Editors and Affiliations

  1. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Thomas Carraro

  2. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Michael Geiger

  3. Heidelberg University, Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

    Stefan Körkel

  4. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Rolf Rannacher

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Quirynen, R., Vukov, M., Diehl, M. (2015). Multiple Shooting in a Microsecond. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_7

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