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Lifted collocation integrators for direct optimal control in ACADO toolkit

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This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of the multiple shooting method for direct optimal control. We establish the mathematical equivalence between the Newton iteration based on direct collocation and the proposed approach, and we discuss the computational advantages of a lifted collocation integrator. In addition, we investigate different inexact versions of the proposed scheme and study their convergence and computational properties. The presented algorithms are implemented as part of the open-source ACADO code generation software for embedded optimization. Their performance is illustrated on a benchmark case study of the optimal control for a chain of masses. Based on these results, the use of lifted collocation within direct multiple shooting allows for a computational speedup factor of about 10 compared to a standard collocation integrator and a factor in the range of 10–50 compared to direct collocation using a general-purpose sparse NLP solver.

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Authors and Affiliations

  1. Department ESAT-STADIUS, KU Leuven University, 3001, Louvain, Belgium

    Rien Quirynen

  2. Department of Signals and Systems, Chalmers University of Technology, Göteborg, Sweden

    Sébastien Gros

  3. Freiburg Institute for Advanced Studies (FRIAS), 79104, Freiburg, Germany

    Sébastien Gros

  4. School of Information Science and Technology, ShanghaiTech University, Shanghai, China

    Boris Houska

  5. Department IMTEK, University of Freiburg, 79110, Freiburg, Germany

    Moritz Diehl

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  1. Rien Quirynen
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  2. Sébastien Gros
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  4. Moritz Diehl
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Correspondence to Rien Quirynen.

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The software that was reviewed as part of this submission has been issued the Digital Object Identifier doi:10.5281/zenodo.321691.

This research was supported by the EU via ERC-HIGHWIND (259 166), FP7-ITN-TEMPO (607 957), H2020-ITN-AWESCO (642 682) and by the DFG in context of the Research Unit FOR 2401. At the time of initial submission, the first author held a Ph.D. fellowship of the Research Foundation—Flanders (FWO) and he is currently affiliated with the Mitsubishi Electric Research Laboratories (MERL) in Cambridge, MA.

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Quirynen, R., Gros, S., Houska, B. et al. Lifted collocation integrators for direct optimal control in ACADO toolkit. Math. Prog. Comp. 9, 527–571 (2017). https://doi.org/10.1007/s12532-017-0119-0

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