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Exploiting Chordality in Optimization Algorithms for Model Predictive Control

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Large-Scale and Distributed Optimization

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2227))

Abstract

In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.

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Notes

  1. 1.

    In case the graph is not chordal we make a chordal embedding, i.e., we add edges to the graph until it becomes chordal. This corresponds to saying that some of the \(\bar F_i\) depend on variables that they do not depend on.

  2. 2.

    Here we use a super-node for all components of a state and a control signal, respectively. In case there is further structure in the dynamic equations such that not all components of the control signal and the states are coupled, then more detailed modeling could potentially be beneficial.

  3. 3.

    Notice that the assumptions are not necessary for the block matrix to be positive definite. Moreover, for the case when it is only positive semidefinite, we still have a solution u N−1, but it is not unique. One may use pseudo inverse to obtain one solution. This follows from the generalized Schur complement formula. The full row rank assumption is equivalent to (A, B) not having any uncontrollable modes corresponding to zero eigenvalues. The positive definiteness of Q on the null-space of is equivalent to C(zI − A)−1B + D not having any zeros at the origin where is a full rank factorization.

  4. 4.

    The added edges corresponds to saying that terms in the objective function are functions of variables which they are actually not.

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Acknowledgements

The authors want to thank Daniel Axehill and Isak Nielsen for interesting discussions regarding parallel computations for Riccati recursions. Shervin Parvini Ahmadi has contributed with a figure. This research has been supported by WASP, which is gratefully acknowledged.

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Authors and Affiliations

  1. Division of Automatic Control, Linkoping University, Linkoping, Sweden

    Anders Hansson

  2. C3 IoT, Redwood city, CA, USA

    Sina Khoshfetrat Pakazad

Authors
  1. Anders Hansson
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  2. Sina Khoshfetrat Pakazad
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Corresponding author

Correspondence to Anders Hansson .

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Editors and Affiliations

  1. Department of Automatic Control, Lund University, Lund, Sweden

    Pontus Giselsson

  2. Department of Automatic Control, Lund University, Lund, Sweden

    Anders Rantzer

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Hansson, A., Pakazad, S.K. (2018). Exploiting Chordality in Optimization Algorithms for Model Predictive Control. In: Giselsson, P., Rantzer, A. (eds) Large-Scale and Distributed Optimization. Lecture Notes in Mathematics, vol 2227. Springer, Cham. https://doi.org/10.1007/978-3-319-97478-1_2

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