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Explicit (Offline) Optimization for MPC

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Handbook of Model Predictive Control

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Abstract

In this chapter, we present the fundamentals of multi-parametric programming and its application to explicit model predictive control (MPC), i.e. the offline solution of MPC problems for both continuous and hybrid systems. In particular, we first show how MPC problems can be reformulated as multi-parametric programming problems, then we show how explicit/multi-parametric solutions are derived and the key underlying theoretical properties. Finally, we present solution procedures for these type of problems and discuss applicability issues and potential future research directions.

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Notes

  1. 1.

    Problem (10) can be viewed as a special case of problem (9) with Q = 0n×n and H = 0n×q, which is inherently positive semi-definite.

  2. 2.

    For an introduction into the concept of Lagrangian multipliers and duality in general, the reader is referred to the excellent textbook by C. A. Floudas [28].

  3. 3.

    Assuming no degeneracy, in the case of mp-LP problems, the cardinality of the active set k is \(\text{card}\left (k\right ) = n\) and thus the parametric solution is directly given as \(u\left (\theta \right ) = G_{k}^{-1}\left (W_{k} + S_{k}x\right )\).

  4. 4.

    In general, the term “parametric” refers to the case where a single parameter is considered, while “multi-parametric” suggests the presence of multiple parameters.

  5. 5.

    It is convex, if the following z-transformation is applied \(u = z -\frac{1} {2}H^{-1}Zx\), based on the nomenclature of problem (9).

  6. 6.

    In an excellent technical note from 2002, Baotić actually already commented on the connected graph nature of the problem, however without providing a formal proof or further discussion on the topic [7].

  7. 7.

    This does not consider problems arising from scaling, round-off computational errors or the presence of identical constraints in the problem formulation.

  8. 8.

    Consider Figure 1a: if the constraint which only coincides at the single point with the feasible space is chosen as part of the active set, the corresponding parametric solution will only be valid in that point.

  9. 9.

    A similar approach was presented in 2006 by Olaru and Dumur [56].

  10. 10.

    In other words: if k is infeasible, so is its powerset.

  11. 11.

    In the case of mp-MILP problems, the MINLP becomes a mixed-integer linear programming problem.

  12. 12.

    The limiting iteration number ρ limit is the maximum number of iterations performed on a single machine before the result is returned to the main algorithm.

  13. 13.

    Note that in the case of the envelopes of solutions approach to the mp-MIQP problem the critical regions are polytopes but a comparison procedure between alternative solutions is necessary.

  14. 14.

    Note that this claim refers mainly to alleviating the necessity of optimization hardware equipment for the application of optimization based MPC as the explicit solution enables the use of MPC-on-a-chip as described in [60].

  15. 15.

    If the sampling time of a system is 1μs, but the point location of the explicit MPC controller may require up to 5μs, the explicit MPC controller cannot be applied in practice.

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Diangelakis, N.A., Oberdieck, R., Pistikopoulos, E.N. (2019). Explicit (Offline) Optimization for MPC. In: Raković, S., Levine, W. (eds) Handbook of Model Predictive Control. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77489-3_16

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