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Robust Optimization for MPC

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Handbook of Model Predictive Control

Part of the book series: Control Engineering ((CONTRENGIN))

Abstract

This chapter aims to give a concise overview of numerical methods and algorithms for implementing robust model predictive control (MPC). We introduce the mathematical problem formulation and discuss convex approximations of linear robust MPC as well as numerical methods for nonlinear robust MPC. In particular, we review and compare generic approaches based on min-max dynamic programming and scenario-trees as well as Tube MPC based on set-propagation methods. As this chapter has a strong focus on numerical methods and their practical implementation, we also review a number of existing software packages for set computations, which can be used as building blocks for the implementation of robust MPC solvers.

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Notes

  1. 1.

    the set of compact and convex and compact sets in \(\mathbb{R}^{n}\) are denoted, respectively, by \(\mathbb{K}^{n}\) and \(\mathbb{K}_{\mathrm{C}}^{n}\).

  2. 2.

    We denote with L 2 n the set of n-dimensional L 2-integrable functions. Similarly, W 1,2 n denotes the associated Sobolev space of weakly differentiable and L 2-integrable functions on [0, T] with L 2-integrable weak derivatives.

  3. 3.

    The symbols used for continuous-time models will also be used for discrete-time. Since hybrid models are not considered in this chapter, no confusion should arise from this abuse of notation.

  4. 4.

    In this chapter, \(\mathcal{E}(Q):=\{ Q^{\frac{1} {2} }v\ \vert \ v^{\intercal }v \leq 1\}\) denotes an n-dimensional ellipsoid with positive semidefinite shape matrix Q.

  5. 5.

    The set of symmetric positive semidefinite matrices in \(\mathbb{R}^{n\times n}\) is denoted by \(\mathbb{S}_{+}^{n}\).

  6. 6.

    We use the notation co(S) for the convex hull of a set S.

  7. 7.

    The idea to use affine disturbance feedback parametrization in order to approximate robust MPC can be found, in different variants, in [7, 12, 44] as well as in an early article by J. Löfberg [48].

  8. 8.

    Details about this equivalence statement together with a formal proof in the discrete-time setting can be found in an article by Goulart and Kerrigan [28].

  9. 9.

    The boundary of \(Z \subset \mathbb{R}^{n}\) is denoted by \(\mathop{\mathrm{bd}}\nolimits Z\), while \(\mathcal{S}^{n-1}\) denotes the n-dimensional unit sphere.

  10. 10.

    In practical implementations, the memory allocation policies for function evaluations depend on the compiler and hardware. For MPC applications on embedded hardware one often uses static memory. For example, in modern code-generation based MPC solvers for small-scale systems the memory for all components of a of all online function evaluations is pre-allocated [34].

  11. 11.

    Early set-valued integrators, as, for example, developed by Nedialkov and Jackson [60], are not based on direct algorithmic differentiation based Taylor expansion of the solution trajectory, but more advanced Hermite-Obreschkoff integration schemes, which have the advantage that they can deal more efficiently with stiff dynamic systems. Some of the above-mentioned software packages are also using more advanced integration schemes, but the basic ideas for bounding the reachable set enclosures are, nevertheless, very similar to the easy-to-implement Taylor expansion based method, which has been outlined in this section.

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

    Boris Houska & Mario E. Villanueva

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    Saša V. Raković

  2. Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA

    William S. Levine

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Houska, B., Villanueva, M.E. (2019). Robust 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_18

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