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Direct Optimal Control and Model Predictive Control

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Optimal Control: Novel Directions and Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2180))

Abstract

Model predictive control is a feedback control technique based on repeatedly solving optimal control problems. Direct methods for optimal control have gained popularity especially for practical applications, due to their flexibility. In this chapter we first present the state of the art in MPC stability theory. Then, we introduce the numerical methods used for direct optimal control and some variants specifically tailored to MPC. We conclude the chapter with five application examples.

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Notes

  1. 1.

    A continuous function \(\rho: \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}_{0}^{+}\) is a \(\mathcal{K}\)-function if ρ(0) = 0 and is strictly increasing. ρ is a \(\mathcal{K}_{\infty }\)-function if it is a \(\mathcal{K}\)-function that is unbounded. A continuous function \(\beta: \mathbb{R}_{0}^{+} \times \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}_{0}^{+}\) is a \(\mathcal{KL}\)-function if for each r ≥ 0, β(r, ⋅ ) is decreasing and lim t → ∞ β(r, t) = 0 and for each \(t \geq 0,\ \beta (\cdot,t) \in \mathcal{ K}_{\infty }\). A continuous function \(\beta: \mathbb{R}_{0}^{+} \times \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}_{0}^{+}\) is a \(\mathcal{KL}_{0}\)-function if for each r ≥ 0, β(r, ⋅ ) is decreasing and lim t → ∞ β(r, t) = 0 and for each t ≥ 0 we either have \(\beta (\cdot,t) \in \mathcal{ K}_{\infty }\) or β(⋅ , t) ≡ 0.

  2. 2.

    A set \(S \subset \mathbb{X}\) is said to be forward invariant or viable for (3.1) if, for every x ∈ S there exists \(u \in \mathbb{U}(x)\) such that f(x, u) ∈ S.

  3. 3.
    1. (i)

      The optimal value function V N is said to be uniformly continuous on a set \(A \subseteq \mathbb{X}\) if there exists a \(\mathcal{K}\)-function \(\omega _{V _{N}}\) such that for all x 1, x 2 ∈ A

      $$\displaystyle{\vert V _{N}(x_{1}) - V _{N}(x_{2})\vert \leq \omega _{V _{N}}\left (\|x_{1} - x_{2}\|\right ).}$$
    2. (ii)

      The cost functional J N is said to be uniformly continuous on \(A \subseteq \mathbb{X}\) uniformly in \(u \in \mathbb{U}^{N}\) if there exists a function \(\omega _{J_{N}} \in \mathcal{ K}\) such that for all x 1, x 2 ∈ A and all \(u \in \mathbb{U}^{N}\)

      $$\displaystyle{\vert J_{N}(x_{1},u) - J_{N}(x_{2},u)\vert \leq \omega _{J_{N}}\left (\|x_{1} - x_{2}\|\right ).}$$

    The functions \(\omega _{V _{N}}\) and \(\omega _{J_{N}}\) are called moduli of continuity. Analogous uniform continuity definitions can be defined for f, ℓ and B K with the corresponding moduli of continuity.

  4. 4.

    We say that f is uniformly bounded on each ball \(\overline{\mathcal{B}}_{\varDelta }(x_{s})\) if for any Δ > 0 the value \(\sup _{\|x\|_{x_{ s}}\leq \varDelta,u\in \mathbb{U}(x)}\|\,f(x,u)\|\) is finite.

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Authors and Affiliations

  1. Chalmers University of Technology, Göteborg, Sweden

    Mario Zanon

  2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrea Boccia

  3. Universität Bayreuth, Bayreuth, Germany

    Vryan Gil S. Palma

  4. ESG Elektroniksystem- und Logistik-GmbH, Fürstenfeldbruck, Germany

    Sonja Parenti

  5. Volkswagen, Wolfsburg, Germany

    Ilaria Xausa

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  1. Mario Zanon
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  2. Andrea Boccia
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  3. Vryan Gil S. Palma
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  4. Sonja Parenti
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  5. Ilaria Xausa
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Correspondence to Mario Zanon .

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  1. CEREMADE, UMR CNRS 7534, Université Paris Dauphine, PSL Research University, Paris, France

    Daniela Tonon

  2. Escola de Matemática Aplicada, Fundação Getúlio Vargas, Rio de Janeiro, Brazil

    Maria Soledad Aronna

  3. Department of Mathematics, Imperial College London, London, United Kingdom

    Dante Kalise

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Zanon, M., Boccia, A., Palma, V.G.S., Parenti, S., Xausa, I. (2017). Direct Optimal Control and Model Predictive Control. In: Tonon, D., Aronna, M., Kalise, D. (eds) Optimal Control: Novel Directions and Applications. Lecture Notes in Mathematics, vol 2180. Springer, Cham. https://doi.org/10.1007/978-3-319-60771-9_3

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