Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle

Cots, Olivier

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1715-1749.

In this article, the minimum time control problem of an electric vehicle is modeled as a Mayer problem in optimal control, with affine dynamics with respect to the control and with state constraints. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle. An analysis, with the techniques of geometric control, is used first to reduce the set of candidates and then to construct the numerical methods. This leads to a numerical investigation based on indirect methods using the HamPath software. Multiple shooting and homotopy techniques are used to build a synthesis with respect to the bounds of the boundary sets.

Classification : 49K15, 49M05, 90C90, 80M50
Mots clés : Geometric optimal control, state constraints, shooting and homotopy methods, electric car

A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint. Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004) xiv+412. | MR | Zbl

E. Allgower and K. Georg, Introduction to numerical continuation methods. Vol. 45 of Classics in Applied Mathematic. SIAM, Philadelphia, PA, USA (2003) xxvi+388. | MR | Zbl

V.G. Boltyanskiĭ, R.V. Gamkrelidze, E.F. Mishchenko and L.S. Pontryagin, The Mathematical Theory of Optimal Processes. Classics of Soviet Mathematics. Gordon and Breach Science Publishers, New York (1986) xxiv+360. | Zbl

F.J. Bonnans, P. Martinon and V. Grélard, Bocop - A collection of examples. Technical report, INRIA (2012). RR-8053.

B. Bonnard and M. Chyba, Singular trajectories and their role in control theory. Vol. 40 of Mathematics & Applications. Springer-Verlag, Berlin (2003) xvi+357. | MR | Zbl

B. Bonnard, M. Claeys, O. Cots and P. Martinon, Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance. Acta Appl. Math. 135 (2014) 5–45. | DOI | MR | Zbl

B. Bonnard and O. Cots, Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance. Math. Models Methods Appl. Sci. 24 (2012) 187–212. | DOI | MR | Zbl

B. Bonnard, L. Faubourg, G. Launay and E. Trélat, Optimal Control With State Constraints And The Space Shuttle Re-entry Problem. J. Dyn. Control Syst. 9 (2003) 155–199. | DOI | MR | Zbl

A.E. Bryson, W.F. Denham and S.E. Dreyfus, Optimal programming problems with inequality constraints I: necessary conditions for extremal solutions. AIAA Journal 1 (1963) 2544–2550. | DOI | MR | Zbl

R. Bulirsch and J. Stoer, Introduction to numerical analysis, vol. 12 of Texts in Applied Mathematics. Springer-Verlag, New York, 2${}^{nd}$ edition (1993) xvi+744. | MR | Zbl

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems. Optim. Methods Softw. 27 (2011) 177–196. | DOI | MR | Zbl

J.-B. Caillau and B. Daoud, Minimum time control of the restricted three-body problem. SIAM J. Control Optim. 50 (2012) 3178–3202. | DOI | MR | Zbl

E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Vol. 8 of Springer Serie Comput. Math. Springer-Verlag, 2nd edn. (1993). | MR | Zbl

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Vol. 14 of Springer Serie in Computational Mathematics, Springer-Verlag, 2nd edn. (1996). | MR | Zbl

R.F. Hartl, S.P. Sethi and R.G. Vickson, A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37 (1995) 181–218. | DOI | MR | Zbl

L. Hascoët and V. Pascual, The Tapenade Automatic Differentiation tool: principles, model, and specification. Rapport de recherche RR-7957, INRIA (2012). | MR | Zbl

D.H. Jacobson, M.M. Lele and J.L. Speyer, New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints. J. Math. Anal. Appl. 35 (1971) 255–284. | DOI | MR | Zbl

J. Gergaud and T. Haberkorn, Homotopy Method for minimum consumption orbit transfer problem. ESAIM: COCV 12 (2006) 294–310. | Numdam | MR | Zbl

B. De Jager, T. Van Keulen and J. Kessels, Optimal Control of Hybrid Vehicles, in Advances in Industrial Control. Springer Verlag (2013). | MR

S. Jan, Minimizing the fuel consumption of a vehicle from the Shell Eco-marathon: a numerical study. ESAIM: COCV 19 (2013) 516–532. | Numdam | MR | Zbl

C. Kirches, H. Bock, J. Schlöder and S. Sager,Mixed-integer NMPC for predictive cruise control of heavy-duty trucks, In Europ. Control Conf. Zurich. Switzerland (2013) 4118–4123.

I. Kupka, Geometric theory of extremals in optimal control problems. i. the fold and maxwell case. Trans. Amer. Math. Soc. 299 (1987) 225–243. | MR | Zbl

H. Maurer, On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim. 15 (1971) 345–362. | DOI | MR | Zbl

A. Merakeb, F. Messine and M. Aidène, A Branch and Bound algorithm for minimizing the energy consumption of an electrical vehicle. 4OR 12 (2014) 261–283. | DOI | MR | Zbl

J.J. Moré, B.S. Garbow and K.E. Hillstrom, User Guide for MINPACK-1, ANL-80-74, Argonne National Laboratory (1980).

S. Sager, M. Claeys and F. Messine, Efficient upper and lower bounds for global mixed-integer optimal control. J. Global Optimiz. 61 (2015) 721–743. | DOI | MR | Zbl

H. Schättler and U. Ledzewicz, Geometric optimal control: theory, methods and examples. Vol 38 of Interdisciplinary applied mathematics. Springer Science and Business Media, New York (2012) xiv+640. | MR | Zbl

A. Sciarretta and L. Guzzella, Control of hybrid electric vehicles. IEEE Control Syst. Mag. 27 (2007) 60–70. | DOI