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

Cots, Olivier

doi:10.1051/cocv/2016070

Cited by 9 publications

(9 citation statements)

References 23 publications

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“…Many examples can be found in mechanics and aerospace engineering (e.g., an engine may overheat or overload). We refer to [11,25,43,61,62] and references therein for other examples. State constrained optimal control problems are also important in management and economics (e.g., an inventory level may be limited in a production model).…”

Section: Introductionmentioning

confidence: 99%

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Bourdin

Dhar

2020

Math. Program.

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Sampled-data control systems have steadily been gaining interest for their applications in Engineering and Automation. In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. In particular we obtain a nonpositive averaged Hamiltonian gradient condition associated to an adjoint vector being a function of bounded variations. As a well-known challenge, theoretical and numerical difficulties may arise in optimal permanent control problems with state constraints due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the trajectory in contact with the boundary of the restricted state space). However, in our case of sampled-data controls, we find that, under certain general hypotheses, the optimal trajectory only contacts the running inequality state constraints at most at the sampling times. In that context the adjoint vector only experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Hence, taking advantage of this bouncing trajectory phenomenon, we are able to implement a numerical indirect method which we use to solve three simple examples.

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Section: Introductionmentioning

confidence: 99%

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Bourdin

Dhar

2020

Math. Program.

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“…Likewise the singular case, in the state constrained case, we have the following result excerpted from [17,Prop. 4.5] which is useful to define the numerical shooting method, see Section 4.1.…”

Section: Parameterization Of the Boundary Extremalsmentioning

confidence: 98%

“…Another contribution of this article is to analyze the influence of the bounds ϕ max and ψ max on the structure of the trajectories for the minimum time-to-climb problem. We propose a methodology introduced in [17], based on differential homotopy [12] and geometry techniques [10] to classify the different structures with respect to ϕ max and ψ max . Remarkably, the CAS/MACH procedure appears in our classification.…”

Section: ')mentioning

confidence: 99%

Singular versus boundary arcs for aircraft trajectory optimization in climbing phase

Cots¹,

Gergaud²,

Goubinat³

et al. 2023

ESAIM: M2AN

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In this article, we are interested in optimal aircraft trajectories in climbing phase. We consider the cost index criterion which is a convex combination of the time-to-climb and the fuel consumption. We assume that the thrust is constant and we control the air slope of the aircraft. This optimization problem is modeled as a Mayer optimal control problem with a single-input affine dynamics in the control and with two pure state constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The candidates as minimizers are selected among a set of extremals given by the maximum principle. We first analyze the minimum time-to-climb problem with respect to the bounds of the state constraints, combining small time analysis, indirect multiple shooting and homotopy methods with monitoring. This investigation emphasizes two strategies: the common CAS/Mach procedure in aeronautics and the classical Bang-Singular-Bang policy in control theory. We then compare these two procedures for the cost index criterion.

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“…Many applications can be found in mechanics and aerospace engineering, management and economics, etc. We refer to [14,29,56,63,70] and references therein for examples. A version of the PMP for state constrained continuous-time optimal control problems was given by Gamkrelidze et al (see, e.g., [38] and [61,Theorem 25 p. 311]).…”

Section: State Constrained Optimal Control Problemsmentioning

confidence: 99%

Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales

Bettiol

Bourdin

2021

ESAIM: COCV

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In this paper we consider optimal sampled-data control problems on time scales with inequality state constraints. A Pontryagin maximum principle is established, extending to the state constrained case existing results in the time scale literature. The proof is based on the Ekeland variational principle and on the concept of implicit spike variations adapted to the time scale setting. The main result is then applied to continuous-time min-max optimal sampled-data control problems and a maximal velocity minimization problem for the harmonic oscillator with sampled-data control is numerically solved for illustration.

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

Cited by 9 publications

References 23 publications

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Singular versus boundary arcs for aircraft trajectory optimization in climbing phase

Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales

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