View in EDS

Minimal-time nonlinear control via semi-infinite programming

Saved in:
Bibliographic Details
Title: Minimal-time nonlinear control via semi-infinite programming
Authors: Oustry, Antoine, Tacchi, Matteo
Contributors: Optimization at LIX (OptimiX), Laboratoire d'informatique de l'École polytechnique Palaiseau (LIX), École polytechnique (X), Institut Polytechnique de Paris (IP Paris)-Institut Polytechnique de Paris (IP Paris)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Institut Polytechnique de Paris (IP Paris)-Institut Polytechnique de Paris (IP Paris)-Centre National de la Recherche Scientifique (CNRS), École nationale des ponts et chaussées (ENPC), GIPSA - Modelling and Optimal Decision for Uncertain Systems (GIPSA-MODUS), GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA)
Source: https://hal.science/hal-04145402 ; 2025.
Publisher Information: CCSD
Publication Year: 2025
Collection: Université Grenoble Alpes: HAL
Subject Terms: Nonlinear control, Minimal time control, Weak formulation, Semi-infinite programming, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Description: We address the problem of computing a control for a time-dependent nonlinear system to reach a target set in a minimal time. To solve this minimal time control problem, we introduce a hierarchy of linear semi-infinite programs, the values of which converge to the value of the control problem. These semi-infinite programs are increasing restrictions of the dual of the nonlinear control problem, which is a maximization problem over the subsolutions of the Hamilton-Jacobi-Bellman (HJB) equation. Our approach is compatible with generic dynamical systems and state constraints. Specifically, we use an oracle that, for a given differentiable function, returns a point at which the function violates the HJB inequality. We solve the semi-infinite programs using a classical convex optimization algorithm with a convergence rate of O(1/k), where k is the number of calls to the oracle. This algorithm yields subsolutions of the HJB equation that approximate the value function and provide a lower bound on the optimal time. We study the closed-loop control built on the obtained approximate value functions, and we give theoretical guarantees on its performance depending on the approximation error for the value function. We show promising numerical results for three non-polynomial systems with up to 6 state variables and 5 control variables.
Document Type: report
Language: English
Relation: info:eu-repo/semantics/altIdentifier/arxiv/2307.00857; ARXIV: 2307.00857
Availability: https://hal.science/hal-04145402
https://hal.science/hal-04145402v2/document
https://hal.science/hal-04145402v2/file/oustry_tacchi_mintimecontrol_230706.pdf
Rights: http://creativecommons.org/licenses/by/ ; info:eu-repo/semantics/OpenAccess
Accession Number: edsbas.CD32D3BC
Database: BASE
Description
Abstract:We address the problem of computing a control for a time-dependent nonlinear system to reach a target set in a minimal time. To solve this minimal time control problem, we introduce a hierarchy of linear semi-infinite programs, the values of which converge to the value of the control problem. These semi-infinite programs are increasing restrictions of the dual of the nonlinear control problem, which is a maximization problem over the subsolutions of the Hamilton-Jacobi-Bellman (HJB) equation. Our approach is compatible with generic dynamical systems and state constraints. Specifically, we use an oracle that, for a given differentiable function, returns a point at which the function violates the HJB inequality. We solve the semi-infinite programs using a classical convex optimization algorithm with a convergence rate of O(1/k), where k is the number of calls to the oracle. This algorithm yields subsolutions of the HJB equation that approximate the value function and provide a lower bound on the optimal time. We study the closed-loop control built on the obtained approximate value functions, and we give theoretical guarantees on its performance depending on the approximation error for the value function. We show promising numerical results for three non-polynomial systems with up to 6 state variables and 5 control variables.