Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Coent, Adrien Camille le; Fribourg, Laurent

doi:10.1007/978-3-030-41131-2_9

Cited by 5 publications

(6 citation statements)

References 44 publications

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“…The optimization task is to find a control pattern π P U k which guarantees that all states in a given set S " r0, 1s n Ă R n 2 are steered at time t end " kτ as closely as possible to an end state y end P S. Let us explain the principle of the method based on DP and Euler integration method used in [CF19a;CF19b]. We consider the cost function: J k : S Û k Ñ R ě0 defined by: J k py, πq " }Y π y pkτ q ´yend }, where } ¨} denotes the Euclidean norm in R n3…”

Section: B Finite Horizon and Dynamic Programmingmentioning

confidence: 99%

“…We then discretize the space S by means of a grid X such that any point y 0 P S has an "ε-representative" z 0 P X with }y 0 ´z0 } ď ε, for a given value ε ą 0. As explained in [CF19b], it is easy to construct via DP a procedure PROC ε k which, for any y P S, takes its representative z P X as input, and returns a pattern π ε k P U k corresponding to an approximate optimal value of v k pyq.…”

Section: B Finite Horizon and Dynamic Programmingmentioning

confidence: 99%

“…We keep the idea of "tube" used in [Hou+09; SHD12; Ste+12], but we make use of recent results related to approximate solutions by Euler's method (see [CF19a;CF19b]). Our method makes a preliminary use of a dynamic programming (DP) method for generating a finite sequence of control π which solves a finite horizon optimal problem in the absence of perturbation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Robust optimal periodic control using guaranteed Euler's method

Jerray,

Fribourg,

André

2021

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In this paper, we consider the application of optimal periodic control sequences to switched dynamical systems. The control sequence is obtained using a finite-horizon optimal method based on dynamic programming. We then consider Euler approximate solutions for the system extended with bounded perturbations. The main result gives a simple condition on the perturbed system for guaranteeing the existence of a stable limit cycle of the unperturbed system. An illustrative numerical example is provided which demonstrates the applicability of the method.

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Section: B Finite Horizon and Dynamic Programmingmentioning

confidence: 99%

Section: B Finite Horizon and Dynamic Programmingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Robust optimal periodic control using guaranteed Euler's method

Jerray,

Fribourg,

André

2021

Preprint

Self Cite

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“…The optimization task is to find a control pattern π P U k which guarantees that all states in a given set S " r0, 1s M Ă R M 1 are steered at time t end " kτ as closely as possible to an end state y end P S. Let us explain the principle of the method based on DPP and Euler integration method used in [CF19a;CF19b]. We consider the cost function: J k : S ˆU k Ñ R ě0 defined by:…”

Section: Finite Horizon Control Problemsmentioning

confidence: 99%

Robust optimal control using dynamic programming and guaranteed Euler's method

Jerray,

Fribourg,

André

2020

Preprint

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Set-based integration methods allow to prove properties of differential systems, which take into account bounded disturbances. The systems (either time-discrete, time-continuous or hybrid) satisfying such properties are said to be "robust". In the context of optimal control synthesis, the set-based methods are generally extensions of numerical optimal methods of two classes: first, methods based on convex optimization; second, methods based on the dynamic programming principle. Heymann et al. have recently shown that, for certain systems of low dimension, the second numerical method can give better solutions than the first one. They have built a solver (Bocop) that implements both numerical methods. We show in this paper that a set-based extension of a method of the second class which uses a guaranteed Euler integration method, allows us to find such good solutions. Besides, these solutions enjoy the property of robustness against uncertainties on initial conditions and bounded disturbances. We demonstrate the practical interest of our method on an example taken from the numerical Bocop solver. We also give a variant of our method, inspired by the method of Model Predictive Control, that allows us to find more efficiently an optimal control at the price of losing robustness.

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“…We now give an upper bound to the error between the exact solution of the ODE and its Euler approximation on S (see [10,9]). Definition 1.…”

Section: Explicit Euler Time Integration and Error Boundsmentioning

confidence: 99%

ORBITADOR: A tool to analyze the stability of periodical dynamical systems

Jerray¹

EPiC Series in Computing

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Tool Presentation: We present ORBITADOR, a tool for stability analysis of dynamical systems. ORBITADOR uses a method that generates a bounded invariant set of a differential system with a given set of initial conditions around a point x0 to prove the existence of a limit cycle. This invariant has the form of a tube centered on the Euler approximate solution starting at x0, which has for radius an upper bound on the distance between the approximate solution and the exact ones. The method consists in finding a real T > 0 such that the “snapshot” of the tube at time t = (i+1)T is included in the snapshot at t = iT , for some integer i with adding a small bounded uncertainty. This uncertainty allows using an approximate value T of the exact period. We successfully applied ORBITADOR to several classical examples of periodical systems.

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Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Cited by 5 publications

References 44 publications

Robust optimal periodic control using guaranteed Euler's method

Robust optimal periodic control using guaranteed Euler's method

Robust optimal control using dynamic programming and guaranteed Euler's method

ORBITADOR: A tool to analyze the stability of periodical dynamical systems

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