• Corpus ID: 117076969

Higher order variational time discretization of optimal control problems

@article{Campos2012HigherOV,
  title={Higher order variational time discretization of optimal control problems},
  author={C{\'e}dric M. Campos and Oliver Junge and Sina Ober-Blobaum},
  journal={arXiv: Optimization and Control},
  year={2012}
}
We reconsider the variational integration of optimal control problems for mechanical systems based on a direct discretization of the Lagrange-d'Alembert principle. This approach yields discrete dynamical constraints which by construction preserve important structural properties of the system, like the evolution of the momentum maps or the energy behavior. Here, we employ higher order quadrature rules based on polynomial collocation. The resulting variational time discretization decreases the… 

Tables from this paper

High order variational integrators in the optimal control of mechanical systems
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class
Higher Order Variational Integrators for Multibody System Dynamics with Constraints
The continuous and discrete Euler-Lagrangian equations with holonomic constraints are presented based on continuous and discrete Hamiltonian Principle. Using Lagrangian polynomial to interpolate
Higher Order Variational Integrators: a polynomial approach
TLDR
A family of variational integrators are developed which are referred to as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta schemes, which are easily applicable to optimal control problems based on mechanical systems.
The variational discretizaton of the constrained higher-order Lagrange-Poincaré equations
TLDR
It is studied how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space).
Construction and analysis of higher order Galerkin variational integrators
TLDR
This work derives and analyzes variational integrators of higher order for the structure-preserving simulation of mechanical systems and investigates which combination of space of polynomials and quadrature rules provide optimal convergence rates.
Variational integrators for mechanical control systems with symmetries
TLDR
The variational formalism for the class of underactuated mechanicalControl systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems are discussed.
Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators
  • O. Kosmas
  • Mathematics, Computer Science
    Applied Mechanics
  • 2021
TLDR
This work introduces different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials of split potential systems, namely, to address cases when the potential function can be decomposed into several components.
Optimality Conditions (in Pontryagin Form)
This chapter aims at being a friendly presentation of various results related to optimality conditions of Optimal Control problems. Different classes of systems are considered, such as equations with
Methods for the Design and Development
TLDR
This chapter introduces new system optimization and design methods to develop reconfigurations of the software and the microelectronics, and proposes new testing methods and formal methods to ensure safety-properties of theSoftware.

References

SHOWING 1-9 OF 9 REFERENCES
Discrete mechanics and variational integrators
This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of
DISCRETE MECHANICS AND OPTIMAL CONTROL: AN ANALYSIS ∗
TLDR
The DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and this fact is employed in order to give a proof of convergence.
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how
Introduction to Model Based Optimization of Chemical Processes on Moving Horizons
TLDR
This contribution provides a concise introduction into problem formulation and standard numerical techniques commonly found in the context of moving horizon optimization using nonlinear differential algebraic process models.
Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
The C++ package ADOL-C described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting
On the construction and analysis of higher order variational integrators
  • Diploma thesis, Paderborn,
  • 2012
Integrable systems with discrete time, and difference operators. Funktsional
  • Anal. i Prilozhen., 22(2):1–13,
  • 1988
Integrable systems with discrete time , and difference operators
  • Funktsional . Anal . i Prilozhen
  • 1988
Hamiltonian methods of Runge-Kutta type and their variational interpretation
  • Mat. Model.,
  • 1990