The Adjoint Method for Time-Optimal Control Problems

Journal of Computational and Nonlinear Dynamics - Tập 16 Số 2 - 2021
Philipp Eichmeir1, Thomas Lauß2, Stefan Oberpeilsteiner2, Karin Nachbagauer2, Wolfgang Steiner2
1Institute of Mechanics and Mechatronics, Vienna University of Technology, Getreidemarkt 9/E325, Wien 1060, Austria; Josef Ressel Center for Advanced Multibody Dynamics, University of Applied Sciences Upper Austria, Stelzhamerstraße 23, Wels 4600, Austria
2Josef Ressel Center for Advanced Multibody Dynamics, University of Applied Sciences Upper Austria, Stelzhamerstraße 23, Wels 4600, Austria

Tóm tắt

Abstract In this article, we discuss a special class of time-optimal control problems for dynamic systems, where the final state of a system lies on a hyper-surface. In time domain, this endpoint constraint may be given by a scalar equation, which we call transversality condition. It is well known that such problems can be transformed to a two-point boundary value problem, which is usually hard to solve, and requires an initial guess close to the optimal solution. Hence, we propose a new gradient-based iterative solution strategy instead, where the gradient of the cost functional, i.e., of the final time, is computed with the adjoint method. Two formulations of the adjoint method are presented in order to solve such control problems. First, we consider a hybrid approach, where the state equations and the adjoint equations are formulated in time domain but the controls and the gradient formula are transformed to a spatial variable with fixed boundaries. Second, we introduce an alternative approach, in which we carry out a complete elimination of the time coordinate and utilize a formulation in the space domain. Both approaches are robust with respect to poor initial controls and yield a shorter final time and, hence, an improved control after every iteration. The presented method is tested with two classical examples from satellite and vehicle dynamics. However, it can also be extended to more complex systems, which are used in industrial applications.

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Journal of Computational and Nonlinear Dynamics

Tập 16 Số 2

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