TY - JOUR

T1 - The adjoint method for time-optimal control problems

AU - Eichmeir, Philipp

AU - Lauß, Thomas

AU - Oberpeilsteiner, Stefan

AU - Nachbagauer, Karin

AU - Steiner, Wolfgang

N1 - open access (Nov. 2020)

PY - 2021/2/1

Y1 - 2021/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85096931547&partnerID=8YFLogxK

U2 - 10.1115/1.4048808

DO - 10.1115/1.4048808

M3 - Article

AN - SCOPUS:85096931547

VL - 16

JO - Journal of Computational and Nonlinear Dynamics

JF - Journal of Computational and Nonlinear Dynamics

SN - 1555-1423

IS - 2

M1 - 021003

ER -