Analytical and Numerical Methods for Optimal Control Problems on Manifolds and Lie Groups.

Optimal Control;Aerospace Engineering;Engineering;Aerospace Engineering

Gupta, RohitBernstein, Dennis S ;

University of Michigan

关键词: Optimal Control;    Aerospace Engineering;    Engineering;    Aerospace Engineering;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/120767/rohitgpt_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The dissertation extends some of the existing analytical and numerical methods for optimal control problems (OCPs) on manifolds and Lie groups. The dissertation is divided into four parts. In the first part of the dissertation, we develop a method for obtaining sub-optimal control in OCPs defined on a Euclidean space using the combination of two techniques, homotopy and neighboring extremal optimal control (NEOC). The idea is to start with a simpler problem by creating a homotopy on the dynamic constraint and then using NEOC to iteratively update the optimal control as the homotopy parameter changes. In the second part of the dissertation, we develop a numerical solver for nonlinear model predictive control (NMPC) of spacecraft attitude. The numerical solver is based on solving the necessary conditions for optimality in a discrete-time OCP defined over each prediction horizon, where the discrete-time spacecraft dynamics are based on the Lie group variational integrator (LGVI) model. In the third part of the dissertation, we extend NEOC, which is well established for OCPs defined on a Euclidean space, to the setting of Riemannian manifolds. In particular, we consider an OCP for a class of mechanical systems on Riemannian manifolds. In the fourth part of the dissertation, we investigate the reduction for OCPs on Lie groups with symmetry breaking cost functions. Reduction is an indispensable tool in the study of Lagrangian/Hamiltonian systems (which include OCPs), as it allows the dynamics associated with the Lagrangian/Hamiltonian to be described on a quotient space, e.g., in the case of a Lie group G, the dynamics associated with a G-invariant Lagrangian/Hamiltonian can be described on the Lie algebra g/dual space of g (g*) instead of the tangent bundle TG/cotangent bundle T*G. From the Lagrangian point of view, we obtain the Euler-Poincare equations and from the Hamiltonian point of view, we obtain the Lie-Poisson equations. We also study the relationship between both formalisms. A variational integrator for OCPs on Lie groups with symmetry breaking cost functions is also developed.