The Hamiltonian Structure-Preserving Control and Some Applications to Nonlinear Astrodynamics

Abstract

A systematic research on the structure-preserving controller is investigated in this paper, including its applications to the second-order, first-order, time-periodic, or degenerated astrodynamics, respectively. The general form of the controller is deduced for the typical Hamiltonian system in full feedback and position-only feedback modes, which is successful in changing the hyperbolic equilibrium to an elliptic one. With the poles assigned at any different positions on imaginary axis, the controlled Hamiltonian system is Lyapunov stable. The Floquet multiplier is employed to measure the stability of time-dependent Hamiltonian system, because the equilibrium of periodic system may be unstable even though the equilibrium is always elliptic. One type of periodic orbits is achieved by the resonant conditions of control gains, and another type is making judicious choice in the foundational motions with different frequencies. The control gains are selected from the viewpoint of both the local and global optimizations on fuel cost. This controller is applied to some astrodynamics to achieve some interesting conclusions, including stable lissajous orbits in solar sail’s three-body problem and degenerated two-body problem, quasiperiodic formation flying on a $J_2$-perturbed mean circular orbit, and controlled frozen orbits for a spacecraft with a high area-to-mass ratio.