Lyapunov Differential Equation Approach to Elliptical Orbital Rendezvous with Constrained Controls

Abstract

and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner-Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controllersemigloballystabilizesthesystem.Equivalently,forany fixedinitialconditions,themagnitudeandenergy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations,thenewapproachrequiresonlythesolutionoflinearperiodicLyapunovdifferentialequations,whichare investigated inthepaperbyusingtheperiodicgenerator approach.Numericalsimulationsofthe nonlinearmodelof the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily.