On the Application of Recursive Nonlinear State Transformations and Nonlinear Feedback to Enable Continuous Thrust Spacecraft Orbital Rendezvous

This dissertation presents a method to address exact feedback linearization problems based on the recursive application of approximate feedback linearization techniques. An approximate feedback linearization method is applied recursively to systems which are known to be an exact feedback linearizable to obtain a family of exact solutions up to the ρ-th degree utilizing the null space that appears as part of the computations. The coordinate transformation and feedback parameters are computed symbolically using a one-step approach in multi-stage form. This algorithm is algebraic and computationally simpler than solving the set of nonlinear partial differential equations. We employ the approximate feedback linearization method to address the problem of circular orbit transfer to establish a family of exact solution up to the ρ-th degree for continuous thrust circular orbit transfers. During the recursive steps, patterns were detected in the approximate solutions as they evolved illuminating a family of exact solutions to the circular orbit nonlinear feedback control problem utilizing the null space. It is shown that applying higher-degree feedback improves the closed-loop system stability for the orbital transfer problem. The relationship between the ρ-th degree exact solution obtained through the recursive approximations and a known exact solution is illustrated where it is shown that two different exact solutions can have different performance in terms of fuel usage leading to the possibility of optimization considerations in selecting the desired exact solution. We also apply the approximate linearization method to the more realistic elliptical orbit transfer problem and through simulation established closed-loop system stability