Nonlinear Optimal Approach to Spacecraft Formation Flying Using Lorentz and Impulsive Actuation

This paper proposes an optimal control design framework for hybrid nonlinear time-dependent dynamical systems involving an interacting mixture of continuous-time and discrete-time dynamics. Aiming to extend hybrid linear optimal control synthesis to nonlinear non-quadratic formulation, a hybrid version of the Hamilton–Jacobi–Bellman (HJB) equation with time-dependency is first presented. A hybrid computational framework, which alternates between continuous-time and discrete-time subsystems at an appropriate sequence of time instants, is then developed to solve the resultant set of the HJB equations in an interacting manner. The proposed control scheme is subsequently applied to spacecraft formation flying establishment with a hybrid source of actuation, namely Lorentz force and impulsive thrusting. By optimal combination of the Lorentz force and impulsive thrusting proposed in this paper, not only are the uncontrollability issues pertinent inherently to the Lorentz-actuated control systems effectively resolved, but use of both Lorentz and impulsive control inputs is also optimized. As a consequence, the expendable chemical fuels required to actuate thrusters are significantly reduced, thereby extending the duration of such space missions.