Optimal satellite attitude control: a geometric approach

Optimal nonlinear control remains one of the most challenging subjects in control theory despite a long research history. In this paper, we present a geometric optimal control approach, which circumvents the tedious task of numerically solving online the Hamilton Jacobi Bellman (HJB) partial differential equation, which represents the dynamic programming formulation of the nonlinear global optimal control problem. Our approach makes implementation of nonlinear optimal attitude control practically feasible with low computational demand onboard a satellite. Optimal stabilizing state feedbacks are obtained from the construction of a control Lyapunov function. Based on a phase space analysis, two natural dual optimal control objectives are considered to illustrate the application of this approach to satellite attitude control: Minimizing the norm of the control torque subject to a constraint on the convergence rate of a Lyapunov function, then maximizing the convergence rate of a Lyapunov function subject to a constraint on the control torque. Both approaches provide ease of implementation and achieve robust optimal trade-offs between attitude control rapidity and torque expenditure, without computational issues.