Powered Descent Funnel Control Synthesis Based on the Sum of Squares Programming

Powered descent guidance is an enabling technology for extraterrestrial soft landing and reusable rocket recovery. Although the focus of guidance research has tilted towards trajectory optimization recently, feedback tracking control is still necessary to suppress the effects of flight perturbations and uncertainties. In this paper, the behavior of the powered descent closed-loop system is considered and a feedback control synthesis method based on the funnel (finite-time invariant set) is proposed. Firstly, the basic model of the powered descent problem is given, and the state feedback closed-loop system is constructed with uncertainties of thrust and drag. Subsequently, the invariance condition is described as sum-of-squares constraints, and the backward funnel model is established considering closed-loop thrust saturation constraints. Further, to address the multivariate and non-convex features of the funnel model, a point-by-point solution scheme for funnel control synthesis is formulated integrating the techniques of bilinear search, line search, and heuristic search. The simulation results show that for the considered bounded uncertainties, the states within the funnel inlet will evolve to the specific target set. Compared with LQR guidance, the controller obtained by funnel synthesis has a stronger stabilization capability with a certification, which ensures terminal accuracy while enhancing the adaptability to initial state deviations.