Backward Reachability for Polynomial Systems on a Finite Horizon

A method is presented to obtain an inner-approximation of the backward reachable set a given target tube, along with admissible controller that maintains trajectories inside this tube. The proposed optimization algorithms are formulated as nonlinear problems, which decoupled into tractable subproblems and solved by iterative algorithm using polynomial S-procedure sum-of-squares techniques. This framework also extended uncertain systems $\mathcal {L}_2$ disturbances {L}_{\infty }$ parametric uncertainties. effectiveness demonstrated on several robotics aircraft control saturation.