Backward Reachability for Polynomial Systems on a Finite Horizon

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