Finite Horizon Backward Reachability Analysis and Control Synthesis for Uncertain Nonlinear Systems

We present a method for synthesizing controllers to steer trajectories from an initial set to a target set on a finite time horizon. The proposed control synthesis problem is decomposed into two steps. The first step under-approximates the backward reachable set (BRS) from the target set, using level sets of storage functions. The storage function is constructed with an iterative algorithm to maximize the volume of the under-approximated BRS. The second step obtains a control law by solving a pointwise min-norm optimization problem using the pre-computed storage function. A closed-form solution of this min-norm optimization can be computed through the KKT conditions. This control synthesis framework is then extended to uncertain nonlinear systems with parametric uncertainties and L2 disturbances. The computation algorithm for all cases is derived using sum-of-squares (SOS) programming and the S-procedure. The proposed method is applied to several examples, including robotic systems.