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

TL;DR

This paper introduces a method for finite horizon backward reachability analysis and control synthesis for uncertain nonlinear systems, utilizing level set approximations, sum-of-squares programming, and extending to systems with uncertainties.

Contribution

It develops a novel approach combining level set methods, iterative algorithms, and SOS programming for control synthesis in uncertain nonlinear systems.

Findings

Successfully applied to robotics and aircraft examples.

Provides a closed-form control law via KKT conditions.

Effectively handles parametric uncertainties and disturbances.

Abstract

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 L_2 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 robotics and aircraft examples.