Incremental Affine Abstraction of Nonlinear Systems

In this letter, we propose an incremental abstraction method for dynamically over-approximating nonlinear systems in a bounded domain by solving a sequence of linear programs, resulting in a sequence of affine upper and lower hyperplanes with expanding operating regions. Although the affine abstraction problem can be solved using a single linear program, existing approaches suffer from a computation space complexity that grows exponentially with the state dimension. Thus, the motivation for incremental abstraction is to reduce the space complexity of abstraction algorithms for high-dimensional systems or systems with limited on-board resources. Specifically, we start with an operating region that is a subregion of the state space and compute a pair of affine hyperplanes that bracket the nonlinear function locally. Then, by incrementally expanding the operating region, we dynamically update the two affine hyperplanes such that we eventually yield hyperplanes that are guaranteed to over-approximate the nonlinear system over the entire domain. Finally, the effectiveness of the proposed approach is demonstrated using several numerical examples.