  In this paper, 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 offline using a single linear program, existing
approaches suffer from a computation space complexity that grows exponentially
with the state dimension. Hence, the motivation for incremental abstraction is
to reduce the space complexity for high-dimensional systems, but at the cost of
yielding potentially worse abstractions/overapproximations. Specifically, we
start with an operating region that is a subregion of the state space and
compute two 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
numerical examples of high-dimensional nonlinear systems.
