  Piecewise Barrier Tubes (PBT) is a new technique for flowpipe
overapproximation for nonlinear systems with polynomial dynamics, which
leverages a combination of barrier certificates. PBT has advantages over
traditional time-step based methods in dealing with those nonlinear dynamical
systems in which there is a large difference in speed between trajectories,
producing an overapproximation that is time independent. However, the existing
approach for PBT is not efficient due to the application of interval methods
for enclosure-box computation, and it can only deal with continuous dynamical
systems without uncertainty. In this paper, we extend the approach with the
ability to handle both continuous and hybrid dynamical systems with uncertainty
that can reside in parameters and/or noise. We also improve the efficiency of
the method significantly, by avoiding the use of interval-based methods for the
enclosure-box computation without loosing soundness. We have developed a C++
prototype implementing the proposed approach and we evaluate it on several
benchmarks. The experiments show that our approach is more efficient and
precise than other methods in the literature.
