  Model mismatches prevail in real-world applications. Ensuring safety for
systems with uncertain dynamic models is critical. However, existing robust
safe controllers may not be realizable when control limits exist. And existing
methods use loose over-approximation of uncertainties, leading to conservative
safe controls. To address these challenges, we propose a control-limits aware
robust safe control framework for bounded state-dependent uncertainties. We
propose safety index synthesis to find a robust safe controller guaranteed to
be realizable under control limits. And we solve for robust safe control via
Convex Semi-Infinite Programming, which is the tightest formulation for convex
bounded uncertainties and leads to the least conservative control. In addition,
we analyze when and how safety can be preserved under unmodeled uncertainties.
Experiment results show that our robust safe controller is always realizable
under control limits and is much less conservative than strong baselines.
