  Controller synthesis, including reset controller, feedback controller, and
switching logic controller, provides an essential mechanism to guarantee the
correctness and reliability of hybrid systems in a correct-by-construction
manner. Unfortunately, reset controller synthesis is still in an infant stage
in the literature, although it makes theoretical and practical significance. In
this paper, we propose a convex programming based method to synthesize reset
controllers for polynomial hybrid systems subject to safety, possibly together
with liveness. Such a problem essentially corresponds to computing an initial
set of continuous states in each mode and a reset map associated with each
discrete jump such that any trajectory starting from any computed initial state
keeps safe if only safety constraints are given or reaches the target set
eventually and keeps safe before that if both safety and liveness are given,
through the computed reset maps. Both cases can be reduced to reach-avoid
and/or differential invariant generation problems, further encoded as convex
optimization problems. Finally, several examples are provided to demonstrate
the efficiency and effectiveness of our method.
