  To deal with stochastic hybrid systems with general state-dependent
switching, we propose an approximation method by a sequence of stochastic
hybrid systems with piecewise constant type switching. The convergence rate in
the Wasserstein distance is estimated in terms of the difference between
transition rate matrices. Our method is based on an elaborate construction of
coupling processes in terms of Skorokhod's representation theorem for jumping
processes. Moreover, we establish explicit criteria on the ergodicity and
stability for stochastic hybrid systems with piecewise constant type switching.
Some examples are given to illustrate the sharpness of these criteria.
