  Delays are ubiquitous in modern hybrid systems, which exhibit both continuous
and discrete dynamical behaviors. Induced by signal transmission, conversion,
the nature of plants, and so on, delays may appear either in the continuous
evolution of a hybrid system such that the evolution depends not only on the
present state but also on its execution history, or in the discrete switching
between its different control modes. In this paper we come up with a new model
of hybrid systems, called \emph{delay hybrid automata}, to capture the dynamics
of systems with the aforementioned two kinds of delays. Furthermore, based upon
this model we study the robust switching controller synthesis problem such that
the controlled delay system is able to satisfy the specified safety properties
regardless of perturbations. To the end, a novel method is proposed to
synthesize switching controllers based on the computation of differential
invariants for continuous evolution and backward reachable sets of discrete
jumps with delays. Finally, we implement a prototypical tool of our approach
and demonstrate it on some case studies.
