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.