  Temporal Logic (TL) guided control problems have gained interests in recent
years. By using the TL, one can specify a wide range of temporal constraints on
the system and is widely used in cyber-physical systems. On the other hand,
Control Barrier Functions have also gained interests in the context of safety
critical applications. However, most of the existing approaches only focus on
discrete-time dynamical systems. In this paper, we propose an offline
trajectory planner for linear systems subject to safety and temporal
specifications. Such specifications can be expressed as logical junctions or
disjunctions of linear CBFs, or as STL specifications with linear predicates.
Our planner produces trajectories that are valid in continuous time, while
assuming only discrete-time control updates and arbitrary time interval in the
STL formula. Our planner is based on a Mixed Integer Quadratic Programming
(MIQP) formulation, where the linear STL predicates are encoded as set of
linear constraints to guarantee satisfaction at on a finite discrete set of
time instants, while we use CBFs to derive constraints that guarantee
continuous satisfaction between time instants. Moreover, we have shown the
predicates can be encoded as time-based CBF constraints for system with any
relative degrees. We validate our theoretical results and formulation through
numerical simulations.
