  Temporal logic has been widely used to express complex task specifications
for cyber-physical systems (CPSs). One way to synthesize a controller for CPS
under temporal logic constraints is to first abstract the CPS as a discrete
transition system, and then apply formal methods. This approach, however, is
computationally demanding and its scalability suffers due to the curse of
dimensionality. In this paper, we propose a control barrier function (CBF)
approach to abstraction-free control synthesis under a linear temporal logic
(LTL) constraint. We first construct the deterministic Rabin automaton of the
specification and compute an accepting run. We then compute a sequence of LTL
formulae, each of which must be satisfied during a particular time interval,
and prove that satisfying the sequence of formulae is sufficient to satisfy the
LTL specification. Finally, we compute a control policy for satisfying each
formula by constructing an appropriate CBF. We present a quadratic program to
compute the controllers, and show the controllers synthesized using the
proposed approach guarantees the system to satisfy the LTL specification,
provided the quadratic program is feasible at each time step. A numerical case
study is presented to demonstrate the proposed approach.
