  This paper presents two new control approaches for guaranteed safety
(remaining in a safe set) subject to actuator constraints (the control is in a
convex polytope). The control signals are computed using real-time
optimization, including linear and quadratic programs subject to affine
constraints, which are shown to be feasible. The first control method relies on
a soft-minimum barrier function that is constructed using a finite-time-horizon
prediction of the system trajectories under a known backup control. The main
result shows that the control is continuous and satisfies the actuator
constraints, and a subset of the safe set is forward invariant under the
control. Next, we extend this method to allow from multiple backup controls.
This second approach relies on a combined soft-maximum/soft-minimum barrier
function, and it has properties similar to the first. We demonstrate these
controls on numerical simulations of an inverted pendulum and a nonholonomic
ground robot.
