  Safety is one of the fundamental challenges in control theory. Recently,
multi-step optimal control problems for discrete-time dynamical systems were
formulated to enforce stability, while subject to input constraints as well as
safety-critical requirements using discrete-time control barrier functions
within a model predictive control (MPC) framework. Existing work usually focus
on the feasibility or the safety for the optimization problem, and the majority
of the existing work restrict the discussions to relative-degree one control
barrier functions. Additionally, the real-time computation is challenging when
a large horizon is considered in the MPC problem for relative-degree one or
high-order control barrier functions. In this paper, we propose a framework
that solves the safety-critical MPC problem in an iterative optimization, which
is applicable for any relative-degree control barrier functions. In the
proposed formulation, the nonlinear system dynamics as well as the safety
constraints modeled as discrete-time high-order control barrier functions
(DHOCBF) are linearized at each time step. Our formulation is generally valid
for any control barrier function with an arbitrary relative-degree. The
advantages of fast computational performance with safety guarantee are analyzed
and validated with numerical results.
