  We address the problem of optimizing the performance of a dynamic system
while satisfying hard safety constraints at all times. Implementing an optimal
control solution is limited by the computational cost required to derive it in
real time, especially when constraints become active, as well as the need to
rely on simple linear dynamics, simple objective functions, and ignoring noise.
The recently proposed Control Barrier Function (CBF) method may be used for
safety-critical control at the expense of sub-optimal performance. In this
paper, we develop a real-time control framework that combines optimal
trajectories generated through optimal control with the computationally
efficient CBF method providing safety guarantees. We use Hamiltonian analysis
to obtain a tractable optimal solution for a linear or linearized system, then
employ High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) to
account for constraints with arbitrary relative degrees and to track the
optimal state, respectively. We further show how to deal with noise in
arbitrary relative degree systems. The proposed framework is then applied to
the optimal traffic merging problem for Connected and Automated Vehicles (CAVs)
where the objective is to jointly minimize the travel time and energy
consumption of each CAV subject to speed, acceleration, and speed-dependent
safety constraints. In addition, when considering more complex objective
functions, nonlinear dynamics and passenger comfort requirements for which
analytical optimal control solutions are unavailable, we adapt the HOCBF method
to such problems. Simulation examples are included to compare the performance
of the proposed framework to optimal solutions (when available) and to a
baseline provided by human-driven vehicles with results showing significant
improvements in all metrics.
