  Quadratic Program(QP) based state-feedback controllers, whose inequality
constraints bound the rate of change of control barrier(CBFs) and lyapunov
function with a class-$\mathcal{K}$ function of their values, are sensitive to
the parameters of these class-$\mathcal{K}$ functions. The construction of
valid CBFs, however, is not straightforward, and for arbitrarily chosen
parameters of the QP, the system trajectories may enter states at which the QP
either eventually becomes infeasible, or may not achieve desired performance.
In this work, we pose the control synthesis problem as a differential policy
whose parameters are optimized for performance over a time horizon at high
level, thus resulting in a bi-level optimization routine. In the absence of
knowledge of the set of feasible parameters, we develop a Recursive Feasibility
Guided Gradient Descent approach for updating the parameters of QP so that the
new solution performs at least as well as previous solution. By considering the
dynamical system as a directed graph over time, this work presents a novel way
of optimizing performance of a QP controller over a time horizon for multiple
CBFs by (1) using the gradient of its solution with respect to its parameters
by employing sensitivity analysis, and (2) backpropagating these as well as
system dynamics gradients to update parameters while maintaining feasibility of
QPs.
