  Providing an execution time certificate and handling possible infeasibility
in closed-loop are two pressing requirements of Model Predictive Control (MPC).
To simultaneously meet these two requirements, this paper uses an $\ell_1$
penalty-based soft-constrained MPC formulation and innovatively transforms the
resulting non-smooth QP into a box-constrained QP, which is solved by our
previously proposed direct and execution-time certified algorithm with only
dimension-dependent (data-independent), simple-calculated and exact number of
iterations (Wu and Braatz (2023)). This approach not only overcomes the
limitation of our previously proposed algorithm (Wu and Braatz (2023)), only
applicable to input-constrained MPC, but also enjoys exact recovery feature
(exactly recover the same solution when the original problem is feasible) of
$\ell_1$ penalty-based soft-constrained MPC formulation without suffering
numerical difficulty of the resulting non-smoothness. Other various real-time
QP applications, not limited to MPC, would also benefit from our QP algorithm
with execution-time certificate and global feasibility.
