Online Constraint Tightening in Stochastic Model Predictive Control: A
  Regression Approach
Alexandre Capone Tim Brüdigam Sandra Hirche
Solving chance-constrained stochastic optimal control problems is a
significant challenge in control. This is because no analytical solutions exist
for up to a handful of special cases. A common and computationally efficient
approach for tackling chance-constrained stochastic optimal control problems
consists of reformulating the chance constraints as hard constraints with a
constraint-tightening parameter. However, in such approaches, the choice of
constraint-tightening parameter remains challenging, and guarantees can mostly
be obtained assuming that the process noise distribution is known a priori.
Moreover, the chance constraints are often not tightly satisfied, leading to
unnecessarily high costs. This work proposes a data-driven approach for
learning the constraint-tightening parameters online during control. To this
end, we reformulate the choice of constraint-tightening parameter for the
closed-loop as a binary regression problem. We then leverage a highly
expressive \gls{gp} model for binary regression to approximate the smallest
constraint-tightening parameters that satisfy the chance constraints. By tuning
the algorithm parameters appropriately, we show that the resulting
constraint-tightening parameters satisfy the chance constraints up to an
arbitrarily small margin with high probability. Our approach yields
constraint-tightening parameters that tightly satisfy the chance constraints in
numerical experiments, resulting in a lower average cost than three other
state-of-the-art approaches.