Proportional-Integral Projected Gradient Method for Model Predictive Control

Recently there has been an increasing interest in primal-dual methods for model predictive control (MPC), which require minimizing the (augmented) Lagrangian at each iteration. We propose a novel first order primal-dual method, termed \emph{proportional-integral projected gradient method}, for MPC where the underlying finite horizon optimal control problem has both state and input constraints. Instead of minimizing the (augmented) Lagrangian, each iteration of our method only computes a single projection onto the state and input constraint set. Our method ensures that, along a sequence of averaged iterates, both the distance to optimum and the constraint violation converge to zero at a rate of $O(1/k)$ if the objective function is convex, where $k$ is the iteration number. If the objective function is strongly convex, this rate can be improved to $O(1/k^2)$ for the distance to optimum and $O(1/k^3)$ for the constraint violation. We compare our method against existing methods via a trajectory-planning example with convexified keep-out-zone constraints.