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 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/k²) for the distance to optimum and O(1/k³) for the constraint violation. We compare our method against existing methods via a trajectory-planning example with convexified keep-out-zone constraints.