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. We prove that our method ensures convergence to optimal solutions at \(O(1/k)\) and \(O(1/k^2)\) rate if the objective function is convex and, respectively, strongly convex. We demonstrate our method via a trajectory-planning example with convexified keep-out-zone constraints.