Proportional-Integral Projected Gradient Method for Model Predictive Control

TL;DR

This paper introduces a novel primal-dual gradient method for model predictive control that efficiently handles constraints by using projections, achieving faster convergence rates than existing methods under convexity assumptions.

Contribution

The proposed proportional-integral projected gradient method simplifies computations by avoiding Lagrangian minimization and guarantees improved convergence rates for constrained MPC problems.

Findings

Convergence rate of O(1/k) for convex objectives

Enhanced rates of O(1/k^2) and O(1/k^3) for strongly convex objectives

Successful comparison with existing methods on a trajectory-planning example

Abstract

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.