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 <italic>proportional-integral projected gradient method</italic>, 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 <inline-formula> <tex-math notation="LaTeX">$O{(}1/k{)}$ </tex-math></inline-formula> if the objective function is convex, where <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is the iteration number. If the objective function is strongly convex, this rate can be improved to <inline-formula> <tex-math notation="LaTeX">$O{(}1/k^{2}{)}$ </tex-math></inline-formula> for the distance to optimum and <inline-formula> <tex-math notation="LaTeX">$O{(}1/k^{3}{)}$ </tex-math></inline-formula> for the constraint violation. We compare our method against existing methods via a trajectory-planning example with convexified keep-out-zone constraints.

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