Improved Fast Dual Gradient Methods for Embedded Model Predictive Control

Abstract

Recently, several authors have suggested the use of first order methods, such as fast dual ascent and the alternating direction method of multipliers, for embedded model predictive control. The main reason is that they can be implemented using simple arithmetic operations only. In this paper, we present results that enable for significant improvements when using fast dual ascent for embedded model predictive control. These improvements rely on a new characterization of the set of matrices that can be used to describe a quadratic upper bound to the negative dual function. For many interesting formulations, it is shown that the provided bound cannot be improved and that it is tighter than bounds previously presented in the literature. The improved quadratic upper bound is used in fast dual gradient methods as an approximation of the dual function. Since it better approximates the dual function than previous upper bounds, the convergence is improved. The performance enhancement is most significant for ill-conditioned problems. This is illustrated by a numerical evaluation on a AFTI-16 aircraft model where the algorithm outperforms previous proposals of first order methods for embedded control with one to three orders of magnitude.