Pseudo-Transient Continuation for Enhanced Quadratic Programming and Optimal Control

Quadratic programming (QP) solvers that join effectiveness with a simple implementation are becoming essential in the field of optimal control, specifically when dealing with real-time applications with strict timing constraints and limited computational resources. To address this need, we present a novel high-performance QP solution method based on pseudo-transient continuation (PTC). PTC is a numerical technique that transforms multivariate nonlinear equations into autonomous systems that converge to the solution sought. In our approach, we recast the general QP Karush-Kuhn-Tucker (KKT) conditions into a system of equations and employ PTC to solve the latter to attain the optimal solution. Importantly, we provide theoretical guarantees demonstrating the global convergence of our PTC-based solver to the optimal solution of any given QP. To showcase the effectiveness of PTC, we employ it within the domain of Model Predictive Control (MPC). Specifically, numerical simulations are carried out on the MPC control of a quadrotor - a demanding dynamical system - highlighting excellent results in accurately executing the control task and ensuring lower computational times compared to conventional QP solvers.