Continuous-Time-Constrained Model Predictive Control with a Parallel Solver

In this article, we address the model predictive control (MPC) problem for continuous-time linear time-invariant systems, with both state and input constraints. For computational efficiency, existing approaches typically discretize both dynamics and constraints, which potentially leads to constraint violations in between discrete-time instants. In contrast, to ensure strict constraint satisfaction, we equivalently replace the differential equations with linear mappings between state, input, and flat output, leveraging the differential flatness property of linear systems. By parameterizing the flat output with piecewise polynomials and employing Markov-Lukács theorem, the original MPC problem is then transformed into a semidefinite programming (SDP) problem, which guarantees the strict constraints satisfaction at all time. Furthermore, exploiting the fact that the proposed SDP contains numerous small-sized positive semidefinite (PSD) matrices as optimization variables, we propose a primal-dual hybrid gradient (PDHG) algorithm that can be efficiently parallelized, expediting the optimization procedure with GPU parallel computing. The simulation and experimental results demonstrate that our approach guarantees rigorous adherence to constraints at all time, and our solver exhibits superior computational speed compared to existing solvers for the proposed SDP problem.