An Analysis of Closed-Loop Stability for Linear Model Predictive Control Based on Time-Distributed Optimization

Time-distributed optimization (TDO) is an approach for reducing the computational burden of model predictive control (MPC) and a generalization of the real-time iteration scheme. When using TDO, optimization iterations are distributed over time by maintaining a running solution estimate and updating it at each sampling instant. In this article, TDO applied to input-constrained linear-quadratic MPC is studied in detail, and an analytic bound for the number of optimization iterations per sampling instant required to guarantee closed-loop stability is derived. Further, it is shown that the closed-loop stability of TDO-based MPC can be guaranteed using multiple mechanisms, including increasing the number of solver iterations, preconditioning the optimal control problem, adjusting the MPC cost matrices, and reducing the length of the receding horizon. These results in a linear system setting also provide insights and guidelines that could be more broadly applicable, for example, to nonlinear MPC.