Approximation of Continuous-Time Infinite-Horizon Optimal Control Problems Arising in Model Predictive Control - Supplementary Notes

TL;DR

This paper investigates two approximation methods for infinite-horizon linear optimal control problems in model predictive control, providing bounds on the optimal cost and conditions for convergence as the approximation improves.

Contribution

It introduces two basis function-based approximation schemes that yield bounds on the optimal cost and analyzes their convergence properties.

Findings

Bounds on the optimal cost are obtained using the two approximations.

Tighter bounds are achieved with an increasing number of basis functions.

Conditions for convergence to the true optimal cost are established.

Abstract

These notes present preliminary results regarding two different approximations of linear infinite-horizon optimal control problems arising in model predictive control. Input and state trajectories are parametrized with basis functions and a finite dimensional representation of the dynamics is obtained via a Galerkin approach. It is shown that the two approximations provide lower, respectively upper bounds on the optimal cost of the underlying infinite dimensional optimal control problem. These bounds get tighter as the number of basis functions is increased. In addition, conditions guaranteeing convergence to the cost of the underlying problem are provided.