Linear-quadratic optimal sampled-data control problems: convergence result and Riccati theory

TL;DR

This paper investigates linear-quadratic control problems with sampled-data controls, proving convergence of sampled controls to continuous controls as sampling gets finer, and extending Riccati theory to this framework for optimal control synthesis.

Contribution

It establishes the convergence of sampled-data controls to continuous controls and extends Riccati theory to sampled-data systems using two different approaches.

Findings

Optimal sampled-data controls converge pointwise to continuous controls as sampling periods tend to zero.

Two approaches extend Riccati theory to sampled-data control problems.

Derived closed-loop expressions for optimal sampled-data controls in linear-quadratic settings.

Abstract

We consider a general linear control system and a general quadratic cost, where the state evolves continuously in time and the control is sampled, i.e., is piecewise constant over a subdivision of the time interval. This is the framework of a linear-quadratic optimal sampled-data control problem. As a first result, we prove that, as the sampling periods tend to zero, the optimal sampled-data controls converge pointwise to the optimal permanent control. Then, we extend the classical Riccati theory to the sampled-data control framework, by developing two different approaches: the first one uses a recently established version of the Pontryagin maximum principle for optimal sampled-data control problems, and the second one uses an adequate version of the dynamic programming principle. In turn, we obtain a closed-loop expression for optimal sampled-data controls of linear-quadratic problems.