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

Loïc Bourdin (1) , Emmanuel Trélat (2)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: sampled-data control; linear-quadratic problems; optimal control; digital control; convergence; Pontryagin maximum principle; Riccati theory; dynamic programming principle

Domaines: Optimisation et contrôle [math.OC]

Loïc Bourdin, Emmanuel Trélat. Linear-quadratic optimal sampled-data control problems: convergence result and Riccati theory. Automatica, 2017, 79, pp.273--281. ⟨hal-01304197⟩