Discrete Control Systems

TL;DR

This paper explores discrete control systems based on geometric integrators that preserve the geometric structure of continuous systems, aiming to improve nonlinear control algorithms by maintaining essential properties like energy and momentum.

Contribution

It develops a control theory for discrete-time models using geometric integrators, ensuring numerical schemes preserve the underlying geometric properties without additional approximation.

Findings

Preservation of symplectic form, momentum, and energy in discrete control models

Control algorithms that maintain geometric structures of continuous systems

Enhanced stability and accuracy in nonlinear control simulations

Abstract

Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control system aims to address this deficiency by restricting the approximation to choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure.