Pole placement for overdetermined 2D systems

TL;DR

This paper develops a novel pole placement method for overdetermined 2D systems using algebraic geometry and operator theory, extending classical results and addressing open questions in control theory.

Contribution

It introduces a new approach to pole placement for overdetermined 2D systems modeled by operator vessels, connecting it to algebraic geometry and meromorphic bundle maps.

Findings

Provides a solution to pole placement using meromorphic bundle maps.

Reduces the 2D problem to a 1D case, linking to classical theorems.

Answers a question posed by Ball and Vinnikov.

Abstract

We formulate and solve a pole placement problem by state feedback for overdetermined 2D systems modeled by commutative operator vessels. In this setting, the transfer function of the system is given by a meromorphic bundle map between two holomorphic vector bundles of finite rank over the normalization of a projective plane algebraic curve. The obstruction for a solution is given by an existence of a certain meromorphic bundle map on the input bundle. Reducing to the 1D case, this gives a functional obstruction which is equivalent to the classical pole placement theorem. Our result improves on, and gives a new approach to pole placement even in the classical case, and answers a question of Ball and Vinnikov.