Complex static skew-symmetric output feedback control

TL;DR

This paper investigates static skew-symmetric output feedback control for specific classes of transfer functions, extending previous symmetric control work, and provides conditions for pole placement and feedback law enumeration using algebraic geometry.

Contribution

It introduces a geometric framework for skew-symmetric control systems, deriving pole placement conditions and counting feedback laws via Schubert calculus.

Findings

Necessary and sufficient conditions for pole placement by static skew-symmetric feedback.

Count of feedback laws using Schubert calculus for orthogonal Grassmannian.

Construction of real systems with real feedback for any real poles.

Abstract

We study the problem of feedback control for skew-symmetric and skew-Hamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke, et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, give necessary and sufficient conditions for pole placement by static skew-symmetric complex feedback, and use Schubert calculus for the orthogonal Grassmannian to count the number of complex feedback laws when there are finitely many of them. Finally, we also construct a real skew-symmetric linear system with only real feedback for any set of real poles.