Symmetrizable systems

TL;DR

The paper addresses transforming a linear system into a symmetric form to leverage symmetry-based control tools. It introduces symmetrizable systems, characterized by a static gain $K$ that yields $H(s)=K^{-1}G(s)K$ symmetric, and provides both necessary and sufficient conditions and practical methods (via a block-factorization and the Khatri-Rao product) to certify and construct such transformations; it also shows that complete symmetry leads to a convex semidefinite program to find $Q\succ0$ with $PQ=QP^T$ and a structured zero-off-diagonal constraint. The work extends analytic results for symmetric systems to the broader class of symmetrizable systems, including an explicit extension of an optimal control problem and demonstrations through classical and numerical examples. It also proves that linear systems are generically neither symmetric nor symmetrizable, motivating future work on approximating arbitrary systems by nearest symmetrizable ones for practical control design.

Abstract

Transforming an asymmetric system into a symmetric system makes it possible to exploit the simplifying properties of symmetry in control problems. We define and characterize the family of symmetrizable systems, which can be transformed into symmetric systems by a linear transformation of their inputs and outputs. In the special case of complete symmetry, the set of symmetrizable systems is convex and verifiable by a semidefinite program. We show that a Khatri-Rao rank needs to be satisfied for a system to be symmetrizable and conclude that linear systems are generically neither symmetric nor symmetrizable.

Theorems & Definitions (13)

• Proposition 1: LMU
• Proposition 2: willems1976sym
• Definition 1
• Theorem 1
• proof
• Remark 1
• Remark 2
• Proposition 3: uh1974
• Theorem 2
• proof