Symmetrizable systems

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.