Convex invertible cones of state space systems

In the paper [CL1] the notion of a convex invertible cone, cic, of matrices was introduced and its geometry was studied. In that paper close connections were drawn between this cic structure and the algebraic Lyapunov equation. In the present paper the same geometry is extended to triples of matrices and cics of minimal state space models are defined and explored. This structure is then used to study balancing, Hankel singular values, and simultaneous model order reduction for a set of systems. State space cics are also examined in the context of the so-called matrix sign function algorithm commonly used to solve the algebraic Lyapunov and Riccati equations.