Minimal strongly eventually positive realization for a class of externally positive systems

The minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state-space matrices. In this paper, the minimal strongly eventually positive realization, which relaxes the nonnegativity constraint on the state-space model and only requires that the state trajectory is nonnegative after a certain number of steps, is proposed for an externally positive system. It is shown that a discrete-time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization. A constructive proof is provided. The continuous-time counterpart is also addressed by transforming it into a corresponding discrete-time case.