The roots of unity and a direct method for the computation of stable Internal Positive Representations of linear systems

This paper presents a new technique for the construction of Internal Positive Representations (IPRs) of discrete time linear systems. The proposed method overcomes the limitations of a previously proposed technique, which provides stable IPRs of systems under a restrictive assumption on the position of the eigenvalues in the complex plane. The new method here presented exploits a suitable representation of complex vectors and matrices by means of nonnegative combinations of the roots of unity, and provides a stable IPR for any stable system. The position of the eigenvalues in the complex plane only affects the state-space dimension of the IPR.