Stable Internally Positive Representations of Continuous Time Systems

An Internally Positive Representation (IPR) of a non-positive system is a positive system that, under suitable input, state and output transformations, exactly replicates the behavior of the original system. Any construction method of an IPR necessarily introduces additional natural modes, which in some cases are unstable. In a previous paper the authors have presented a method that provides a stable IPR if and only if the eigenvalues of the original system lie in a specific sector of the open left-half complex plane. In this technical note a new technique is proposed that overcomes such a limitation, and provides a stable IPR for any stable system, although in some cases the dimension of the IPR must be large in order to guarantee its stability. Although, for simplicity, the method is only illustrated for single-input single-output systems with distinct eigenvalues, it also applies to multi-input multi-output systems with multiple eigenvalues.