Differentiation similarities in fractional pseudo-state space representations and the subspace-based methods

The paper starts by presenting a new concept of differentiation similarity transformations for commensurate pseudo-states-space representations. It is proven that a pseudo-state-space representation with a commensurate differentiation order nu and a dimension of the transition matrix n can be similar to a pseudo-state-space representation with a commensurate order nu/k and a dimension of the transition matrix kn, where k is an integer number. A direct consequence of the aforementioned concept in fractional subspace-based identification methods for MIMO systems is that an overestimated pseudo-state-space representation has multiple minimums at commensurate differentiation orders over the integral number k. This result is especially visible when deterministic input/output signals are considered and less visible in the stochastic case due to overestimation.