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Abstract: We discuss identifiability of dynamic SISO errors-in-variables (EIV) models with white measurement errors. Although this class of models turns out to be generically identifiable, it has been pointed out that in certain circumstances there may be two EIV models which are indistinguishable from external input–output experiments. This lack of (global) identifiability may be prejudicial to identification and needs better understanding. The identifiability conditions found in the literature guarantee uniqueness under certain coprimality assumptions on the (rational) transfer function of the ideal “true” system and the spectral density of the noiseless “true” input. Unfortunately these conditions are not testable since they concern precisely the unknowns of the problem which are not available to the experimenter. We provide new identifiability conditions which are instead expressible in terms of the external description of the observable signals, namely their joint power spectral densities. [Copyright &y& Elsevier]

In this paper, a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented. Various extensions of the positive real lemma, as well as of other classical factorization results, both in the continuous and discrete-time cases, are established. In the case of square factorizations, the map between solutions to an asymmetric algebraic Riccati equation and pairs of factors is shown to be a homeomorphism. In this framework, we also derive a geometric characterization of non-square factorizations. [ABSTRACT FROM AUTHOR]

It is a well-known, yet poorly understood fact that, contrary to the continuous-time case, the same discrete-time process y can be represented by minimal linear models (see (1.1a), (1.1b) below) which may either have a non-singular or a singular D matrix. In fact, models with D=0 have been commonly used in the statistical literature. On the other hand, for models with a singular D matrix the Riccati difference equation of Kalman filtering involves in general the pseudo-inversion of a singular matrix. This “cheap filtering” problem, dual to the better known “cheap control” problem, has been studied for several decades in connection with the so-called “invariant directions” of the Riccati equation. For a singular D, a reduction of the order of the Riccati equation is in general possible. The reasons for such a reduction do not seem to be completely clear either. In this paper we provide an explanation of this phenomenon from the classical point of view of “zero flipping” among minimal spectral factors. Changing D''s occurs whenever zeros are “flipped” from z=∞ to their reciprocals at z=0. It is well known that for finite zeros, the zero-flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all-pass matrix function. For infinite zeros, zero flipping is implemented by a dual version of the Silverman structure algorithm. Using this interpretation, we derive a new algorithm for filtering of non-regular processes, based on a reduced-order Riccati equation. We also obtain a precise characterization of the reduction of the order of the Riccati equation which is afforded by zeros either at z=∞ or at the origin. This order reduction has traditionally been associated with the study of invariant directions, a point of view which, as we show, does not capture the essence of the phenomenon. [Copyright &y& Elsevier]