From structurally independent local LTI models to LPV model

Abstract

The local approach to linear parameter varying (LPV) system identification consists in interpolating individually estimated local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, individually estimated local state-space LTI models do not contain su cient information for determining similarity transformations making them coherent. It is possible to estimate these similarity transformations from input-output data under appropriate excitation conditions.

On Local LTI Model Coherence for LPV Interpolation

“…It is thus (implicitly) assumed that a collection of local models are "coherent" if they are in the same canonical form. This assumption is obviously in contradiction with the aforementioned result reported in [22]. Nevertheless, satisfactory results have been reported in the above cited examples.…”

“…It was shown in the last section that, for most practically significant LPV systems, Definition 2 is the only relevant definition of coherent local models. It is then important to recall that, as reported in [22], without any global structural assumption, locally estimated LTI models do not contain sufficient information to make themselves coherent, in the sense of Definition 2.…”

“…However, the accurate meaning of the term "coherent" or "coherence" in this context is rarely defined. To our knowledge, two different definitions have been proposed in [20] and in [22], which will be respectively recalled as Definitions 2 and 3 in Section III. The purpose of this paper is to analyze the relevance of these two definitions and to point out the consequence of this analysis on the practice of LPV system identification.…”

“…It is shown in [22] that, without any global structural assumption, locally estimated LTI models do not contain sufficient information to make themselves coherent. This result is in the sense of the local model coherence definition formulated in the same cited paper.…”

“…As a matter of fact, this result is in contradiction with the practice of interpolating local models in some canonical form as recalled above. This paper will carefully analyze the relevance of the coherence definition formulated in [22] and its consequence. Here let us make a related remark: the reported works based on different canonical forms naturally raise the compatibility issue: given a set of local LTI state-space models, the interpolations based on different canonical forms lead certainly to different LPV models, but do they have (closely) the same input-output (I-O) behavior?…”