Structure Identifiability of an NDS with LFT Parametrized Subsystems

Abstract

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the NDS is structurally identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.

Global Structure Identifiability and Reconstructibility of an NDS with Descriptor Subsystems

“…Note that when the regularity assumption is satisfied, both the TFM I mv − ΦG zv (λ) and the matrix pencil λE(i) − A xx (i) with i ∈ {1, 2, • • • , N } are invertible. Then arguments similar to the proof of Theorem 1 in [21] lead to the conclusions. This completes the proof.…”

“…On the basis of these TFMs, the following results are obtained which take completely the same form as those of Theorem 1 in [21].…”

“…The latter has been attracting extensive research attentions for a long time in various fields, including engineering, finance, biology, etc. Nevertheless, it is still a challenging issue even for a lumped LTI system [1,8,9,14,21]. As structure information is widely recognized to be quite important in NDS analysis and synthesis, structure identifiability is used throughout this paper, in order to reflect this importance and to distinguish it from the traditional parameter identifiability, which is also adopted in other works, such as [13,14,16,17] and the references therein.…”

“…One treats each measured signal as a node and a transfer function from one measured signal to another measured signal as an edge [4,13,14]. The other treats each subsystem as a node and subsystem interactions as edges [12,17,21]. In either of these two approaches, the set of the edges connecting different nodes of an NDS is called its structure, which can be associated to a graph that reveals direct influences among different signals or subsystems.…”

“…It has also been proven there that under some special situations, this condition becomes also sufficient. When the dynamics of each subsystem is described by a state space model and its inputs/outputs are divided into internal and external ones, [21] proves that if in each subsystem, the TFM from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the structure of the NDS with arbitrary subsystem connections is identifiable. These restrictions have been partly removed in [19], which gives a necessary and sufficient condition for NDS structure identifiability when only one of the aforementioned two assumptions is satisfied, as well as an explicit description for the set of subsystem interactions that can not be differentiated from a particular subsystem connection matrix (SCM) through only experiment data.…”