Errors-in-variables identification in dynamic networks — Consistency results for an instrumental variable approach

Abstract

In this paper we consider the identification of a linear module that is embedded in a dynamic network using noisy measurements of the internal variables of the network. This is an extension of the errors-in-variables (EIV) identification framework to the case of dynamic networks. The consequence of measuring the variables with sensor noise is that some prediction error identification methods no longer result in consistent estimates. The method developed in this paper is based on a combination of the instrumental variable philosophy and closed-loop prediction error identification methods, and leads to consistent estimates of modules in a dynamic network. We consider a flexible choice of which internal variables need to be measured in order to identify the module of interest. This allows for a flexible sensor placement scheme. We also present a method that can be used to validate the identified model.

Signal Selection for Estimation and Identification in Networks of Dynamic Systems: A Graphical Model Approach

“…On the other hand, if the network topology is fully known, there is a wealth of closed loop parametric identification techniques which have been extended to general networks in order to identify individual transfer functions. These techniques include the Direct Method, the Joint IO method, the Two-Stage identification [18], and the instrumental variable method [19]. Most of these approaches can address scenarios where inputs are non manipulable if observed internal signals in the network can be used as predictors and/or instrumental variables.…”

“…They result in sufficient conditions on how to select these signals to arrive at consistent estimation of an individual transfer function [20]. However, these techniques often require some form of knowledge about the strict causality or the degree of delay embedded in some operators (especially if feedback loops are present [19]).…”

Identification of modules in dynamic networks: An empirical Bayes approach

“…This is the idea underlying the method presented in [2], which generalizes the two-stage method, originally developed for closed-loop systems, to dynamic networks [12]. Instrumental variable methods for closed-loop systems [13] are adapted to networks in [9]. Similarly, the methodology proposed in [14] for the identification of cascaded systems is generalized to the context of dynamic networks in [8].…”

“…Some of the recent papers deal with both the aforementioned problems [4], [5] [1], [6], whereas others are mainly focused on the identification of a single module, see [7] [8], [9], [10], [11]. In particular, [7], [2] study the problem of understanding which of the available output measurements should be used to obtain a consistent estimate of a target module.…”

“…In particular, [7], [2] study the problem of understanding which of the available output measurements should be used to obtain a consistent estimate of a target module. In [9] instead, errors-in-variables dynamic networks are considered, and methods that lead to consistent module estimates are proposed. As observed in [2], dynamic networks with known topology can be seen as a generalization of simple compositions, such as cascaded systems or closed-loop systems.…”