Abstractions of linear dynamic networks for input selection in local module identification

TL;DR

The paper develops a generalized framework for abstracting linear dynamic networks by removing unmeasured nodes while preserving a designated local module. It unifies immersion and indirect-input approaches through a flexible transformation-based abstraction that relies on partitioning nodes into retained, indirect-observing, and abstracted groups and on full-rank observability conditions. It provides explicit invariance conditions (including a generalized theorem) that guarantee the target module remains unchanged in the abstracted network, enabling consistent identification from a reduced set of measurements. The authors also discuss identifiability considerations, node-selection strategies, external-excitation requirements, and how the resulting abstracted model can be parameterized for standard identification methods. The work offers practical guidance for measurement design in large-scale networks and lays the groundwork for data-efficient, robust local-module identification in complex dynamical systems.

Abstract

In abstractions of linear dynamic networks, selected node signals are removed from the network, while keeping the remaining node signals invariant. The topology and link dynamics, or modules, of an abstracted network will generally be changed compared to the original network. Abstractions of dynamic networks can be used to select an appropriate set of node signals that are to be measured, on the basis of which a particular local module can be estimated. A method is introduced for network abstraction that generalizes previously introduced algorithms, as e.g. immersion and the method of indirect inputs. For this abstraction method it is shown under which conditions on the selected signals a particular module will remain invariant. This leads to sets of conditions on selected measured node variables that allow identification of the target module.

Figures (7)

• Figure 1: Example network with $\mathcal{V}=\{v_1,v_2\}$, $\tilde{\mathcal{Z}}=\{z\}$, $\mathcal{L}=\{l_1,l_2,l_3\}$, where the full rank condition is satisfied.
• Figure 2: Example network to illustrate abstraction.
• Figure 3: Network obtained after abstracting node $w_4$ through immersion.
• Figure 4: Modification of the network depicted in Figure \ref{['fig:netw_abs']}, obtained after removal of node $w_4$ by the indirect inputs method.
• Figure 5: Networks to illustrate issues with parallel paths when abstracting.