Data-informativity conditions for structured linear systems with implications for dynamic networks

TL;DR

The paper addresses consistent identification of a single target module in dynamic networks under a prediction-error framework by introducing path-based, graph-theoretic data-informativity conditions that leverage network topology and known zeros. It relaxes full MIMO data-informativity requirements by focusing on the target module and by incorporating structured input generation via graph connectivity, ensuring $\Phi_{\kappa^{[j]}}(\omega) \succ 0$ (or equivalent spectral conditions) through vertex-disjoint paths and disconnecting sets. Key contributions include a structured data-informativity theory for single rows of the predictor, methods to embed network structure into input signal generation, and connections to generic single-module identifiability. The results reduce experimental burden while preserving consistency, and link graph-based path concepts to identifiability when identifying modules in dynamic networks. The work provides practical guidelines for sensor/actuator placement and offers theoretical bridges between identifiability and data-informativity for networked systems.

Abstract

When estimating a single subsystem (module) in a linear dynamic network with a prediction error method, a data-informativity condition needs to be satisfied for arriving at a consistent module estimate. This concerns a condition on input signals in the constructed, possibly MIMO (multiple input multiple output) predictor model being persistently exciting, which is typically guaranteed if the input spectrum is positive definite for a sufficient number of frequencies. Generically, the condition can be formulated as a path-based condition on the graph of the network model. The current condition has two elements of possible conservatism: (a) rather than focussing on the full MIMO model, one would like to be able to focus on consistently estimating the target module only, and (b) structural information, such as structural zero elements in the interconnection structure or known subsystems, should be taken into account. In this paper relaxed conditions for data-informativity are derived addressing these two issues, leading to relaxed path-based conditions on the network graph. This leads to experimental conditions that are less strict, i.e. require a smaller number of external excitation signals. Additionally, the new expressions for data-informativity in identification are shown to be closely related to earlier derived conditions for (generic) single module identifiability.

Figures (5)

• Figure 1: Six node example with target module $G_{12}(q)$ and correlated disturbances $v_1, v_2$.
• Figure 2: Five node example with target module $G_{j1}(q)$, and dependence structure among the input components $u$ as in (\ref{['equ']}), where $v_j$ is the $j$-th component of $H^0(q)e(t)$ in (\ref{['eq:1']}).
• Figure 3: Network situation with target module $G_{ji}$. Module $G_{j8}$ is known and therefore not parametrized. The disconnecting set $w_{\! { \mathcal{D}} ^c}$ is disconnecting from node $w_i$ to the remaining inputs to parametrized modules. $w_{\! { \mathcal{D}} ^c}$ and $w_i$ need sufficient excitation from external signals, either $e$- or $r-$ signals, while external signals that have a parametrized link to the output $w_j$ are excluded. Note that $u = R^0 r$ with $R^0$ a binary matrix.
• Figure 4: Two node example with correlated disturbances and target module $G_{21}(q)$.
• Figure 5: Six node example with target module $G_{12}(q)$ and correlated disturbances $v_1, v_2$.

Theorems & Definitions (22)

• Definition 1
• Proposition 1
• Proposition 2
• Remark 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 3
• Theorem 4