Maximum Likelihood Estimation for System Identification of Networks of Dynamical Systems

This paper investigates maximum likelihood estimation for direct system identification in networks of dynamical systems. We establish that the proposed approach is both consistent and efficient. In addition, it is more generally applicable than existing methods, since it can be employed even when measurements are unavailable for all network nodes, provided that network identifiability is satisfied. Finally, we demonstrate that the maximum likelihood problem can be formulated without relying on a predictor, which is key to achieving computationally efficient numerical solutions.

💡 Research Summary

The paper addresses the problem of identifying the dynamics of a network of interconnected linear systems directly via maximum‑likelihood estimation (MLE). Unlike most existing approaches, the proposed method does not require measurements at every node; it only needs a subset of node signals provided that the network satisfies a generic identifiability condition. The authors first model each node as a single‑input single‑output ARMAX system and describe the interconnection through binary matrices Υ (internal coupling) and Ω (exogenous input routing). The observed output is a linear selection of the node outputs and inputs, represented by a full‑row‑rank matrix Tₒ.

By converting each ARMAX module into an observer‑form state‑space realization, the whole network is expressed as a block‑diagonal closed‑loop system with state ξₖ, input rₖ (reference/excitation), and process noise eₖ. The resulting state‑space equations are ξₖ₊₁ = F_c ξₖ + G_r rₖ + G_e eₖ, xₒ,ₖ = H_o ξₖ + J_{ro} rₖ + J_{eo} eₖ, where the matrices F_c, G_r, G_e, H_o, J_{ro}, J_{eo} are constructed from the ARMAX parameters and the interconnection matrices.

A Kalman‑filter‑based innovation form is then introduced. Solving the associated algebraic Riccati equation yields the steady‑state Kalman gain K, the state covariance Σ, and the innovation covariance Σ_ε. Lemma 1 shows that a unique stabilising solution exists under two technical assumptions: (A2) J_{eo} has full row rank, and (A3) the noise polynomials C_i(z) have no zeros on the unit circle. These conditions are mild and can be enforced by discarding redundant observed signals.

A crucial contribution is the identification of a special case where the Riccati equation admits the trivial solution Σ = 0, K = S R⁻¹, Σ_ε = R (Theorem 2). This occurs when the noise covariance Σ_e is diagonal and positive definite and the noise polynomials satisfy C_i(z) ≠ 0 for |z| ≥ 1. In this situation the predictor transfer functions W(z) become affine, θ‑independent, zero‑one transformations of the open‑loop transfer function G(z). For output‑error (OE) models (a = c) an additional structural condition (13) guarantees the same simplification (Theorem 4, Corollary 4.1). Consequently, the ML problem collapses to a linear prediction‑error‑method (PEM) problem, explaining why linear PEM formulations fail to solve the full ARMAX ML problem.

The paper defines “informativity” of the data (xₒ, r) as the requirement that the joint spectrum Φ_z(ω) be positive definite for almost all frequencies. Under this condition, distinct parameter vectors produce distinct transfer functions, ensuring identifiability.

With the predictor in hand, the log‑likelihood of the observed data is written in the usual Gaussian form. However, instead of the conventional predictor‑based formulation (which leads to a highly non‑convex optimization), the authors recast the ML problem as a latent‑variable (missing‑data) estimation problem. This reformulation eliminates the explicit predictor, allowing the use of efficient unconstrained optimisation techniques or EM‑type algorithms. The authors argue that this approach is computationally more tractable than the predictor‑based methods, especially when only a subset of nodes is measured.

The main contributions are:

1. Derivation of consistency and efficiency conditions for ML identification of dynamic networks under partial observations, extending earlier results for single‑module and full‑measurement cases.
2. An unconstrained, predictor‑free formulation of the ML problem that leverages latent‑variable theory to obtain numerically efficient solutions.
3. Insight into the relationship between the ML estimator and linear PEM, showing that they coincide only in the trivial Riccati‑solution case.

The paper concludes by highlighting the broader applicability of the method to large‑scale networks where full sensor coverage is impractical, and suggests future work on scalable algorithms for very large networks and extensions to nonlinear or time‑varying dynamics.