Distributed Parameter Estimation in Sensor Networks: Nonlinear Observation Models and Imperfect Communication (0809.0009v2)

Abstract: The paper studies distributed static parameter (vector) estimation in sensor networks with nonlinear observation models and noisy inter-sensor communication. It introduces \emph{separably estimable} observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation. It studies two distributed estimation algorithms in separably estimable models, the $\mathcal{NU}$ (with its linear counterpart $\mathcal{LU}$) and the $\mathcal{NLU}$. Their update rule combines a \emph{consensus} step (where each sensor updates the state by weight averaging it with its neighbors' states) and an \emph{innovation} step (where each sensor processes its local current observation.) This makes the three algorithms of the \textit{consensus + innovations} type, very different from traditional consensus. The paper proves consistency (all sensors reach consensus almost surely and converge to the true parameter value,) efficiency, and asymptotic unbiasedness. For $\mathcal{LU}$ and $\mathcal{NU}$, it proves asymptotic normality and provides convergence rate guarantees. The three algorithms are characterized by appropriately chosen decaying weight sequences. Algorithms $\mathcal{LU}$ and $\mathcal{NU}$ are analyzed in the framework of stochastic approximation theory; algorithm $\mathcal{NLU}$ exhibits mixed time-scale behavior and biased perturbations, and its analysis requires a different approach that is developed in the paper.

• The paper introduces separably estimable observation models that extend classical observability to nonlinear, distributed sensor networks.
• The paper develops and analyzes LU, NU, and NLU algorithms, proving consistency and asymptotic normality under noisy conditions.
• The paper’s findings enable robust parameter estimation in real-world applications such as smart grids facing imperfect communication.

Overview of "Distributed Parameter Estimation in Sensor Networks: Nonlinear Observation Models and Imperfect Communication"

The paper under consideration comprehensively investigates distributed static parameter estimation within the context of sensor networks, particularly focusing on scenarios involving nonlinear observation models and communication challenges such as noise and unreliability between sensors. The primary contribution lies in the introduction and examination of separably estimable observation models, which extend the notion of observability from linear centralized systems to nonlinear distributed contexts.

Summary of Distributed Estimation Algorithms

The paper presents and analyzes three distributed estimation algorithms: LU, NU, and NLU. These algorithms are integral to the consensus + innovations type, differing markedly from conventional consensus approaches:

1. LU (Linear Update): Suitable for linear observation models, this algorithm ensures consensus and parameter convergence under the linear structure and incorporates innovations in the observation process.
2. NU (Nonlinear Update): Extends the framework to nonlinear models. It leverages stochastic approximation to establish the asymptotic normality of the estimate sequences.
3. NLU (Nonlinear Linear Update): Tailored for more complex nonlinear models that may not satisfy Lipschitz conditions. The NLU algorithm exhibits a dual time-scale property, necessitating specialized analytical treatment beyond conventional approaches.

Theoretical Foundations and Results

The paper rigorously establishes the consistency of these algorithms under various conditions:

• Consistency is characterized by the almost sure convergence of the parameter estimates to the true parameter values across all sensors.
• Asymptotic normality is proven for the LU and NU algorithms, where the residue of the estimates follows a normal distribution as the number of iterations approaches infinity.

Implications and Applications

In practical terms, the separably estimable models accommodate realistic scenarios in sensor networks where individual sensors cannot independently estimate the entire parameter vector due to computational or dimensionality constraints. The theoretical guarantees provided by these algorithms open the door for their deployment in systems with nonlinear interactions and noisy communication states, such as smart grid networks for distributed phase estimation as highlighted by the static phase estimation example in power grids.

Conclusion

This paper situates itself within a foundational framework for distributed parameter estimation, offering robust algorithms with proofs that address the complexity of real-world sensor networks. Looking ahead, exploration into mixed time-scale analyses and further optimization of the consensus + innovations paradigm remains a critical future direction to enhance the efficacy of distributed estimation in dynamically varying environments.