Distributed Observers for LTI Systems (1608.01429v2)

Abstract: We consider the problem of distributed state estimation of a linear time-invariant (LTI) system by a network of sensors. We develop a distributed observer that guarantees asymptotic reconstruction of the state for the most general class of LTI systems, sensor network topologies and sensor measurement structures. Our analysis builds upon the following key observation - a given node can reconstruct a portion of the state solely by using its own measurements and constructing appropriate Luenberger observers; hence it only needs to exchange information with neighbors (via consensus dynamics) for estimating the portion of the state that is not locally detectable. This intuitive approach leads to a new class of distributed observers with several appealing features. Furthermore, by imposing additional constraints on the system dynamics and network topology, we show that it is possible to construct a simpler version of the proposed distributed observer that achieves the same objective while admitting a fully distributed design phase. Our general framework allows extensions to time-varying networks that result from communication losses, and scenarios including faults or attacks at the nodes.

• The paper presents novel distributed observers for LTI systems in sensor networks, utilizing multi-sensor decomposition for state estimation without augmentation.
• The proposed observers work across diverse systems and networks, with a fully distributed design possible under specific detectability conditions.
• The research provides insights for robust monitoring in large-scale infrastructures like power grids, supports time-varying networks, and is validated via simulation.

Distributed Observers for LTI Systems: An Analytical Approach

This paper addresses the problem of distributed state estimation in linear time-invariant (LTI) systems using sensor networks. The authors introduce a new class of distributed observers that aim to reconstruct the state asymptotically across a broad class of system dynamics, measurement structures, and sensor network topologies. The approach builds upon Luenberger observer design principles, enabling nodes to estimate detectable portions of the state locally while employing consensus dynamics for parts that are not directly observable.

Key Contributions and Methodology

The paper presents two observer designs tailored for different conditions:

1. Condition 1: Applies to systems and graphs where every source component is detectable.
2. Condition 2: Specifically suited for scenarios where each unstable and marginally stable eigenvalue can be detected by at least one node within each source component.

Central to the paper is the multi-sensor observable canonical decomposition, a novel method that extends the classical Kalman observable decomposition to a network with multiple sensors. This technique efficiently reveals which portions of the state each sensor can independently monitor, allowing for a distributed observer design without state augmentation often needed in existing approaches.

For systems satisfying Condition 1, the proposed scheme involves a centralized design phase. It establishes a transformation to identify sub-states associated with each node and designs consensus-based estimation protocols for nodes unable to detect certain eigenvalues locally. In contrast, systems under Condition 2 benefit from a fully distributed estimator design phase, granting practical ease for real-world implementations. Notably, this scheme leverages the Jordan canonical form to identify observable and unobservable eigenvalues collectively, resulting in efficient consensus dynamics for state portions not locally observable.

Theoretical and Practical Implications

The research has profound implications for distributed monitoring in large-scale infrastructures such as power grids, transportation networks, and manufacturing ecosystems. Emphasizing robustness, the framework accounts for time-varying networks due to communication losses, ensuring that the distributed observers can adaptively maintain state estimation accuracy despite intermittent connectivity.

Furthermore, the work provides insights into the selection and optimization of consensus weights, ensuring stability even in the presence of marginally stable eigenvalues. The approach holds promise for future work in extending distributed observer designs to scenarios involving node failures or attacks, as briefly explored through potential network robustness conditions.

Simulation and Validation

Through illustrative examples, including simulations, the authors demonstrate the efficacy of their proposed schemes. They validate how their distributed observers maintain asymptotic stability and accuracy in state estimation across all nodes, emphasizing their framework's comprehensive applicability to diverse network configurations.

Conclusion

This paper contributes significantly to distributed state estimation by offering a versatile, and theoretically grounded approach to managing LTI systems within sensor networks. Future work may consider exploring low-complexity distributed algorithms for observer design, stochastic system models, and functional observer designs tailored for large-scale systems with specific local state estimation needs. As advances in communication technologies unfold, particularly in decentralization and synchronization mechanisms, these insights pave the way for enhancing robustness and operational efficiency in complex networked environments.