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Distributed Observers for LTI Systems
Aritra Mitra, Shreyas Sundaram
TL;DR
The paper addresses distributed reconstruction of LTI system states when individual sensors have limited measurements and communication is constrained. It designs observers that combine local Luenberger estimation with consensus, guaranteeing asymptotic reconstruction under broad system and graph conditions while also supporting simpler distributed designs and communication losses. The framework additionally identifies requirements for handling node failures or malicious behavior.
Problem
Distributed estimation must reconstruct an entire LTI state at every node despite partial local measurements and restricted communication over sensor networks.
Method
The approach estimates locally reconstructible state portions with Luenberger observers and uses consensus among nodes for portions that are not locally detectable.
Results
The proposed observers guarantee asymptotic state reconstruction for the broadest class of LTI systems, directed communication graphs, and linear sensor measurement structures, with simpler fully distributed schemes for subclasses.
Takeaways & Limitations
Distributed observers can combine local estimation and network consensus while avoiding state augmentation and exchanging only local state estimates in the general scheme.
Takeaways & Limitations
Node failures or malicious behavior require globally detectable collective measurements, sufficient network and sensor redundancy, and estimation algorithms that exploit that redundancy.
Abstract
from arXiv · showhide
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.
1 Introduction
Distributed state estimation addresses how sensor nodes can reconstruct an LTI system's state using limited local measurements and neighbor information over a directed communication network.
- Large-scale systems are monitored by spatially distributed sensors connected through sparse networks with limited communication.
- Each node estimates the entire state from its partial measurements and information exchanged with neighboring nodes.
- The paper targets asymptotic reconstruction of the full state at every sensor node.
- The main development focuses on LTI systems and time-invariant directed communication graphs.
- The framework is extended to certain time-varying networks caused by intermittent communication link failures.
2 System Model
The paper formalizes distributed observers for LTI systems monitored over directed sensor networks, where nodes combine local measurements with neighbor information to achieve asymptotic state estimation.
- Nodes update state estimates using information from neighbors and any available local measurements over a predefined directed graph.
- A distributed observer requires every node's estimate to converge asymptotically to the plant state.
- If a node's local pair is not detectable, information exchange with other nodes is necessary for estimating the true state.
- The proposed approach lets nodes reconstruct locally detectable state portions and use consensus for the remaining portions.
- For the broadest system and graph class, the scheme offers asymptotically stable estimators, single-time-scale operation, no state augmentation, local estimate exchange, and broad applicability.
- The journal version adds full proofs, efficient distributed implementations, communication-loss robustness analysis, and simulations beyond the conference version.
3 Preliminaries
The preliminaries classify eigenvalues by detectability and organize system-graph pairs through root nodes, source components, and two increasingly restrictive conditions.
- For each node, the state dynamics are partitioned into detectable and undetectable eigenvalue sets.
- Root nodes are nodes whose measurements can detect a specified unstable or marginally stable eigenvalue.
- A source component is a strongly connected component with no incoming edges from outside the component.
- Condition 1 requires the subsystem associated with every source component to be detectable.
- Condition 2 requires at least one root node in every source component for each unstable or marginally stable plant eigenvalue.
- Condition 2 implies Condition 1, but Condition 1 can hold when a source component lacks a root node for a particular unstable eigenvalue.
4 Illustrative Example
A scalar unstable example illustrates the observer principle: a node estimates locally detectable state information, while other nodes propagate it through a directed acyclic graph.
- The example uses the unstable scalar plant x[k + 1] = 1.5x[k], with node 1 measuring the state and nodes 2 and 3 having no measurements.
- Node 1 can detect the plant eigenvalue λ = 1.5 from its own measurement and runs a Luenberger observer.
- Nodes 2 and 3 update their estimates from node 1's estimate, so their estimates also converge to the true state.
- Consensus is needed only for state portions that are not locally detectable; locally detectable portions can be estimated with Luenberger observers.
- A DAG rooted at the locally informed node transmits state information through the network without requiring that node to listen to neighbors.
5 Estimation Scheme for systems and graphs satisfying Condition
The scheme combines local observers with consensus-based estimation after decomposing the state into sensor-associated sub-states and an unobservable sub-state. Under the stated conditions, every node asymptotically reconstructs the full state, while simpler distributed design is possible under additional structure.
- Assumptions and network structure: The development assumes a strongly connected graph, while general directed networks can be handled by applying the strategy within each source component.Under strong connectivity, the aggregate pair (A, C) is detectable.
- Multi-sensor decomposition: The multi-sensor observable canonical decomposition sequentially identifies each sensor’s observable portion within the remaining unobservable subspace.It applies observable canonical decompositions in sensor order and produces a transformed system with sensor-associated blocks and an unobservable block.
- Convergence analysis: There exist observer gains and consensus weights such that the update rules form a distributed observer and every node’s estimates asymptotically converge to the full state.The sub-state errors converge asymptotically, and stability of the transformed unobservable dynamics completes reconstruction of x[k].
- Implementation trade-offs: The resulting estimator has dimension n at each node and avoids state augmentation, but the general design is most naturally centralized because decomposition requires all sensor measurement matrices.Under additional system and graph structure, the decomposition can be avoided for a more efficiently distributed design, potentially at the cost of increasing observer dimension.
6 Estimation Scheme for systems and graphs satisfying Condition
The scheme decomposes the transformed state into locally detectable and undetectable eigenvalue components. Each node estimates the former with a local Luenberger observer and the latter through consensus driven by nodes that can detect those components.
- Local Luenberger observers: Each node identifies locally detectable eigenvalues and estimates their associated sub-states using a locally designed Luenberger observer.The detectable sub-states are recovered asymptotically under a suitable observer gain.
- State decomposition: The state is permuted into detectable and undetectable components, with local coordinate transformations separating observable and unobservable portions of each node’s measurements.The decomposition groups Jordan blocks and measurement columns according to each node’s detectable eigenvalues.
- Consensus estimation: For eigenvalues a node cannot detect, it updates estimates using consensus information from neighboring nodes, while root nodes provide detectable estimates.Consensus weights are assigned separately for the relevant undetectable sub-states.
- Distributed observer: The distributed observer combines local observer updates, consensus updates, state permutations, and the global transformation x[k] = T z[k].The resulting update rules are specified by the distributed observer construction.
- Guarantee and design: There exist local observer gains and consensus weights making the update rules a distributed observer under Condition 2.The construction uses directed trees rooted at nodes that detect each relevant eigenvalue; distributed tree algorithms such as BFS can build them.
- Guarantee and limitation: Every node asymptotically estimates the complete transformed state and therefore the true state, although the general observer can have dimension larger than the plant state.The dimension increase results from redundant consensus estimation of the undetectable component.
7 Robustness to Communication Losses and Sensor Failures
The framework extends distributed estimation to communication losses and discusses the additional redundancy and detectability requirements posed by node failures or attacks.
- Communication Losses: Communication-loss extensions use spanning DAGs rooted at source nodes, preserving redundant links during observer design.
- Communication Losses: When only some parent nodes are reachable, observers redistribute weights across the available subset while preserving stochasticity.
- Communication Losses: Assumption 2 requires every non-source node to receive information from at least one parent for each sub-state within every interval of length T.
- Communication Losses: Under Condition 1, a strongly connected baseline graph, and Assumption 2, the time-varying observer achieves the stated distributed-observer construction.
- Node Attacks: Node failures and attacks require collective global detectability plus network and sensor redundancy that prevents compromised nodes from becoming bottlenecks.
- Node Attacks: Robust reconstruction under attacks has been preliminarily established for distinct-real-eigenvalue systems, while broader dynamics remain future work.
8 Example
The example applies the Condition 1 scheme to a network whose system is not detectable at any individual node. The resulting error plots validate estimation of all three states.
- The example network and associated system and measurement matrices instantiate the distributed-observer scheme for Condition 1.
- The system is not detectable from any individual node, while the source component formed by nodes 1 and 2 is observable.
- The multi-sensor observable canonical decomposition produces two sub-states and a zero-dimensional unobservable sub-state.
- Figure 5 plots error dynamics for the first, second, and third states, and the plots validate the Condition 1 scheme.
- Nodes 1 and 2 maintain dimension-3 observers using local measurement updates and exchanged estimates, while node 3 runs pure consensus.
9 Conclusions and Future Work
The paper introduces distributed observers for general LTI systems, directed graphs, and linear measurements, while separating locally detectable estimation from consensus-based recovery.
- The paper introduces a new class of distributed observers for the most general LTI systems, directed communication graphs, and linear sensor measurement structures.
- The multi-sensor observable canonical decomposition supports the general design, while certain dynamics and networks admit a simpler fully distributed design phase.
- The unifying scheme uses local Luenberger observers for locally detectable state portions and consensus dynamics for the remaining portions.
- The framework is extended to certain time-varying networks, while stochastic systems, functional observers, delays, and asynchronicities remain future-work topics.
A Proof of Proposition 1
The proof constructs a sequence of sensor-specific observable canonical transformations, recursively identifying portions of each sensor's previously unobservable subspace.
- The construction begins with an observable canonical decomposition of the pair (A, C1) at sensor 1.
- At sensor 2, a transformation identifies the portion of sensor 1's unobservable subspace that sensor 2 can observe.
- The transformed dynamics and output equations express the newly identified observable sub-state and its measurement contribution.
- The same design procedure continues one sensor at a time until the N-th sensor, producing the final similarity transformation.
B Proof of Theorem 4
The proof uses induction across sub-states and analyzes error dynamics over switching intervals of length T. Schur stability of the interval transition matrix, together with vanishing upstream errors and ISS, yields asymptotic convergence.
- Inductive error analysis: The proof inducts on the sub-state number, assuming earlier sub-state errors converge to zero before analyzing sub-state p.The composite error dynamics for sub-state p are considered over intervals of length T.
- Switched dynamics: Over each interval of length T, the composite transition matrix is formed by multiplying the switched matrices M_p[k] across that interval.The same construction defines the interval matrix for the first sub-state.
- Stability: For switching signals satisfying Assumption 2, the interval matrix M̄_p(k) is Schur stable, as established analogously for M̄_1(k).This stability property applies to the class of switching signals considered in the theorem.
- Convergence: Because the induction hypothesis gives v^(p)[k] → 0, input-to-state stability implies E(p)[k] → 0.The proof invokes ISS and Proposition 2.9 to obtain this convergence.
- Unobservable component: The unobservable-state update is unaffected by network changes, so the remaining argument proceeds as in the proof of Theorem 1.This transfers the convergence argument to the unobservable component.