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A Round-Robin Type Protocol for Distributed Estimation with $H_\infty$ Consensus
V. Ugrinovskii, E. Fridman
TL;DR
Distributed observers may need to combine partial measurements to track a plant, but their interconnections can be sampled rather than continuous. The paper develops a Round-Robin observer design with sufficient LMI conditions; it reports H∞ consensus without combinatorial gain scheduling, at the expense of moderately deteriorated performance.
Problem
Distributed estimation must address plant tracking when individual nodes may lack observability or detectability, while networked observers communicate intermittently.
Source: §1 · p. 2Method
Results
The proposed protocol achieves the H∞ consensus objective with sampled communications and without combinatorial gain scheduling, at the expense of moderately deteriorated performance.
Source: §1 · p. 2, §5 · p. 11Takeaways & Limitations
The design permits sampled-data communication between observers while keeping observer gains unchanged, supporting scalability without combinatorial gain scheduling.
Source: §1 · p. 2, §5 · p. 11In high-dimensional problems, the free LMI variables can burden the solver; constraining them reduces the number of variables at the expense of a more conservative design.
Source: Lemma 3 · p. 10
Abstract
from arXiv · showhide
The paper considers a distributed robust estimation problem over a network with directed topology involving continuous time observers. While measurements are available to the observers continuously, the nodes interact according to a Round-Robin rule, at discrete time instances. The results of the paper are sufficient conditions which guarantee a suboptimal $H_\infty$ level of consensus between observers with sampled interconnections.
1. Introduction
The paper develops robust distributed estimation for networks whose nodes have partial measurements and communicate intermittently under a Round-Robin schedule. It gives sufficient LMI conditions for H∞ consensus with sampled interconnections, while keeping observer gains fixed and accounting for disturbances.
1. Introduction
- Distributed estimation matters when spatially distributed sensors make joint decisions from partial measurements, particularly when process and measurement noise or disturbances raise robustness concerns.Digital communication constraints also limit the information nodes can transmit and receive.Source: §1 · p. 1
- The protocol adapts Round-Robin communication to distributed estimation: nodes exchange information at sampling times and poll one neighbour at a time.The sampling schedule is assumed known and agreed upon at each node.Source: §1 · p. 2
- The Round-Robin design yields non-switching observers: input channels switch, but observer gains stay unchanged, avoiding combinatorially complex gain scheduling.The paper identifies fixed gains as important for scalability in large networks.Source: §1 · p. 2
- Sufficient LMI conditions compute filter and interconnection gains that guarantee consensus between sampled-data observers tracking the plant, including under disturbances.The conditions also address transient H∞ consensus performance.Source: §1 · p. 2
- Generalized supply rates reflect sampled interconnections and vanish as the maximum sampling period approaches zero, recovering continuous-interconnection vector dissipativity properties.The conditions are sufficient and potentially conservative; continuous-interconnection feasibility is a preliminary, not conclusive, test.Source: §1 · p. 2
2. The problem formulation
The formulation considers observers on a directed graph, with continuous local measurements and inter-node information exchange at scheduled discrete times. It defines an H∞ consensus objective for disturbed estimation and a Round-Robin protocol that polls one neighbour at a time.
2.1. Graph theory
- The network is a directed graph G=(V,E), with incoming-neighbour sets and node in- and out-degrees; it is assumed weakly connected.The paper motivates this assumption by reducing consensus optimization over disconnected graphs to the corresponding problems on weakly connected components.Source: §2.1 · p. 3
2.2. Distributed estimation with H∞consensus
- The plant disturbance is assumed to be L2-integrable, and the plant is taken to have been at its initial state for all t≤0.The continuous-measurement interpretation focuses the formulation on sampling and intermittent interconnections.Source: §2.2 · p. 3
- Each observer estimates the plant from its measurements and neighbour feedback, with observer and coupling gains to be determined.Observers start with zero initial conditions, and the sampled terms before time zero are set to zero.Source: §2.2 · pp. 3–4
- Observers use local measurements continuously, while inter-node exchanges occur at shared discrete times and each node polls one neighbour according to the Round-Robin rule.Sampling times are assumed known to all participants; the sampled neighbour information is stored and used until the next polling cycle.Source: §2.2 · pp. 3–4
- The formulation measures transient consensus performance under disturbance through a disagreement cost based on the observers’ estimation errors.The cost expression is the same as for the corresponding continuous-time observer network.Source: §2.2 · pp. 4–5
- The design objective requires exponential stability without uncertainty and a specified transient consensus performance level for the observer network.The gains are selected to satisfy the conditions defining the distributed estimation problem.Source: §2.2 · p. 5
- Unlike earlier formulations, this problem targets internal stability and H∞ performance with observer inputs sampled according to the Round-Robin rule.The problem is a Round-Robin modification of the distributed H∞ consensus-based estimation problem.Source: §2.2 · p. 5
3. The main results
The main results give sufficient conditions for observer gains to meet the distributed estimation requirements, using a vector-dissipation analysis of sampled interconnections. The synthesis conditions are built through an LMI derivation, with a trade-off between design flexibility and solver burden.
3. The main results
- The analysis follows the time-delay approach of, representing the closed loop as a switched system with multiple ordered time-varying delays.The maximum communication delay is the longest interval between polls of the same channel; the network maximum is τ = max_i τ_i.Source: §3 · p. 5
- Theorem 1 states that when gains Ki and Li and matrices Wi satisfy the stated vector dissipation inequality for every i = 1,...,N, system (5) satisfies conditions (i) and (ii) of Definition 1.Source: §3 · p. 6
Lemma 3
- The descriptor method is used to derive LMIs, with the error equation driven by current and sampled interconnection inputs ordered by neighborhood.The free matrices are introduced in the derivation of the LMI conditions.Source: Lemma 3 · p. 8
- Theorem 2 provides a basis for Theorem 3, a sufficient condition for synthesizing observer networks that solve the distributed estimation problem.The corresponding network is stated to solve Definition 1 when the stated conditions hold.Source: Lemma 3 · pp. 9–10
- The free variables X_i, Z_i, and Q_i reduce LMI conservatism but increase solver unknowns; added constraints, such as diagonal X_i, reduce variables at the cost of conservatism.The trade-off is noted for high-dimensional problems where solver burden becomes excessive.Source: Lemma 3 · p. 10
4. Example
The example tests a three-node observer network in which sampled neighbor information supports nodes whose own measurements are insufficient for detectability. Its guaranteed H∞ disagreement is close to the continuous-time comparison at a high sampling rate, but rises sharply as the sampling period increases.
4. Example
- The three-node example uses a plant whose nominal part describes a regime of the Chua electronic circuit.The network is the three-node observer network shown in Figure 1.Source: §4 · p. 10
- Nodes 1 and 2 are not detectable from their own measurements, while node 3 is observable and acts as a conventional continuous-time H∞ filter.Nodes 1 and 2 use sampled neighbor data to overcome difficulties due to unstable unobservable modes of A.Source: §4 · pp. 10–11
- The example uses a constant sampling period, with τ1=Δ, τ2=2Δ, and τ3=0, and computes observer and interconnection gains by solving LMIs.The optimization minimizes γ2 subject to LMI constraints (11) and (21).Source: §4 · p. 11
- At Δ=0.0001, γ2=0.2274 for the proposed method versus γ2=0.2299 for the continuous-time method in.The paper attributes the slight discrepancy as likely due to numerical errors and/or conservative parameter selection.Source: §4 · p. 11
- At Δ=0.1, γ2=0.5537; at Δ=0.2, γ2=39.6506; at Δ=0.22, γ2=896.9248.The gain L3 at Δ=0.1 is [17.9083 13.1006 −19.6797]′, comparable with the estimator of at that γ2.Source: §4 · p. 11
5. Conclusions
The paper presents a sufficient LMI condition for designing a Round-Robin interconnection protocol for distributed observers. The protocol uses sampled communications without combinatorial gain scheduling, and the example achieves the H∞ consensus objective with moderately deteriorated performance.
5. Conclusions
- The paper presents a sufficient LMI condition for designing a Round-Robin interconnection protocol for networks of distributed observers.Source: §5 · p. 11
- The protocol uses sampled-data communications and does not require combinatorial gain scheduling.Source: §5 · p. 11
- The node observers achieve the H∞ consensus objective introduced in, at the expense of moderately deteriorated performance in the example.Source: §5 · p. 11
6. Appendix
The appendix proof of Theorem 1 partitions time according to held neighbor samples and applies an inequality to bound the resulting terms. The proof of Lemma 2 uses error increments and a matrix inequality.
6.1. Proof of Theorem 1
- Theorem 1 proof defines a vector of node error functions V and a matrix M with diagonal entries −2αi and neighbor entries πj.Source: §6.1 · p. 12
- Theorem 1 proof partitions time into intervals on which each neighbor-error input is held at a sampled value, then uses Wirtinger’s inequality.Source: §6.1 · pp. 12–13
6.2. Proof of Lemma 2
- Lemma 2 proof defines error increments δν from successive error samples and concludes using the matrix Ψ inequality.Source: §6.2 · p. 14