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Distributed robust estimation over randomly switching networks using $H_\infty$ consensus

V. Ugrinovskii

arXiv:1512.01294v1eess.SY

TL;DR

The paper addresses robust distributed estimation when network topology and sensing conditions switch randomly but nodes lack global network information. It designs locally selected observer gains through an auxiliary non-fragile H∞ consensus problem, with LMIs for globally known states and rank constraints for local information. The resulting estimator provides guaranteed suboptimal H∞ disagreement while accommodating dependent links and additional random events.

  • Problem

    Distributed estimators must handle Markovian topology variation without requiring every node to know the complete network state or assuming independent communication links.

  • Method

    The paper uses a two-step robust observer design: solve an auxiliary non-fragile H∞ consensus problem, then condition gains on locally available communication and sensing states.

  • Results

    The constructed estimator guarantees suboptimal H∞ disagreement using local network information; globally informed gains use LMIs, while local information introduces additional rank constraints.

  • Takeaways & Limitations

    The framework supports robust distributed estimation without global topology broadcasts or independent-link assumptions and can incorporate sensor failures, recoveries, and other random network events.

  • Takeaways & Limitations

    The local-information design is non-convex because of rank constraints, making the resulting matrix-inequality problem difficult to solve in general.

Abstract

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The paper considers a distributed robust estimation problem over a network with Markovian randomly varying topology. The objective is to deal with network variations locally, by switching observer gains at affected nodes only. We propose sufficient conditions which guarantee a suboptimal $H_\infty$ level of relative disagreement of estimates in such observer networks. When the status of the network is known globally, these sufficient conditions enable the network gains to be computed by solving certain LMIs. When the nodes are to rely on a locally available information about the network topology, additional rank constraints are used to condition the gains, given this information. The results are complemented by necessary conditions which relate properties of the interconnection graph Laplacian to the mean-square detectability of the plant through measurement and interconnection channels.

1 Introduction

The paper develops robust distributed estimation for randomly switching networks without global topology broadcasts or independent-link assumptions. It combines local-state gain selection with H∞ consensus objectives and graph-based detectability conditions.

  • 1 Introduction: Markovian network models capture memory-dependent communication losses and other random events, but conventional analyses usually assume every node knows the complete Markov state.Independent two-state link models avoid that assumption only at the cost of potentially exponential design complexity and questionable practicality under congestion.
  • 1 Introduction: The method enables distributed filtering using only locally available connectivity information, without global topology broadcasts or Markovian segmentation.It also accommodates dependent communication links and random sensor failures or recoveries.
  • 1 Introduction: Relative H∞ consensus targets disagreement among node estimates despite uncertain plant, measurement, and interconnection models.Vector storage functions and supply rates support mean-square robust convergence and transient relative-consensus performance analysis.
  • 1 Introduction: A two-step design first solves an auxiliary globally informed problem, then modifies its non-fragile estimators to satisfy the local information constraint while retaining robust performance.The approach addresses the non-Markovian nature of local network-state information.
  • 1 Introduction: The communication topology must satisfy more than a simple zero Laplacian eigenvalue: detectability of observer–interconnection matrix pairs is also required.These conditions connect graph structure with mean-square detectability through measurement and interconnection channels.
  • 1 Introduction: The paper is organized around problem formulation, an unconstrained auxiliary estimation problem, the constrained main result, topology requirements, and an illustrative example.Notation includes Euclidean norms, weighted norms, Kronecker products, block-diagonal matrices, and L2 signal spaces.

2 Problem formulation

The formulation models communication and sensing as a stationary Markovian global process assembled from node-local states, while observer gains depend only on each node’s local state. It defines robust estimation and relative-disagreement objectives for uncertain plants, measurements, and channels.

  • 2.1 Networks with Markovian switching topology: Random link dropouts, recoveries, sensor adjustments, and sensor failures or recoveries jointly vary each node’s neighborhood and sensing regime.Local state processes need not be Markov, and link statuses may be statistically dependent.
  • 2.1 Networks with Markovian switching topology: Each node’s local communication and sensing state combines its neighborhood with its measurement-matrix triplet, and feasible local states are indexed separately.Global configurations are formed from local states, including dependencies that make not every combination feasible.
  • 2.1 Networks with Markovian switching topology: The network is a directed weakly connected graph whose topology switches among subgraphs represented by a stationary Markov process, with every active graph assumed weakly connected.The global state determines adjacency, degrees, and the corresponding Laplacian.
  • 2.2 Distributed estimation with H∞ consensus: The plant and node measurements include deterministic disturbances and uncertain coefficients, while communication channels carry uncertainty affecting transmitted information.The disturbance assumptions include L2-integrability and mean-square L2-integrability where specified.
  • 2.2 Distributed estimation with H∞ consensus: Observers process local measurements and received neighbor signals through switching innovation and interconnection gains, with gains constrained to depend on local rather than global states.The problem is to choose these gain functions to satisfy robust performance criteria.
  • 2.2 Distributed estimation with H∞ consensus: A node may fail to receive or accept a broadcast because of random congestion, even when the sender always transmits it.This motivates allowing channel reception uncertainty independently of the receiving node’s local sensing state.
  • 2.2 Distributed estimation with H∞ consensus: The disagreement function measures average total disagreement between each estimate and neighboring estimates for the current network state.It defines the transient consensus performance metric used in the distributed estimation problem.
  • 2.2 Distributed estimation with H∞ consensus: The formulation seeks switching observer and coupling gains that ensure mean-square and almost-sure convergence properties for node estimators under the stated robust criteria.The conditions distinguish exponential mean-square convergence, almost-sure asymptotic convergence, and convergence in mean square and with probability one.

3 An auxiliary global distributed estimation problem

This section develops an auxiliary distributed estimation problem without locality constraints, assuming every node knows the global network state. Feasible coupled LMIs provide observer gains guaranteeing robust consensus and estimation properties under admissible perturbations.

  • 3 An auxiliary global distributed estimation problem: The auxiliary problem estimates the uncertain plant state using a network of uncertain estimators subject to norm-bounded estimator perturbations.The estimation errors and perturbations are modeled jointly with the Markovian network state.
  • 3 An auxiliary global distributed estimation problem: Under the theorem’s conditions, the estimator-error subsystem is mean-square exponentially stable and asymptotically stable with probability 1.These stability guarantees apply for all estimator perturbations satisfying the stated constraints.
  • 3 An auxiliary global distributed estimation problem: The same conditions guarantee the mean-square consensus performance requirement in the presence of admissible exogenous disturbances.The guarantee concerns the relative disagreement performance metric used for distributed estimation.
  • 3 An auxiliary global distributed estimation problem: All estimators converge in mean square and with probability 1 when the auxiliary conditions hold.The convergence statement corresponds to the paper’s conditions (8) and (9).
  • 3 An auxiliary global distributed estimation problem: Feasible coupled LMIs yield observer gains that solve the auxiliary distributed consensus estimation problem.Theorem 1 establishes this result and gives the corresponding matrix P for the performance condition.
  • 3 An auxiliary global distributed estimation problem: The global-state solution directly supports the globally informed distributed H∞ consensus estimator through an LMI-based corollary.The corollary replaces the auxiliary perturbation specialization with coupled LMIs for the globally available Markov state.

4 The main result

The paper constructs locally implementable robust distributed observers for Markovian switching networks by conditioning auxiliary gains on local network states. The resulting network solves the robust consensus estimation problem, while rank constraints make the synthesis non-convex.

  • 4 The main result: Feasible gains are characterized by LMIs together with rank constraints indexed by global network states and local node information.The theorem requires matrices satisfying inequalities (14), (15), (20), and rank constraint (21).
  • 4 The main result: The local-gain construction preserves the auxiliary H∞ consensus guarantee because the induced estimator mismatch satisfies the required admissible perturbation conditions.The proof links the conditional-expectation gains to perturbations satisfying condition (12).
  • 4 The main result: Local gains obtained from asymptotic conditional expectations of auxiliary gains yield a distributed estimator that solves the robust consensus estimation problem.The gains depend on each node’s local communication and sensing state process.
  • 4 The main result: The rank constraints make the solution set non-convex, so solving the synthesis problem is generally difficult.The paper notes that numerical algorithms are available for such problems.

5 Requirements on the communication graph and interconnections

The paper relates mean-square detectability of the distributed observer to properties of the switching network’s graph Laplacians and measurement/interconnection channels. Under a simple zero-eigenvalue condition, the stated Laplacian properties provide necessary and sufficient conditions.

  • 5 Requirements on the communication graph and interconnections: The topology requirements concern mean-square stabilizability of estimation-error dynamics via output injection in the absence of perturbations.This places the graph analysis within stochastic mean-square detectability for linear jump-parameter systems.
  • 5 Requirements on the communication graph and interconnections: A necessary observability requirement is that every combination of undetectable measurement states across nodes forms an observable vector through the aggregate measurement and interconnection model.This result is stated for the observer problem under the simplifying homogeneous-channel assumptions introduced in the section.
  • 5 Requirements on the communication graph and interconnections: The graph results do not use the locality information-structure constraint and therefore also apply to the auxiliary distributed estimation problem.The section notes specializations for balanced graphs and graphs containing a spanning tree.

6 Example

The example evaluates robust distributed observers on a five-node network whose topology and sensing regimes switch according to a two-state Markov chain. It illustrates locally conditioned gains, including a nonswitching observer at node 1, and verifies the design through simulation.

  • 6 Example: The five-node observer network operates intermittently in two regimes governed by a two-state Markov chain, with switching topologies shown in Figure 1.The plant is observed over a network whose graph configurations vary over time.
  • 6 Example: The example uses coefficient values C∗1 = 10^-3 × [3.1923 − 4.6597 1] and C∗2 = [−0.8986 0.1312 −1.9703].These coefficients are listed in Table 1 for the two network regimes.
  • 6 Example: Nodes 3, 4, and 5 have varying neighbourhoods, node 2 changes sensor parameters, and node 1 therefore uses a nonswitching observer gain.All nodes except node 1 track the global state through their local state processes.
  • 6 Example: Nodes 1 and 4 include undetectable modes and node 2 can switch between detectable and undetectable pairs, so estimation relies on neighbour communication.The example sets D_i^k = 0.025, H_ij = I_3×3, and G_ij = 0.5 × I_3×3.
  • 6 Example: Both network instances have spanning trees rooted at detectable nodes 3 and 5, satisfying the necessary global-detectability condition.The pair (H, A) is also observable.
  • 6 Example: Numerical simulations use random initial conditions and decaying sinusoidal perturbations to examine errors at nodes 1, 2, and 5 under different observer configurations.The plotted realization includes the global state process η(t) and first-coordinate estimation errors.

7 Conclusions

The paper establishes robust distributed consensus-estimator conditions for Markovian networks using only locally available communication and sensing information. Global-state designs reduce to LMIs, whereas local information introduces rank constraints and associated algorithmic challenges.

  • 7 Conclusions: The proposed estimator guarantees a suboptimal H∞ disagreement level while using only locally available communication and sensing-state information.The synthesis conditions support robust distributed consensus estimation over Markovian networks.
  • 7 Conclusions: When the network’s global state is available, the feasibility problem is convex and the LMIs can be solved using methods such as decentralized gradient descent.The LMI structure is partitioned to enable decentralized solution approaches.
  • 7 Conclusions: Removing global network-state broadcast requires rank constraints in addition to the LMI conditions.Developing numerical algorithms that exploit the partitioned LMIs and rank constraints remains future work.

8 Appendix: Proof of Theorem 1.

The appendix proves Theorem 1 by applying a vector Lyapunov analysis to the interconnected Markov-switching system. A dissipation inequality, Dynkin’s formula, and a continuous-time Robbins–Siegmund argument yield almost-sure convergence and stability results.

  • 8 Appendix: Proof of Theorem 1.: The continuous-time Robbins–Siegmund lemma ensures that the relevant nonnegative process has an almost-sure finite limit under the stated integrability conditions.The lemma assumes right continuity, local integrability, finite accumulated φ, and an almost-sure inequality.
  • 8 Appendix: Proof of Theorem 1.: A vector Lyapunov candidate with quadratic components is constructed for the interconnected subsystems, and its infinitesimal generator is analyzed.The generator is used to derive the dissipation inequality underlying Theorem 1.
  • 8 Appendix: Proof of Theorem 1.: Completing the squares and applying the LMI produce an inequality that bounds the coupled subsystem dynamics and uncertainty terms.The proof verifies the required generator inequalities for arbitrary admissible signals.
  • 8 Appendix: Proof of Theorem 1.: Dynkin’s formula converts the dissipation inequality into conditional expectation bounds for the Markov process formed by the errors and network state.The proof then applies the convergence lemma to the resulting processes.
  • 8 Appendix: Proof of Theorem 1.: For arbitrary square-integrable disturbances, each estimation error converges to zero with probability one.This follows from finite accumulated error energy and existence of the limiting error norm.
  • 8 Appendix: Proof of Theorem 1.: With disturbances set to zero, the same argument establishes almost-sure internal stability and internal exponential mean-square stability.The latter follows from the nonnegativity of Ψ^k.
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