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14-11-2018

Resilient distributed state estimation with mobile agents: overcoming Byzantine adversaries, communication losses, and intermittent measurements

Authors: Aritra Mitra, John A. Richards, Saurabh Bagchi, Shreyas Sundaram

Published in: Autonomous Robots | Issue 3/2019

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Abstract

Applications in environmental monitoring, surveillance and patrolling typically require a network of mobile agents to collectively gain information regarding the state of a static or dynamical process evolving over a region. However, these networks of mobile agents also introduce various challenges, including intermittent observations of the dynamical process, loss of communication links due to mobility and packet drops, and the potential for malicious or faulty behavior by some of the agents. The main contribution of this paper is the development of resilient, fully-distributed, and provably correct state estimation algorithms that simultaneously account for each of the above considerations, and in turn, offer a general framework for reasoning about state estimation problems in dynamic, failure-prone and adversarial environments. Specifically, we develop a simple switched linear observer for dealing with the issue of time-varying measurement models, and resilient filtering techniques for dealing with worst-case adversarial behavior subject to time-varying communication patterns among the agents. Our approach considers both communication patterns that recur in a deterministic manner, and patterns that are induced by random packet drops. For each scenario, we identify conditions on the dynamical system, the patrols, the nominal communication network topology, and the failure models that guarantee applicability of our proposed techniques. Finally, we complement our theoretical results with detailed simulations that illustrate the efficacy of our algorithms in the presence of the technical challenges described above.

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In the absence of any constraints placed on the sensing capabilities or movement patterns of an agent, one can just have each mobile agent patrol all the sensing locations. However, such an assumption would in general be impractical, thereby necessitating inter-agent communication. Note that it is precisely the need for inter-agent communication that makes the issues of communication losses and adversarial attacks studied in this paper relevant.

We resort to such a notation here since the superscript on the z[k] states are reserved for eigenvalues, and the subscripts are reserved for mobile agents. Thus, we introduce the notation \({\mathbf {v}}[k]\), with a superscript on \({\mathbf {v}}[k]\) pointing to a location number.

The gains \({\mathbf {L}}^{(i_r)}_i\) are agent-specific, since different agents might visit the same location with different frequencies.

Essentially, an odd period ensures that eigenvalues that are equal in magnitude, but opposite in sign in \({\mathbf {A}}\), remain so in \({\mathbf {A}}^{{\bar{T}}}\). Thus, if the eigenvalues of \({\mathbf {A}}\) are distinct in magnitude, then clearly no restrictions need to be imposed on the time-period \({\bar{T}}\).

Since we are considering system matrices with distinct eigenvalues, an eigenvalue is detectable w.r.t. the pair \(({\mathbf {A}},{\mathbf {C}}^{({\mathcal {P}}_i)})\) if and only if it is detectable w.r.t. \(({\mathbf {A}},{\mathbf {C}}^{(i_r)})\), for some \(i_r \in {\mathcal {P}}_i.\) The ‘only if’ part of the statement may not be true for system matrices with repeated eigenvalues.

This is one of the key differences of our present formulation with the resilient consensus literature. In the latter setting, there is no external state that needs to be tracked, and Sundaram and Hadjicostis (2011) and Pasqualetti et al. (2012) have shown that making the network sufficiently connected suffices to facilitate resilient consensus.

Details of such an attack strategy can be found in Mitra and Sundaram (2018c). For centralized systems where f sensors are compromised, Fawzi et al. (2014) and Chong et al. (2015) have shown that for recovering the state of the system asymptotically, the system must remain detectable after the removal of any 2f sensors.

For notational simplicity, while considering the eigenvalue \(\lambda _j\), we drop the superscript ‘j’ on the time-stamp \(\phi _{il}[k]\) and the delay \(\tau _{il}[k]\).

If agent i receives an estimate without a time-stamp from some agent in \({\mathcal {N}}^{(j)}_i \cap {\mathcal {A}}\), it simply assigns a value of 0 to such an estimate (without loss of generality). Note that based on Assumption 2, agent i is guaranteed to receive a time-stamped estimate from every regular agent l in \({\mathcal {N}}^{(j)}_i\) at least once over every interval of the form \([k-T,k], \, \forall k \ge T\), i.e., for each \(l \in {\mathcal {N}}^{(j)}_i \cap {\mathcal {R}}\), \({\bar{z}}^{(j)}_{il}[k]\) will necessarily be of the form \( {\lambda _j}^{\tau _{il}[k]}{\hat{z}}^{(j)}_l[k-\tau _{il}[k]]\), \(\, \forall k\ge T\).

In other words, due to false time-stamp information, the quantity \({\hat{z}}^{(j)}_l[k-\tau _{il}[k]]\) may not represent the true estimate of an adversarial agent l at time \((k-\tau _{il}[k])\). Thus, we resort to a slight abuse of notation here.

Explicit dependence of u, v on the parameters represented by i, j, l and k is not shown to avoid cluttering of the exposition.

Although we only establish asymptotic stability of the error dynamics in Proposition 1, verifying exponential stability is fairly straightforward, and hence, not explicitly proven.

Unlike the SW-LFRE algorithm developed in Sect. 5, the algorithm we propose here is memoryless, i.e., at each time-step, an agent acts only on the information that it acquires (via measurements and from neighboring agents) at that time-step. We do this primarily to simplify the analysis.

The choice of \(m \ge 3\) is justified later in Remark 13.

To avoid cluttering the exposition, we drop the superscript ‘j’ on \({\mathcal {I}}_i[k]\) and certain other terms throughout the proof, since they can be inferred from context.

The result continues to hold for the general update rule (21).

The set \(\mathcal {M}^{(j)}_i[k]\) is not well-defined when \(\mathcal {I}_i[k]=1\). For such a case, l can be taken to be any node in the set \(\mathcal {N}^{(j)}_i\cap \mathcal {R}\).

The need for strong \((3f+1)\)-robustness in the baseline network was provided in Remark 13, and will also be justified explicitly via simulations.

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Title: Resilient distributed state estimation with mobile agents: overcoming Byzantine adversaries, communication losses, and intermittent measurements
Authors: Aritra Mitra
John A. Richards
Saurabh Bagchi
Shreyas Sundaram
Publication date: 14-11-2018
Publisher: Springer US
Published in: Autonomous Robots / Issue 3/2019
Print ISSN: 0929-5593
Electronic ISSN: 1573-7527
DOI: https://doi.org/10.1007/s10514-018-9813-7

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