Distributed Estimation Problem Research Articles

Event-based distributed state estimation for linear systems under unknown input and false data injection attack

This paper investigates an event-based distributed state estimation problem for a linear system subject to unknown input and false data injection attack. The unknown input is treated as a process with a non-informative prior. An event-triggered transmission scheme is designed to reduce exchange of unreliable state estimate information between sensor nodes. In order to defend against false data injection attack, a novel event-based distributed state estimator is proposed by using state estimate information that comes from individual node and neighboring nodes, where filter gains are derived by minimizing an upper bound of estimation error covariance. Then sufficient conditions are established to ensure asymptotic boundedness of the designed estimation error covariance. Finally, effectiveness of the proposed technique is demonstrated by a numerical example.

On Bayesian Fisher Information Maximization for Distributed Vector Estimation

In this paper, we consider the problem of bandwidth-constrained distributed estimation of a Gaussian vector with linear observation model. Each sensor makes a scalar noisy observation of the unknown vector, employs a multi-bit scalar quantizer to quantize its observation, and maps it to a digitally modulated symbol. Sensors transmit their symbols over orthogonal-power-constrained fading channels to a fusion center (FC). The FC is tasked with fusing the received signals from sensors and estimating the unknown vector. We derive the Bayesian Fisher Information Matrix (FIM) for three types of receivers: (i) coherent receiver; (ii) noncoherent receiver with known channel envelopes; and (iii) noncoherent receiver with known channel statistics only. We also derive the Weiss–Weinstein bound (WWB). We formulate two constrained optimization problems, namely maximizing trace and log-determinant of Bayesian FIM under network transmit power constraint, with sensors’ transmit powers being the optimization variables (we refer to as FIM-max schemes). We show that for coherent receiver, these problems are concave. However, for noncoherent receivers, they are not necessarily concave. The solution to the trace of Bayesian FIM maximization problem can be implemented in a distributed fashion, in the sense that each sensor calculates its own transmit power using its local parameters. On the other hand, the solution to the log-determinant of Bayesian FIM maximization problem cannot be implemented in a distributed fashion and the FC needs to find the powers (using parameters of all sensors) and inform the active sensors of their transmit powers. We numerically investigate how the FIM-max power allocation across sensors depends on the sensors observation qualities and physical layer parameters as well as the network transmit power constraint. Moreover, we evaluate the system performance in terms of mean square error (MSE) using the solutions of FIM-max schemes, and compare it with the solution obtained from minimizing the MSE of the LMMSE estimator (MSE-min scheme), and that of uniform power allocation. These comparisons illustrate that, although the WWB is tighter than the inverse of Bayesian FIM, it is still suitable to use FIM-max schemes, since the performance loss in terms of the MSE of the LMMSE estimator is not significant. Furthermore, comparing the performance of different receivers, our numerical results reveal that coherent receiver and noncoherent receiver with known channel statistics have the best and the worst performance, respectively.

Distributed state estimation by a network of observers under communication and measurement delays

In this paper we study the distributed state estimation of autonomous dynamic systems, under arbitrarily large communication and measurement delays. In the distributed estimation problem, we consider a network of observer agents that cannot independently determine all the states of the observed plant from their own collected measurements, but they must collaborate with neighboring agents to agree on the correct state estimation. Our current work considers the presence of arbitrarily large time delays in the communication and measurements of the observer agents, and we propose a framework for distributed observers following the low gain methodology. Sufficient conditions for the stability of the corresponding observation error dynamics are derived, including an upper bound for the low gain parameter of the observer equations. In the presence of bounded noisy measurements from the plant, we derive the upper-bound on the local estimation errors. The results of the distributed state estimation is then applied to propose a solution to the leader–follower synchronization problem for multi-agent systems under leader-agent and inter-agent communication delays. An illustrative example is also presented to verify the effectiveness of the theoretical analysis.

Distributed Interpolatory Algorithms for Set Membership Estimation

This work addresses the distributed estimation problem in a set membership framework. The agents of a network collect measurements which are affected by bounded errors, thus implying that the unknown parameters to be estimated belong to a suitable feasible set. Two distributed algorithms are considered, based on projections of the estimate of each agent onto its local feasible set. The main contribution of the paper is to show that such algorithms are asymptotic interpolatory estimators, i.e. they converge to an element of the global feasible set, under the assumption that the feasible set associated to each measurement is convex. The proposed techniques are demonstrated on a distributed linear regression estimation problem.

Distributed Estimation of Gaussian Correlations

We study a distributed estimation problem in which two remotely located parties, Alice and Bob, observe an unlimited number of i.i.d. samples corresponding to two different parts of a random vector. Alice can send k bits on average to Bob, who in turn wants to estimate the cross-correlation matrix between the two parts of the vector. In the case where the parties observe jointly Gaussian scalar random variables with an unknown correlation ρ, we obtain two constructive and simple unbiased estimators attaining a variance of (1 - ρ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> )/(2k ln 2), which coincides with a known but non-constructive random coding result of Zhang and Berger. We extend our approach to the vector Gaussian case, which has not been treated before, and construct an estimator that is uniformly better than the scalar estimator applied separately to each of the correlations. We then show that the Gaussian performance can essentially be attained even when the distribution is completely unknown. This in particular implies that in the general problem of distributed correlation estimation, the variance can decay at least as O(1/k) with the number of transmitted bits. This behavior, however, is not tight: we give an example of a rich family of distributions for which local samples reveal essentially nothing about the correlations, and where a slightly modified estimator attains a variance of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-Ω(k)</sup> .

Distributed Kalman Filters With State Equality Constraints: Time-Based and Event-Triggered Communications

In this paper, we investigate a distributed estimation problem for multi-agent systems with state equality constraints (SEC). First, under a time-based consensus communication protocol, applying a modified projection operator and the covariance intersection fusion method, we propose a distributed Kalman filter with guaranteed consistency and satisfied SEC. Furthermore, we establish the relationship between consensus step, SEC and estimation error covariance in dynamic and steady processes, respectively. Employing a space decomposition method, we show the error covariance in the constraint set can be arbitrarily small by setting a sufficiently large consensus step. Besides, we propose an extended collective observability (ECO) condition based on SEC, which is milder than existing observability conditions. Under the ECO condition, through utilizing a technique of matrix approximation, we prove the boundedness of error covariance and the exponentially asymptotic unbiasedness of state estimate, respectively. Moreover, under the ECO condition for linear time-invariant systems with SEC, we provide a novel event-triggered communication protocol by employing the consistency, and give an offline design principle of triggering thresholds with guaranteed boundedness of error covariance. More importantly, we quantify and analyze the communication rate for the proposed event-triggered distributed Kalman filter, and provide optimization based methods to obtain the minimal (maximal) successive non-triggering (triggering) times. Two simulations are provided to demonstrate the developed theoretical results and the effectiveness of the filters.

Distributed adaptive node-specific signal estimation in a wireless sensor network with noisy links

We consider a distributed signal estimation problem in a wireless sensor network where each node aims to estimate a node-specific desired signal using all sensor signals available in the network. In this setting, the distributed adaptive node-specific signal estimation (DANSE) algorithm is able to learn optimal fusion rules with which the nodes fuse their sensor signals, as the fused signals are then transmitted between the nodes. Under the assumption of transmission without errors, DANSE achieves the performance of centralized estimation. However, noisy communication links introduce errors in these transmitted signals, e.g., due to quantization or communication errors. In this paper we show fusion rules which take additive noise in the transmitted signals into account at almost no increase in computational complexity, resulting in a new algorithm denoted as ‘noisy-DANSE’ (N-DANSE). As the convergence proof for DANSE cannot be straightforwardly generalized to the case with noisy links, we use a different strategy to prove convergence of N-DANSE, which also proves convergence of DANSE without noisy links as a special case. We validate the convergence of N-DANSE and compare its performance with the original DANSE through numerical simulations, which demonstrate the superiority of N-DANSE over the original DANSE in noisy links scenarios.

A Scalable Algorithm for Event-Triggered State Estimation With Unknown Parameters and Switching Topologies Over Sensor Networks

An event-triggered distributed state estimation problem is investigated for a class of discrete-time nonlinear stochastic systems with unknown parameters over sensor networks (SNs) subject to switched topologies. An event-triggered communication strategy is employed to govern the information broadcast and reduce the unnecessary resource consumption. Based on the adopted communication strategy, a distributed state estimator is designed to estimate the plant states and also identify the unknown parameters. In the framework of input-to-state stability, sufficient conditions with an average dwell time are established to ensure the boundedness of estimation errors in mean-square sense. In addition, the gains of the designed estimators are dependent on the solution of a set of matrix inequalities whose dimensions are unrelated to the scale of underlying SNs, thereby fulfill the scalability requirement. Finally, an illustrative simulation is utilized to verify the feasibility of the proposed design scheme.

Distributed State Estimation Using Intermittently Connected Robot Networks

This paper considers the problem of distributed state estimation using multi-robot systems. The robots have limited communication capabilities and, therefore, communicate their measurements intermittently only when they are physically close to each other. To decrease the distance that the robots need to travel only to communicate, we divide them into small teams that can communicate at different locations to share information and update their beliefs. Then, we propose a new distributed scheme that combines (i) communication schedules that ensure that the network is intermittently connected, and (ii) sampling-based motion planning for the robots in every team with the objective to collect optimal measurements and decide a location for those robots to communicate. To the best of our knowledge, this is the first distributed state estimation framework that relaxes all network connectivity assumptions, and controls intermittent communication events so that the estimation uncertainty is minimized. We present simulation results that demonstrate significant improvement in estimation accuracy compared to methods that maintain an end-to-end connected network for all time.

Distributed energy‐efficient target tracking algorithm based on event‐triggered strategy for sensor networks

The distributed estimation problem for wireless sensor networks with limited communication/sensing ranges and observability is studied. A novel sensor measuring activation scheme based on a fully distributed event-triggered strategy is proposed to make each node achieve a better trade-off between estimation error and energy saving. The strategy depends on both the predicted synthetic performance index and the predicted position of the target. A distributed Kalman filtering algorithm based on the minimum trace fusion principle is proposed. It is proved that comparing with the time-triggered strategy, the proposed event-triggered measuring strategy has better performance. Although the event-triggered measuring topology is time-varying and each sensor is not observable, it is proved that as long as there exists at least one collaboratively observable sensor in the available distance-based sensing network at each time instant, the estimation errors are bounded in mean square sense. Simulation examples are given to illustrate the validity of the algorithm.

Non-Fragile Distributed Fault Estimation for a Class of Nonlinear Time-Varying Systems Over Sensor Networks: The Finite-Horizon Case

This paper is concerned with a new non-fragile distributed fault estimation problem for a class of time-varying systems through sensor networks over a finite horizon. Both randomly occurring nonlinearities and randomly occurring gain variations, whose occurrences are governed by random variables obeying certain probabilistic distributions on the interval $[0,1]$ are taken into simultaneous consideration. The fault estimation is based on the information not only from the individual sensor, but also from its neighboring ones according to the given topology of the sensor network. The aim of this paper is to design a non-fragile distributed fault estimator such that, in the presence of certain drifts/variations/perturbations of the gain parameters during the implementation, a prescribed average $H_{\infty }$ performance constraint on the estimation error dynamics is satisfied. Based on stochastic analysis techniques, the gain parameters are calculated by solving a series of recursive linear matrix inequalities over a finite horizon. Moreover, a simulation example is provided to show the effectiveness of the proposed fault estimation scheme.

A Simple Approach to Distributed Observer Design for Linear Systems

This note investigates the distributed estimation problem for continuous-time linear time-invariant (LTI) systems observed by a network of observers. Each observer in the network has access to only part of the output of the observed system, and communicates with its neighbors according to a given network graph. In this note we recover the known result that if the observed system is observable and the network graph is a strongly connected digraph, then a distributed observer exists. Moreover, the estimation error can be made to converge to zero at any a priori given decay rate. Our approach leads to a relatively straightforward proof of this result, using the mirror of the balanced graph associated with the original network graph. The numerical design of our distributed observer is reduced to solving linear matrix inequalities (LMI's). Each observer in the network has state dimension equal to that of the observed plant.

Common Criterion of Privacy Metrics and Parameters Analysis Based on Error Probability for Randomized Response

Privacy security issues under the classic randomized response (RR) model proposed by Warner and its extended $K$ -RR model are studied. First, in order to provide references for the accuracy of the private distribution estimation problem under RR mechanism, the lower bounds of the differential privacy parameter and the number of participates are deduced for an accuracy objective of $({\alpha }, {\delta })$ – accurate in statistics. Second, when the prior distribution is nonuniform, the data utility has a ceiling effect in the high privacy region by taking the prior into account. In this case, the average distortion, which is defined as the expected Hamming distance between the input and output data, is no longer feasible to measure the data utility. Motivated by this, the error probability is proposed as a measure of data utility for unifying different privacy metrics, where the error probability is defined to be the expected Hamming distance between the input and reconstructed data based on maximum a posteriori estimation. Third, under a unified privacy-preserving framework using RR mechanism based on error probability criterion, the relationship among differential privacy, identifiability privacy, and mutual information privacy is established. Given a maximum allowable error probability $P_{E}^{\mathrm {max}}$ , the optimal privacy parameters of these three privacy notions are derived with the full consideration of the prior distribution. Then, a Bayes-based utility function, which corresponds to the converse of the Bayes risk, is constructed to measure the degree of privacy leakage. Given a maximum allowable correct probability $P_{C}^{\mathrm {max}}$ , the accuracy objective of the statistical estimate is considered to derive the range of local differential privacy parameter from the perspective of security. Fourth, all the research results above are further extended to $K$ -RR mechanism too. Finally, the correctness and effectiveness are further verified by simulation experiments. The results reveal that the error probability can be applied to any prior distribution case, while the average distortion criterion is only a special case with uniform distribution. Therefore, the error probability proposed in the paper is more reasonable to be used as a common criterion to measure the data utility for the RR model so as to unify different privacy metrics.

UIF-based cooperative tracking method for multi-agent systems with sensor faults

For maneuvering target tracking with sensor faults, consensus-based distributed state estimation problems are studied herein. The communication status of the nonlinear system composed of multiple agents is described using the graph theory. Considering the impacts caused by sensor failures on measurement equations, a weighted average consensus-based unscented information filter (UIF) algorithm is proposed to improve tracking accuracy. Moreover, the estimation error for the investigated nonlinear system has been analyzed based on the stochastic boundedness theory to evaluate the proposed algorithms performance and feasibility. Finally, simulation results are presented to assert the validity of the method.

Fully Distributed Dynamic State Estimation With Uncertain Process Models

This paper studies the problem of distributed state estimation in the scenario where the process model of interest is not certain to any local agent. By extending the previously proposed distributed hybrid information fusion (DHIF) algorithm to this considered scenario, two algorithms are proposed by following the frameworks of two well-known multiple-model (MM) approaches, namely, the first-order generalized pseudo-Bayesian and interacting MM approaches. The extended algorithms inherit the advantages of the original DHIF algorithm and are hence fully distributed, robust against agents not directly sensing the target, and only require a single communication iteration among agents during each sampling interval. It is also shown in the case when the unknown underlying model/mode is fixed; all local agents are able to asymptotically identify the true underlying model/mode and estimate the state of interest simultaneously. Sufficient conditions are formulated. Simulations are shown to illustrate the analytical results as well as the performances of the proposed algorithms.

Distributed Estimation in Periodically Switching Sensor Networks

This paper studies the distributed estimation problem of sensor networks, in which each node is periodically sensing and broadcasting in order. A consensus estimation algorithm is applied, and a weight design approach is proposed. The weights are designed based on an adjusting parameter and the nodes’ lengths of their shortest paths to the target node. By introducing a (T+2)-partite graph of the time-varying networks over a time period [0,T] and studying the relationships between the product of the time-sequence estimation error system matrices and the sequences of edges in the (T+2)-partite graph, a sufficient condition in terms of the observer gain and the adjusting parameter for the stability of the estimation error system is proposed. A simulation example is given to illustrate the results.

Distributed Observers for LTI Systems

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.

Distributed state estimation for stochastic discrete-time sensor networks with redundant channels

This paper investigates the distributed state estimation problem for a class of sensor networks described by discrete-time stochastic systems with stochastic switching topologies. In the sensor network, the redundant channel and the randomly varying nonlinearities are considered. The stochastic Brownian motions affect both the dynamical plant and the sensor measurement outputs. Through available output measurements from each individual sensor, the distributed state estimators are designed to approximate the states of the networked dynamic system. Sufficient conditions are established to guarantee the convergence of the estimation error systems for all admissible stochastic disturbances and randomly varying nonlinearities. Then, the distributed state estimator gain matrices are derived using the linear matrix inequality method. Moreover, a numerical example is given to verify the effectiveness of the theoretical results.

Distributed H∞ Estimation in Sensor Networks With Two-Channel Stochastic Attacks.

This paper is concerned with the distributed estimation problem in sensor networks subjected to unknown attacks. Network attacks are considered to exist in two classes of channels: 1) communication channels from the plant to sensors and 2) communication channels among sensors. The status of an attack is viewed as a stochastic phenomenon, and the transmitted information will be affected when the attacker successfully carries out an attack on the related data packet. Based on the sensors' own measurements and their neighbors' local information, a novel distributed estimation model against two-channel stochastic attacks is presented. A sufficient condition on the existence of the desired distributed H ∞ estimators is derived and the distributed estimator gains are designed by solving a linear matrix inequality. Two illustrative examples are provided to demonstrate the effectiveness of the new design techniques.

Distributed sparse diffusion estimation with reduced communication cost

The issue considered in the current study is the problem of adaptive distributed estimation based on diffusion strategy which can exploit sparsity in improving estimation error and reducing communications. It has been shown that distributed estimation leads to a good performance in terms of the error value, convergence rate, and robustness against node and link failures in wireless sensor networks. However, the main focus of many works in the field of distributed estimation research is on convergence speed and estimation error, neglecting the fact that communications among the nodes require a lot of transmissions. In this work, the focus is on a solution based on sparse diffusion least mean squares (LMS) algorithm, and a new version of sparse diffusion LMS algorithm is proposed which takes both communications and error cost into account. Also, the computation complexity and communication cost for every node of the network, as well as performance analysis of the proposed strategy, is provided. The performance of the proposed method in comparison with the existing methods is illustrated by means of simulations in terms of computational and communicational cost, and flexibility to signal changes.