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State estimation in networked systems

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State fusion with unknown correlation: Ellipsoidal intersection
This article focuses on the problem of fusing two prior Gaussian estimates into a single estimate, when the correlation is unknown. Existing solutions either lead to a conservative fusion result, as the chosen parametrization focuses on the fusion formulas instead of correlations, or they are computationally expensive. The contribution of this article is a novel parametrization, in which the correlation is explicitly characterized a priori to deriving the fusion formulas. Then, maximizing the correlation ensures that the fusion result is based on independent parts of the prior estimates and, simultaneously, addresses the fact that the correlation is unknown. In addition, a guaranteed improvement of the accuracy after fusion is attained. An illustrative example demonstrates the benefits of the proposed method compared to an existing fusion method.

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On the dispersion of the Kalman filtering algorithm
Distributed Kalman filtering (DKF) is a rapidly emerging highly relevant research topic within the control systems community, mainly due to recent advances in (wireless) sensor networks. Application examples of DKF vary from estimating the position and speed of each vehicle on a highway, to condition monitoring of a bridge via accelerometers.

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Empirical casestudies of state fusion via ellipsoidal intersection
This article presents a practical assessment of the recently developed state fusion method ellipsoidal intersection and focusses on distributed state estimation in sensor networks. It was already proven that this fusion method combines strong fundamental properties with attractive features in accuracy and computational requirements. However, these features were derived for linear processes with observability of the state vector in at least one of the local measurements. Therefore, several empirical casestudies are performed to assess ellipsoidal intersection with respect to three reallife limitations. A scenario of cooperative adaptive cruise control is used to analyze the absence of observability in any local measurement. Furthermore, the VanderPol oscillator and a benchmark application of tracking shockwaves on highways assess the fusion method for nonlinear process models. The latter example is also used in a setup where the employed state estimation methodology differs per node, so to meet different computational requirements per node.

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A distributed Kalman filter with global covariance
Most distributed Kalman filtering (DKF) algorithms for sensor networks calculate a local estimate of the global statevector in each node. An important challenge within distributed estimation is that all sensors in the network contribute to the local estimate in each node. In this paper, a novel DKF algorithm is proposed with the goal of attaining the above property, which is denoted as global covariance. In the considered DKF setup each node performs two steps iteratively, i.e., it runs a standard Kalman filter using local measurements and then fuses the resulting estimates with the ones received from its neighboring nodes. The distinguishing aspect of this setup is a novel statefusion method, i.e., ellipsoidal intersection (EI). The main contribution consists of a proof that the proposed DKF algorithm, in combination with EI for statefusion, enjoys the desired property under similar conditions that should hold for observability of standard Kalman filters. The advantages of developed DKF with respect to alternative DKF algorithms are illustrated for a benchmark example of cooperative adaptive cruise control. © 2011 AACC American Automatic Control Council.

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Event based state estimation with time synchronous updates
To reduce the amount of data transfer in networked systems, measurements are usually taken only when an event occurs rather than at each synchronous sample instant. However, this complicates estimation problems considerably, especially in the situation when no measurement is received anymore. The goal of this paper is therefore to develop a state estimator that can successfully cope with event based measurements and attains an asymptotically bounded errorcovariance matrix. To that extent, a general mathematical description of event sampling is proposed. This description is used to set up a state estimator with a hybrid update, i.e., when an event occurs the estimated state is updated using the measurement, while at synchronous instants the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space. Furthermore, to minimize computational complexity of the estimator, the algorithm is implemented using a sum of Gaussians approach. The benefits of this implementation are demonstrated by an illustrative example of state estimation with event sampling.

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Towards selforganizing Kalman filters
Distributed Kalman filtering is an important signal processing method for state estimation in largescale sensor networks. However, existing solutions do not account for unforeseen events that are likely to occur and thus dramatically changing the operational conditions (e.g. node failure, communication deterioration). This article presents an integration solution for distributed Kalman filtering with distributed selforganization to cope with these events. An overview of existing methods on both topics is presented, followed by an empirical case study of a selforganizing sensor network for observing the contaminant distribution process across a large area in time.

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Self organizing distributed stateestimators
Distributed solutions for signal processing techniques are important for establishing largescale monitoring and control applications. They enable the deployment of scalable sensor networks for particular application areas. Typically, such networks consists of a large number of vulnerable components connected via unreliable communication links and are sometimes deployed in harsh environment. Therefore, dependability of sensor network is a challenging problem. An efficient and cost effective answer to this challenge is provided by employing runtime reconfiguration techniques that assure the integrity of the desired signal processing functionalities. Runtime reconfigurability has thorough impact both on system design, implementation, testing/validation and deployment. The presented research focuses on the widespreaded signal processing method known as state estimation with Kalman filtering in particular. To that extent, a number of distributed state estimation solutions that are suitable for networked systems in general are overviewed, after which robustness of the system is improved according to various runtime reconfiguration techniques

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On event based state estimation
To reduce the amount of data transfer in networked control systems and wireless sensor networks, measurements are usually taken only when an event occurs, rather than at each synchronous sampling instant. However, this complicates estimation and control problems considerably. The goal of this paper is to develop a state estimation algorithm that can successfully cope with event based measurements. Firstly, we propose a general methodology for defining event based sampling. Secondly, we develop a state estimator with a hybrid update, i.e. when an event occurs the estimated state is updated using measurements; otherwise the update makes use of the knowledge that the monitored variable is within a bounded set that defines the event. A sum of Gaussians approach is employed to obtain a computationally tractable algorithm.

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Visionbased control of the Manus using SIFT
The rehabilitation robot Manus is an assistive device for severely motor handicapped users. The executing of all day living tasks with the Manus, can be very complex and a visionbased controller can simplify this. The lack of existing visionbased controlled systems, is the poor reliability of the computer vision in unstructured environments. In this paper, a computer vision solution is presented, which can estimate realtime the pose of an object and cooperate with a visionbased controller. The computer vision is robust to illumination changes, a varying scale and rotation and is robust to occlusion. The computer vision is mainly based on the SIFTalgorithm and the usage of a 3Dmodel of an object. Important steps to create this 3Dmodel are discussed. The detection and recognition of the required SIFTkeypoints, has become realtime, by exchanging redundancy against calculation time. With a positionbased visual servoing controller, the Manus can be positioned with respect to an object. © 2007 IEEE.

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Relevant sampling applied to eventbased stateestimation
To reduce the amount of data transfer in networked control systems and wireless sensor networks, measurements are usually sampled only when an event occurs, rather than synchronous in time. Today's event sampling methodologies are triggered by the current value of the sensor. Stateestimators are designed to cope with such methods. In this paper we propose a sampling method in which an event is triggered depending on the reduction of the estimator's uncertainty and estimationerror. As such, communication requirements are minimized while attaining a certain errorcovariance matrix and estimation error at the stateestimator. Furthermore, it is proven that the errorcovariance matrix is asymptotically bounded in case the designed sampling protocol is combined with an eventbased stateestimator. An illustrative example shows that the developed protocol provides an improved state estimation, while minimizing communication between sensor and stateestimator. © 2010 IEEE.

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An Empirical Method to Fuse Partially Overlapping State Vectors for Distributed State Estimation
State fusion is a method for merging multiple estimates of the same state into a single fused estimate. Dealing with multiple estimates is one of the main concerns in distributed state estimation, where an estimated value of the desired state vector is computed in each node of a networked system. Most solutions for distributed state estimation currently available assume that every node computes an estimate of the (same) global state vector. This assumption is impractical for systems observing largearea processes, due to the sheer size of the process state. A feasible solution is one where each node estimates a part of the global state vector, allowing different nodes in the network to have overlapping state elements. Although such an approach should be accompanied by a corresponding state fusion method, existing solutions cannot be employed as they merely consider fusion of two different estimates with equal state representations. Therefore, an empirical solution is presented for fusing two state estimates that have partially overlapping state elements. A justification of the proposed fusion method is presented, along with an illustrative case study for observing the temperature profile of a large rod, though a formal derivation is future research.

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Nonlinear estimation with a network of heterogenous algorithms
Centralized stateestimation algorithms, such as the original Kalman filter, are no longer feasible in large scale sensor networks, due to practical limitations on communication bandwidth and spatial distribution of resources. To cope with these limitations, various distributed estimation algorithms have been proposed that estimate the state of a process in each sensor node using local measurements. State fusion of this local estimate with the estimates obtained in neighboring nodes ensures that the difference between local estimates is reduced. A common perspective in distributed stateestimation is that each individual node performs the same algorithm locally. This paper investigates whether it is beneficial to have some nodes that can perform a different, more accurate estimation method, i.e., heterogenous. To that extent, a networked system where each node employs the same local stateestimator is compared to a similar system where different nodes can perform different types of local estimation algorithms. Their performances are assessed on a VanderPol oscillator and on a benchmark application to estimate speed profiles in traffic shockwaves. The results of these examples encourage further investigation of heterogeneous, distributed stateestimation. © 2011 IEEE.

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Eventbased State Estimation with Negative Information
To reduce the amount of data transfer in networked systems, measurements are usually taken only when an event occurs rather than periodically in time. However, this complicates estimation problems considerably as it is not guaranteed that new sensor measurements will be sampled. In order to cope with such event sampled measurements, an existing state estimator is modified so that any divergent behavior in estimation results will be curtailed. To start, a general formulation of event sampling is proposed, which is then used to set up a state estimator combining stochastic as well as setmembership measurement information according to a hybrid update: when an event occurs the estimated state is updated using the stochastic measurement received (positive information), while at periodic time instants no new measurement is received (negative information) and the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space. An illustrative example further shows that the developed estimator has an improved representation of estimation errors compared to purely stochastic estimators for various event sampling strategies.

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Sensor network for realtime vehicle tracking on road networks
Due to the continuously increasing mobility demand the efficiency of the existing road infrastructure should be improved. Advanced control and management techniques offer promising, cost effective and environmentally friendly solutions to the problem. As traffic and vehicle control/management applications become more advanced they require more and more detailed and accurate realtime observations about the surrounding relevant world. The paper describes a road infrastructure based sensing and data interpretation system, which is scalable, robust and delivers the motion state estimation of the individual vehicles in realtime. The system enables a wide spectrum of intelligent transport system functionalities ranging from traffic management to fully autonomous driving. Distinguishing features of the solution that it does not require instrumentation on the vehicles to become detected and it is insensitive to ambient environmental conditions (weather, light, etc.). The paper describes the event driven object tracking algorithm, and its "mapping" to distributed computing platform resulting in an inherently robust implementation. © 2009 IEEE.

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Fusion strategies for unequal state vectors in distributed Kalman filtering
Distributed implementations of state estimation algorithms generally have in common that each node in a networked system computes an estimate on the entire global state. Accordingly, each node has to store and compute an estimate of the same state vector irrespective of whether its sensors can only observe a small part of it. In particular, the task of monitoring largescale phenomena renders such distributed estimation approaches impractical due to the sheer size of the corresponding state vector. In order to reduce the workload of the nodes, the state vector to be estimated is subdivided into smaller, possibly overlapping parts. In this situation, fusion does not only refer to the computation of an improved estimate but also to the task of reassembling an estimate for the entire state from the locally computed estimates of unequal state vectors. However, existing fusion methods require equal state representations and, hence, cannot be employed. For that reason, a fusion strategy for estimates of unequal and possibly overlapping state vectors is derived that minimizes the mean squared estimation error. For the situation of unknown crosscorrelations between local estimation errors, also a conservative fusion strategy is proposed. © IFAC.

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Inverse covariance intersection: New insights and properties
Decentralized data fusion is a challenging task. Either it is too difficult to maintain and track the information required to perform fusion optimally, or too much information is discarded to obtain informative fusion results. A wellknown solution is Covariance Intersection, which may provide too conservative fusion results. A less conservative alternative is discussed in this paper, and generalizations are proposed in order to apply it to a wide class of fusion problems. The Inverse Covariance Intersection algorithm is about finding the maximum possible common information shared by the estimates to be fused. A bound on the possibly shared common information is derived and removed from the fusion result in order to guarantee consistency. It is shown that the conditions required for consistency can be significantly relaxed, and also other causes of correlations, such as common process noise, can be treated. © 2017 International Society of Information Fusion (ISIF). China Gezhouba Group No.3 Engineering Co., Ltd (CGGC); Energy China; et al.; Hangzhou Dianzi Univeristy; LIFT; SATPRO

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Algebraic analysis of data fusion with ellipsoidal intersection
For decentralized fusion problems, ellipsoidal intersection has been proposed as an efficient fusion rule that provides less conservative results as compared to the wellknow ovariance intersection method. Ellipsoidal intersection relies on the computation of a common estimate that is shared by the estimates to be fused. In this paper, an algebraic reformulation of ellipsoidal intersection is discussed that circumvents the computation of the common estimate. It is shown that ellipsoidal intersection corresponds to an internal ellipsoidal approximation of the intersection of covariance ellipsoids. An interesting result is that ellipsoidal intersection can be computed with the aid of the BarShalom/Campo fusion formulae. This is achieved by assuming a specific correlation structure between the estimates to be fused.

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A study on event triggering criteria for estimation
To reduce the amount of data transfer in networked systems measurements are usually taken only when an event occurs rather than periodically in time. However, a fundamental assessment on the response of estimation algorithms receiving event sampled measurements is not available. This research presents such an analysis when new measurements are sampled at welldesigned events and sent to a Luenberger observer. Conditions are then derived under which the estimation error is bounded, followed by an assessment of two event sampling strategies when the estimator encounters two different types of disturbances: an impulse and a step disturbance. The sampling strategies are compared via four performance measures, such as estimationerror and communication resources. The result is a clear insight of the estimation response in an eventbased setup.

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State fusion with unknown correlation: ellipsoidal intersection
Some crucial challenges of estimation over sensor networks are reaching consensus on the estimates of different systems in the network and separating the mutual information of two estimates from their exclusive information. Current fusion methods of two estimates tend to bypass the mutual information and directly optimize the fused estimate. Moreover, both the mean and covariance of the fused estimate are fully determined by optimizing the covariance only. In contrast to that, this paper proposes a novel fusion method in which the mutual information results in an additional estimate, which defines a mutual mean and covariance. Both variables are derived from the two initial estimates. The mutual covariance is used to optimize the fused covariance, while the mutual mean optimizes the fused mean. An example of decentralized state estimation, where the proposed fusion method is applied, shows a reduction in estimation error compared to the existing alternatives. © 2010 AACC.

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