Local Kalman Filter Research Articles

Distributing the Kalman Filter for Large-Scale Systems

This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale, n -dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> -dimensional subsystems, n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> Lt n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-order approximation by the divergence, which decreases as L increases. The order of the approximation L leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of n-dimensional vectors and matrices is required; only n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> Lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.

Consensus Based Overlapping Decentralized Estimation With Missing Observations and Communication Faults

In this paper a new algorithm for discrete-time overlapping decentralized state estimation of large scale systems is proposed in the form of a multi-agent network based on a combination of local Kalman filters and a dynamic consensus strategy, assuming intermittent observations and communication faults. Conditions are derived for the algorithm to provide, under general conditions concerning the agent resources and the network topology, asymptotic stability in the sense of bounded mean-square estimation error. It is also demonstrated how the consensus gains can be chosen by minimizing the total steady-state mean-square estimation error. Numerical examples illustrate some properties of the proposed algorithm.

Assimilating non-local observations with a local ensemble Kalman filter

Assimilating non-local observations with a local ensemble Kalman filter

Assessing Predictability with a Local Ensemble Kalman Filter

Abstract In this paper, the spatiotemporally changing nature of predictability is studied in a reduced-resolution version of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS), a state-of-the-art numerical weather prediction model. Atmospheric predictability is assessed in the perfect model scenario for which forecast uncertainties are entirely due to uncertainties in the estimates of the initial states. Uncertain initial conditions (analyses) are obtained by assimilating simulated noisy vertical soundings of the “true” atmospheric states with the local ensemble Kalman filter (LEKF) data assimilation scheme. This data assimilation scheme provides an ensemble of initial conditions. The ensemble mean defines the initial condition of 5-day deterministic model forecasts, while the time-evolved members of the ensemble provide an estimate of the evolving forecast uncertainties. The observations are randomly distributed in space to ensure that the geographical distribution of the analysis and forecast errors reflect predictability limits due to the model dynamics and are not affected by inhomogeneities of the observational coverage. Analysis and forecast error statistics are calculated for the deterministic forecasts. It is found that short-term forecast errors tend to grow exponentially in the extratropics and linearly in the Tropics. The behavior of the ensemble is explained by using the ensemble dimension (E dimension), a spatiotemporally evolving measure of the evenness of the distribution of the variance between the principal components of the ensemble-based forecast error covariance matrix. It is shown that in the extratropics the largest forecast errors occur for the smallest E dimensions. Since a low value of the E dimension guarantees that the ensemble can capture a large portion of the forecast error, the larger the forecast error the more certain that the ensemble can fully capture the forecast error. In particular, in regions of low E dimension, ensemble averaging is an efficient error filter and the ensemble spread provides an accurate prediction of the upper bound of the error in the ensemble-mean forecast.

SUBOPTIMAL STATE ESTIMATION FOR UNCERTAIN MULTISENSOR DISCRETE-TIME LINEAR STOCHASTIC SYSTEMS

The problem of recursive estimation for uncertain multisensor linear discrete-time systems is considered. A new suboptimal filtering algorithm is herein proposed. It is based on the fusion formula for an arbitrary number of local Kalman filters. Each local Kalman filter is fused by the minimum mean square error criterion. The suboptimal gains and weights do not depend on current observations; therefore the proposed filter can easily be implemented in real-time. The examples demonstrate the efficiency and high-accuracy of the proposed filter.

Decentralized sigma-point information filters for target tracking in collaborative sensor networks

Tracking a target in a cluttered environment is a representative application of sensor networks and a benchmark for collaborative signal processing algorithms. This paper presents a strictly decentralized approach to Bayesian filtering that is well fit for in-network signal processing. By combining the sigma-point filter methodology and the information filter framework, a class of algorithms denoted as sigma-point information filters is developed. These techniques exhibit the robustness and accuracy of the sigma-point filters for nonlinear dynamic inference while being as easily decentralized as the information filters. Furthermore, the computational cost of this approach is equivalent to a local Kalman filter running in each active node while the communication burden can be made linearly growing in the number of sensors involved. The proposed algorithms are then adapted to the specific problem of target tracking with data association ambiguity. Making use of a local probabilistic data association, we formulate a decentralized tracking scheme that significantly outperforms the existing schemes with similar computational and communication complexity.

Assessing a local ensemble Kalman filter: perfect model experiments with the National Centers for Environmental Prediction global model

The accuracy and computational efficiency of the recently proposed local ensemble Kalman filter (LEKF) data assimilation scheme is investigated on a state-of-the-art operational numerical weather prediction model using simulated observations. The model selected for this purpose is the T62 horizontal- and 28-level vertical-resolution version of the Global Forecast System (GFS) of the National Center for Environmental Prediction. The performance of the data assimilation system is assessed for different configurations of the LEKF scheme. It is shown that a modest size (40-member) ensemble is sufficient to track the evolution of the atmospheric state with high accuracy. For this ensemble size, the computational time per analysis is less than 9 min on a cluster of PCs. The analyses are extremely accurate in the mid-latitude storm track regions. The largest analysis errors, which are typically much smaller than the observational errors, occur where parametrized physical processes play important roles. Because these are also the regions where model errors are expected to be the largest, limitations of a real-data implementation of the ensemble-based Kalman filter may be easily mistaken for model errors. In light of these results, the importance of testing the ensemble-based Kalman filter data assimilation systems on simulated observations is stressed.

An Efficient Motion Estimation Algorithm for Video Coding Using Kalman Filter

This paper presents a new motion estimation algorithm to improve the performance of the existing searching algorithms at a relatively low computational cost. We try to amend the incorrect and/or inaccurate estimate of motion with higher precision by using Kalman filter. We first obtain a measurement of motion vector of a block by using the existing searching scheme. We then generate the predicted motion vector utilizing the inter-block correlation in both spatial and temporal directions. Based on these two motion information, a three-dimensional (3D) motion model is developed and then a 3D local Kalman filter is designed to obtain the optimal estimate of motion vector.

Adaptive training and pruning in feedforward networks

A local extended Kalman filter training and pruning approach is proposed to train feedforward networks with the goal of reducing the computational complexity and storage requirement in large-scale practical problems. The performance of the proposed algorithm is demonstrated for the problem of handwritten digit recognition.

ADAPTIVE TRAINING AND PRUNING FOR NEURAL NETWORKS:ALGORITHMS AND APPLICATION

Finding an optimal network size is one of the major concerns when building a neural network. In using the local extended Kalman filter (EKF) algorithm, we propose an efficient approach that combines EKF training and pruning as a whole. In particular, the covariance matrix obtained along with the local EKF training can be utilized to indicate the importance of the network weights. As a result, the network size can be determined adaptively to keep pace with the changes in input characteristics. The effectiveness of this algorithm is demonstrated on generalized XOR logic function and handwritten digit recognition.

Gain fusion algorithm for decentralised parallel Kalman filters

A new gain fusion algorithm is proposed for application to decentralised sensor systems. The proposed algorithm gives computer-efficient suboptimal estimation results, such that it reconstructs the global estimate and covariance from local Kalman filter gains and estimates without significant loss of accuracy. Compared to the conventional algorithm, the smaller communication requirement and the removal of the calculation requirement of inverse covariances make the proposed algorithm more suitable for real time applications. A numerical example shows that the proposed algorithm provides a convincing suboptimal decentralised algorithm. In addition, the proposed gain fusion algorithm can be easily extended to accommodate local Kalman filters with reduced order.

Stochastic Optimal and Suboptimal Control of Irrigation Canals

The Saint-Venant equations of open-channel flow were linearized using the Taylor series and a finite-difference approximation of the original nonlinear, partial differential equations. Using the linear optimal control theory, a proportional-plus-integral (PI) controller was developed for an irrigation canal with five pools. Since the order of the controller gain matrix was large, the Kalman filter was designed to estimate values for the state variables that were not measured. For this problem, there were a total of 45 state and five control variables. With two flow depth measurements per pool, values for the remaining 35 state variables were estimated using the Kalman filter. The simulated canal dynamics with the regional PI controllers along with the local Kalman filter were compared with the performance of the global control algorithms for achieving a constant-volume control and a constant-level control of an example irrigation canal. The performance of the regional constant-volume control algorithm was found to be as good as the performance of the global control algorithm, whereas the performance of the regional constant-level control algorithm was marginally acceptable.

Periodic activation function for fast on-line EKF training and pruning

In the use of the local extended Kalman filter algorithm for network training and pruning, the authors propose replacing the monotonical increasing sigmoid activation function with a periodic function. Computer simulations show that the rate of convergence is substantially improved and that more insignificant weights can be pruned.

Fully decentralised algorithm for multisensor Kalman filtering

An algorithm that achieves complete decentralisation of the Kalman filter algorithm amongst the sensing nodes of a multisensor system is preserved. This algorithm does not require any form of central processing facility or centralised communications medium. Each sensing node implements its own local Kalman filter to arrive at a partial decision which it then broadcasts to every other node. Each node then assimilates this received information to arrive at its own local estimate of the system state. The algorithm guarantees that each local estimate thus obtained is identical to the estimate that would be obtained if a conventional, completely centralised, Kalman filter were used. Because there is no single central processing node the algorithm is highly resilient to loss of one or more sensing nodes. The fully decentralised nature ensures that it is ideal for implementation on a parallel processing array, such as a transputer network. The algorithm has been tested using a simulated multiperson pursuitevasion game to check the viability. The algorithm has also been implemented on a transputer based array of cameras to perform tracking of an object moving around a room.

A Method of Hierarchical Multiple Model Adaptive Estimation and Control

A method of hierarchical multiple model adaptive estimation and control is proposed for discrete-time distributed systems with unknown parameters, in which the decentralized structure consists of a central station and of two local stations which do not communicate between each other. It is assumed that the global and local hypotheses on the unknown parameters are introduced to the decentralized multiple model structure. This modeling provides a flexible design algorithm for passively adaptive control problems in steady-state. Instead of utilizing the local a posteriori probabilities, the probability density functions of the innovation processes for the local Kalman filters are adopted to simply reconstruct the global a posteriori probability in the central station. The effectiveness of the present method is illustrated through a numerical simulation of the FDI (failure detection and idetification) system for a hydrofoil boat.