Showing papers on "Alpha beta filter published in 2008"

Nonlinear Complementary Filters on the Special Orthogonal Group

Abstract: This paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO (3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on SO(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle.

Sliding mode observers: a survey

Abstract: Sliding mode observers have unique properties, in that the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. It is also the case that analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. These unique properties, coupled with the fact that the discontinuous injection signals which were perceived as problematic for many control applications have no disadvantages for software-based observer frameworks, have generated a ground swell of interest in sliding mode observer methods in recent years. This article presents an overview of both linear and non-linear sliding mode observer paradigms. The use of the equivalent injection signal in problems relating to fault detection and condition monitoring is demonstrated. A number of application specific results are also described. The literature in the area is presented and qualified in the context of continuing developments in the broad areas of the theory and application of sliding mode observers.

Distributing the Kalman Filter for Large-Scale Systems

Abstract: 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 nl-dimensional subsystems, nl 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 nl 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.

Higher‐order sliding‐mode observer for state estimation and input reconstruction in nonlinear systems

Abstract: In this paper, a higher-order sliding-mode observer is proposed to estimate exactly the observable states and asymptotically the unobservable ones in multi-input–multi-output nonlinear systems with unknown inputs and stable internal dynamics. In addition the unknown inputs can be identified asymptotically. Numerical examples illustrate the efficacy of the proposed observer. Copyright © 2007 John Wiley & Sons, Ltd.

Adaptive estimation of multiple fading factors in Kalman filter for navigation applications

Abstract: Kalman filter is the most frequently used algorithm in navigation applications. A conventional Kalman filter (CKF) assumes that the statistics of the system noise are given. As long as the noise characteristics are correctly known, the filter will produce optimal estimates for system states. However, the system noise characteristics are not always exactly known, leading to degradation in filter performance. Under some extreme conditions, incorrectly specified system noise characteristics may even cause instability and divergence. Many researchers have proposed to introduce a fading factor into the Kalman filtering to keep the filter stable. Accordingly various adaptive Kalman filters are developed to estimate the fading factor. However, the estimation of multiple fading factors is a very complicated, and yet still open problem. A new approach to adaptive estimation of multiple fading factors in the Kalman filter for navigation applications is presented in this paper. The proposed approach is based on the assumption that, under optimal estimation conditions, the residuals of the Kalman filter are Gaussian white noises with a zero mean. The fading factors are computed and then applied to the predicted covariance matrix, along with the statistical evaluation of the filter residuals using a Chi-square test. The approach is tested using both GPS standalone and integrated GPS/INS navigation systems. The results show that the proposed approach can significantly improve the filter performance and has the ability to restrain the filtering divergence even when system noise attributes are inaccurate.

A New Approximate Solution of the Optimal Nonlinear Filter for Data Assimilation in Meteorology and Oceanography

Abstract: This paper introduces a new approximate solution of the optimal nonlinear filter suitable for nonlinear oceanic and atmospheric data assimilation problems. The method is based on a local linearization in a low-rank kernel representation of the state's probability density function. In the resulting low-rank kernel particle Kalman (LRKPK) filter, the standard (weight type) particle filter correction is complemented by a Kalman-type correction for each particle using the covariance matrix of the kernel mixture. The LRKPK filter's solution is then obtained as the weighted average of several low-rank square root Kalman filters operating in parallel. The Kalman-type correction reduces the risk of ensemble degeneracy, which enables the filter to efficiently operate with fewer particles than the particle filter. Combined with the low-rank approximation, it allows the implementation of the LRKPK filter with high-dimensional oceanic and atmospheric systems. The new filter is described and its relevance demonstrated through applications with the simple Lorenz model and a realistic configuration of the Princeton Ocean Model (POM) in the Mediterranean Sea.

Convergence properties of a decentralized Kalman filter

Abstract: We consider the problem of decentralized Kalman filtering in a sensor network. Each sensor node implements a local Kalman filter based on its own measurements and the information exchanged with its neighbors. It combines the information received from other sensors through using a consensus filter as proposed in [14]. For a time-invariant process and measurement model, we show that this algorithm guarantees that the local estimates of the error covariance matrix converge to the centralized error covariance matrix and that the local estimates of the state converge in mean to the centralized Kalman filter estimates. However, due to the use of the consensus filter, the local estimates of the state do not converge to the least-squares estimate that would be obtained from a centralized Kalman filter.

Optimal Full-Order and Reduced-Order Estimators for Discrete-Time Systems With Multiple Packet Dropouts

Abstract: This paper is concerned with the estimation problem for discrete-time stochastic linear systems with multiple packet dropouts. Based on a recently developed model for multiple-packet dropouts, the original system is transferred to a stochastic parameter system by augmentation of the state and measurement. The optimal full-order linear filter of the form of employing the received outputs at the current and last time instants is investigated. The solution to the optimal linear filter is given in terms of a Riccati difference equation governed by packet arrival rate. The optimal filter is reduced to the standard Kalman filter when there are no packet dropouts. The steady-state filter is also studied. A sufficient condition for the existence of the steady-state filter is given and the asymptotic stability of the optimal filter is analyzed. At last, a reduced-order filter is investigated.

Constrained state estimation using the Unscented Kalman Filter

Abstract: A simple procedure to include state inequality constraints in the unscented Kalman filter is proposed. With this procedure, the information of active state constraints influences the state covariance matrix, resulting in better estimates. In a numerical example, the approach outperforms the extended Kalman filter implemented with constraint handling via ldquoclippingrdquo.

Gain-Constrained Kalman Filtering for Linear and Nonlinear Systems

Abstract: This paper considers the state-estimation problem with a constraint on the data-injection gain. Special cases of this problem include the enforcing of a linear equality constraint in the state vector, the enforcing of unbiased estimation for systems with unknown inputs, and simplification of the estimator structure for large-scale systems. Both the one-step gain-constrained Kalman predictor and the two-step gain-constrained Kalman filter are presented. The latter is extended to the nonlinear case, yielding the gain-constrained unscented Kalman filter. Two illustrative examples are presented.

Electric power system static state estimation through Kalman filtering and load forecasting

Abstract: Static state estimation in electric power systems is normally accomplished without the use of time-history data or prediction. This paper presents preliminary work on the use of the discrete-time Kalman filter to incorporate time history and power demand prediction into state estimators. The problem of state estimation combined with the knowledge of the forecasted load is posed as a Kalman filtering problem using a novel discrete-time model. The model relates current and previous states using the electric power flow equations. An IEEE 14-bus test system example is used to illustrate the potential for enhanced performance of such Kalman filter-based state estimation.

The Kalman Filter

Abstract: The Kalman filter and its variants are some of the most popular tools in statistical signal processing and estimation theory. In this chapter, we introduce the Kalman filter, providing a succinct, yet rigorous derivation thereof, which is based on the orthogonality principle. We also introduce several important variants of the Kalman filter, namely various Kalman smoothers, a Kalman predictor, a nonlinear extension (the extended Kalman filter), and adaptation to cases of temporally correlated measurement noise.

Erratum to "Implementation Issues and Performance Evaluation of Sinusoidal, Surface-Mounted PM Machine Drives With Hall-Effect Position Sensors and a Vector-Tracking Observer"

Abstract: This paper presents the implementation of the Hall-effect sensor-based, vector-tracking observer and discusses its performance when used in an ac brushless drive for surface-mounted PM machines. First, the tuning of the observer is presented. Then, decoupling is used to improve the performance of the observer. Various decoupling strategies are investigated. Stability analysis is also carried out that leads to a maximum amount of position estimation error for the observer to track properly. This paper also demonstrates the benefits of harmonic decoupling with respect to position estimation and disturbance rejection. Both simulation and experimental testing are used to illustrate the performance and limits of the proposed observer topology and of the drive when this observer is used for state feedback.

High-gain observers in the presence of measurement noise: A nonlinear gain approach

Abstract: This paper studies a high-gain observer with a nonlinear gain. The nonlinearity is chosen to have a higher observer gain during the transient period and a lower gain afterwards, thus overcoming the tradeoff between fast state reconstruction and measurement noise attenuation. The observer is designed such that the behavior of the innovation process can be controlled separately from the other system states. This is accomplished by assigning one fast eigenvalue, with the remaining eigenvalues chosen relatively slow. Without this key step, the stability analysis for the proposed observer is unattainable. Recently, a switched observer approach has been investigated, but the nonlinear gain approach bypasses the complications associated with switching, with little to no appreciable degradation in performance. This paper presents a new model for the nonlinear observer, accompanied by a discussion focusing on the main ideas behind the proof.

Estimation of Vehicle Lateral Velocity

Abstract: For the performance of the Haldex Active Yaw Control, accurate information about vehicle's lateral dynamic is important. It is for practical reasons not possible to measure the vehicle's lateral velocity, wherefore this state has to be estimated. Previous work [1] with an observer based on a single track bicycle model show promising results but with limited accuracy at high lateral acceleration, therefore was the approach in this thesis to expand the single track model into a two track model to at a more extensive level capture the chassis dynamics. An alternative tire model was developed, because the well known Magic Formula was for this application too computational expensive and the alternative, Exponential tire model, which was previously used in [1] have several disadvantages. Two observers have been evaluated, the Extended Kalman Filter (EKF) and an Averaging Observer. The EKF is a well known observer that is able to perform well but with the disadvantage to require much calculation power. The Averaging Observer is on the other hand light on calculations, which in this application are desired. Therefore it is tested how well the Averaging Observer performs compared to the EKF. The evaluation was done by comparing the estimated states with the states from both a more complex vehicle model and also real world measurements. The observers performed well in both the cases. The EKF and the Averaging Observer performed almost similar results, which is a favor for the Averaging Observer to achieve same accuracy with less computational effort. Brief tests to do road friction estimations were done and showed promising results if the lateral acceleration sensor signal is reliable. (Less)

Vehicle Tracking by non-Drifting Mean-shift using Projective Kalman Filter

Abstract: Robust vehicle tracking is essential in traffic monitoring because it is the groundwork to higher level tasks such as traffic control and event detection. This paper describes a new technique for tracking vehicles with mean-shift using a projective Kalman filter. The shortcomings of the mean-shift tracker, namely the selection of the bandwidth and the initialization of the tracker, are addressed with a fine estimation of the vehicle scale and kinematic model. Indeed, the projective Kalman filter integrates the non-linear projection of the vehicle trajectory in its observation function resulting in an accurate localization of the vehicle in the image. The proposed technique is compared to the standard Extended Kalman filter implementation on traffic video sequences. Results show that the performance of the standard technique decreases with the number of frames per second whilst the performance of the projective Kalman filter remains constant.

Implicit triangular observer form dedicated to a sliding mode observer for systems with unknown inputs

Abstract: It has been presented in previous works that every uniformly observable single-output system can be put on a triangular observation form. For this structure a special kind of sliding mode observer has been designed by authors, which ensures a finite-time state reconstruction using a step by step observation algorithm. In this paper, we show that the multi-output case is more delicate to study especially when the system has some unknown inputs. Thus, in order to generalizes the triangular observer form, from single to multi-output case, we define an Implicit Triangular Observer (ITO) form. For such a form, two results are given. Firstly, we design a finite time converging observer for all values of the unknown inputs. Secondly, we give the necessary and sufficient condition, including a matching condition, for the existence of a coordinate change to put the system into this form. It is also shown that this class of systems is a subset of the uniform observable class of systems.

State estimation in distribution grids

Abstract: Problems and techniques for estimating the state of an observable distribution grid are investigated. A distribution grid is observable if the state of the grid can be fully determined. For the simulations, the modified 34-bus IEEE test feeder is used. The measurements needed for the state estimation are generated by the ladder iterative technique. Two methods for the state estimation are analyzed: weighted least squares and extended Kalman filter. Both estimators try to find the most probable state based on the available measurements. The result is that the Kalman filter mostly needs less iterations and calculation time. The disadvantage of the Kalman filter is that it needs some foreknowlegde about the state.

Comparison of Statistical Dynamical, Square Root and Ensemble Kalman Filters

Abstract: We present a statistical dynamical Kalman filter and compare its performance to deterministic ensemble square root and stochastic ensemble Kalman filters for error covariance modeling with applications to data assimilation. Our studies compare assimilation and error growth in barotropic flows during a period in 1979 in which several large scale atmospheric blocking regime transitions occurred in the Northern Hemisphere. We examine the role of sampling error and its effect on estimating the flow dependent growing error structures and the associated effects on the respective Kalman gains. We also introduce a Shannon entropy reduction measure and relate it to the spectra of the Kalman gain.

Non-linear DSGE Models, The Central Dierence Kalman Filter, and The Mean Shifted Particle Filter

Abstract: This paper shows how non-linear DSGE models with potential non-normal shocks can be estimated by Quasi-Maximum Likelihood based on the Central Dierence Kalman Filter (CDKF). The advantage of this estimator is that evaluating the quasi log-likelihood function only takes a fraction of a second. The second contribution of this paper is to derive a new particle …lter which we term the Mean Shifted Particle Filter (MSPFb). We show that the MSPFb outperforms the standard Particle Filter by delivering more precise state estimates, and in general the MSPFb has lower Monte Carlo variation in the reported log-likelihood function.

On Kalman Smoothing With Random Packet Loss

Abstract: This correspondence studies the performance of Kalman fixed lag smoothers with random packet losses and its comparison with the Kalman filter with packet loss. In terms of estimator stability via boundedness of the expectation of the error covariance, we show that smoothing does not provide any benefit over filtering. On the other hand, it is demonstrated that using a probabilistic notion of performance, smoothing can provide significant gains when compared to Kalman filtering. An analysis of Kalman filtering using two simple retransmission schemes and its comparison with Kalman smoothing is also made.

Test Statistics in Kalman Filtering

Abstract: Many estimation problems can be modeled using a Kalman filter. One of the key requirements for Kalman filtering is to characterize various error sources, essentially for the quality assurance and quality control of a system. This characterization can be evaluated by applying the principle of multivariate statistics to the system innovations and the measurement residuals. This manuscript will systematically examine the test statistics in Kalman filter on the ground of the normal, 2χ-, t- and F- distributions, and the strategies for global, regional and local statistical tests as well. It is hoped that these test statistics can generally help better understand and perform the statistical analysis in specific applications using a Kalman filter.

Two-Stage Neural Observer for Mechanical Systems

Abstract: This paper proposes a novel velocity observer which uses neural network and sliding mode for unknown mechanical systems. The neural observer in this paper has two stages: 1) a dead-zone neural observer assures that the observer error is bounded and 2) a super-twisting second-order sliding-mode is used to guarantee finite time convergence of the observer. With sliding mode compensation, the two-stage neural observer ensures finite time convergence, and reduces the chattering during its discrete realization.