Filter Recursions Research Articles

Posterior Linearization Filter: Principles and Implementation Using Sigma Points

This paper is concerned with Gaussian approximations to the posterior probability density function (PDF) in the update step of Bayesian filtering with nonlinear measurements. In this setting, sigma-point approximations to the Kalman filter (KF) recursion are widely used due to their ease of implementation and relatively good performance. In the update step, these sigma-point KFs are equivalent to linearizing the nonlinear measurement function by statistical linear regression (SLR) with respect to the prior PDF. In this paper, we argue that the measurement function should be linearized using SLR with respect to the posterior rather than the prior to take into account the information provided by the measurement. The resulting filter is referred to as the posterior linearization filter (PLF). In practice, the exact PLF update is intractable but can be approximated by the iterated PLF (IPLF), which carries out iterated SLRs with respect to the best available approximation to the posterior. The IPLF can be seen as an approximate recursive Kullback-Leibler divergence minimization procedure. We demonstrate the high performance of the IPLF in relation to other Gaussian filters in two numerical examples.

Two implementations of marginal distribution Bayes filter for nonlinear Gaussian models

The marginal distribution Bayes (MDB) filter is an efficient approach for tracking an unknown and time-varying number of targets in the presence of clutter, noise, data association uncertainty, and detection uncertainty. This filter propagates the marginal distributions and existence probabilities of each target in the filter recursion, and it admits a closed-form solution for a linear Gaussian multi-target model. However, this closed-form solution is not general enough to accommodate nonlinear multi-target models. In this paper, we propose two implementations of the MDB filter to accommodate nonlinear multi-target models. The first is the first-order Taylor approximation MDB (FTA-MDB) filter which is based on the linearization technique of nonlinear function, and the second is the unscented transform MDB (UT-MDB) filter which is based on the unscented transform technique. Simulation results demonstrate that the proposed implementations are better on multiple targets tracking than the UK-PHD filter.

Sequential measurement-driven multi-target Bayesian filter

Bayesian filter is an efficient approach for multi-target tracking in the presence of clutter. Recently, considerable attention has been focused on probability hypothesis density (PHD) filter, which is an intensity approximation of the multi-target Bayesian filter. However, PHD filter is inapplicable to cases in which target detection probability is low. The use of this filter may result in a delay in data processing because it handles received measurements periodically, once every sampling period. To track multiple targets in the case of low detection probability and to handle received measurements in real time, we propose a sequential measurement-driven Bayesian filter. The proposed filter jointly propagates the marginal distributions and existence probabilities of each target in the filter recursion. We also present an implementation of the proposed filter for linear Gaussian models. Simulation results demonstrate that the proposed filter can more accurately track multiple targets than the Gaussian mixture PHD filter or cardinalized PHD filter.

Refined GM-PHD tracker for tracking targets in possible subsequent missed detections

Gaussian mixture probability hypothesis density (GM-PHD) filter is proposed as an alternative of PHD filter to estimate the first-order moment of the multi-target posterior density. Theoretically, the GM-PHD filter handles missed detection in its filtering recursion. However, in practice, the performance of this filter might be degraded in subsequent miss-detection, when all factors that affect miss-detection cannot be appropriately incorporated in the filtering recursions. In this paper, we propose a heuristic method called Refined GM-PHD (RGM-PHD) tracker which aims at improving the performance of GM-PHD filter in subsequent missed detections. To accomplish this purpose, we define a survival model and introduce probability of confirm for each Gaussian component and each track which are adaptively calculated in time. Accordingly, we propose a state refinement step and a novel state extraction step to improve the tracking performance of the filter. Comprehensive Monte Carlo simulations in terms of various probabilities of detection, false alarm rates and subsequent miss-detection rates are performed to investigate the effectiveness of the method.

Input–Output Linearizing Control of the Underactuated Hovercraft Using the Derivative-Free Nonlinear Kalman Filter

The paper proposes a nonlinear control approach for the underactuated hovercraft model based on differential flatness theory and uses a new nonlinear state vector and disturbances estimation method under the name of derivative-free nonlinear Kalman filter. It is proven that the nonlinear model of the hovercraft is a differentially flat one. It is shown that this model cannot be subjected to static feedback linearization, however it admits dynamic feedback linearization which means that the system's state vector is extended by including as additional state variables the control inputs and their derivatives. Next, using the differential flatness properties it is also proven that this model can be subjected to input–output linearization and can be transformed to an equivalent canonical (Brunovsky) form. Based on this latter description the design of a state feedback controller is carried out enabling accurate maneuvering and trajectory tracking. Additional problems that are solved in the design of this feedback control scheme are the estimation of the nonmeasurable state variables in the hovercraft's model and the compensation of modeling uncertainties and external perturbations affecting the vessel. To this end, the application of the derivative-free nonlinear Kalman filter is proposed. This nonlinear filter consists of the Kalman Filter's recursion on the linearized equivalent model of the vessel and of an inverse nonlinear transformation based on the differential flatness features of the system which enables to compute estimates for the state variables of the initial nonlinear model. The redesign of the filter as a disturbance observer makes possible the estimation and compensation of additive perturbation terms affecting the hovercraft's model. The efficiency of the proposed nonlinear control and state estimation scheme is confirmed through simulation experiments.

SYNCHRONIZATION OF CIRCADIAN OSCILLATORS AND PROTEIN SYNTHESIS CONTROL USING THE DERIVATIVE-FREE NONLINEAR KALMAN FILTER

The paper proposes a new method for synchronization of coupled circadian cells and for nonlinear control of the associated protein synthesis process using differential flatness theory and the derivative-free nonlinear Kalman filter. By proving that the dynamic model of the FRQ protein synthesis is a differentially flat one, its transformation to the linear canonical (Brunovsky) form becomes possible. For the transformed model, one can find a state feedback control input that makes the oscillatory characteristics in the concentration of the FRQ protein vary according to desirable setpoints. To estimate nonmeasurable elements of the state vector, the derivative-free nonlinear Kalman filter is used. The derivative-free nonlinear Kalman filter consists of the standard Kalman filter recursion on the linearized equivalent model of the coupled circadian cells and on computation of state and disturbance estimates using the diffeomorphism (relations about state variables transformation) provided by differential flatness theory. Moreover, to cope with parametric uncertainties in the model of the FRQ protein synthesis and with stochastic disturbances in measurements, the derivative-free nonlinear Kalman filter is redesigned in the form of a disturbance observer. The efficiency of the proposed Kalman filter-based control scheme is tested through simulation experiments.

Sensorless Control of Distributed Power Generators With the Derivative-Free Nonlinear Kalman Filter

A control method for distributed interconnected power generation units is developed. The power system comprises permanent-magnet synchronous generators (PMSGs), which are connected to each other through transformers and tie-lines. A derivative-free nonlinear Kalman filtering approach is introduced aiming at implementing sensorless control of the distributed power generators. In the proposed derivative-free Kalman filtering method, the generator's model is first subjected to a linearization transformation that is based on differential flatness theory and next state estimation is performed by applying the standard Kalman filter recursion to the linearized model. Unlike Lie algebra-based estimator design methods, the proposed approach provides estimates of the state vector of the PMSG without the need for derivatives and Jacobian calculation. Moreover, by redesigning the proposed derivative-free nonlinear Kalman filter as a disturbance observer, it is possible to estimate at the same time the nonmeasurable elements of each generator's state vector, the unknown input power (torque), and the disturbance terms induced by interarea oscillations. The efficient real-time estimation of the aggregate disturbance that affects each local generator makes possible to introduce a counterdisturbance control term, thus maintaining the power system on its nominal operating conditions.

Detection of parametric changes in the Peyrard-Bishop- Dauxois model of DNA using nonlinear Kalman filtering

The derivative-free nonlinear Kalman filter is proposed for state estimation and fault diagnosis in distributed parameter systems of the wave-type and particularly in the Peyrard-Bishop-Dauxois model of DNA dynamics. At a first stage, a nonlinear filtering approach is introduced for estimating the dynamics of the Peyrard-Bishop-Dauxois 1D nonlinear wave equation, through the processing of a small number of measurements. It is shown that the numerical solution of the associated partial differential equation results in a set of nonlinear ordinary differential equations. With the application of a diffeomorphism that is based on differential flatness theory it is shown that an equivalent description of the system is obtained in the linear canonical (Brunovsky) form. This transformation enables to obtain local estimates about the state vector of the DNA model through the application us of the standard Kalman filter recursion. At a second stage, the local statistical approach to fault diagnosis is used to perform fault diagnosis for this distributed parameter system by processing with statistical tools the differences (residuals) between the output of the Kalman filter and the measurements obtained from the distributed parameter system. Optimal selection of the fault threshold is succeeded by using the local statistical approach to fault diagnosis. The efficiency of the proposed filtering approach in the problem of fault diagnosis for parametric change detection, in nonlinear wave-type models of DNA dynamics, is confirmed through simulation experiments.

Robust control of valves in ship diesel engines with the use of the Derivative-free nonlinear Kalman filter

This article studies robust nonlinear control for gas exchange valves in ship diesel engines, with the use of the derivative-free nonlinear Kalman filter. Robust control of gas exchange valves is important for improving the efficiency in the operation of diesel engines. Due to the relatively wide range of typical internal combustion engine operating conditions imposed on a variable valve timing actuator, the control scheme must compensate for model uncertainties, parametric variations and rapid external disturbances. By applying differential flatness theory, the initial nonlinear model of the system is transformed into the linear canonical (Brunovsky) form. For the latter model, it is possible to design a state feedback controller that enables accurate tracking of the valve’s reference setpoints. To estimate the non-measurable state variables of the model and the unknown external disturbances, the derivative-free nonlinear Kalman filter is used as a disturbance observer. The derivative-free nonlinear Kalman filter consists of the standard Kalman filter recursion on the linearized equivalent model of the valve and of computation of state and disturbance estimates using the diffeomorphism (relations about state variables’ transformation) provided by differential flatness theory. Evaluation tests are performed for assessing the performance of the proposed control scheme.

Fast Bayesian parameter estimation for stochastic logistic growth models.

The transition density of a stochastic, logistic population growth model with multiplicative intrinsic noise is analytically intractable. Inferring model parameter values by fitting such stochastic differential equation (SDE) models to data therefore requires relatively slow numerical simulation. Where such simulation is prohibitively slow, an alternative is to use model approximations which do have an analytically tractable transition density, enabling fast inference. We introduce two such approximations, with either multiplicative or additive intrinsic noise, each derived from the linear noise approximation (LNA) of a logistic growth SDE. After Bayesian inference we find that our fast LNA models, using Kalman filter recursion for computation of marginal likelihoods, give similar posterior distributions to slow, arbitrarily exact models. We also demonstrate that simulations from our LNA models better describe the characteristics of the stochastic logistic growth models than a related approach. Finally, we demonstrate that our LNA model with additive intrinsic noise and measurement error best describes an example set of longitudinal observations of microbial population size taken from a typical, genome-wide screening experiment.

Statically Fused Converted Position and Doppler Measurement Kalman Filters

The work presented in this paper makes two contributions for exploiting Doppler (range rate) measurements in tracking systems. First, a new linear filter, the converted Doppler measurement Kalman filter (CDMKF), is presented to extract nonlinear pseudostates from converted Doppler measurements (i.e., the product of the range measurements and Doppler measurements). The pseudostates are constructed from the converted Doppler and its derivatives. The linearly evolving equations of the pseudostates are derived for common target motion models. The second contribution of this paper is using the CDMKF along with the converted position measurement Kalman filter (CPMKF), in which only the position measurements are used, to establish a new filtering structure, statically fused converted measurement Kalman filters (SF-CMKF). The resulting states of CPMKF and CDMKF are combined by a static minimum mean squared error (MMSE) estimator, where the nonlinearity and correlation between the pseudostates and the Cartesian states are handled simultaneously, to yield the final state estimates. The dynamic nonlinear estimation problem is converted into dynamic linear estimation followed by static nonlinear fusion. The estimation accuracy can be enhanced by incorporating the Doppler measurements via the linear CDMKF, while the filtering stability can be improved by dealing with nonlinearity outside the filtering recursions. Monte Carlo simulations and comparison with the posterior Cramer-Rao bound demonstrate the effectiveness of the CDMKF and SF-CMKF.

SLAM With Dynamic Targets via Single-Cluster PHD Filtering

This paper presents the first algorithm for simultaneous localization and mapping (SLAM) that can estimate the locations of both dynamic and static features in addition to the vehicle trajectory. We model the feature-based SLAM problem as a single-cluster process, where the vehicle motion defines the parent, and the map features define the daughter. Based on this assumption, we obtain tractable formulae that define a Bayesian filter recursion. The novelty in this filter is that it provides a robust multi-object likelihood which is easily understood in the context of our starting assumptions. We present a particle/Gaussian mixture implementation of the filter, taking into consideration the challenges that SLAM presents over target tracking with stationary sensors, such as changing fields of view and a mixture of static and dynamic map features. Monte Carlo simulation results are given which demonstrate the filter's effectiveness with high measurement clutter and non-linear vehicle motion.

Decoupled target tracking with bearings-only measurements

Bearings-only tracking has been the focus of distinct research interest in the last few decades. To attack target maneuvering and measuring nonlinearity, most of the existing methods adopt an interactive multiple model (IMM) with a bank of nonlinear filtering recursions, such as an extended Kalman filter, an unscented Kalman filter (UKF), or a particle filter. However, these nine-state coupled-tracking methods usually suffer from low tracking accuracy due to inseparable model probability and high computational burden because of the calculating inverse of 9×9 matrices. This present work focuses on target tracking by two platforms's bearings-only measurements in an axial decoupled way. First, the passive ranging formula is derived with the conversion error proven to be Gaussian noises. Then to reduce the computational burden and maintain high tracking accuracy, the diagonal least upper bound for the covariance matrix of conversion error is given. This makes the decoupled tracking algorithm applicable. To attack the target maneuvering, the present decoupled IMM that uses a bank of passive-ranging-based Kalman filters is adopted along each axis separately. Finally, the decoupled tracking algorithm is compared with conventional coupled IMM in Cartesian coordinates by criterions such as tracking accuracy, model probability, and computational burden.

Detecting structural breaks in multivariate financial time series: evidence from hedge fund investment strategies

This paper extends the class of asset-based style factor models with multiple structural breaks to the multivariate setting. We propose a model that allows for the presence of common breaks in a system of factor models for individual hedge fund investment strategies, which share common investment characteristics. We develop a Bayesian approach to inference for the unknown number and positions of the structural breaks, based on a set of filtering recursions similar to those of the forward–backward algorithm. Furthermore, we identify relevant risk factors, common among the series of hedge funds, using a Bayesian model comparison approach. We apply our method to a set of correlated hedge fund strategies, which are mainly characterized by equity related bets. Multiple common breaks are identified, consistent with well-known market events, which reveal evidence for structural changes in the risk exposures as well as in the correlation structure of the analysed series.

A phd Filter for Tracking Multiple Extended Targets Using Random Matrices

This paper presents a random set based approach to tracking of an unknown number of extended targets, in the presence of clutter measurements and missed detections, where the targets' extensions are modeled as random matrices. For this purpose, the random matrix framework developed recently by Koch is adapted into the extended target phd framework, resulting in the Gaussian inverse Wishart phd (giw-phd) filter. A suitable multiple target likelihood is derived, and the main filter recursion is presented along with the necessary assumptions and approximations. The particularly challenging case of close extended targets is addressed with practical measurement clustering algorithms. The capabilities and limitations of the resulting extended target tracking framework are illustrated both in simulations and in experiments based on laser scans.

인지로봇 청각시스템을 위한 의사최적 이동음원 도래각 추적 필터

This paper proposes a quasi-optimal linear DOA (Direction-of-Arrival) estimator which is necessary for the development of a real-time robot auditory system tracking moving acoustic source. It is well known that the use of conventional nonlinear filtering schemes may result in the severe performance degradation of DOA estimation and not be preferable for real-time implementation. These are mainly due to the inherent nonlinearity of the acoustic signal model used for DOA estimation. This motivates us to consider a new uncertain linear acoustic signal model based on the linear prediction relation of a noisy sinusoid. Using the suggested measurement model, it is shown that the resultant DOA estimation problem is cast into the NCRKF (Non-Conservative Robust Kalman Filtering) problem [12]. NCRKF-based DOA estimator provides reliable DOA estimates of a fast moving acoustic source in spite of using the noise-corrupted measurement matrix in the filter recursion and, as well, it is suitable for real-time implementation because of its linear recursive filter structure. The computational efficiency and DOA estimation performance of the proposed method are evaluated through the computer simulations.

Multitarget Tracking using Probability Hypothesis Density Smoothing

In general, for multitarget problems where the number of targets and their states are time varying, the optimal Bayesian multitarget tracking is computationally demanding. The Probability Hypothesis Density (PHD) filter, which is the first-order moment approximation of the optimal one, is a computationally tractable alternative. By evaluating the PHD, the number of targets as well as their individual states can be extracted. Recent sequential Monte Carlo (SMC) implementations of the PHD filter have paved the way to its application to realistic nonlinear non-Gaussian problems. It is observed that the particle implementation of the PHD filter is dependent on current measurements, especially in the case of low observable target problems (i.e., estimates are sensitive to missed detections and false alarms). In this paper a PHD smoothing algorithm is proposed to improve the capability of PHD-based tracking system. It involves forward multitarget filtering using the standard PHD filter recursion followed by backward smoothing recursion using a novel recursive formula. Smoothing, which produces delayed estimates, results in better estimates for target states and a better estimate for the number of targets. Multiple model PHD (MMPHD) smoothing, which is an extension of the proposed technique to maneuvering targets, is also provided. Simulations are performed with the proposed method on a multitarget scenario. Simulation results confirm improved performance of the proposed algorithm.

Diffusion Strategies for Distributed Kalman Filtering and Smoothing

We study the problem of distributed Kalman filtering and smoothing, where a set of nodes is required to estimate the state of a linear dynamic system from in a collaborative manner. Our focus is on diffusion strategies, where nodes communicate with their direct neighbors only, and the information is diffused across the network through a sequence of Kalman iterations and data-aggregation. We study the problems of Kalman filtering, fixed-lag smoothing and fixed-point smoothing, and propose diffusion algorithms to solve each one of these problems. We analyze the mean and mean-square performance of the proposed algorithms, provide expressions for their steady-state mean-square performance, and analyze the convergence of the diffusion Kalman filter recursions. Finally, we apply the proposed algorithms to the problem of estimating and tracking the position of a projectile. We compare our simulation results with the theoretical expressions, and note that the proposed approach outperforms existing techniques.

Parameter Estimation for a Bidimensional Partially Observed Ornstein-Uhlenbeck Process with Biological Application

We consider a bidimensional Ornstein-Uhlenbeck process to describe the tissue microvascularisation in anti-cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is the estimation of the SDE parameters. We use the main advantage of a one-dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion. We also propose a recursive computation of the gradient and hessian of the log-likelihood based on Kalman filtering, which allows to implement an easy numerical maximisation of the likelihood and the exact maximum likelihood estimator (MLE). Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the MLE. We show that this ARMA property can be generalised to a higher dimensional underlying Ornstein-Uhlenbeck diffusion. We compare this estimator with the one obtained by the well-known EM algorithm on simulated data.

Non‐Gaussian state space models in decomposition of ice core time series in long and short time‐scales

AbstractStatistical modelling of six time series of geological ice core chemical data from Greenland is discussed. We decompose the total variation into long time‐scale (trend) and short time‐scale variations (fluctuations around the trend), and a pure noise component. Too heavy tails of the short‐term variation makes a standard time‐invariant linear Gaussian model inadequate. We try non‐Gaussian state space models, which can be efficiently approximated by time‐dependent Gaussian models. In essence, these time‐dependent Gaussian models result in a local smoothing, in contrast to the global smoothing provided by the time‐invariant model. To describe the mechanism of this local smoothing, we utilise the concept of a local variance function derived from a heavy‐tailed density. The time‐dependent error variance expresses the uncertainty about the dynamical development of the model state, and it controls the influence of observations on the estimates of the model state components. The great advantage of the derived time‐dependent Gaussian model is that the Kalman filter and the Kalman smoother can be used as efficient computational tools for performing the variation decomposition. One of the main objectives of the study is to investigate how the distributional assumption on the model error component of the short time‐scale variation affects the decomposition. Copyright © 2009 John Wiley & Sons, Ltd.