Nonlinear Estimation Problem Research Articles

A color-based particle filter for multiple object tracking in an outdoor environment

Tracking multiple objects is more challenging than tracking a single object. Some problems arise in multiple-object tracking that do not exist in single-object tracking, such as object occlusion, the appearance of a new object and the disappearance of an existing object, updating the occluded object, etc. In this article, we present an approach to handling multiple-object tracking in the presence of occlusions, background clutter, and changing appearance. The occlusion is handled by considering the predicted trajectories of the objects based on a dynamic model and likelihood measures. We also propose target-model-update conditions, ensuring the proper tracking of multiple objects. The proposed method is implemented in a probabilistic framework such as a particle filter in conjunction with a color feature. The particle filter has proven very successful for nonlinear and non-Gaussian estimation problems. It approximates a posterior probability density of the state, such as the object's position, by using samples or particles, where each state is denoted as the hypothetical state of the tracked object and its weight. The observation likelihood of the objects is modeled based on a color histogram. The sample weight is measured based on the Bhattacharya coefficient, which measures the similarity between each sample's histogram and a specified target model. The algorithm can successfully track multiple objects in the presence of occlusion and noise. Experimental results show the effectiveness of our method in tracking multiple objects.

A comparison of linear and nonlinear optimal estimators in nonlinear navigation problem

A nonlinear estimation problem often encountered in processing navigation data is considered. The linear optimal estimator (LOE) that minimizes the root mean-square (RMS) criterion in the class of linear estimates for this nonlinear estimation problem is investigated. The equivalent linear models for nonlinear measurements have been introduced for the LOE and the Cramer-Rao Bound. The features of the LOE in comparison with the Bayesian nonlinear optimal estimator (NOE), which corresponds to the conditional mean, are studied. Some examples are considered to illustrate the results obtained.

Satellite orbit determination using a batch filter based on the unscented transformation

In this paper, the non-recursive batch filter has been presented and utilized for satellite orbit determination. Using the unscented transformation, the non-recursive batch filter is developed without any traditional linearization process. For the orbit determination system, the range, azimuth, and elevation angles of the satellite measured from ground tracking stations are used for observations. For evaluation and verification of the presented batch filter's performance, the results are compared with those of the batch least squares filter for various initial errors in position and velocity, measurement sampling periods and measurement errors. For relatively small initial errors or short measurement sampling periods or small measurement errors, the accuracy of the orbit determination is similar in both the filters. Under large initial errors or long measurement sampling periods or large measurement errors, the presented non-recursive batch filter yields more robust and stable convergence than the existing batch least squares filter. The non-recursive batch filter based on the unscented transformation is effectively applicable for highly nonlinear batch estimation problems.

Probabilistic Set-Membership Approach for Robust Regression

Interval constraint propagation methods have been shown to be efficient and reliable to solve difficult nonlinear bounded-error estimation problems. However they are considered as unsuitable in a probabilistic context, where the approximation of a probability density function by a set cannot be accepted as reliable. This paper shows how probabilistic estimation problems can be transformed into a set estimation problem by assuming that some rare events will never happen. Since the probability of occurrence of those rare events can be computed, we can give some prior lower bounds for the probability associated to solution set of the corresponding set estimation problem. The approach will be illustrated on a parameter estimation problem.

Estimation of parameters in rational reaction rates of molecular biological systems via weighted least squares

The models of gene regulatory networks are often derived from statistical thermodynamics principle or Michaelis–Menten kinetics equation. As a result, the models contain rational reaction rates which are nonlinear in both parameters and states. It is challenging to estimate parameters nonlinear in a model although there have been many traditional nonlinear parameter estimation methods such as Gauss–Newton iteration method and its variants. In this article, we develop a two-step method to estimate the parameters in rational reaction rates of gene regulatory networks via weighted linear least squares. This method takes the special structure of rational reaction rates into consideration. That is, in the rational reaction rates, the numerator and the denominator are linear in parameters. By designing a special weight matrix for the linear least squares, parameters in the numerator and the denominator can be estimated by solving two linear least squares problems. The main advantage of the developed method is that it can produce the analytical solutions to the estimation of parameters in rational reaction rates which originally is nonlinear parameter estimation problem. The developed method is applied to a couple of gene regulatory networks. The simulation results show the superior performance over Gauss–Newton method.

Nonlinear Control of an Induction Motor Using a Reduced-Order Extended Sliding Mode Observer for Rotor Flux and Speed Sensorless Estimation

This article proposes an innovative strategy to the problem of non-linear estimation of states for electrical machine systems. This method allows the estimation of variables that are difficult to access or that are simply impossible to measure. Thus, as compared with a full-order sliding mode observer, in order to reduce the execution time of the estimation, a reduced-order discrete-time Extended sliding mode observer is proposed for on-line estimation of rotor flux, speed and rotor resistance in an induction motor using a robust feedback linearization control. Simulations results on Matlab-Simulink environment for a 1.8 kW induction motor are presented to prove the effectiveness and high robustness of the proposed nonlinear control and observer against modeling uncertainty and measurement noise.

Nonlinear Control of an Induction Motor Using a Reduced-Order Extended Sliding mode Observer for Rotor Flux and Speed Sensorless Estimation

This article proposes an innovative strategy to the problem of non-linear estimation of states for electrical machine systems. This method allows the estimation of variables that are difficult to access or that are simply impossible to measure. Thus, as compared with a full-order sliding mode observer, in order to reduce the execution time of the estimation, a reduced-order discrete-time Extended sliding mode observer is proposed for on-line estimation of rotor flux, speed and rotor resistance in an induction motor using a robust feedback linearization control. Simulations results on Matlab-Simulink environment for a 1.8 kW induction motor are presented to prove the effectiveness and high robustness of the proposed nonlinear control and observer against modeling uncertainty and measurement noise.

Optimal minimum variance estimation for non-linear discrete-time multichannel systems

A non-linear operator approach to estimation in discrete-time multivariable systems is described. It involves inferential estimation of a signal which enters a communication channel that contains non-linearities and transport delays. The measurements are assumed to be corrupted by a coloured noise signal correlated with the signal to be estimated. The solution of the non-linear estimation problem is obtained using non-linear operators. The signal and noise channels may be grossly non-linear and are represented in a very general non-linear operator form. The resulting so-called Wiener non-linear minimum variance estimation algorithm is relatively simple to implement. The optimal non-linear estimator is derived in terms of the non-linear operators and can be implemented as a recursive algorithm using a discrete-time non-linear difference equation. In the limiting case of a linear system, the estimator has the form of a Wiener filter in discrete-time polynomial matrix system form. A non-linear channel equalisation problem is considered for the design example.

A SURE Approach for Digital Signal/Image Deconvolution Problems

In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is twofold. First, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein's unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein's risk estimate are also provided in this work. Simulations carried out on natural images show the good performance of our method with respect to conventional wavelet-based restoration methods.

A new approach for adaptive blind equalization of chaotic communication: The optimal linearization technique

Together with the optimal linearization technique, a blind-channel equalization for the extended-Kalman-filter-based chaotic communication is proposed in this paper. First, the optimal linearization technique is utilized to find the exact linear models of the chaotic system at operating states of interest. The proposed blind-channel equalization is formulated as a mixed nonlinear parameter and state estimation problem by an autoregressive (AR) model. The channel coefficients of a fading and multipath channel can be represented by an AR process. Then, an extended Kalman filter algorithm is utilized to reduce the effect of channel noise. By using the extended Kalman filter, the channel coefficients and the state of the system, which is the signal before going through the channel, can be estimated. The stability problem of the proposed blind-channel equalization is also addressed. Numerical examples and simulations are given to show the effectiveness and speed of convergence for the proposed methodology.

A moving object tracking based on color information employing a particle filter algorithm

In this article, we present a new algorithm to track a moving object based on color information employing a particle filter algorithm. Recently, a particle filter has been proven very successful for nonlinear and non-Gaussian estimation problems. It approximates a posterior probability density of the state, such as the object position, by using samples which are called particles. The probability distribution of the state of the tracked object is approximated by a set of particles, where each state is denoted as the hypothetical state of the tracked object and its weight. The particles are propagated according to a state space model. Here, the state is treated as the position of the object. The weight is considered as the likelihood of each particle. For this likelihood, we consider the similarity between the color histogram of the tracked object and the region around the position of each particle. The Bhattacharya distance is used to measure this similarity. Finally, the mean state of the particles is treated as the estimated position of the object. Experiments were performed to confirm the effectiveness of this method to track a moving object.

Adaptive Grid-Voltage Sensorless Control Scheme for Inverter-Based Distributed Generation

This paper presents an adaptive grid-voltage sensorless control scheme for inverter-based distributed generation units based on an adaptive grid-interfacing model. An adaptive grid-interfacing model is proposed to estimate, in real time, the interfacing parameters seen by the inverter and the grid-voltage vector simultaneously. A reliable solution to the present nonlinear estimation problem is presented by combining a grid-voltage estimator with an interfacing parameter estimator in a parallel structure. Both estimators adjust the grid-interfacing model in a manner that minimizes the current error between the grid model and the actual current dynamics, which acts as a reference model. The estimated quantities are utilized within the inner high-bandwidth current control loop and the outer power controller to realize an adaptive grid-voltage sensorless interfacing scheme. Theoretical analysis and simulation results are provided to demonstrate the validity and usefulness of the proposed interfacing scheme.

Unscented Kalman filter approach to tracking a moving interfacial boundary in sedimentation processes using three-dimensional electrical impedance tomography

The monitoring of solid-fluid suspensions under the influence of gravity is widely used in industrial processes. By considering sedimentation layers with different electrical properties, non-invasive methods such as electrical impedance tomography (EIT) can be used to estimate the settling curves and velocities. In recent EIT studies, the problem of estimating the locations of phase interfaces and phase conductivities has been treated as a nonlinear state estimation problem and the extended Kalman filter (EKF) has been successfully applied. However, the EKF is based on a Gaussian assumption and requires a linearized measurement model. The linearization (or derivation of the Jacobian) is possible when there are no discontinuities in the system. Furthermore, having a complex phase interface representation makes derivation of the Jacobian a tedious task. Therefore, in this paper, we explore the unscented Kalman filter (UKF) as an alternative approach for estimating phase interfaces and conductivities in sedimentation processes. The UKF uses a nonlinear measurement model and is therefore more accurate. In order to justify the proposed approach, extensive numerical experiments have been performed and a comparative analysis with the EKF is provided.

Solving nonlinear estimation problems using splines [Lecture Notes

We describe the use of splines for solving nonlinear model estimation problems, in which nonlinear functions with unknown shapes and values are involved, by converting the nonlinear estimation problems into linear ones at a higher- dimensional space. This contrasts with the typical use of the splines for function interpolation where the functional values at some input points are given and the values corresponding to other input points are sought for via interpolation. The technique described in this column applies to arbitrary nonlinear estimation problems where one or more one-dimensional nonlinear functions are involved and can be extended to cases where higher-dimensional nonlinear functions are used.

Fixed‐order optimal deconvolution filter with irregular missing data

AbstractIn general, the reconstruction performance of the conventional deconvolution filter is deteriorated by the missing data. In this paper, a fixed‐order deconvolution filter design method is proposed for the signal reconstruction from received signal with irregular missing data. The missing data model is based on a probabilistic structure. The probability of occurrence of missing data is unknown a prior. In this situation, the deconvolution filter design problem becomes a complicated nonlinear estimation problem. In this study, a design method based on genetic algorithms is proposed to treat the signal reconstruction design problem with irregular missing data. Finally, two examples are given to illustrate the simulation results of the proposed deconvolution filter. The results show that the reconstruction performance is improved significantly if the missing probability is considered in the deconvolution filter design procedure. Copyright © 2009 John Wiley & Sons, Ltd.

Cubature Kalman Filters

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter. Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters.

Nonlinear Estimation for a Class of Systems

This paper considers nonlinear estimation problems for classes of models, and employs relative entropy to describe the uncertainty classes. Two optimization problems are formulated on general Banach spaces, and their solutions are sought: 1) when the transition probability between the signal to be estimated X and the measurement Y or stochastic kernel is unknown, and 2) when the joint probability induced by the random variables (RVs) X, Y is unknown. For both problems, the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution. The results include existence of the optimal measures using weak convergence techniques, and properties associated with the estimate of the true distribution. Classical examples are chosen to illustrate the applicability of the results.

On the Estimation of Parameters of Takagi–Sugeno Fuzzy Filters

This study derives a class of filtering algorithms for Takagi-Sugeno fuzzy models via solving a nonlinear parameters estimation problem. The considered estimation problem is related to the problem of minimizing the expected value of the exponential of filtering errors energy. Under some stochastic assumptions, the filtering criteria (which involve an expectation operator) are replaced by the deterministic quadratic optimization problems whose solutions provide a class of fuzzy filtering algorithms. From a viewpoint of errors in the estimation of linear parameters of the fuzzy filter, the derived filtering algorithms were analyzed with emphasis on stability, robustness, and steady-state error issues. The stability and robustness analyses have been made deterministically without making any assumption.

Fuzzy and Bayesian algorithms for nonlinear estimation problem

A problem of nonlinear estimation is considered. An approach to designing estimation algorithms based on fuzzy logic is offered. The approach suggested is compared with the well-known Bayesian approach. The conditions are determined such that the algorithms derived within these two approaches coincide. It is shown, in particular, that the fuzzy algorithm based on the Takagi-Sugeno inference that uses Gaussian membership functions and the defuzzification procedure as the centroid, will coincide with the point-mass algorithm that corresponds to the problem solution within the Bayesian approach with the Gaussian measurement errors and the vector to be estimated. The results derived are illustrated by an example.

Recursive state estimation for linear non-causal descriptor systems with uncertain parameters

This paper describes a state estimation approach for a uncertain dynamical system described by linear operator equation in Hilbert space. The uncertainty is supposed to admit a set-membership description. The approach is based on the notion of linear minimax a posteriori estimation. We introduce a new notion of minimax directional observability. It is used to derive the new equations describing the dynamics of the minimax recursive estimator for discrete-time non-causal DAEs in the time domain. The algorithm has the same structure as celebrated Kalman filter if the plant equation is causal. We illustrate the benefits of non-causality of the plant applying our approach to scalar nonlinear set-membership state estimation problem.