Nonlinear State Estimation

The use of dynamical system techniques, optimization methods and statistical algorithms to estimate the characteristics of brain electrical activity are explored. A system approach for characterizing EEG (electroencephalogram) signals, based on nonlinear estimation of dynamical characteristics and modeling the evolution of dynamical processes over time is applied. The dynamical characteristics can be used to better visualize the “state vector” of epileptic EEG signals and for the purpose of pattern recognition. An optimization method for reconstructing parameter spaces of dynamical systems is applied to systems with one or more hidden variables, and can be used to reconstruct maps or differential equations of the brain dynamics. The methods are illustrated by using numerically generated data and EEG data from epileptic patients.

Bioethanol production from fermentation of a sub- strate using biomass as catalyst is considered. Four alternative reaction rate models with dier- ent levels of details are derived and implemented in Modelica. The problem of parameter estima- tion of models using state/parameter estimation techniques in a Modelica-Dymola/Matlab setup is discussed. Practical aspects concerning the dier- ent implementations of nonlinear estimators are

This paper investigates the cubature Kalman filtering (CKF) for nonlinear dynamic systems. This third-degree rule based filter employs a spherical-radial cubature rule to numerically compute the integrals encountered in nonlinear filtering problems, thereby removing the requirements of explicitly computing the Jacobians. The cubature rule, however, requires computing the intractable integrals over a high-dimensional spherical region for multidimensional applications. Moreover, the cubature formula that has been used to construct the spherical cubature formula has some demerits, most notably its inconvenient properties in computation and low estimation accuracy. Aimed at these issues, a general class of CKFs that uses only cubature rules is derived in this paper. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes higher-degree CKFs, especially two representative types of the fifth-degree CKFs. Performance of the proposed algorithms is demonstrated via two target tracking problems. The experimental results, presented herein, illustrate the superior performance of higher-degree CKFs to conventional nonlinear filters.

The information form of the Kalman filter (KF) is preferred over standard covariance filters in multiple sensor fusion problems. Aiming at this issue, two types of cubature information filters (CIF) for nonlinear systems are presented in this article. The two approaches, which we have named the embedded cubature information filter (ECIF) and the fifth-degree cubature information filter (FCIF), are developed from a fifth-degree cubature Kalman filter and a newly proposed embedded cubature KF. Theoretical analysis shows that the proposed filters can achieve higher level estimation accuracy than conventional information filters, such as the CIF and the extended information filter (EIF). Performance comparisons of the proposed information filters with the conventional CIF are demonstrated via two independent multisensor tracking problems. The experimental results, presented herein, demonstrate that the proposed algorithms are more reliable and accurate than the CIF.

We introduce an approach based on state observers to estimate the exponent and the coefficient of a power law that describes the friction in a horizontal pipeline. The main advantage of our approach is twofold: (a) it can be useful when the friction is needed but the pipeline physical parameters required to compute it are unknown, and (b) it can be applied during the pipeline operation. To show the applicability of our proposal, we present some simulation results that include the performance of the observers and a numerical validation of the proposed power law against a more sophisticated model.

High reliability systems generally require individual system components having extremely high reliability over long periods of time. Short product development times require reliability tests to be conducted with severe time constraints. Frequently few or no failures occur during such tests, even with acceleration. Thus, it is difficult to assess reliability with traditional life tests that record only failure times. For some components, degradation measures can be taken over time. A relationship between component failure and amount of degradation makes it possible to use degradation models and data to make inferences and predictions about a failure-time distribution. This article describes degradation reliability models that correspond to physical-failure mechanisms. We explain the connection between degradation reliability models and failure-time reliability models. Acceleration is modeled by having an acceleration model that describes the effect that temperature (or another accelerating variable) has on the rate of a failure-causing chemical reaction. Approximate maximum likelihood estimation is used to estimate model parameters from the underlying mixed-effects nonlinear regression model. Simulation-based methods are used to compute confidence intervals for quantities of interest (e.g., failure probabilities). Finally we use a numerical example to compare the results of accelerated degradation analysis and traditional accelerated life-test failure-time analysis.

In most solutions to state estimation problems like, for example, target tracking, it is generally assumed that the state evolution and measurement models are known a priori. The model parameters include process and measurement matrices or functions as well as the corresponding noise statistics. However, there are situations where the model parameters are not known a priori or are known only partially (i.e., with some uncertainty). Moreover, there are situations where the measurement is biased. In these scenarios, standard estimation algorithms like the Kalman filter and the extended Kalman Filter (EKF), which assume perfect knowledge of the model parameters, are not accurate. In this paper, the problem with uncertain model parameters is considered as a special case of maximum likelihood estimation with incomplete-data, for which a standard solution called the expectation-maximization (EM) algorithm exists. In this paper a new extension to the EM algorithm is proposed to solve the more general problem of joint state estimation and model parameter identification for nonlinear systems with possibly non-Gaussian noise. In the expectation (E) step, it is shown that the best variational distribution over the state variables is the conditional posterior distribution of states given all the available measurements and inputs. Therefore, a particular type of particle filter is used to estimate and update the posterior distribution. In the maximization (M) step the nonlinear measurement process parameters are approximated using a nonlinear regression method for adjusting the parameters of a mixture of Gaussians (MofG). The proposed algorithm is used to solve a nonlinear bearing-only tracking problem similar to the one reported recently with uncertain measurement process. It is shown that the algorithm is capable of accurately tracking the state vector while identifying the unknown measurement dynamics. Simulation results show the advantages of the new technique over standard algorithms like the EKF whose performance degrades rapidly in the presence of uncertain models.

Summary In this paper, we aim to analyze the relationships between the quality and price of secondhand condominiums in the 23 wards of Tokyo. We propose a secondhand condominium price estimation method. Specifically, with a linear model as the starting point, we estimate nonlinear relationships of secondhand condominium unit prices in terms of floor space in square meters, the number

We present a systematic comparison of machine learning methods applied to the problem of fully automatic recognition of facial expressions. We explored recognition of facial actions from the Facial Action Coding System (FACS), as well as recognition of full facial expressions. Each videoframe is first scanned in real-time to detect approximately upright-frontal faces. The faces found are scaled into image patches of equal size, convolved with a bank of Gabor energy filters, and then passed to a recognition engine that codes facial expressions into 7 dimensions in real time: neutral, anger, disgust, fear, joy, sadness, surprise. We report results on a series of experiments comparing recognition engines, including AdaBoost, support vector machines, linear discriminant analysis, as well as feature selection techniques. Best results were obtained by selecting a subset of Gabor filters using AdaBoost and then training Support Vector Machines on the outputs of the filters selected by AdaB...

We present a simple method for estimating kinetic parameters from progress curve analysis of biologically catalyzed reactions that reduce to forms analogous to the Michaelis–Menten equation. Specifically, the Lambert W function is used to obtain explicit, closed-form solutions to differential rate expressions that describe the dynamics of substrate depletion. The explicit nature of the new solutions greatly simplifies nonlinear estimation of the kinetic parameters since numerical techniques such as the Runge–Kutta and Newton–Raphson methods used to solve the differential and integral forms of the kinetic equations, respectively, are replaced with a simple algebraic expression. The applicability of this approach for estimating Vmax and Km in the Michaelis–Menten equation was verified using a combination of simulated and experimental progress curve data. For simulated data, final estimates of Vmax and Km were close to the actual values of 1 μM/h and 1 μM, respectively, while the stand...

Kalman filter and its variants are well known for the static and dynamic state estimation of power systems because of their accuracies. These adaptive filters generally employed for estimation purposes require high computational power when
it comes to real time estimation. Therefore, in this paper we
propose a computationally light yet effective estimation algorithm based on state space model which have not yet been applied to the problem of power system estimation. We propose the use of state space least mean square algorithms for the purpose of state estimation considering the problem of a two phase permanent magnet synchronous motor. The algorithms have been employed successfully in this paper in the state estimation of the highly non linear synchronous motor. We investigate the problem in the presence of Gaussian noise to show the novelty of the algorithms. Moreover, these algorithms are compared with the state estimation performance of the non linear Extended Kalman filter.

This paper shows the applicability of recently-developed Gaussian nonlinear filters to sensor data fusion for positioning purposes. After providing a brief review of Bayesian nonlinear filtering, we specially address square-root, derivative-free algorithms based on the Gaussian assumption and approximation rules for numerical integration, namely the Gauss--Hermite quadrature rule and the cubature rule. Then, we propose a motion model based on the observations taken by an Inertial Measurement Unit, that takes into account its possibly biased behavior, and we show how heterogeneous sensors (using time-delay or received-signal-strength based ranging) can be combined in a recursive, online Bayesian estimation scheme. These algorithms show a dramatic performance improvement and better numerical stability when compared to typical nonlinear estimators such as the Extended Kalman Filter or the Unscented Kalman Filter, and require several orders of magnitude less computational load when compared to Sequential Monte Carlo methods, achieving a comparable degree of accuracy.

This paper presents a new and general nonlinear framework for fMRI data analysis based on statistical learning methodology: support vector machines. Unlike most current methods which assume a linear model for simplicity, the estimation and analysis of fMRI signal within the proposed framework is nonlinear, which matches recent findings on the dynamics underlying neural activity and hemodynamic physiology. The approach utilizes spatio-temporal support vector regression (SVR), within which the intrinsic spatio-temporal autocorrelations in fMRI data are reflected. The novel formulation of the problem allows merging model-driven with data-driven methods, and therefore unifies these two currently separate modes of fMRI analysis. In addition, multiresolution signal analysis is achieved and developed. Other advantages of the approach are: avoidance of interpolation after motion estimation, embedded removal of low-frequency noise components, and easy incorporation of multi-run, multi-subjec...

The purpose of this paper is to describe and assess some methods for accounting for certainty primary sampling units (PSUs) when using a pseudo- replication procedure (specifically balanced repeated replication (BRR) procedure) for variance estimation 1 . We compare these alternatives to the variance estimation procedure using the explicit variance estimation equations with Taylor series linearization of nonlinear estimators. The

ABSTRACT Target tracking, nonlinear control, and fault detection are typically evaluated with only a Root Mean Square (RMS). RMS is an absolute measurement of the system performance and does not provide a statistic as to the tracker, controller, or fault detection algorithmic performance. For this paper, we investigate the non-credibility index (NCI) and average normalized estimation error square (ANEES) for nonlinear estimation for the Kalman Filter (KF), the Central Difference Filter (DD1), the unscented Kalman filter (UKF), and the particle filter (PF). Fault detection and target track performance is dependent on target maneuvers, sensor errors, model parameters, and state estimation which need to be understood relative to the filter performance versus the absolute performance (i.e. root mean square) of the system. Utilizing the developments of the Nonlinear Estimation Framework (NEF) toolbox, we develop methods of nonlinear relative comparison performance between nonlinear filters in a unified scenario.

In this paper, an adaptive nonlinear estimator is developed to identify the Euclidean coordinates of feature points on a moving object using a single fixed camera. No explicit model is used to describe the movement of the object. Homography-based techniques are used in the development of the object kinematics, while Lyapunov design methods are utilized in the synthesis of the adaptive estimator. Simulation results are included to demonstrate the performance of the estimator.

We introduce an approach based on state observers to estimate the exponent and the coefficient of a power law that describes the friction in a horizontal pipeline. The main advantage of our approach is twofold: (a) it can be useful when the friction is needed but the pipeline physical parameters required to compute it are unknown, and (b) it can be applied during the pipeline operation. To show the applicability of our proposal, we present some simulation results that include the performance of the observers and a numerical validation of the proposed power law against a more sophisticated model.