Levent Özbek

Evaluation of convergence rate in the central limit theorem for the Kalman filter

State-space models are used for modeling of many physical and economic processes. An asymptotic distribution theory for the state estimate from a Kalman filter in the absence of the usual Gaussian assumption was presented by Spall and Wall (1984). They proved the central limit theorem for state estimators when the random terms in the model have arbitrary distribution. In this study, some convergence rates in the central limit theorem are given. These convergence rates are used for the development of a nonparametric test of the validity of the model.

An Adaptive Extended Kalman Filter with Application to Compartment Models

Abstract In this paper the ingestion and subsequent metabolism of a drug in a given individual, are investigated through the use of compartmental models. An adaptive formulation of the widely used extended Kalman filter (EKF) has been derived in order to solve the resulting nonlinear estimation problem at the output of the compartments. The adaptive EKF employs a forgetting factor to emphasize artificially the effect of current data by exponentially weighting the observations. The dependence of EKFs performance on the selection of appropriate values of the arbitrary matrices, the measurement covariance R and the process noise covariance Q has been demonstrated through simulations. With the appropriate choice of the matrices the EKF provides a very useful tool for online estimation of both the state and the parameters.

Stability of the Extended Kalman Filter When the States are Constrained

In this note, stability of the projection-based constrained discrete time extended Kalman filter (EKF) when applied to deterministic nonlinear systems has been studied. It is proved that, like the unconstrained case, under certain assumptions, the EKF with state equality constraints is an exponential observer, i.e., it keeps the dynamics of its estimation error exponentially stable. Also, it has been shown that a simple modification to the general definition of the EKF with exponential weighting increases the filters degree of stability and convergence speed with or without state constraints.

Joint Generation of Binary, Ordinal, Count, and Normal Data with Specified Marginal and Association Structures in Monte-Carlo Simulations

This chapter is concerned with building a unified framework for concurrently generating data sets that include all four major kinds of variables (i.e., binary, ordinal, count, and normal) when the marginal distributions and a feasible association structure are specified for simulation purposes. The simulation paradigm has been commonly employed in a wide spectrum of research fields including the physical, medical, social, and managerial sciences. A central aspect of every simulation study is the quantification of the model components and parameters that jointly define a scientific process. When this quantification cannot be performed via deterministic tools, researchers resort to random number generation (RNG) in finding simulation-based answers to address the stochastic nature of the problem. Although many RNG algorithms have appeared in the literature, a major limitation is that they were not designed to concurrently accommodate all variable types mentioned above. Thus, these algorithms provide only an incomplete solution, as real data sets include variables of different kinds. This work represents an important augmentation of the existing methods as it is a systematic attempt and comprehensive investigation for mixed data generation. We provide an algorithm that is designed for generating data of mixed marginals, illustrate its logistical, operational, and computational details; and present ideas on how it can be extended to span more complicated distributional settings in terms of a broader range of marginals and associational quantities.

Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

Abstract In this paper, the stability of the adaptive fading extended Kalman filter with the matrix forgetting factor when applied to the state estimation problem with noise terms in the non–linear discrete–time stochastic systems has been analysed. The analysis is conducted in a similar manner to the standard extended Kalman filter’s stability analysis based on stochastic framework. The theoretical results show that under certain conditions on the initial estimation error and the noise terms, the estimation error remains bounded and the state estimation is stable. The importance of the theoretical results and the contribution to estimation performance of the adaptation method are demonstrated interactively with the standard extended Kalman filter in the simulation part.

Dual Kalman filter approach for colored noise corrupted speech enhancement

In this paper, Kalman and Least Mean Square based filters are used for colored noise corrupted speech enhancement. Unlike previous studies a second speech signal has been utilized as colored noise which represents the situation where two persons are talking concurrently. Such a setup will help analyse the performance of speech enhancement algorithms when there are more than one speech components in the signal to be analysed and main speech signal has to be recovered. Final Prediction Error method has been employed for determining the model parameters, Speech was modeled with AR model and selected methods has been tested for their performance in terms of mean square error. The experimental results show that dual Kalman filter, which estimates both state and parameters concurently, has produced lower mean square error values when compared to joint and single Kalman filters. Joint Kalman filter, on the other hand, produced lower mean square error than single Kalman filter. Finally, it was observed that, the performance of LMS based filters was not adequate for the enhancement of colored noise corrupted speech.

A Fuzzy Model Based Control of Well Mixed Reactor with Cooling Jacket

Fuzzy logic control uses linguistic rather than crisp numerical rules to control the industrial processes. Fuzzy logic control may be attractive when the process is either difficult to control or to model by conventional methods. Fuzzy control of processes is an alternative when systems cannot be well controlled by classical or modern control techniques which are based on crisp mathematical models and also reply heavily on measurements to indicate variation in process conditions. For many processes, models and measurements are very difficult to obtain correctly and they show nonlinear behavior. Chemical reactors, pH neutralization process and waste water treatment are the examples for these processes. In such a systems, control rules can be developed depending on whether process is to be controllable. If the operators knowledge and experience can be explained in words, then linguistic rules can be written easily. In such cases, fuzzy model refers to the description of the operators input/output control actions using fuzzy implications.