RLS Wiener Fixed-Point Smoother and Filter with Randomly Delayed or Uncertain Observations in Linear Discrete-Time Stochastic Descriptor Systems

The purpose of this paper is to design the recursive least-squares (RLS) Wiener fixed-point smoother and filter in linear discrete-time descriptor systems. The signal process is observed with additional observation noise. The observed value is randomly delayed by multiple sampling intervals or has the possibility of uncertainty that the observed value does not include the signal and contains the observation noise only. It is assumed that the probability of the observation delay and the probability that the observation does not contain the signal are known. The delayed or uncertain measurements are characterized by the Bernoulli random variables. The characteristic of this paper is that the RLS Wiener estimators are proposed from the randomly delayed, by multiple sampling intervals, or uncertain observations particularly for the descriptor systems in linear discrete-time stochastic systems

RLS Wiener Filter and Fixed-Point Smoother with Randomly Delayed or Uncertain Observations in Linear Discrete-Time Stochastic Systems

This paper designs the recursive least-squares (RLS) Wiener fixed-point smoother and filter from randomly delayed observed values by multiple sampling times or uncertain observations in linear discrete-time stochastic systems. The observed value is generated in terms of the delayed observed values or uncertain observed values. In the case of the observed value with delay or without delay, their probabilities are assigned. Here, each observation includes signal plus white observation noise. Related to the uncertain observed value with delay or without delay, the probability that the observation consists of only observation noise is allocated, according to the time delayed or not delayed. It is assumed that the delay and uncertain measurements are characterized by the Bernoulli random variables. The RLS Wiener estimators use the following information. (1) The system matrix. (2) The observation matrix. (3) The variance of the state vector. (4) The probabilities concerned with the delayed observation and the uncertain observation. (5) The variance of white observation noise

Design of H-infinity Extended Recursive Wiener Estimators in Discrete-Time Stochastic Systems

This paper designs (1) the H-infinity RLS Wiener fixed-point smoother and filter for the observation equation with the linear modulation and (2) the extended H-infinity recursive Wiener fixed-point smoother and filter in discrete-time wide-sense stationary stochastic systems. ln the extended estimatars, it is assumed that the signal is observed with the nonlinear modulation and with additional white observation noise. In the estimators, the system matrix Φ for the state vector x(k), the observation vector C for the state vector, the variance K(k,k) = K(0) of the state vector, the nonlinear observation function and the variance of the white observation noise are used. Φ, C and K(0) axr calculated from the auto covariance data of the signal. A simulation example, on the estimation of a speech signal in the phase demodulation problem, is demonstrated to show the estimation characteristics of the proposed extended H-infinity recursive Wiener estimatoxs

Bibliographic Review on Distributed Kalman Filtering

In recent years, a compelling need has arisen to understand the effects of distributed information structures on estimation and filtering. In this paper, a bibliographical review on distributed Kalman filtering (DKF) is provided.\ud The paper contains a classification of different approaches and methods involved to DKF. The applications of DKF are also discussed and explained separately. A comparison of different approaches is briefly carried out. Focuses on the contemporary research are also addressed with emphasis on the practical applications of the techniques. An exhaustive list of publications, linked directly or indirectly to DKF in the open literature, is compiled to provide an overall picture of different developing aspects of this area

Design of discrete time controllers and estimators.

This thesis considers optimal linear least-squares filtering smoothing prediction and regulation for discrete-time processes. A finite interval smoothing filter is derived in the z domain giving a transfer function solution. The resulting time-invariant smoother can be applied to problems where, a time varying solution using matrix Riccati equations would diverge if the process is modelled inaccurately. A self-tuning algorithm is given for the filtering and fixed lag smoothing problems as applied to square multi-variable ARMA processes when only the order of the process is assumed known. The dynamics of the process can also be slowly time varying. If the dynamics remain constant and unknown, it is shown how the self-tuning filter or smoother algorithm converges asymptotically to the optimal Wiener solutions. LQG self-tuning regulation is considered. The LQG algorithms rely on input-output data rather than from the conventional state-space approach employing the Kalman filter. An explicit algorithm is given which is similar to certain pole placement self-tuning regulators, requiring the solution of a diophantine equation. Following this, an implicit algorithm is shown to overcome the problem of solving a diophantine equation by estimating the regulator parameters directly using recursive least squares. The LQG algorithms are shown to be able to cope with processes which are non-minimum phase, open loop unstable and with an unknown time delay

Nested filtering methods for Bayesian inference in state space models

Mención Internacional en el título de doctorA common feature to many problems in some of the most active fields of science is the need to calibrate (i.e., estimate the parameters) and then forecast the time evolution of high-dimensional dynamical systems using sequentially collected data. In this dissertation we introduce a generalised nested filtering methodology that is structured in (two or more) intertwined layers in order to estimate the static parameters and the dynamic state variables of nonlinear dynamical systems. This methodology is essentially probabilistic. It aims at recursively computing the sequence of posterior probability distributions of the unknown model parameters and its (time-varying) state variables conditional on the available observations. To be specific, in the first layer of the filter we approximate the posterior probability distribution of the static parameters and in the consecutive layers we employ filtering (or data assimilation) techniques to track and predict different conditional probability distributions of the state variables. We have investigated the use of different Monte Carlo-based methods and Gaussian filtering techniques in each of the layers, leading to a wealth of algorithms. In a first approach, we have introduced a nested filtering methodology of two layers that aims at recursively estimating the static parameters and the dynamical state variables of a state space model. This probabilistic scheme uses Monte Carlo-based methods in the first layer of the filter, combined with the use of Gaussian filters in the second layer. Different from the nested particle filter (NPF) of [25], the use of Gaussian filtering techniques in the second layer allows for fast implementations, leading to algorithms that are better suited to high-dimensional systems. As each layer uses different types of methods, we refer to the proposed methodology as nested hybrid filtering. We specifically explore the combination of Monte Carlo and quasi–Monte Carlo approximations in the first layer, including sequential Monte Carlo (SMC) and sequential quasi-Monte Carlo (SQMC), with standard Gaussian filtering methods in the second layer, such as the ensemble Kalman filter (EnKF) and the extended Kalman filter (EKF). However, other algorithms can fit naturally within the framework. Additionally, we prove a general convergence result for a class of procedures that use SMC in the first layer and we show numerical results for a stochastic two-scale Lorenz 96 system, a model commonly used to assess data assimilation (filtering) procedures in Geophysics. We apply and compare different implementations of the methodology to the tracking of the state and the estimation of the fixed parameters. We show estimation and forecasting results, obtained with a desktop computer, for up to 5000 dynamic state variables. As an extension of the nested hybrid filtering methodology, we have introduced a class of schemes that can incorporate deterministic sampling techniques (such as the cubature Kalman filter (CKF) or the unscented Kalman filter (UKF)) in the first layer of the algorithm, instead of the Monte Carlo-based methods employed in the original procedure. As all the methods used in this scheme are Gaussian, we refer to this class of algorithms as nested Gaussian filters. One more time, we reduce the computational cost with the proposed scheme, making the resulting algorithms potentially better-suited for high-dimensional state and parameter spaces. In the numerical results, we describe and implement a specific instance of the new method (a UKF-EKF algorithm) and evaluate its average performance in terms of estimation errors and running times for nonlinear stochastic models. Specifically, we present numerical results for a stochastic Lorenz 63 model using synthetic data, as well as for a stochastic volatility model with real-world data. Finally, we have extended the proposed methodology in order to estimate the static parameters and the dynamical variables of a class of heterogeneous multi-scale state-space models [1]. This scheme combines three or more layers of filters, one inside the other. Each of the layers corresponds to the different time scales that are relevant to the dynamics of this kind of state-space models, allocating the variables with the greatest time scales (the slowest ones) in the outer-most layer and the variables with the smallest time scales (the fastest ones) to the inner-most layer. In particular, we describe a three-layer filter that approximates the posterior probability distribution of the parameters in a first layer of computation, in a second layer we approximate the posterior probability distribution of the slow state variables, and the posterior probability distribution of the fast state variables is approximated in a third layer. To be specific, we describe two possible algorithms that derive from this scheme, combining Monte Carlo methods and Gaussian filters at different layers. The first method uses SMC methods in both first and second layers, together with a bank of UKFs in the third layer (i.e., a SMC-SMC-UKF algorithm). The second method employs a SMC in the first layer, EnKFs at the second layer and introduces the use of a bank of EKFs in the third layer (i.e., a SMC-EnKF-EKF algorithm). We present numerical results for a two-scale stochastic Lorenz 96 model with synthetic data.Programa de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Víctor Elvira Arregui.- Secretario: Stefano Cabras.- Vocal: David Luengo Garcí

Review of Unbiased FIR Filters, Smoothers, and Predictors for Polynomial Signals

Extracting an estimate of a slowly varying signal corrupted by noise is a common task. Examples can be found in industrial, scientific and biomedical instrumentation. Depending on the nature of the application the signal estimate is allowed to be a delayed estimate of the original signal or, in the other extreme, no delay is tolerated. These cases are commonly referred to as filtering, prediction, and smoothing depending on the amount of advance or lag between the input data set and the output data set. In this review paper we provide a comprehensive set of design and analysis tools for designing unbiased FIR filters, predictors, and smoothers for slowly varying signals, i.e. signals that can be modeled by low order polynomials. Explicit expressions of parameters needed in practical implementations are given. Real life examples are provided including cases where the method is extended to signals that are piecewise slowly varying. A critical view on recursive implementations of the algorithms is provided