Boundary Value Problems Arising in Kalman Filtering

Optim Control Appl Methods

Mazhar

Etikan

et al. 2012

Self Cite

SUMMARYThe paper presents a mathematical theory of handling and working with wideband noises. We demonstrate that a wideband noise can be represented as a distributed delay of a white noise. From this, we deduce that the behavior of a wideband noise is the same as the behavior of an infinite dimensional colored noise along the boundary line. All these are used to deduce a complete set of formulae for the Kalman-type optimal filter and also to derive nonlinear filtering equation for wideband-noise-driven linear and nonlinear systems.

Section: Discussionmentioning

confidence: 99%

Delay structure of wideband noises with application to filtering problems

Optim Control Appl Methods

Mazhar

Etikan

et al. 2012

Self Cite

“…Thus ϕ becomes a delayed (multiply and time-dependent) white noise. This kind of relaxing functions has been studied in Bashirov et al [34,35,36,37] by approximation of them with relaxing functions from C(0, ∞; W 1,2 0 (−ε, 0; R n×k )).…”

Section: Wide Band Noisesmentioning

confidence: 99%

Linear Filtering for Wide Band Noise Driven Observation Systems

Circuits Syst Signal Process

2016

Filtering of wide band noise driven systems accounts the following problem. Given an autocovariance function, there are infinitely many wide band noise processes, which have this autocovariance function. Each of them produces its own best estimate. The problem is a selection of the best one of these best estimates. A similar problem arises in control theory as a selection of optimal one of the optimal controls. In this paper we investigate this problem for a wide class of wide band noises. It is proved that in the case of independent wide band and white noises corrupting, respectively, the signal and observations, the best estimates and the optimal controls in the linear filtering and LQG problems are independent of the respective wide band noises. We present a complete set of formulae for the best estimate and, respectively, for the optimal control in terms of the system parameters and autocovariance function of the wide band noise disturbing the signal system.

“…In [15][16][17][18][19][20][21][22][23] wide band noise driven systems are investigated by a method of approximation. A different method by integral representation was suggested in [24][25][26] which leads to modelling wide band noises as a distributed delay of white noises [27,28].…”

Section: Introductionmentioning

confidence: 99%

Invariant Kalman Filter for Correlated Wide Band Noises

Asian Journal of Control

Abuassba

2018

Kalman filtering is a powerful estimation method. One of its weaknesses is related to the white or coloured nature of the disturbing noises in the Kalman filtering model. At the same time, real noises are rarely white or coloured. They are mostly wide band. In this regard, white or coloured noise Kalman filtering makes concessions on adequacy. This pushes system scientists to develop mathematical methods of estimation for systems corrupted by wide band noises. In applications, wide band noises are detected by their autocovariance and cross‐covariance functions, which do not allow to model them uniquely. Therefore, it becomes important to develop estimation methods which are independent of a class of wide band noises, but dependent on the unique autocovariance and cross‐covariance functions. Such results are called invariant results. In this paper, we prove a complete set of invariant equations for Kalman type filter for a linear signal‐observation system corrupted by correlated wide band noises. This filter has a ready form to be used in applications, just respective numerical methods must be developed.