Optimal linear filtering in systems with noise in observations dependent on signal and estimate
type: Item
instance of: scholarly article
title: Optimal linear filtering in systems with noise in observations dependent on signal and estimate
zbMATH Open document ID: 1062.93040
author: Yanyan Li
published in: Automation and Remote Control
publication date: 2004-10-19T00:00:00Z
review text: The usual underlying idea of traditional approaches to observation control is the conditional Gaussian filtering. In practice, however, there are several problems which do not fall into this class of conditionally Gaussian models, among them namely the problems in which the noises in the observations depend on the state of the non-observable process and its estimate. This class of problems is studied here. To be more precise, let a partially observable system be described by the multi-dimensional Itô equations \(dX_{t}=X_{t}dA_{t}+d\xi_{t}\), \(dY_{t}=H_{t}X_{t}dt+d\eta_{t}\), where \(X_{0}=x_{0}\) is Gaussian, \(\xi\) is a Gaussian martingale and \(A\) is a matrix-valued bounded process of bounded variation; \(X\) is interpreted as the non-observable process. Given a Brownian motion \(W\) independent of \(\xi\) and \(x_{0}\), the observation process \(Y\) is driven by the Itô integral process \(\eta_{t}=\int_{0}^{t}G(s,X_{s},\widehat{X}_{s})dW_{s}\), depending on the non-observable process \(X\) and its estimate \(\widehat{X}\). The authors of the paper study linear estimates \(\widehat{X}\) of the type \(\widehat{X}_{t} =F_{t}+\int_{0}^{t}L(t,s)dY_{s}\), where \(F\) and \(L\) are deterministic functions. They show that the linear estimate \(\widehat{X}\) minimizing the quadratic estimation error matrix is a Kalman-type estimate and they determine an explicit form of the optimal coefficient of this Kalman-type estimate. After that, the authors study the above partially observable system but now supplemented with a control term in the state equation, and they associate a quadratic cost function. Its minimal value and the optimal control are characterized with the help of Riccati-type equations.
Mathematics Subject Classification ID: 93E11
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 2108722
zbMATH Keywords: optimal linear filtering
zbMATH Keywords: Kalman estimate
zbMATH Keywords: Zakai equation
zbMATH Keywords: Riccati equation
zbMATH Keywords: state dependent observation noise
zbMATH Keywords: stochastic optimal control
zbMATH Keywords: partially observable system
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2108722