The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise
type: Item
instance of: scholarly article
title: The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise
zbMATH Open document ID: 0617.93054
DOI: 10.1016/0167-6911(87)90106-X
author: Wassim M. Haddad
author: Dennis S. Bernstein
published in: Systems \& Control Letters
publication date: 1987-01-01T00:00:00Z
review text: Given the n-th-order observed system \[ x(k+1)=(A+\sum^{p}_{i=1}v_ i(k)A_ i)x(k)+W_ 1(k),\quad y(k)=(C+\sum^{p}_{i=1}v_ i(k)C_ i)x(k)+W_ 2(k) \] where \(W_ 1(k)\), \(W_ 2(k)\) are n-dimensional and \(\ell\)-dimensional white noise processes, \(v_ i(k)\) is an unit variance white noise, A, \(A_ i\) are \(n\times n\) matrices and C, \(C_ i\) are \(\ell \times n\) matrices, \(i=1,...,p\). The problem is to design an \(n_ e\)-th \((1\leq n_ e\leq n)\) reduced-order state estimator \[ x_ e(k+1)=A_ ex_ e(k)+B_ ey(k),\quad y_ e(k)=C_ ex_ e(k)+D_ ey(k) \] (x\({}_ e\), \(y_ e\) denote \(n_ e\)-dimensional and q- dimensional vectors, \(A_ e\), \(B_ e\), \(C_ e\), \(D_ e\) are \(n_ e\times n_ e\), \(n_ e\times \ell\), \(q\times n_ e\), \(q\times \ell\) matrices) which minimizes the state-estimation error criterion \[ J(A_ e,B_ e,C_ e,D_ e)=\lim_{k\to \infty}E[Lx(k)-y_ e(k)]^ T=R[Lx(k)-y_ e(k)] \] where L and R are some matrices. The optimality conditions for the \(n_ e\)-th reduced-order state estimator for the matrices \(A_ e\), \(B_ e\), \(C_ e\), \(D_ e\) are given. The optimal estimator is characterized by three matrix equations (a modified Riccati equation and two modified Lyapunov equations).
Mathematics Subject Classification ID: 93E10
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93E11
Mathematics Subject Classification ID: 93E12
Mathematics Subject Classification ID: 62F35
zbMATH DE Number: 4001995
zbMATH Keywords: reduced-order state estimator
zbMATH Keywords: optimality conditions
zbMATH Keywords: optimal estimator
zbMATH Keywords: modified Riccati equation
zbMATH Keywords: modified Lyapunov equations
reviewed by: Ryszarda Rempała
MaRDI profile type: Publication
cites work: The optimal projection equations for reduced-order state estimation
cites work: A survey of specific optimal techniques in control and estimation†
cites work: Design of low-order observers for linear feedback control laws
cites work: Bias, variance, and estimation error in reduced order filters
cites work: Information and distortion in reduced-order filter design
cites work: Reduced order state estimators for discrete-time stochastic systems
cites work: On stochastic observer estimators for continuous-time systems
cites work: Optimal and suboptimal results in full- and reduced-order linear filtering
cites work: Q3874080
cites work: Design of low order estimators using reduced models
cites work: Design of multifunctional reduced order observers†
cites work: Method of synthesizing suboptimal filters of reduced order for digital linear dynamic systems
cites work: Reduced order observer design for a linear map of the state
cites work: Q5184791
cites work: Optimal linear filtering for linear systems with state-dependent noise
cites work: An approximate method of state estimation for non-linear dynamical systems with state-dependent noise†
cites work: Linear minimum variance filters applied to carrier tracking
cites work: Q4106051
cites work: Observers for optimal estimation of the state of linear stochastic discrete systems
cites work: Weiner and Kalman filters for systems with random parameters
cites work: Sensitivity analysis of discrete Kalman filters
cites work: The Kalman filter: A robust estimator for some classes of linear quadratic problems
cites work: Robust bayesian estimation for the linear model and robustifying the Kalman filter
cites work: Bounds on estimation errors of discrete-time filters under modeling uncertainty
cites work: Reducing the sensitivity to parameter variations of a minimum-order reduced-order observer
cites work: Robust controller synthesis using the maximum entropy design equations
cites work: Optical projection equations for reduced-order modelling, estimation, and control of linear systems with multiplicative white noise
cites work: Robust static and dynamic output-feedback stabilization: Deterministic and stochastic perspectives
cites work: Kronecker products and matrix calculus in system theory
cites work: Q5645536
full work available at URL: 0167-6911(87)90106-x
OpenAlex ID: W2071924116
description: scientific article; zbMATH DE number 4001995