New Results in Linear Filtering and Prediction Theory

By R. Kálmán, R. Bucy. 1961

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation" completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data (2016, methodology): “Under strong assumptions on the system, one can derive optimal Bayesian filters, such as the classical Kalman filter [7] for linear Gaussian models (LGMs).”
Real-time freeway traffic state estimation based on extended Kalman filter: a general approach (2005, methodology): “For traffic state estimation, a limited number of research works produced and proposed corresponding estimation algorithms that were almost exclusively based on the seminal methodology of Kalman filtering ( Kalman and Bucy, 1961 ) and its extensions for nonlinear systems (see, e.g.”
Kalman filtering and smoothing solutions to temporal Gaussian process regression models (2010, methodology): “Advantages of the proposed approach are illustrated with two numerical experiments.”
Maximum Likelihood Ensemble Filter: Theoretical Aspects (2005, methodology): “Since early 1960s, data assimilation in atmospheric and oceanographic applications is based on the Kalman filtering theory (Kalman and Bucy 1961; Jazwinski 1970 .”
Distributing the Kalman Filter for Large-Scale Systems (2007, background): “C ENTRALIZED implementation of the Kalman filter [1], [2], although possibly optimal, is neither robust nor scalable to complex large-scale dynamical systems with their measurements distributed on a large geographical region.”
Deterministic least squares filtering (2004, methodology): “The Kalman !ltering algorithm originally appeared in Kalman (1960a) for discretetime systems and in Kalman and Bucy (1961) for continuous-time systems.”
The Invariant Extended Kalman Filter as a Stable Observer (2014, methodology): “We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic non-linear observer on Lie groups, for continuoustime systems with discrete observations.”
An Introduction to Hidden Markov Models and Bayesian Networks (2001, unknown): “St+1 P (Yt+2:T |St+1)P (St+1|St)P (Yt+1|St+1) (33)”
Nonlinear Dynamical System Identification from Uncertain and Indirect Measurements (2004, methodology): “The Kalman filter (KF) [Kalman, 1960; Kalman & Bucy, 1960] is one of the most powerful signal analysis tools.”
Approximate robust optimization of nonlinear systems under parametric uncertainty and process noise (2015, background): “It has to be noted that the above expression corresponds to the variance prediction exploited in the classic (continuous) extended Kalman filter (EKF) (Kalman and Bucy, 1961), however, as no measurements are taken during the dynamic optimization, this term is not incorporated.”