On the identification of variances and adaptive Kalman filtering

1970/04/01 by R. Mehra, R.К. Mehra · 30 citations
Computer Science · Decision Sciences · #Target Tracking and Data Fusion in Sensor Networks #Distributed Sensor Networks and Detection Algorithms #Scientific Measurement and Uncertainty Evaluation

paper · doi:10.1109/tac.1970.1099422

Abstract

A Kalman filter requires an exact knowledge of the process noise covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and the measurement noise covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> . Here we consider the case in which the true values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> are unknown. The system is assumed to be constant, and the random inputs are stationary. First, a correlation test is given which checks whether a particular Kalman filter is working optimally or not. If the filter is suboptimal, a technique is given to obtain asymptotically normal, unbiased, and consistent estimates of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> . This technique works only for the case in which the form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> is known and the number of unknown elements in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> is less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n × r</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is the dimension of the state vector and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> is the dimension of the measurement vector. For other cases, the optimal steady-state gain K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">op</inf> is obtained directly by an iterative procedure without identifying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> . As a corollary, it is shown that the steady-state optimal Kalman filter gain K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">op</inf> depends only on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n × r</tex> linear functionals of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> . The results are first derived for discrete systems. They are then extended to continuous systems. A numerical example is given to show the usefulness of the approach.

Cited by

• A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation
• Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters
• Distributionally Robust Kalman Filter
• Distributed Noise Covariance Matrices Estimation in Sensor Networks
• Sparse Kalman Identification for Partially Observable Systems via Adaptive Bayesian Learning
• Convolution-based reformulation and Kalman-filter identification of short-time kernel operators for seat-human vibration modeling
• Correlation-Aware Dual-View Pose and Velocity Estimation for Dynamic Robotic Manipulation
• Trajectory Based Observer Design: A Framework for Lightweight Sensor Fusion
• Introduction to the Kalman Filter and Tuning its Statistics for Near Optimal Estimates and Cramer Rao Bound
• A simple and robust method for noise variance estimation for time-varying signals
• Robust fundamental frequency estimation in sustained vowels: detailed algorithmic comparisons and information fusion with adaptive Kalman filtering. [europepmc]
• Linear theory for filtering nonlinear multiscale systems with model error. [europepmc]
• Fuzzy Neural Network-Based Interacting Multiple Model for Multi-Node Target Tracking Algorithm. [europepmc]
• A Novel Adaptively-Robust Strategy Based on the Mahalanobis Distance for GPS/INS Integrated Navigation Systems. [europepmc]
• A Modified Extended Kalman Filter for a Two-Antenna GPS/INS Vehicular Navigation System. [europepmc]
• Estimating snow mass in North America through assimilation of AMSR-E brightness temperature observations using the Catchment land surface model and support vector machines. [europepmc]
• Measurement Noise Recommendation for Efficient Kalman Filtering over a Large Amount of Sensor Data. [europepmc]
• A Novel Online Approach for Drift Covariance Estimation of Odometries Used in Intelligent Vehicle Localization. [europepmc]
• A Neuron-Based Kalman Filter with Nonlinear Autoregressive Model. [europepmc]
• Robust Outlier-Adaptive Filtering for Vision-Aided Inertial Navigation. [europepmc]
• An FPGA Based Tracking Implementation for Parkinson's Patients. [europepmc]
• Estimation of a Human-Maneuvered Target Incorporating Human Intention. [europepmc]
• Novel Process Noise Model for GNSS Kalman Filter Based on Sensitivity Analysis of Covariance with Poor Satellite Geometry. [europepmc]
• Multi-Agent Reinforcement Learning via Adaptive Kalman Temporal Difference and Successor Representation. [europepmc]
• A CW-CNN regression model-based real-time system for virtual hand control. [europepmc]
• SmartFPS: Neural network based wireless-inertial fusion positioning system. [europepmc]
• Pedestrian Localization with Stride-Wise Error Estimation and Compensation by Fusion of UWB and IMU Data. [europepmc]
• A Novel Approach for Kalman Filter Tuning for Direct and Indirect Inertial Navigation System/Global Navigation Satellite System Integration. [europepmc]
• Real-Time Turning Movement, Queue Length, and Traffic Density Estimation and Prediction Using Vehicle Trajectory and Stationary Sensor Data. [europepmc]
• High-Precision Laser Time-Frequency Synchronization in Space Based on an Improved Kalman Filtering Method. [europepmc]

Related

• Perspective Chapter: Insights from Kalman Filtering with Correlated Noises Recursive Least-Square Algorithm for State and Parameter Estimation
• A Normal Variance Mixture Model for Robust Kalman Filtering
• Introduction to the Kalman Filter and Tuning its Statistics for Near Optimal Estimates and Cramer Rao Bound
• Bridging a gap in Kalman filtering output estimation with correlated noises or with direct feed-through from process noise into measurements
• Outlier-Insensitive Kalman Filtering Using NUV Priors