Quadratic Extended and Unscented Kalman Filter Updates

2025-06-06

AI Summary

Traditional linear estimators—such as the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF)—exhibit limited accuracy in nonlinear filtering due to their reliance on linear approximations of the measurement–state relationship. To address this, we propose an optimal minimum mean-square error (MMSE) estimation framework based on quadratic approximation, yielding the Quadratic EKF (QEKF) and Quadratic UKF (QUKF). Our key contribution is the first explicit incorporation of squared measurement terms into the Kalman filter structure, enabling a principled upgrade from linear to quadratic estimation without altering the underlying algorithmic architecture. The framework integrates quadratic Taylor expansion with the unscented transform and rigorously derives the optimal quadratic estimator. Numerical experiments demonstrate substantial improvements in estimation accuracy across diverse nonlinear scenarios. Moreover, the quadraticization strategy is generalizable to other linear estimators, offering broad applicability beyond the proposed filters.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Common filters are usually based on the linear approximation of the optimal minimum mean square error estimator. The Extended and Unscented Kalman Filters handle nonlinearity through linearization and unscented transformation, respectively, but remain linear estimators, meaning that the state estimate is a linear function of the measurement. This paper proposes a quadratic approximation of the optimal estimator, creating the Quadratic Extended and Quadratic Unscented Kalman Filter. These retain the structure of their linear counterpart, but include information from the measurement square to obtain a more accurate estimate. Numerical results show the benefits in accuracy of the new technique, which can be generalized to upgrade other linear estimators to their quadratic versions.

Problem

• Improves nonlinear state estimation accuracy
• Proposes quadratic Kalman filter variants
• Enhances linear estimators with quadratic terms

Innovation

• Quadratic approximation of optimal estimator
• Includes measurement square information
• Upgrades linear estimators to quadratic versions