[Paper Review] A New Kalman Filter Model for Nonlinear Systems Based on Ellipsoidal Bounding

Ligang Sun, Hamza Alkhatib+3 more|arXiv (Cornell University)|Feb 8, 2018

Target Tracking and Data Fusion in Sensor Networks22 references3 citations

TL;DR

This paper proposes a novel set-membership Kalman filter (SKF) for nonlinear systems that jointly handles random and unknown-but-bounded (UBB) uncertainties using ellipsoidal bounding. By solving an optimization problem at each step to minimize the trace of the state estimation ellipsoid's shape matrix, the method produces more reliable, robust state estimates than the extended Kalman filter (EKF), especially under high angular uncertainty, as demonstrated in trajectory estimation benchmarks.

ABSTRACT

In this paper, a new filter model called set-membership Kalman filter for nonlinear state estimation problems was designed, where both random and unknown but bounded uncertainties were considered simultaneously in the discrete-time system. The main loop of this algorithm includes one prediction step and one correction step with measurement information, and the key part in each loop is to solve an optimization problem. The solution of the optimization problem produces the optimal estimation for the state, which is bounded by ellipsoids. The new filter was applied on a highly nonlinear benchmark example and a two-dimensional simulated trajectory estimation problem, in which the new filter behaved better compared with extended Kalman filter results. Sensitivity of the algorithm was discussed in the end.

Motivation & Objective

• To develop a robust state estimation framework for nonlinear systems that accounts for both random noise and unknown-but-bounded (UBB) uncertainties.
• To improve upon the limitations of the extended Kalman filter (EKF), particularly its sensitivity to model nonlinearity and inaccurate noise assumptions.
• To integrate the recursive structure of the Kalman filter with set-membership methods, preserving computational efficiency while enhancing uncertainty representation.
• To provide a more reliable estimation framework where the state is bounded within an ellipsoid, reflecting conservative and realistic uncertainty bounds.
• To evaluate the performance of the proposed filter under realistic uncertainty conditions, especially where angular errors dominate distance errors.

Proposed method

• The method employs a recursive two-step framework: prediction and correction, both involving solution of a nonlinear optimization problem.
• At each step, the state estimate is represented as an ellipsoid defined by a center vector and a positive-definite shape matrix, derived from minimizing the trace of the shape matrix.
• The prediction step propagates the state ellipsoid through the nonlinear system dynamics, accounting for process uncertainties bounded in ellipsoids.
• The correction step incorporates measurement data by solving an optimization problem that minimizes the trace of the updated shape matrix, yielding a tighter, more accurate ellipsoidal bound.
• The Minkowski sum of ellipsoidal uncertainties is approximated by solving a convex optimization problem to maintain convexity and boundedness.
• The algorithm uses a trace-minimization criterion to balance ellipsoid volume and shape, avoiding extreme oblateness or elongation.

Experimental results

Research questions

• RQ1 How can a Kalman filter be adapted to simultaneously handle random and unknown-but-bounded (UBB) uncertainties in nonlinear systems?
• RQ2 What is the impact of angular uncertainty versus distance uncertainty on the performance of state estimation in nonlinear systems?
• RQ3 Can a set-membership Kalman filter outperform the extended Kalman filter (EKF) in terms of estimation reliability under bounded uncertainty?
• RQ4 How does the choice of optimization criterion (e.g., trace, determinant, largest eigenvalue) affect the shape and size of the estimated state ellipsoid?
• RQ5 To what extent are the filter results sensitive to the initial bounds of the set-membership uncertainties?

Key findings

• The proposed set-membership Kalman filter (SKF) produces more reliable and robust state estimates than the extended Kalman filter (EKF) in highly nonlinear systems with bounded uncertainties.
• In a two-dimensional trajectory estimation problem, the SKF significantly outperformed the EKF, particularly when angular uncertainty dominated—e.g., 1° angle error caused a 1.48 m positional uncertainty at 85 m distance.
• The filter's performance is highly sensitive to the initial bounds of set-membership uncertainties; overly large initial bounds led to unrealistically large output ellipsoids, even with correct measurement data.
• The trace-minimization criterion in the optimization problem produced well-conditioned ellipsoids, avoiding extreme oblateness that can occur with determinant minimization.
• When initial uncertainty bounds were reduced (e.g., 0.01 m distance, 1° angle), the SKF produced realistic, tight ellipsoidal estimates, demonstrating its practical applicability.
• The algorithm's stability and convergence require further analysis, especially since state estimation is inherently ill-posed as an inverse problem.