Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems

Ligang Sun,Ingo Neumann,Boris Kargoll,Vladik Kreinovich,Hamza Alkhatib

doi:10.3390/math7121168

Abstract

In this paper, we propose a new technique—called Ellipsoidal and Gaussian Kalman filter—for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system’s state (and the optimal ellipsoid for describing the systems’s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better—the state estimation technique usually applied to such nonlinear problems.

Highlights

State estimation is important for virtually all areas of engineering and science
We need to take into account that measurements are never absolutely accurate, the measurement results contain inaccuracy (“noise”)—e.g., due to inevitable imperfection of the measuring instruments
Techniques for decreasing the effect of noise—i.e., for separating (“filtering”) signal from noise—are known as filtering; because of this, the state estimation problem can be viewed as an important particular case of filtering

Summary

Introduction

State estimation is important for virtually all areas of engineering and science. Every discipline which uses the mathematical modeling of its systems needs state estimation; this includes electrical engineering, mechanical engineering, chemical engineering, aerospace engineering, robotics, dynamical systems’ control and many others.We estimate the system’s state based on the measurement results. State estimation is important for virtually all areas of engineering and science. Every discipline which uses the mathematical modeling of its systems needs state estimation; this includes electrical engineering, mechanical engineering, chemical engineering, aerospace engineering, robotics, dynamical systems’ control and many others. We estimate the system’s state based on the measurement results. In addition to the internal factors (which are described by the system’s state) and the known external factors, there are usually many other factors that affect the system—and which, from the viewpoint of the model, constitute the noise. Techniques for decreasing the effect of noise—i.e., for separating (“filtering”) signal from noise—are known as filtering; because of this, the state estimation problem can be viewed as an important particular case of filtering

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