Estimation II 1 Discrete-time Kalman filter (original) (raw)

Kalman Filter: A Simple Derivation

Mathematics and Statistics

The Kalman filter is a recursive estimator and plays a fundamental role in statistics for filtering, prediction and smoothing. The key element in any recursive estimator is the estimate of the current state, x k , at time k, based on observations up to and including observation k and the Kalman filter enables the estimate of the state to be updated as new observations become available. In this paper we have tried to derive the Kalman filter as simple as possible.

A Note on Kalman Filtering

—The purpose of this paper is to point out a confusing phenomenon in the teaching of Kalman filtering. Students are often confused by noting that the a posteriori error covariance of the discrete Kalman filter (DKF) is smaller than the error covariance of the continuous Kalman filter (CKF), which would mean that the DKF is better than CKF since it gives a smaller error covariance. However, simulation results show that CKF gives estimates much closer to the true states. We will provide a simple qualitative argument to explain this phenomenon.

A The Kalman Filter

2006

The Kalman Filter developed in the early sixties by R.E. Kalman is a recursive state estimator for partially observed non-stationary stochastic processes. It gives an optimal estimate in the least squares sense of the actual value of a state vector from noisy observations.

An Introduction to the Kalman Filter

In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation.

An Elementary Introduction to Kalman Filtering

Kalman filtering is a classic state estimation technique used widely in engineering applications such as statistical signal processing and control of vehicles. It is now being used to solve problems in computer systems, such as controlling the voltage and frequency of processors to minimize energy while meeting throughput requirements.

A Step by Step Mathematical Derivation and Tutorial on Kalman Filters

arXiv: Other Statistics, 2019

We present a step by step mathematical derivation of the Kalman filter using two different approaches. First, we consider the orthogonal projection method by means of vector-space optimization. Second, we derive the Kalman filter using Bayesian optimal filtering. We provide detailed proofs for both methods and each equation is expanded in detail.

Kalman Filters: Theory and Implementation

We focus primarily on the theory of Discrete Kalman Filters, and have implemented the algorithm in MATLAB using simulations technique. We also have applied the algorithm on a simpli ed model of the "navigation and control" problem.