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# Linear Estimation

## 1. Kalman Filter Tutorial

Nikhil Kumar Singh, Jadavpur University, Kolkata

### 1.1 Introduction

Kalman filter, which estimates the instantaneous state of a dynamic system is an optimal, linear, unbiased estimator. It works based on linear quadratic Gaussian (LQG) theory, where we assume the system is linear, and sum of square of estimation error is minimum. The filter is optimal in the sense that it minimizes the squared error. We also assume that the process and measurement noise are white Gaussian, and both the noises are independent and uncorrelated to each other.

The Kalman filter provides best possible estimation of state of a linear system corrupted by additive, white Gaussian noise. The process equation describes the evaluation of the states with time. If the process could not be modeled with certainty, process noise is incorporated to compensate that. Similarly the measurement noise is considered to model the noisy measurements. In order to estimate all the states the system must be observable.

We mentioned earlier that the Kalman filter is a recursive filter. It recursively predicts and updates the states with the help of process model, measurement model and initial guess of the state variables. In this article, we briefly describe the Kalman filter algorithm, and it's proof and some important properties. Also we discuss the importance of innovation sequence and few tests to validate the performance of the filter.