A Kalman filter is a powerful algorithm used in statistics and control theory for estimating the state of a system from a series of noisy measurements.

## Definition and Purpose

The Kalman filter, also known as linear quadratic estimation, is an algorithm that:

• Estimates unknown variables or system states
• Uses a series of measurements observed over time
• Accounts for statistical noise and other inaccuracies
• Produces more accurate estimates than those based on single measurements

## Key Characteristics

• Recursive: It operates in real-time, using only the present input measurements and the previously calculated state
• Efficient: It minimizes the mean squared error of the estimated parameters
• Versatile: It can be applied to linear dynamic systems in various fields

## How It Works

The Kalman filter operates in a two-phase process:

1. Prediction Phase:
• Produces estimates of current state variables
• Calculates their uncertainties
2. Update Phase:
• Updates the estimates using weighted averages
• Gives more weight to estimates with higher certainty

## Applications

Kalman filters are used in a wide range of applications, including:

• Guidance, navigation, and control of vehicles (aircraft, spacecraft, ships)
• Object tracking and radar
• Computer vision
• Signal processing
• Econometrics
• Robotic motion planning and control

## Mathematical Framework

The Kalman filter uses:

• A systemâ€™s dynamic model (e.g., physical laws of motion)
• Known control inputs to the system
• Measurements from various sensors
• Statistical models of system noise and measurement errors

• Can estimate parameters that cannot be directly measured
• Optimally reduces measurement errors
• Can incorporate dynamic relationships between system variables
• Suitable for real-time applications due to its mathematical structure

## Variations

Several variations of the Kalman filter exist, including:

• Kalman-Bucy filter
• Extended Kalman filter
• Unscented Kalman filter
• Information filter
• Square-root filters

The Kalman filterâ€™s ability to provide accurate estimates in the presence of noise and uncertainty makes it a fundamental tool in many technological applications and scientific fields.