Definition
The Kalman Filter is an iterative mathematical tool used to estimate the state of a dynamic system from a series of noisy measurements. It bridges linear algebra and probability by using a recursive least-squares approach.
Why It Matters
Measurements are always noisy; the world is always uncertain. The Kalman filter is the “golden standard” for navigation and control, allowing us to estimate the true state of a system (like a landing rocket) by mathematically weighing imperfect data against a known model.
Core Concepts
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State Estimation: Predicts the current state based on the previous state and a process model.
- How to read: “The estimated state vector x hat k.”
- Meaning / when to use: Estimated state at time step —best guess of the true system state after combining the model prediction with past information.
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Measurement Update: Corrects the prediction using new data .
- How to read: “The measurement vector b k.”
- Meaning: Measurement at step —observed sensor reading used to pull the estimate toward reality.
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Recursive Weights: The “Kalman Gain” determines how much to trust the new measurement vs. the prediction based on their respective covariance matrices.
- How to read: “The Kalman gain matrix K k.”
- Meaning: Kalman gain at step —high gain trusts measurements more; low gain trusts the model more, weighted by uncertainty.
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Linearity: Relies on the framework, updated to .
- How to read: “The equation A times x equals b, and the state update equation x k plus one equals the transition matrix F k times x k plus noise.”
- Meaning: State evolves by linear transition matrix with process noise; measurements relate states to observations via .