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Understanding Kalman's Beautiful Filter: An Introduction

This introduction to Kalman's Filter explains its application in noise smoothing and state estimation for linear dynamical systems. It details the recursive process of prediction, correction, and update steps. The text offers a geometric interpretation, emphasizing estimation through Gaussian distribution and covariance matrix. It also discusses the implementation of a better state observer with the need to estimate the covariance matrix. In addressing output noise, the Kalman Filter is shown to solve optimization problems effectively.

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Understanding Kalman's Beautiful Filter: An Introduction

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  1. Kalman’s Beautiful Filter(an introduction) George Kantor presented to Sensor Based Planning Lab Carnegie Mellon University December 8, 2000

  2. What does a Kalman Filter do, anyway? Given the linear dynamical system: the Kalman Filter is a recursion that provides the “best” estimate of the state vector x.

  3. What’s so great about that? • noise smoothing (improve noisy measurements) • state estimation (for state feedback) • recursive (computes next estimate using only most recent measurement)

  4. How does it work? 1. prediction based on last estimate: 2. calculate correction based on prediction and current measurement: 3. update prediction:

  5. Finding the correction (no noise!)

  6. A Geometric Interpretation

  7. A Simple State Observer System: 1. prediction: 2. compute correction: Observer: 3. update:

  8. Estimating a distribution for x Our estimate of x is not exact! We can do better by estimating a joint Gaussian distribution p(x). where is the covariance matrix

  9. Finding the correction (geometric intuition)

  10. A new kind of distance

  11. Finding the correction (for real this time!)

  12. A Better State Observer We can create a better state observer following the same 3. steps, but now we must also estimate the covariance matrix P. We start with x(k|k)and P(k|k) Step 1: Prediction What about P? From the definition: and

  13. Continuing Step 1 To make life a little easier, lets shift notation slightly:

  14. Step 2: Computing the correction For ease of notation, define W so that

  15. Step 3: Update (just take my word for it…)

  16. Just take my word for it…

  17. Better State Observer Summary System: 1. Predict 2. Correction Observer 3. Update

  18. Finding the correction (with output noise) Since you don’t have a hyperplane to aim for, you can’t solve this with algebra! You have to solve an optimization problem. That’s exactly what Kalman did! Here’s his answer:

  19. LTI Kalman Filter Summary System: 1. Predict Kalman Filter 2. Correction 3. Update

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