1. EE 495 Modern Navigation Systems Kalman Filtering – Part I Friday, March 28 EE 495 Modern Navigation Systems Slide 1 of 11

2. Kalman Filtering – Part I Basic Estimation – Estimating a Fixed Constant Friday, March 28 EE 495 Modern Navigation Systems CASE 1: A Fixed Constant  Estimate an unknown constant (x) given that we measure the truth + (white) noise  Simplest solution: o An non-finite memory averaging filter A recursive filter Slide 2 of 11

3. Kalman Filtering – Part I Basic Estimation – Estimating a Fixed Constant Friday, March 28 EE 495 Modern Navigation Systems Slide 3 of 11

4. Kalman Filtering – Part I Basic Estimation – Estimating a Time Varying Quantity Friday, March 28 EE 495 Modern Navigation Systems CASE 2: A Slowly Time Varying Quantity  Estimate a time varying quantity (x) given that we measure the truth + (white) noise  Simplest solution: o A fading memory filter (i.e., ~ fixed memory length) o where A recursive filter x is no longer a constant!! Slide 4 of 11

5. Kalman Filtering – Part I Basic Estimation – Estimating a Time Varying Quantity Friday, March 28 EE 495 Modern Navigation Systems CASE 2: A Slowly Time Varying Quantity  A simulation example Essentially a low-pass filter Slide 5 of 11 Mem Len = 10

6. Kalman Filtering – Part I Beyond Basic Estimation Friday, March 28 EE 495 Modern Navigation Systems Can we do better? What if we know something about the noise levels in the measurement?  e.g., the standard deviation of the noise in the measurement o Maybe more => A Gauss-Markov model with correlation time? What if we know something about how the quantity we are estimating evolves over time?  e.g., dynamic model of a object being tracked A Kalman Filter can use all of this type of information (and more)!! Slide 6 of 11

7. Kalman Filtering – Part I Beyond Basic Estimation Friday, March 28 EE 495 Modern Navigation Systems Slide 7 of 11

8. Kalman Filtering – Part I Beyond Basic Estimation Friday, March 28 EE 495 Modern Navigation Systems A recursive filter Slide 8 of 11

9. Kalman Filtering – Part I Beyond Basic Estimation - The Kalman Filter Friday, March 28 EE 495 Modern Navigation Systems Slide 9 of 11

10. Kalman Filtering – Part I Beyond Basic Estimation - The Kalman Filter Friday, March 28 EE 495 Modern Navigation Systems Slide 10 of 11

11. Kalman Filtering – Part I Beyond Basic Estimation – The Kalman Filter Algorithm Friday, March 28 EE 495 Modern Navigation Systems Step 1: Prediction Step 2: Gain Calculation Step 3: UpdateStep 0: Initialize Slide 11 of 11