KALMAN FILTERS STRIKE BACK KALMAN FILTERS STRIKE BACK MATTHIEU - - PowerPoint PPT Presentation

KALMAN FILTERS STRIKE BACK KALMAN FILTERS STRIKE BACK

MATTHIEU BLOCH April 16, 2020

1 / 14

RECAP: GAUSS-MARKOV MODEL RECAP: GAUSS-MARKOV MODEL

• Definition. (Gauss-Markov model)

A Gauss-Markov model is a Gaussian driven linear model where and are Gaussian white processes (assumed independent). We assume that all variables are real-valued for simplicity

2 / 14

RECAP: PROPERTIES OF GAUSSIAN DISTRIBUTIONS RECAP: PROPERTIES OF GAUSSIAN DISTRIBUTIONS

Lemma (Joint Gaussian distribution) Let and be jointly distributed random variables Then , and Lemma. Let and let (e.g., ). Then

3 / 14

PROPERTIES OF GAUSSIAN DISTRIBUTIONS PROPERTIES OF GAUSSIAN DISTRIBUTIONS

Lemma. Let and let . Then . These properties lead to the following crucial observation Lemma (Gaussianity preserved) All distributions are jointly Gaussian in a Gauss-Markov model.

4 / 14

5 / 14

KALMAN FILTER REVISITED KALMAN FILTER REVISITED

Theorem (Kalman filtering) The Kalman filter is the Bayesian optimal filter of the Gauss Markov model

• 1. Initialization:
• 2. Prediction:

with

• 3. Update:

with Remarks:

• 1. Our initial analysis did not assume Gaussian distribution
• 2. Strictly speaking our derivation does not prove least square optimality
• 3. We’re very lucky that there is a closed form solution

6 / 14

7 / 14

8 / 14

EXTENDED KALMAN FILTERING EXTENDED KALMAN FILTERING

• Definition. (EKF model)

Consider model of the form where and are Gaussian white processes (assumed independent). This is a special case of general probabilistic state space model. No closed form solution for non-linear and Solution: linearize non linear functions where and are the Jacobian matrices of , , respectively Can be understood from the perspective of approximate transformation of variables

9 / 14

LINEAR APPROXIMATIONS OF NON-LINEAR TRANSFORMS LINEAR APPROXIMATIONS OF NON-LINEAR TRANSFORMS

Lemma (Approximate transformation of Gaussian) Consider and . The probability density of is not Gaussian but can be approximated as where is the Jacobian matrix of Lemma. Consider , and . The joint probability density of is not Gaussian but can be approximated as

10 / 14

11 / 14

EXTENDED KALMAN FILTER REVISITED EXTENDED KALMAN FILTER REVISITED

Theorem (Extended Kalman filtering) The Kalman filter is an approximate Bayesian optimal filter

• 1. Initialization:
• 2. Prediction:

with

• 3. Update:

with

12 / 14

13 / 14

EKF REMARKS EKF REMARKS

Advantages of EKF Almost same as basic Kalman filter, easy to use Intuitive, engineering way of constructing approximations Works very well in practical estimation problems. Computationally efficient Limitations of EKF Does not work with strong non-linearities Only Gaussian noise processes are allowed Measurement model and dynamic model functions need to be differentiable Computation and programming of Jacobian matrices can be quite error prone

14 / 14



    