An Introduction To The Kalman Filter By, Santhosh Kumar

The Problem System state cannot be measured directly Need to estimate “optimally” from measurements Measuring Devices Estimator Measurement Error Sources System State (desired but not known) External Controls Observed Measurements Optimal Estimate of System State System Error Sources System Black Box

What is a Kalman Filter? The Kalman Filter is essentially a set of mathematical equations that implement a predictor – corrector type estimator that is OPTIMAL – when some presumed conditions are met. Optimal?  For linear system and white Gaussian errors, Kalman filter is “best” estimate based on all previous measurements  For non-linear system optimality is ‘qualified’

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

Discrete Kalman Filter Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement

Components of a Kalman Filter Matrix (nxn) that relates the state at the previous time step k-1 to k without controls or noise. Matrix (nxl) that describes how the control u changes the state from k-1 to k. Matrix (mxn) that describes how to map the state x k to a measurement z k. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance R and Q respectively.

Estimates and Errors is the priori state estimate at step k. is the posteriori state estimate at step k given measurement Errors: Error covariance matrices: Kalman Filter’s task is to find

Residual and Kalman Gain Expected value ◦ innovation is The optimal Kalman gain K k is

Discrete Kalman Filter Algorithm Prediction (Time Update) (1) Project the state ahead (2) Project the error covariance ahead Correction (Measurement Update) (1) Compute the Kalman Gain (2) Update estimate with measurement z k (3) Update Error Covariance

Extended Kalman Filter Suppose the state-estimation and measurement equations are non-linear: ◦ process noise w is drawn from N(0,Q), with covariance matrix Q. ◦ measurement noise v is drawn from N(0,R), with covariance matrix R.

Jacobian Matrix Recap For a scalar function y=f(x), For a vector function y=f(x),

Linearize the Non-Linear The equations that linearize a kalman estimate are Where, and are actual state and measurement vectors. and are approx. state and measurement vectors. andare process and measurement noise. (Cont.)

Linearize the Non-Linear(Cont.) Let A be the Jacobian of f with respect to x. Let W be the Jacobian of h with respect to w. Let H be the Jacobian of h with respect to x. Let V be the Jacobian of h with respect to v.

Extended Kalman Filter Algorithm Prediction (Time Update) (1) Project the state ahead (2) Project the error covariance ahead Correction (Measurement Update) (1) Compute the Kalman Gain (2) Update estimate with measurement z k (3) Update Error Covariance

Quick Example – Constant Model Measuring Devices Estimator Measurement Error Sources System State External Controls Observed Measurements Optimal Estimate of System State System Error Sources System Black Box

Quick Example – Constant Model Time Update Equation Measurement Update Equation

Quick Example – Constant Model

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