Page 861 - The Mechatronics Handbook

0066_Frame_C28 Page 1 Wednesday, January 9, 2002 7:19 PM

28

Kalman Filters

as Dynamic System

State Observers

28.1 The Discrete-Time Linear Kalman Filter
Linearization of Dynamic and Measurement System
Models • Linear Kalman Filter Error Covariance
Propagation • Linear Kalman Filter Update
28.2 Other Kalman Filter Formulations
The Continuous–Discrete Linear Kalman Filter
• The Continuous–Discrete Extended Kalman Filter
Timothy P. Crain II 28.3 Formulation Summary and Review
NASA Johnson Space Center 28.4 Implementation Considerations

28.1 The Discrete-Time Linear Kalman Filter

Distilled to its most fundamental elements, the Kalman filter [1] is a predictor-corrector estimation
algorithm that uses a dynamic system model to predict state values and a measurement model to correct
this prediction. However, the Kalman filter is capable of a great deal more than just state observation in
such a manner. By making certain stochastic assumptions, the Kalman filter carries along an internal metric
of the statistical confidence of the estimate of individual state elements in the form of a covariance matrix.
The essential properties of the Kalman filter are derived from the requirements that the state estimate be

• a linear combination of the previous state estimate and current measurement information
• unbiased with respect to the true state
• and optimal in terms of having minimum variance with respect to the true state.
Starting with these basic requirements an elegant and efficient formulation for the implementation of
the Kalman filter may be derived.
The Kalman filter processes a time series of measurements to update the estimate of the system state
and utilizes a dynamic model to propagate the state estimate between measurements. The observed
measurement is assumed to be a function of the system state and can be represented via

(
Y t() = h X t(),ββ ββ,t) + v t() (28.1)

where Y(t) is an m dimensional observable, h is the nonlinear measurement model, X(t) is the n
dimensional system state, ββ ββ is a vector of modeling parameters, and v(t) is a random process accounting
for measurement noise.

©2002 CRC Press LLC