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Data Fusion via Kalman Filter 103

where x(t) is the n-element state vector of the system, F(t) is the system matrix,
G(t) is the input distribution matrix, y(t) is the measurement vector, H(t) is the
measurement matrix, and v(t) is the measurement noise vector.
The vectors ω(t) and v(t) are assumed to be white and Gaussian with
T
E[ω(t)]= 0, E[ω(t)ω (t + τ)]= Q w (t)δ(τ) (3.3)
T
E[v(t)]= 0, E[v(t)v (t + τ)]= R(t)δ(τ) (3.4)
where Q w is the power spectral density (PSD) of the white noise ω(t) and R(t)
is the covariance matrix of the measurement noise process v(t). If either ω(t)
or v(t) is not white, then it may be possible to append linear dynamics to the
model of Equation (3.1) and Equation (3.2) to still utilize the model of a linear
system driven by white noise, see Reference 19. For the state estimation design
discussions of this chapter, unless otherwise stated, assume that the system
model has been manipulated into the form of Equation (3.1) and Equation (3.2)
with white process and measurement noise.
In applications, such as those involving GPS, where the measurements
occur at discrete instants of time, it is convenient to utilize a discrete-time
formulation of the KF. If we denote the sequence of measurement times by
t 1 , ... , t k , t k+1 , ... , then implementation of the discrete-time KF requires a
model for propagating the state between measurement times and a model for the
relation between the state and the measurement that is valid at the measurement
time. Using linear system theory [20,21], the state transition valid between t k
and t k+1 is

x(t k+1 ) = (t k+1 , t k )x(t k ) + w(t k ) (3.5)

where

t k+1
w(t k ) = (t k+1 , τ)G(τ)ω(τ) dτ (3.6)
t k
and (t k+1 , t) is the state transition matrix from t to t k+1 . The measurement
model valid at t k is

y(t k ) = H(t k )x(t k ) + v(t k ) (3.7)

To simplify notation, these equations will be written as

x k+1 = k x k + w k (3.8)
y k = H k x k + v k (3.9)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 103 — #5