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APPENDIX B
TRACKING
This appendix reviews the Kalman .lter and unscented Kalman .lter and discusses the horizontal and
vertical
motion
trackers
used
in
the
evaluation
of
the
dynamic
programming
logic
in
Section
7.
B.1 KALMAN FILTER
The Kalman .lter is a minimum mean square error (MMSE) estimator that is optimal if the initial state of the system as well as all the noise entering the system are Gaussian. Otherwise, the Kalman .lter is the best estimator among the class of linear MMSE state estimators. This section brie.y derives
the
Kalman
.lter
equations.
The
following
discussion
is
adapted
from
[44],
pp.
200–207.
Consider the following discrete-time linear dynamic system:
x(k +1) = F (k)x(k)+ G(k)u(k)+ B(k)v(k), (B-1)
where x(k) ∈ Rn is the state vector at time k, u(k) is a known input (i.e., control) vector, and v(k) is zero-mean white Gaussian process noise with covariance Q(k). The matrix F (k) is the (possibly) time-varying system dynamics matrix, G(k) is the input gain matrix, and B(k) is the state-noise coupling matrix. Additionally, any measurements z(k) ∈ Rm of the system are modeled by
z(k)= H(k)x(k)+ w(k), (B-2)
where w(k) is zero-mean white Gaussian measurement noise with covariance R(k), and H(k) is the measurement matrix describing the transformation from the state space in Rn to the measurement space in Rm. Let the initial state x(0) be modeled as a Gaussian random variable with known mean and covariance, and let the noise sequences v(k) and w(k) be mutually independent.
It can be shown that estimators that minimize the mean square error (the di.erence between the true value of the state and the estimate) have as their optimal estimate at every time k the conditional mean of the state at time k given the measurements up to and including time k. That is,
x.(k | k) ≡ E[x(k) | Zk], (B-3)
where x.(k | k) denotes the state estimate at time k given all information up to and including time k (i.e., a posterior state estimate) and Zk is the sequence of measurements up to and including time k. For jointly distributed vectors x and z, the conditional mean is given by
x.= E[x | z]=ˉx + PxzP .1(z . zˉ) (B-4)
zz
and the covariance by
x)T P .1
Pxx|z = E[(x . x.)(x . .| z]= Pxx . Pxzzz Pzx, (B-5)
where xˉis the mean of x, Pxx is the covariance of x, Pzz is the covariance of z, and Pxz (= P T ) is
zx
the cross-covariance between x and z.
In the context of the Kalman .lter, x and z would be the state and measurement vectors, respectively. The conditional mean x.of
Eq.
(B-4)
would
represent
the
updated
state
estimate
at
time k + 1, x.(k +1 | k + 1), and the mean xˉwould represent the one-step predicted state at time k + 1 from time k, x.(k +1 | k). Moreover, z would be the actual measurement at time k + 1, z(k + 1), and zˉ, the predicted measurement at time k + 1, z.(k +1 | k). The product PxzP .1, often
zz
called the .lter gain, would then be a measure of the amount of correction that would have to be applied to x.(k +1 | k) due to the measurement prediction error z(k + 1) . z.(k +1 | k).
The Kalman .lter fundamentally consists of two steps: a predict step and an update step. In the predict step, the state estimate at time k + 1, x.(k +1 | k), is predicted based on the posterior state estimate at time k, x.(k | k):
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