of wt is denoted by Qt, and the
noise terms in different time
steps are assumed to be uncorrelated to each other (such as,
E[wi wj]=0 if i≠j) and to x0.
For estimation, we have a random
variable x0|0 that captures our belief
about the likelihood of different states
at time t=0, and two random variables
xt|t− 1 and xt|t at each time step t = 1, 2, …
that capture our beliefs about the likelihood of different states at time t before
and after fusion with the measurement,
respectively. The mean and covariance
matrix of a random variable xi|j are
denoted by and ∑i|j, respectively. We
assume (no bias).
Prediction essentially uses xt− 1|t− 1 as
a proxy for xt− 1 in Equation 32 to determine xt|t− 1 as shown in Equation 33.
a priori estimate and denoted by .
The a priori estimate is then fused with
zt, the state estimate obtained from the
measurement at time t, and the result is
the a posteriori state estimate at time t,
denoted by . This a posteriori estimate
is used by the model to produce the a
priori estimate for the next time step
and so on. As described here, the a priori
and a posteriori estimates are the means
of certain random variables; the covariance matrices of these random variables
are shown within parentheses each estimate in Figure 6b, and these are used to
weight estimates when fusing them.
We first present the state evolution
model and a priori state estimation.
Then we discuss how state estimates
are fused if an estimate of the entire
state can be obtained by measurement.
Finally, we discuss how to address this
problem when only a portion of the
state can be measured directly.
State evolution model and prediction.
The evolution of the state over time is
described by a series of random vari-
ables x0, x1, x2,…
• The random variable x0 captures
the likelihood of different initial
• The random variables at succes-
sive time steps are related by the
following linear model:
Here, ut is the control input, which
is assumed to be deterministic,
and wt is a zero-mean noise term
that models all the uncertainty in
the system. The covariance matrix
Figure 8. Computation of a posteriori estimate.