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Chapter 6 State-Space Design
© National Instruments Corporation 6-19 Xmath Control Design Module
The discrete-time estimator follows from a similar system description,
using the discrete-time difference equation representation of the system,
as shown in the following equations.
You obtain the discrete-time estimator by considering the state estimate at
two separate stages. Begin with the assumption that an estimate of the state
exists prior to each measurement of the output information. This
pre-existing estimate is called . The estimated state value after each
measurement update is denoted by . This method takes into account the
fact that the system’s states change between measurements due to the
system dynamics. The optimization problem, then, consists of minimizing
the estimate error covariance M after each measurement update. This
minimization is performed in [Kai81]. This problem is expressed in the
same manner as in the preceding quadratic expression (for J), except that a
summation sign replaces the integral as you are working with discrete data
and you replace the variable J with M, to denote that this covariance follows
each measurement update.
In determining the state values x
k
from each measured y
yk
, consider the time
just prior to a new measurement for y
k
. At this point and are the
current estimates for the state and covariance. is derived from the
previous measurement and M
k
is derived from the previous
post-measurement error covariance, P
k –1
, as shown in Equation 6-6.
(6-6)
This is referred to as the time update, because you are propagating the state
forward in time until the next measurement arrives.
Then, at the time immediately following the measurement, you effect the
measurement update. This reflects the new information in an improved
state estimate and a somewhat smaller covariance, P
k
. The equations for
x
k 1+
Ax
k
Bu
k
Gω++=
y
k
Cx
k
Du
k
ν++=
x
k
x
ˆ
k
x
k
M
k
x
k
x
ˆ
k 1–
,
x
k
Ax
ˆ
k 1–
G
k
+=
M
k
AP
k 1–
A' GQ
xx
G'+=
x
ˆ
k