National Instruments 370753C-01

Chapter 6 State-Space Design

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 xk from each measured yyk, consider the time

just prior to a new measurement for yk. At this point and are the

current estimates for the state and covariance. is derived from the

previous measurement and Mk is derived from the previous

post-measurement error covariance, Pk–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, Pk. The equations for

xk1+AxkBukGω++=

ykCxkDukν++=

xkMk

k1–,

xkAx

ˆk1–Gk

MkAPk1–A'GQxxG'+=