Page 184 - Modern Control of DC-Based Power Systems

148 Modern Control of DC-Based Power Systems

The Kalman filter estimates the state of a plant based on the input sig-
nals and measurements. However, the plant is not just driven by a known
control signal uðtÞ but also by an unknown plant noise wðtÞ. Furthermore,
the measurements yðtÞ include a noise signal vtðÞ which is called measure-
ment noise. This leads to the following continuous linear time invariant
plant model:

_ x tðÞ 5 Ax tðÞ 1 Bu tðÞ 1 Fw tðÞ
(5.97)
ytðÞ 5 Cx tðÞ 1 HvðtÞ
Typically, the noise signals are assumed to be uncorrelated white
Gaussian noise with zero mean and the following covariance matrices S w
and S v .
For an infinite time horizon the optimal state estimation problem can
be formulated such that the cost J of the mean square estimation error is
minimized:
T
8 9
1
< ð =
T
J 5 lim E ð x tðÞ 2 ~x tðÞÞC Cðx tðÞ 2 ~x tðÞ dt (5.98)
T
T-N : ;
0
Where E .. . is the expected value. Under the assumption that CA is
observable the solution of (5.99) applied to the system of (5.98) will lead
to the Kalman filter [33]. The structure of the Kalman filter can be
depicted in various forms of which one of them is the predictor/corrector
form [45]:
_ (5.99)
^ x tðÞ 5 A^x tðÞ 1 ButðÞ 1 K½ytðÞ 2 C^xðtފ
This terminology appeared first for discrete Kalman filters. It describes
how the derivative of the estimation is first predicted from old data and
then updated using new measurements [33]. Eq. (5.99) clarifies that the
only unknown variable is the Kalman gain K, which is calculated as
follows:
T 21 T 21
v
K 5 PC ðHS H Þ (5.100)
Since the time-varying solution of the differential equation is close to
the steady-state solution when the system is running for a long time, the
derivatives in the Riccati equation are set to zero and where P is the
solution of the algebraic Riccati Eq. (5.101):

T
T 21
T
T
21
AP 1 A P 2 PC ðHS H Þ CP 1 FS w F 5 0 (5.101)
v