Page 314 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

EXTENSIONS OF THE KALMAN FILTER 303

following MATLAB fragment (making use of the Control System
Toolbox):
...
sys ¼ ss(Fn,eye(5),H,0,Ts); % Create a state space model
[Kest,L,P,M,Z] ¼ kalman(sys,Wn,0); % Find the steady state KF
bodemag(Kest(2,:),‘k’); % Plot the Bode diagram

8.5.2 Cross-correlated noise

Another situation occurs when the process and measurement noise are
T
cross-correlated, that is, C wv ¼ E[w(i)v (i)] 6¼ 0. Such might happen if
both the physical process and measurement system is affected by the
same source of disturbance. Examples are changes of temperature and
electrical inference due to induction.
The strategy to bring this situation to the standard estimation problem
is to introduce a modified state equation by including a term T(z(i)
Hx(i) v(i)) 0:

xði þ 1Þ¼ FxðiÞþ wðiÞ
ð
¼ FxðiÞþ wðiÞþ TzðiÞ HxðiÞ vðiÞÞ ð8:59Þ
ð
¼ F THÞxðiÞþ wðiÞ TvðiÞþ TzðiÞ
The factor F TH is the modified transition matrix. The terms
w(i) Tv(i) are regarded as process noise. The term Tz(i) is known
and can be regarded as a control input.
Note that T can be selected arbitrarily because T(z(i) Hx(i) v(i)) 0.
Therefore, we can select T such that the new process noise becomes
uncorrelated with respect to the measurement noise. That is,
T
E[(w(i) Tv(i))v (i)] ¼ 0.Or: C wv TC v ¼ 0.Inother words,if
1
T ¼ C wv C , the modified state equation is in the standard form without
v
cross-correlation.

8.5.3 Smoothing

Up to now only online estimation of continuous states has been con-
sidered. The topic of prediction has been touched on in Section 4.2.1. Here,
we introduce the subject of offline estimation, generally referred to as
smoothing. Assuming a linear-Gaussian model of the process and