Page 281 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

270 STATE ESTIMATION IN PRACTICE

xði þ 1Þ¼ FxðiÞþ LuðiÞ
2
xði þ 2Þ¼ F xðiÞþ FLuðiÞþ Luði þ 1Þ
3 2
xði þ 3Þ¼ F xðiÞþ F LuðiÞþ FLuði þ 1Þþ Luði þ 2Þ
. ð8:19Þ
. .
n 1
n X j
xði þ nÞ¼ F xðiÞþ F Luði þ jÞ
j¼0

or:

2 3
uðiÞ
uði þ 1Þ
6 7
n 1 6 7 n
L FL .. . F L 6 . . 7 ¼ xði þ nÞ F xðiÞ ð8:20Þ
4 . 5
uði þ n 1Þ
The minimum number of steps, n, is at most equal to M, the dimension
of the state vector. Therefore, in order to test the controllability of the
system (F, L) it suffices to check whether the controllability matrix
[ L FL .. . F M 1 L ] has rank M.
The MATLAB functions for creating the controllability matrix and
Gramian are ctrb() and gram(), respectively.

8.2.3 Dynamic stability and steady state solutions

The term stability refers to the ability of a system to resist to and recover
from disturbances acting on this system. A state estimator has to face
three different causes of instabilities: sensor instability, numerical
instability and dynamic instability.
Apart from the usual sensor noise and sensor linearity errors, a sensor
may produce unusual glitches and other errors caused by hard to predict
phenomena, such as radio interference, magnetic interference, thermal
drift, mechanical shocks and so on. This kind of behaviour is sometimes
denoted by sensor instability. Its early detection can be done using
consistency checks (to be discussed in Section 8.4).
Numerical instabilities originate from round-off errors. Particularly,
the inversion of a near singular matrix may cause large errors due to its