Page 101 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

90 STATE ESTIMATION

unknown influences on the system, for instance, formed by disturbances
from the environment. The process noise can also represent an unknown
input/control signal. Sometimes process noise is also used to take care of
modelling errors. The general assumption is that the process noise is a
white random sequence with normal distribution. The term ‘white’ is
used here to indicate that the expectation is zero and the autocorrelation
is governed by the Kronecker delta function:

E wðiފ ¼ 0
½
ð4:12Þ
T

E wðiÞw ð jÞ ¼ C w ðiÞdði; jÞ
C w (i) is the covariance matrix of w(i). Since w(i) is supposed to have
a normal distribution with zero mean, C w (i) defines the density of w(i)
in full.
The initial condition of the state model is given in terms of the
expectation E[x(0)] and the covariance matrix C x (0). In order to find
out how these parameters of the process propagate to an arbitrary time i,
the state equation (4.11) must be used recursively:

E xði þ 1ފ ¼ FðiÞE xðiފ þ LðiÞuðiÞ
½
½
ð4:13Þ
T
C x ði þ 1Þ¼ FðiÞC x ðiÞF ðiÞþ C w ðiÞ
The first equation follows from E[w(i)] ¼ 0. The second equation uses
T
the fact that the process noise is white, i.e. E[w(i)w (j)] ¼ 0 for i 6¼ j. See
(4.12).
If E[x(0)] and C x (0) are known, then equation (4.13) can be used to
calculate E[x(1)] and C x (1). From that, by reapplying the (4.13), the next
values, E[x(2)] and C x (2), can be found, and so on. Thus, the iterative
use of equation (4.13) gives us E[x(i)] and C x (i) for arbitrary i > 0.
In the special case, where neither F(i) nor C w (i) depend on i, the state
space model is time invariant. The notation can be shortened then by
dropping the index, i.e. F and C w . If, in addition, F is stable (the
magnitudes of the eigenvalues of F are all less than one; Appendix
D.3.2), the sequence C x (i), i ¼ 0, 1, .. . converges to a constant matrix.
The balance in (4.13) is reached when the decrease of C x (i) due to F
compensates the increase due to C w . If such is the case, then:

T
C x ¼ FC x F þ C w ð4:14Þ

This is the discrete Lyapunov equation.