Kalman Filtering 63
equations can be represented as a weighted linear combination of the obser-
vations. Additionally, the weights are actually ratios of Fibonacci numbers.
The Fibonacci sequence of numbers is constructed starting from two seed
numbers,F 0 = 0 and F 1 = 1. The next number in the series is generated by
adding the last two numbers in the series, Fn=Fn–1+Fn–2. Applying the for-
mula in an iterative fashion, we obtain the Fibonacci sequence as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34,....There are a variety of situations in which the Fibonacci numbers appear.
Some sources of information on Fibonacci numbers are listed in the refer-
ence section. In any case, the solution to our problem, that is, the estimate
of state x 2 , is given as
Similarly,x 3 is given as
In general, if we are to estimate xT, then
xT=w 0 yT+w 1 yT–1+w 2 yT–2+ .... (4.13)where
(4.14)
Note that the first weight is the ratio of two consecutive Fibonacci numbers.
The ratio approaches the value g. The value gis famously known as
the golden mean ratio. It is given by the formula and has
an approximate value of 1.618. The first weight in the observation is actu-
ally the reciprocal of the ratio^1 ⁄g≈0.618. The subsequent weights are^1 ⁄g^3 ,
(^1) ⁄g^5 , .... and so on. To see that the second ratio is (^1) ⁄g^3 , consider the following:
(4.15)
F
F
F
F
F
F
F
F gT
TT
TT
TT
T213
21213
212212
211211
21
3(+)− 1
(+)( +)−
( +)−(+)−
( +)−( +)−
( +)=≈
g=+ 152 /F
F
T
T21
211()
()+
+−w ww wF
F
F
F
F
F
F
T F
T
TT
TT
TT012211
21213
21215
211
21,,,,... )=...,,,,
( +)−
( +)( +)−
( +)(+)−
( +) ( +)xyyyy 3321013
21
5
21
2
21
1
21
=+++
xyyy 2210