Suboptimal Recursive Methodology for Takagi-Sugeno Fuzzy Models Identification - Fuzzy Modeling and Control: Theory and Applications - page 39

Table 2.2 Average values and standard deviation of 10 runs
EKF(c)

EKF ( c + a )

Online training error 2 . 759294 ± 0 . 128 3 . 007364 ± 0 . 124

Final models error (training) 3 . 304343 ± 0 . 153 3 . 277433 ± 0 . 121

Final models error (validation) 2 . 731736 ± 0 . 130 2 . 965235 ± 0 . 164

RMSE (validation) 3 . 569301 ± 0 . 025 3 . 852766 ± 0 . 032

⎛
⎝

⎞
⎠

Zmf Trapmf1 Trimf1 Trimf2 Trapmf2 Smf
Zmf 10000 0 00 0 00 0 000 0 00
01000 0 00 0 00 0 000 0 00
Trapmf1 1 0 0 00 0 00 0 00 0 000 0 00
0 1 0 0 0000 0 00 0 000 0 00
00001 0 00 0 00 0 000 0 00
00000 1 00 0 00 0 000 0 00
Trimf1 0000 1 0 0 0000 0 000 0 00
00000 1 0 0 0 00 0 000 0 00
00000 0 00 1 00 0 000 0 00
Trimf2 00000 0 0 1 0 0 00000 0 00
00000 0 00 1 0 0 0 000 0 00
00000 0 00 0 00 1 000 0 00
Trapmf2 00000 0 00 0 0 1 0 0 00 0 00
00000 0 00 0 00 1 0 0 0000
00000 0 00 0 00 0 001 0 00
00000 0 00 0 00 0 000 1 00
Smf 00000 0 00 0 00 0 00 1 0 0 0
00000 0 00 0 00 0 000 1 0 0

a =

(2.37)

After run ten times, the average errors are shown in Table 2.2 . As in the previous
case, is taken one of the executions of the algorithm, where the absolute errors of the
final models are shown in Figs. 2.11 and 2.12 , the modeling outputs in Fig. 2.13 ,the
final response from validation data in Fig. 2.14 , and the online evolution of absolute
errors are shown in Fig. 2.15 . Figure 2.10 shows the resulting antecedents, where can
be seen that EKF

( + )

has complied with the antecedents relationship from ( 2.37 ).
Based on the results obtained, it is possible to draw the same conclusions as in the
previous case, but can be seen that the use of the matrix

c a

can impose restrictions

on the adjust of antecedents.

2.4.2 Example 2. Mackey-Glass Chaotic Time Series

In this case, the EKF algorithms will be used to predict 6, 12 and 85 steps ahead of
Mackey-Glass chaotic time series based on the values of the current signal, 6, 12 and
18 steps back ( x

T ). This series is a well-known

=[ v ( t − 18 ), v ( t − 12 ), v ( t − 6 ) v ( t ) ]