Hardware Implementation of Embedded Fuzzy Controllers on FPGAs and ASICs - Fuzzy Modeling and Control: Theory and Applications - page 236

Fig. 13.7 Graphical representation and mathematical definition of functions used to measure the
approximation capability of different fuzzy systems

Table 13.1 MSE for approximation of function F 1 versus implementation option and number of
labels
N. Labels XSG A_ROM_ROM A_LB_LB M_ROM_ROM M_LB_LB

1 . 9533 × 10 − 2 1 . 9469 × 10 − 2 1 . 9469 × 10 − 2 1 . 9499 × 10 − 2 1 . 9499 × 10 − 2 3

10 − 3 10 − 3 10 − 3 10 − 3 10 − 3

5 1 . 2439 × 1 . 2344 × 1 . 2344 × 1 . 2408 × 1 . 2408 ×

10 − 4 10 − 4 10 − 4 10 − 4 10 − 4 7 3 . 7264 × 3 . 8018 × 3 . 8018 × 3 . 7695 × 3 . 7695 ×

1 . 5623 × 10 − 4 1 . 5338 × 10 − 4 1 . 5338 × 10 − 4 1 . 5556 × 10 − 4 1 . 5556 × 10 − 4 9

vided by the Xfuzzy environment ( xfsl ). Then, xfsg and xfvhdl were used to generate
the input files required by the XSG- and VHDL-based design flows. Four options
(Arithmetic- or memory-basedMFCs and ROMor logic block implementation) were
considered in the case of xfvhdl . The resulting VHDL code was encapsulated into
Black Boxes in order to include it in Simulink models to automate and accelerate
the process (Fig. 13.8 ).
The mean square error (MSE) obtained when comparing function F 1 versus the
output of fuzzy controllers generated by xfsg and xfvhdl , with different architectural
options and number of labels, is shown in Table 13.1 . The compared controllers
include Fuzzy Mean as defuzzifier and all the parameters have been codified using
10 bits. As can be observed, the approximation error decreases when grows the
number of labels used in rule antecedents, thus confirming that fuzzy systems are
universal approximators. Data also show that the results are similar for the different
options and design techniques and independent of the use of ROM or logic blocks
to implement the memory components.
The approximation capability of a fuzzy system also depends on the operators
used for implementing the inference mechanism. Data in Table 13.2 allow evaluating
the results provided by different defuzzification methods for function F 2 . All the
controllers use seven normalized triangular membership functions for antecedents
and product as connective operator, thus allowing the elimination of the divider
block in the output stage of Fuzzy Mean and first-order Takagi-Sugeno defuzzifiers.
The parameters have been codified using 10 bits. Obtained values show that fuzzy