Approximation of Optimized Fuzzy Logic Controller for Shunt Active Power Filter - Complex System Modelling and Control Through Intelligent Soft Computations - page 570

µ( δ I max )

NL NM NS ZE PS PM PL

1

0 1/3 2/3 1

-1 -2/3 -1/3

Fig. 8 Triangular membership functions for output variable δ I max of 49-rule FLC

3.2 Approximation Technique

A 4-rule simplest FLC is proposed to approximate the control functionality of its
49-rule counterpart. The outputs of a 49-rule FLC and a 4-rule FLC are compared at
regular intervals in the entire range of input variables to find the deviation in the

responses of two controllers. The approximation is based on evaluating a com-
pensating polynomial such that its series (cascade) combination with simplest FLC,
approximately maps the output of the 49-rule FLC.
The concept of simplest FLC was introduced by Ying ( 2000 ). The term simplest
refers to a minimal possible con guration in terms of number of input variables,

fuzzy sets and fuzzy rules for any properly functional FLC. To realize a simplest
FLC, two membership functions for each input variable, i.e., error (e) and change in
error (ce) are used in the UOD of [ L, L] as shown in Fig. 9 a, b, respectively for

triangular membership functions. Three triangular membership functions are used
for output variable

−

H, H] as shown in Fig. 9 c.
Similarly, simplest FLC can be realized for other membership functions also.
The UOD for input and output variables is taken [

δ I max in the UOD of [

−

1, 1]. The centre of gravity

−

defuzzi cation method is used to obtain the output as crisp value.

Let u(k) and u 1 (k) are the outputs of a 49-rule FLC and 4-rule FLC at kth
sampling instant, respectively. Then the deviation or error can be given as:

ð Þ ¼ ð Þ

u 1 ð

Þ ð 4 Þ

e k u k k

The sum of square error (SSE) can be represented as:

X

N

e 2

SSE ¼ ð k Þ ð 5 Þ

k ¼ 1

This SSE will be used as a cost function, to be minimized for achieving the
accurate approximation. To understand the least square

fitting process, let us
consider N data points to measure the error in responses. To implement the
approximation scheme, an nth order polynomial is used in cascade with the 4-rule