A First Course in FUZZY and NEURAL CONTROL

4.2. MAIN APPROACHES TO FUZZY CONTROL 137

FORCE

C 3 =NS C 4 =PS

Ψ(u)=[0. 6 ∧C 3 (u)]∨[0. 82 ∧C 4 (u)]

To produce an actual control action for the input(θ,θ^0 ), we need to summarize
the fuzzy subsetΨ(u)in some way. The control action depends on the de-
fuzzification technique that is applied to this fuzzy set. A natural choice is the
centroid defuzzification methoddiscussed on page 120, namely, the actual
control value is taken to be

u∗=

R

uΨ(u)du

R

Ψ(u)du

The control methodology described above, with sufficient tuning, will lead to
successful control of the inverted pendulum.
The 36 possible rules determined by the look-up table on page 136 are far
more than are needed. T. Yamakawa [84] successfully balanced an inverted
pendulum on a cart using only seven rules. He used one additional fuzzy set,
namelyapproximately zero(AZ), for each linguistic variable. The seven rules
were as follows:
θθ^0 u
1 AZ AZ AZ
2 PS PS PS
3 PM AZ PM
4 PS NS AZ
5 NM AZ NM
6 NS NS NS
7 NS PS AZ

In practice, a small number of rules will often work as well as a larger number
of rules. Working with a small number of rules has great advantages in terms
of cost, efficiency, and speed of computing.

4.2 Mainapproachestofuzzycontrol..................

The essential knowledge about the dynamics of a plant can be presented in the
form of a linguistic rule base. In order to carry out a control synthesis, it is
necessary to model the linguistic information as well as an inference process,
an aggregation process, and possibly a defuzzification process. The use of fuzzy
logic to transform linguistic knowledge into actual control laws leads to thefield
of fuzzy control. The main idea of fuzzy control is to formulate algorithms for