134 CHAPTER 4. FUZZY CONTROL
To balance an upright pendulum, we know from naive physics that the con-
trol forceu(t)should be chosen according to themagnitudes of the input vari-
ablesθ(t)andθ^0 (t)that measure the angle from the upright position and the
angular velocity. The relation between these variables is linguistic, a much
weaker form than differential equations. That is exactly what happens in a
human mind that processes information qualitatively. Humans chooseu(t)by
using common sense knowledge in the form of ìIf...then...î rules, such as ìIf
θis very small andθ^0 is also small thenushould be small,î or ìIf the pendulum
is in a balanced position, then hold very still, that is, do not apply any force.î
By taking all such rules into account, the inverted pendulum can be successfully
controlled.
Now, in order to create an automatic control strategy duplicating human
ability, it is necessary to be able to translate the above ìIf...then...î rules
into some ìsoftî mathematical forms formachine processing. Looking at these
ìIf...then...î rules, we ask ourselves questions like the following:
ïIs ìIf...then...î an implication operator in logic?ïHow can one model linguistic labels like ìsmall,î ìmedium,î and ìlargeî?ïGiven afinite number of ìIf... then... î rules, how can we handle all pos-
sible numerical inputs that will be measured by machine sensors, in order
to produce actual control actions?The answers to all these basic questions lie in fuzzy logic theory. The term
ìfuzzy controlî refers to the science of building fuzzy controllers based on the
mathematics of fuzzy logic.
Now let us look in more detail at the design of a fuzzy controller for the
inverted pendulum on a cart. The input to the controller is the pair
°
θ,θ^0¢
.Let
us take the input space to beX◊Y,whereXandYare intervals representing
degrees forθand degrees per second forθ^0. The output or control space for
uis an interval representing Newtons foru. The linguistic labels are modeled
as fuzzy subsets of the spacesX,Y,andUby specifying their membership
functions. For example, these linguistic labels could correspond tonegative big
(NB),negative medium(NM),negative small(NS),positive small(PS),positive
medium(PM), andpositive big(PB).
ANGLE
A 1 =NB A 2 =NM A 3 =NS A 4 =PS A 5 =PM A 6 =PB
Fuzzy setsAjfor ìangleî