## Fuzzy data approximation. Fuzzy logic begins with

Fuzzy
logic systems are generally used for system identification, control, and model
recognition problems. To maximize performance, it is often necessary to perform
a project optimization procedure in which the variable parameters of the Fuzzy
systems are tuned to maximize a given performance condition. Some imminent data
are commonly available and produce what is termed a supervised learning
problem. In this regard, we generally want to minimize the sum of error squares
in data approximation.

Fuzzy
logic begins with and builds on a set of human-language rules provided by the
user. The fuzzy system changes these rules to their mathematical equivalents.
This simplifies the work of the system designer and the computer and translates
into a much more accurate representation of how systems behave in the real
world. The additional benefits of fuzzy logic are its simplicity and
flexibility. Fuzzy logic can handle problems with inaccurate and incomplete
data and can model non-linear functions of random complexity. “If you do
not have a good implant model or if the system is changing, then diffusion will
produce a better solution than conventional control techniques,” says Bob
Varley. We can generate a fuzzy system to match any set of input and output
data. Fuzzy Logic Toolbox makes it particularly easy to provide adaptive
techniques such as neuro-diffuse adaptive inference systems (ANFIS) and diffuse
subtractive clustering.

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Fuzzy logic models, called fuzzy inference
systems, consist of a series of “if-then” conditional rules. . For
the designer who understands the system, we can easily write these rules and
you can provide all the rules that are essential for correctly explaining the
system (although generally only a moderate number of rules are required). The
rule-based approach and the flexible membership plan not only simplifies the
creation of fuzzy systems, but also simplifies system design and ensures that
the system can be updated and maintained over time. It is recognized that the prepositional
logic is isomorphic to establish the theory under the association between the
components of these two mathematical systems. Furthermore, both systems are
isomorphic to a Boolean algebra, which is a mathematical system defined by
abstract entities and their axiomatic properties.

The isomorphism between Boolean algebra, set
theory and propositional logic ensures that each theorem in one of these
theories has a counterpart in each of the other two theories. These isomorphism
allow us, in effect, to involve all these theories by scaling only one. We will
not spend much time reviewing the clear logic, but at some point we have to
spend, especially in the concept of commitment, to achieve the concept in fuzzy
logic. Fuzzy rules are the cornerstone of fuzzy logic systems. . Rules are a
form of proposition. A proposition is an ordinary statement that implies, for
example, “The ratio of the buffer is low”. Therefore, we can have the
following rule: “If the damping is low, the impulse response of the system
oscillates long before it goes out”. In traditional propositional logic, a
proposition must be meaningful to call it “true” or
“false”, regardless of whether we know which of these terms applies
correctly.

Logical
reasoning is the process of combining the propositions given in other
propositions, and then doing it repeatedly. The proposition can be combined in
many ways, deriving from three fundamental operations: the conjunction
(indicated by p?q),
where the simultaneous truth of two separate sentences peq is said; Disjunction
(p?q), where to affirm the
truth of one or both the different proposals; the implication (p ? q) which
generally takes the form of an IF-THEN rule (also known as the “production
rule”). The implicit IF part is called antecedent, while the THEN part is
called consequential. Generate propositions using union, disjunction or
implication, a new scam The proposition can be obtained from a given prefix of
the clause “is false that …”; This is the negation operation (~ p).
Furthermore, p ? q is the equivalence relation; means that both p and q are
true or false.

Fuzzy
concepts first introduced by Zadeh in the ’60s and’ 70s the focus is basically
a traditional computational logic and set theory concerns:

·
True or False

·
Zero or One

·
In or Out (in terms of set
membership)

·
Black or White (no grey)

·
Not the case with fuzzy logic and fuzzy
sets

Fuzzy logic allows
conclusions to be true or false. However, there are also proposals with
variable answers, such as those that can be found by asking a group of people
to identify a color. In these cases, the truth appears as the result of
reasoning from an inaccurate or partial knowledge in which the sampled answers
are mapped into a spectrum.

Humans and animals
often operate using fuzzy ratings in most situations. The person does not
calculate the exact values ??for weight, density, distance, direction, height
and width of the container and the air resistance of the object to determine
the force and the projection angle of the object. Instead, the person
impulsively applies “fuzzy” calculations quickly, based on previous
experience, to determine which force output values, direction, and vertical
angle will be used to make the tone. Both degrees of truth and probability
range from 0 to 1 and, therefore, may seem the same at the beginning, but fuzzy
logic uses degrees of truth as a mathematical model of ambiguity, while
probability is a mathematical model of ignorance.

The Basic Concepts of
Fuzzy Logic are:

Approximation (“granulation”): a color can be accurately described
using RGB values, or it can   be roughly
described as “red”, “blue”, etc.

Degree (“graduation”): two different colors can be described as
“red”, but one is considered
more red than the other.

Fuzzy logic tries to reflect the human way of thinking.

·
Approximation (“granulation”):
a color can be accurately described using RGB values, or it can   be roughly described as “red”,
“blue”, etc..

·
different colors can be described as “red”, but one is considered
more red than the other.

·
Fuzzy logic tries to reflect the human
way of thinking.

Implementation of Fuzzy Logic

• Can be implemented in
systems of different sizes and capacities, from small   microcontrollers to workstation-based network
control systems.

• It can be implemented
in software, hardware or in a combination of both.