---

FLex

A toolkit for fuzzy logic, this library includes functions to make fuzzy sets, variables and rules for creating a Fuzzy Logic System (FLS).

The goal of FLex is to easily design and efficiently operate fuzzy logic controllers without relying on external libraries.

Index

• Features

• Installation

• Usage

  • Sets
  • Variables
  • Rules
  • System
  • ANFIS

• Documentation

• Contributing

• License

• TODO

Features

The following list is the current supported backend for each component of the FLS:

• Linguistic Rules:

  • Lambda function syntax
  • Tuple syntax

• Membership functions:

  • Triangular
  • Trapezoidal
  • Saturation
  • Shoulder
  • Gauss
  • Generalized Bell
  • Sigmoid
  • Z-shaped
  • S-shaped
  • Pi-shaped
  • Linear Combination (Takagi-Sugeno, ANFIS only)

• Fuzzy Inference Systems:

  • Mamdani:

    • Inference:

      • Min

    • Output Combination:

      • Root-sum-square

    • Defuzzification:

      • Centroid

  • Takagi-Sugeno:

    • Inference:

      • Max
      • Product

    • Defuzzification:

      • Weighted average

  • ANFIS:

    • Inference:

      • Max
      • Product

    • Defuzzification:

      • Weighted average

    • Optimization Method:

      • Backpropagation.
      • Hybrid (Backpropagation, LSE).

NOTE: All systems are single output.

Installation

The package can be installed by adding flex to your list of dependencies in mix.exs:

def deps do
  [
    {:flex, "~> 0.1.0"}
  ]
end

Usage

---

Sets

Step 1: Define all fuzzy sets with Flex.Set.new/1, the following options are require:

• mf_type - (string) Defines which type of membership function uses the set (e.g., "triangle").
• tag - (string) defines the linguistic name of the fuzzy set (e.g., "too hot"),
• mf_params - The parameters of the membership function, see Membership functions.

t_h = Flex.Set.new(tag: "too hot", mf_type: "saturation", mf_params: [-2, 0, -4])
j_r = Flex.Set.new(tag: "just right", mf_type: "triangle", mf_params: [-2, 0, 2])
t_c = Flex.Set.new(tag: "too cold", mf_type: "shoulder", mf_params: [0, 2, 4])

g_h = Flex.Set.new(tag: "getting hotter", mf_type: "saturation", mf_params: [-5, 0, -10])
n_c = Flex.Set.new(tag: "no change", mf_type: "triangle", mf_params: [-5, 0, 5])
g_c = Flex.Set.new(tag: "getting colder", mf_type: "shoulder", mf_params: [0, 5, 10])

co = Flex.Set.new(tag: "cool", mf_type: "saturation", mf_params: [-50, 0, -100])
d_n = Flex.Set.new(tag: "do nothing", mf_type: "triangle", mf_params: [-50, 0, 50])
he = Flex.Set.new(tag: "heat", mf_type: "shoulder", mf_params: [0, 50, 100])

Variables

Step 2: Define all fuzzy variables with Flex.Variable.new/1, the following options are required:

• :tag - (string) Defines the linguistic name of the fuzzy variable (e.g., "error"),
• :fuzzy_sets - (list) Defines which type of membership function use the set (e.g., "triangle").
• :type - (atom) Defines the type of variable (e.g., :antecedent or :consequent),
• :range - (range) The range in which the variable exists.

fuzzy_sets = [t_h, j_r, t_c]
error = Flex.Variable.new(tag: "error", fuzzy_sets: fuzzy_sets, type: :antecedent, range: -4..4)

fuzzy_sets = [g_h, n_c, g_c]
dt_error = Flex.Variable.new(tag: "dt_error", fuzzy_sets: fuzzy_sets, type: :antecedent, range: -10..10)

fuzzy_sets = [co, d_n, he]
output = Flex.Variable.new(tag: "output", fuzzy_sets: fuzzy_sets, type: :consequent, range: -100..100)

Rules

Currently there are two types of syntax for defining the rules statement:

• Anonymous function syntax:

  r1 = fn [at1, at2, con] ->
      (at1 ~> "too hot" &&& at2 ~> "getting colder") >>> con ~> "cool"
    end

• Tuple syntax:

  r1 = {{{{"error", "too hot", "~>"}, {"dt_error", "getting colder", "~>"}, "&&&"}, "output",
      ">>>"}, "cool", "~>"}

Step 3: Define all Linguistic rules with Flex.Rule.new/1, the following options are required:

• :statement - Defines the rule behavior.
• :antecedent - (list) Defines the input variables.
• :consequent - Defines the output variable.

  import Flex.Rule
  r1 =
    {{{{"error", "too hot", "~>"}, {"dt_error", "getting colder", "~>"}, "&&&"}, "output",
      ">>>"}, "cool", "~>"}

  r2 =
    {{{{"error", "just right", "~>"}, {"dt_error", "getting colder", "~>"}, "&&&"}, "output",
      ">>>"}, "heat", "~>"}

  r3 =
    {{{{"error", "too cold", "~>"}, {"dt_error", "getting colder", "~>"}, "&&&"}, "output",
      ">>>"}, "heat", "~>"}

  r4 =
    {{{{"error", "too hot", "~>"}, {"dt_error", "no change", "~>"}, "&&&"}, "output", ">>>"},
      "cool", "~>"}

  r5 =
    {{{{"error", "just right", "~>"}, {"dt_error", "no change", "~>"}, "&&&"}, "output", ">>>"},
      "do nothing", "~>"}

  r6 =
    {{{{"error", "too cold", "~>"}, {"dt_error", "no change", "~>"}, "&&&"}, "output", ">>>"},
      "heat", "~>"}

  r7 =
    {{{{"error", "too hot", "~>"}, {"dt_error", "getting hotter", "~>"}, "&&&"}, "output",
      ">>>"}, "cool", "~>"}

  r8 =
    {{{{"error", "just right", "~>"}, {"dt_error", "getting hotter", "~>"}, "&&&"}, "output",
      ">>>"}, "cool", "~>"}

  r9 =
    {{{{"error", "too cold", "~>"}, {"dt_error", "getting hotter", "~>"}, "&&&"}, "output",
      ">>>"}, "cool", "~>"}

  rule1 = Flex.Rule.new(statement: r1, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule2 = Flex.Rule.new(statement: r2, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule3 = Flex.Rule.new(statement: r3, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule4 = Flex.Rule.new(statement: r4, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule5 = Flex.Rule.new(statement: r5, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule6 = Flex.Rule.new(statement: r6, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule7 = Flex.Rule.new(statement: r7, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule8 = Flex.Rule.new(statement: r8, consequent: output.tag, antecedent: [error.tag, dt_error.tag])
  rule9 = Flex.Rule.new(statement: r9, consequent: output.tag, antecedent: [error.tag, dt_error.tag])

  rules = [rule1, rule2, rule3, rule4, rule5, rule6, rule7, rule8, rule9]

Note: You need to import Flex.Rule module.

System

Step 4: Define FLS with Flex.System.start_link/1 or Flex.System.start_link/2 if you want to overwrite the GenServer options; the following options are require:

• :rules - Defines the behavior of the system based on a list of rules.
• :antecedent - (list) Defines the input variables.
• :consequent - Defines the output variable.
• :engine_type - Defines the inference engine behavior (default: Mamdini).

  {:ok, s_pid} = Flex.System.start_link(antecedent: [error, dt_error], consequent: output, rules: rules)

Step 5: Fit the FLS with a input vector using Flex.System.compute/2.

  result = Flex.System.compute(s_pid, [-1, -2.5])
  #result ~= -63.4 aprox

In test/system_test.exs there is an example of use, that is based on this example.

ANFIS

An adaptive network-based fuzzy inference system (ANFIS) is a kind of artificial neural network that is based on Takagi–Sugeno fuzzy inference system, this implementation use backpropagation, only Gaussian & Generalized Bell Membership functions are allowed. In examples/anfis_demo1.exs there is an example of use.

Documentation

The docs can be found at https://hexdocs.pm/flex.

Contributing

• Fork our repository on github.
• Fix or add what is needed.
• Commit to your repository.
• Issue a github pull request (fill the PR template).

License

See LICENSE.

TODO

• Add more membership functions.
• Add more inference methods.
• Add more defuzzification methods.
• Add helper functions.
• Add metaprogramming for linguistic rules.