About T-Controller

Fuzzy logic, implemented in Fuzzy Inference Systems (FIS), makes use of human common sense or expert knowledge to build control systems or model data.

Purpose

Extension of traditional methods of data modeling and control.

Common types of FIS

Mamdani FIS – applicable when numerical data basis is incomplete and can be extended by human expert knowledge,
Takagi-Sugeno FIS – applicable when numerical data basis is sufficient. Reduces inference time.

Advantage

Good solution can be created at low cost using expert knowledge.

Problems

  • Good interpretability vs. good accuracy,
  • small number of input variables,
  • time consuming tuning,
  • Mamdani FIS: Dependency on defuzzification method – same rule gives different results depending on defuzzification function (Centre of Gravity, Minimum, First Maximum). Reasoning for defuzzyfication function choice based only on experiments,

*Takagi-Sugeno FIS: requires sufficient data basis.

Traditional: Mamdani-type FIS

Structure of Mamdani-type FIS

Features

  • Defuzzyfication method is chosen based on experiments and expert knowledge,
  • there are no standards for rule building,
  • rule form:Ri: ~IF ~ X1~ IS~ A_{r1}~ AND~ X_{2}~ IS~ A_{r2}~ AND~ X_{n}~ IS~ A_{rn}~ THEN ~Y~ IS~ B_{k},where R_{i} is the rule number i, X_{n} is the input number n (linguistic variable), A_{rn} and B_{k} are linguistic terms, keyword IS marks a clause, keyword AND marks a conjunction.

New: T-Controller FIS v

Structure of T-Controller FIS

Features

  • Peculiar defuzzification method,
  • rule building is driven by expected output, rules are disjunct: each output value correspond only to one rule,
  • rule form:Ri(B): ~X1 ~IS ~A_{r1} ~AND ~X_{2} ~IS ~A_{r2} ~AND ~X_{n} ~IS ~A_{rn} ~OR ~X_{1} ~IS ~A_{s1} ~ANDX_{2} ~IS ~A_{s2} ~AND ~X_{n} ~IS ~A_{sn}, where R_{i} is the rule number i, X_{n} is the input number n (linguistic variable), A_{rn}, A_{sn}, B_{i} are linguistic terms, keyword IS marks a clause, keyword AND marks a conjunction, keyword OR marks a disjunction.

Rule building

  1. Divide output variable space into intervals, each interval corresponds to one rule. Minimum number of rules is 2,
  2. Divide each input variable space into intervals, each interval corresponds to one linguistic variable. Minimum number of linguistic variables is 2,
  3. Define rules: Each valid combination of input variables forms a conjunction (AND) of clauses (INPUT IS LINGUISTIC TERM). Valid conjunctions, corresponding to one rule are connected to a disjunction (OR).

Defuzzification

Defuzzyfication is performed on Geometrical Transformation Machine trained on (ideal) fuzzyfied output values.

Advantages over traditional FIS

  • Logical inference and composition are combined into one specific step,
  • high speed geometrical defuzzification method with zero systematic error,
  • number of rules is driven by features of output variable only,
  • procedure of rule building is intuitive for experts via analysis of possible situations for output variable,
  • fast defuzzification,
  • simple hardware implementation.