Short Notes on Fuzzy Logic

Table of Contents

What is Fuzzy Logic

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Fuzzy Inference Systems (FIS)

Mamdani-Type and Takagi-Sugeno-Kang (TSK)-Type FIS

There are mainly two types of fuzzy inference systems, namely,

  • Mamdani-type [1], [2]
    • fuzzy consequents (also being called type-I)
    • Singleton consequents (also being called type-II)
  • TSK-type (also being called type-III) [3]

Mamdani-type FIS


  • A model-free approach
  • Mainly based on human skills and experience (, or expert/prior knowledge)
  • Possibly achieve the performance with generally satisfactory


  • Lack of systematic approaches for the design of control systems
  • Cannot guarantee the stability, controllability, parametric sensitivity and robustness [4], [5]

TSK-type FIS


  • Any model-based technique (including a nonlinear one) can be applied to the fuzzy dynamic models
  • The FL controller can be considered as a fuzzy system, usually based on a set of local linear models

Principal Design Parameters For FIS

  1. fuzzification strategies and the interpretation of a fuzzification operator (fuzzifier),
  2. data base:

    1. discretization/normalization of universes of discourse,
    2. fuzzy partition of the input and output spaces,
    3. completeness,
    4. choice of the membership function of a primary fuzzy set;
  3. rule base:

    1. choice of process state (input) variables and control (output) variables of fuzzy control rules,
    2. source and derivation of fuzzy control rules,
    3. types of fuzzy control rules,
    4. consistency, interactivity, completeness of fuzzy control rules;
  4. decision making logic:

    1. definition of a fuzzy implication,
    2. interpretation of the sentence connective and,
    3. interpretation of the sentence connective also,
    4. definitions of a compositional operator,
    5. inference mechanism;
  5. defuzzification strategies and the interpretation of a defuzzification operator (defuzzifier).

Adaptive Fuzzy Systems

An adaptive fuzzy system is has a supervisory module, which can learn the data from the process to modify the components of the fuzzy system, such as [5]

  • the size of the membership functions of the fuzzy sets,
  • the position of the membership functions, and
  • the rule weights and/or the link values.

The adaptation of the size of the membership functions is often implemented by monitoring four criteria of error correction:

  1. The error average value (EAV)
  2. Its first derivative (ΔEAV)
  3. Maximum value of the error (V_err)
  4. The average value of the output values (UAV)

The adaptation of the position of the membership functions typically uses a clustering algorithm to identify the data clusters.

The adaptation of the rule base is performed by increasing or decreasing the rule weights between 0 and 1; in addition, the adaptation of the link values is the concept derived from neutral networks and is not natural for fuzzy systems.

Fuzzy Logic applications

Fuzzy Logic controller (FLC) / control systems

Fuzzy control is originally introduced as a model-free control design approach, model-based fuzzy control has gained widespread significance in the past decade. [5]

Lee [6] gave an overview of fuzzy logic controllers by 1990. The review paper summarized the concept and the structure of fuzzy logic controllers. It also provided a survey of the methods of fuzzifying membership functions, defining the rule base, and justification of fuzzy control rules.

Some Mamdani-type FLC are listed below [5]:

  • Control of machining processes
  • Laser tracking systems
  • Plastic injection molding
  • Vibration suppression
  • Manufacturing area related to robotics, such as manipulators and mobile robots

Other FLC:

  • Fuzzy Statistical Process Control [7]

Fuzzy Logic in Python

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  1. E. H. Mamdani and S. Assilian, “Experiment in Linguistic Synthesis with a Fuzzy Logic Controller,” International Journal of Man-Machine Studies, vol. 7, no. 1, pp. 1–13, 1975, doi: 10.1016/S0020-7373(75)80002-2.
  2. E. H. Mamdani, “Advances in Linguistic-Synthesis of Fuzzy Controllers,” International Journal of Man-Machine Studies, vol. 8, no. 6, pp. 669–678, 1976, doi: 10.1016/S0020-7373(76)80028-4.
  3. K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 45, no. 2, pp. 135–156, Jan. 1992, doi: 10.1016/0165-0114(92)90113-I.
  4. M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 201–224, Apr. 1999, doi: 10.1109/91.755401.
  5. R.-E. Precup and H. Hellendoorn, “A survey on industrial applications of fuzzy control,” Computers in Industry, vol. 62, no. 3, pp. 213–226, Apr. 2011, doi: 10.1016/j.compind.2010.10.001.
  6. C.-C. Lee, “Fuzzy logic in control systems: fuzzy logic controller. I,” IEEE Transactions on Systems, Man and Cybernetics, vol. 20, no. 2, pp. 404–418, Apr. 1990, doi: 10.1109/21.52551.
  7. T. J. Ross, Fuzzy logic with engineering applications. Hoboken, NJ: John Wiley, 2004.
Chris F. Author of this blog, M.Phil.
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