Soft Computing

, Volume 22, Issue 15, pp 5147–5161 | Cite as

Modeling and analysis of the simplest fuzzy PID controller of Takagi–Sugeno type with modified rule base

  • Ritu RajEmail author
  • B. M. Mohan
Methodologies and Application


This paper deals with the simplest fuzzy PID controllers of the Takagi–Sugeno (TS) type. The term simplest is coined since minimal (two) number of fuzzy sets is used for fuzzification in these controllers. The mathematical models of these controllers are found using a modified rule base. The rule base consists of two rules which reduce the number of tuning parameters. Two classes of the simplest fuzzy PID controller are defined using algebraic product triangular norm and bounded sum/maximum triangular co-norm. It is shown that the fuzzy PID controller with modified TS rule base is equivalent to a nonlinear variable gain/ structure controller. The BIBO stability of the closed-loop control system is studied using the small gain theorem. The computational aspects of the controllers are investigated. The applicability of the simplest fuzzy PID controllers is demonstrated with the help of examples.


Mathematical model BIBO stability Fuzzy control PID controller Takagi–Sugeno controller Variable gain controller Nonlinear controller 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Electrical EngineeringIndian Institute of TechnologyKharagpurIndia

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