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This paper reveals mathematical models of the simplest Mamdani PI/PD controllers which employ two fuzzy sets (N: negative and P: positive) on the universe of discourse (UoD) of each of two input variables (displacement and velocity) and three fuzzy sets (N: negative, Z: zero, and P: positive) on the UoD of output variable (control output in the case of PD, and incremental control output in the case of PI). The basic constituents of these models are algebraic product/minimum AND, bounded sum/algebraic sum/maximum OR, algebraic product inference, three linear fuzzy control rules, and Center of Sums (CoS) defuzzification. Properties of all these models are investigated. It is shown that all these controllers are different nonlinear PI/PD controllers with their proportional and derivative gains changing with the inputs. The proposed models are significant and useful to control community as they are completely new and qualitatively different from those reported in the literature.

Mathematical modelling of the simplest Takagi−Sugeno fuzzy Two-Input Two-Output (TITO) Proportional-Integral/Proportional-Derivative (PI/PD) controller is presented in this paper. Mathematical model of fuzzy PI/PD controller is proposed using Algebraic Product (AP) t-norm, Bounded Sum (BS) t-co-norm and Center of Gravity (CoG) defuzzifier. The inputs are fuzzified by fuzzy sets having L and type membership functions. The rule base consists of five rules with different linear models in the consequent parts. Both static and dynamic coupling is taken into account while deriving the models. The model of the fuzzy PI/PD controller reveals that it is a variable gain/structure controller. Also, each output of the TITO fuzzy controller is the sum of two nonlinear PI or PD controllers with variable gains. The properties and the gain variations of the controllers are investigated. KEYWORDSfuzzy control; mathematical model; Takagi−Sugeno controller; TITO controller; variable gain controller I ndustrial applications mainly employ conventional controllers due to their ease of implementation. Multiple-Input Multiple-Output (MIMO) systems poses a lot of challenges when it comes to controller design. Tuning of controllers for multi-variable systems is one such task. The literature is rich in methods dealing with the tuning of Two-Input Two-Output (TITO) conventional controllers. In Ref.[1], the coupling information of the TITO system was used to develop a model for tuning the controller for industrial processes. TITO processes were decoupled using a decoupler matrix which was further used to tune Proportional-Integral-Derivative (PID) controllers [2] . Dominant pole placement approach [3] was used for tuning multi-variable PID controllers. A frequency domain approach [4] was applied to design a fractional order proportional integral controller to control the level of water in a two-tank MIMO system.With the advent of fuzzy sets [5] , the area "fuzzy control" became quite popular among the control practitioners. A fuzzy control algorithm was applied to a laboratory built steam engine by Mamdani [6] , where the control strategy (rule base) was in the form of "if-then" statements. Later, the control strategy was named as Mamdani rule base. In 1985 a new fuzzy algorithm was developed, which is now referred to as Takagi-Sugeno (TS) fuzzy model [7] , where the "then" part is a linear function of the premise variables. Mamdani rule base and TS rule base differ in the consequent ("then") part of the rule base, where the former uses fuzzy sets and the latter uses linear functions of premise variables. Mamdani rule base is