Mathematical Modelling and Analysis of the Simplest Fuzzy Two-Input Two-Output Two-Term Controller of Takagi−Sugeno Type

Ritu Raj; Murali Mohan Bosukonda

doi:10.26599/FIE.2023.9270004

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Article | Open Access

Mathematical Modelling and Analysis of the Simplest Fuzzy Two-Input Two-Output Two-Term Controller of Takagi−Sugeno Type

Ritu Raj^¹(), Murali Mohan Bosukonda^²

1Department of Electronics and Communication Engineering, Indian Institute of Information Technology Kota, MNIT Campus, Jaipur 302017, India

2Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India

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Abstract

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 $Γ$ type membership functions. The rule base consists of five rules with different linear models in the consequent parts. Both static and dynamic couplings are 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.

Keywords

fuzzy control mathematical model Takagi−Sugeno controller TITO controller variable gain controller

References

[1]

I. L. Chien, H. P. Huang, and J. C. Yang, A simple TITO PI tuning method suitable for industrial applications, Chem. Eng. Commun., vol. 182, no. 1, pp. 181–196, 2000.

Rule No.	$I_{1}$	$I_{2}$	$I_{3}$	$I_{4}$	$J_{1}$	$J_{2}$
Note: If $e_{1} (k) = I_{1} a n d ∆ e_{1} (k) = I_{2} a n d = ∆ e_{1} (k) = I_{3} a n d ∆ e_{2} (k) = I_{4},$ then $∆ u_{1} (k) = J_{1} a n d ∆ u_{2} (k) = J_{2}, w h e r e J_{1} = 0.5 [(I_{1} + I_{2}) + ∆_{12} (I_{3} + I_{4})]$ and $J_{2} =$ $0.5 [∆_{21} (I_{1} + I_{2}) + (I_{3} + I_{4})]$ .
1	−1	−1	−1	−1	$- 1 - ∆_{12}$	$- ∆_{21} - 1$
2	−1	−1	−1	+1	−1	$- ∆_{21}$
3	−1	−1	+1	−1	−1	$- ∆_{21}$
4	−1	−1	+1	+ 1	$- 1 + ∆_{12}$	$- ∆_{21} + 1$
5	−1	+1	−1	−1	$- ∆_{12}$	−1
6	−1	+1	−1	+1	0	0
7	−1	+1	+1	−1	0	0
8	−1	+1	+1	+1	$∆_{12}$	1
9	+1	−1	−1	−1	$- ∆_{12}$	−1
10	+1	−1	−1	+1	0	0
11	+1	−1	+1	−1	0	0
12	+1	−1	+1	+1	$∆_{12}$	1
13	+1	+1	−1	−1	$1 - ∆_{12}$	$∆_{21} - 1$
14	+1	+1	−1	+1	1	$∆_{21}$
15	+1	+1	+1	−1	1	$∆_{21}$
16	+1	+1	+1	+1	$1 + ∆_{12}$	$∆_{21} + 1$

# Rule	$I_{1}$	$I_{2}$	$I_{3}$	$I_{4}$	$J_{1}$	$J_{2}$
1	−1	−1	−1	−1	$- 1 - δ_{12}$	$- δ_{21} - 1$
2	−1	−1	−1	+1	−1	$- δ_{21}$
2	−1	−1	+1	−1	−1	$- δ_{21}$
3	−1	−1	+1	+1	$- 1 + δ_{12}$	$- δ_{21} + 1$
4	−1	+1	−1	−1	$- δ_{12}$	−1
4	+1	−1	−1	−1	$- δ_{12}$	−1
5	−1	+1	−1	+1	0	0
	−1	+1	+1	−1
	+1	−1	−1	+1
	+1	−1	+1	−1
6	−1	+1	+1	+1	$δ_{12}$	1
6	+1	−1	+1	+1	$δ_{12}$	1
7	+1	+1	−1	−1	$1 - δ_{12}$	$δ_{21} - 1$
8	+1	+1	−1	+1	1	$δ_{21}$
8	+1	+1	+1	−1	1	$δ_{21}$
9	+1	+1	+1	+1	$1 + δ_{12}$	$δ_{21} + 1$

Cell	Resultant membership function
Cell	$μ_{1}$	$μ_{2}$	$μ_{3}$	$μ_{4}$	$μ_{5}$
(1, 1), (1, 3), (3, 1), (3, 3)	0	0	0	0	1
(1, 2), (1, 4), (3, 2), (3, 4)	0	0	0	1	0
(1, 5), (3, 5)	0	0	0	$μ_{- 1} (e_{2})$	$μ_{+ 1} (e_{2})$
(1, 6), (3, 6)	0	0	0	$μ_{- 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{2})$
(1, 7), (3, 7)	0	0	0	$μ_{+ 1} (e_{2})$	$μ_{- 1} (e_{2})$
(1, 8), (3, 8)	0	0	0	$μ_{+ 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{2})$
(1, 9), (3, 9)	0	0	0	$A_{2} + D_{2}$	$B_{2} + C_{2}$
(2, 1), (2, 3), (4, 1), (4, 3)	0	1	0	0	0
(2, 2), (4, 4)	1	0	0	0	0
(2, 4), (4, 2)	0	0	1	0	0
(2, 5)	$μ_{- 1} (e_{2})$	$μ_{+ 1} (e_{2})$	0	0	0
(2, 6)	$μ_{- 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{2})$	0	0	0
(2, 7)	0	$μ_{- 1} (e_{2})$	$μ_{+ 1} (e_{2})$	0	0
(2, 8)	0	$μ_{- 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{2})$	0	0
(2, 9)	$A_{2}$	$B_{2} + C_{2}$	$D_{2}$	0	0
(4, 5)	0	$μ_{+ 1} (e_{2})$	$μ_{- 1} (e_{2})$	0	0
(4, 6)	0	$μ_{+ 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{2})$	0	0
(4, 7)	$μ_{+ 1} (e_{2})$	$μ_{- 1} (e_{2})$	0	0	0
(4, 8)	$μ_{+ 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{2})$	0	0	0
(4, 9)	$D_{2}$	$B_{2} + C_{2}$	$A_{2}$	0	0
(5, 1), (5, 3)	0	$μ_{- 1} (e_{1})$	0	0	$μ_{+ 1} (e_{1})$
(5, 2)	$μ_{- 1} (e_{1})$	0	0	$μ_{+ 1} (e_{1})$	0
(5, 4)	0	0	$μ_{- 1} (e_{1})$	$μ_{+ 1} (e_{1})$	0
(5, 5)	$μ_{- 1} (e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (e_{2})$	0	$μ_{+ 1} (e_{1}) μ_{- 1} (e_{2})$	$μ_{+ 1} (e_{1}) μ_{+ 1} (e_{2})$
(5, 6)	$μ_{- 1} (e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	0	$μ_{+ 1} (e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{+ 1} (e_{1}) μ_{+ 1} (∆ e_{2})$
(5, 7)	0	$μ_{- 1} (e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (e_{2})$
(5, 8)	0	$μ_{- 1} (e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (∆ e_{2})$
(5, 9)	$μ_{- 1} (e_{1}) A_{2}$	$μ_{- 1} (e_{1}) (B_{2} + C_{2})$	$μ_{- 1} (e_{1}) D_{2}$	$μ_{+ 1} (e_{1}) (A_{2} + D_{2})$	$μ_{+ 1} (e_{1}) (B_{2} + C_{2})$
(6, 1), (6, 3)	0	$μ_{- 1} (∆ e_{1})$	0	0	$μ_{+ 1} (∆ e_{1})$
(6, 2)	$μ_{- 1} (∆ e_{1})$	0	0	$μ_{+ 1} (∆ e_{1})$	0
(6, 4)	0	0	$μ_{- 1} (∆ e_{1})$	$μ_{+ 1} (∆ e_{1})$	0
(6, 5)	$μ_{- 1} (∆ e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	0	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (e_{2})$
(6, 6)	$μ_{- 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	0	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$
(6, 7)	0	$μ_{- 1} (∆ e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (e_{2})$
(6, 8)	0	$μ_{- 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$
(6, 9)	$μ_{- 1} (∆ e_{1}) A_{2}$	$μ_{- 1} (∆ e_{1}) (B_{2} + C_{2})$	$μ_{- 1} (∆ e_{1}) D_{2}$	$μ_{+ 1} (∆ e_{1}) (A_{2} + D_{2})$	$μ_{+ 1} (∆ e_{1}) (B_{2} + C_{2})$
(7, 1), (7, 3)	0	$μ_{+ 1} (e_{1})$	0	0	$μ_{- 1} (e_{1})$
(7, 2)	0	0	$μ_{+ 1} (e_{1})$	$μ_{- 1} (e_{1})$	0
(7, 4)	$μ_{+ 1} (e_{1})$	0	0	$μ_{- 1} (e_{1})$	0
(7, 5)	0	$μ_{+ 1} (e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (e_{2})$
(7, 6)	0	$μ_{+ 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (e_{1}) μ_{+ 1} (∆ e_{2})$
(7, 7)	$μ_{+ 1} (e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (e_{2})$	0	$μ_{- 1} (e_{1}) μ_{+ 1} (e_{2})$	$μ_{- 1} (e_{1}) μ_{- 1} (e_{2})$
(7, 8)	$μ_{+ 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (e_{1}) μ_{- 1} (∆ e_{2})$	0	$μ_{- 1} (e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{- 1} (e_{1}) μ_{- 1} (∆ e_{2})$
(7, 9)	$μ_{+ 1} (e_{1}) D_{2}$	$μ_{+ 1} (e_{1}) (B_{2} + C_{2})$	$μ_{+ 1} (e_{1}) A_{2}$	$μ_{- 1} (e_{1}) (A_{2} + D_{2})$	$μ_{- 1} (e_{1}) (B_{2} + C_{2})$
(8, 1), (8, 3)	0	$μ_{+ 1} (∆ e_{1})$	0	0	$μ_{- 1} (∆ e_{1})$
(8, 2)	0	0	$μ_{+ 1} (∆ e_{1})$	$μ_{- 1} (∆ e_{1})$	0
(8, 4)	$μ_{+ 1} (∆ e_{1})$	0	0	$μ_{- 1} (∆ e_{1})$	0
(8, 5)	0	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{- 1} (e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (e_{2})$
(8, 6)	0	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$
(8, 7)	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (e_{2})$	0	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{- 1} (e_{2})$
(8, 8)	$μ_{+ 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{+ 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$	0	$μ_{- 1} (∆ e_{1}) μ_{+ 1} (∆ e_{2})$	$μ_{- 1} (∆ e_{1}) μ_{- 1} (∆ e_{2})$
(8, 9)	$μ_{+ 1} (∆ e_{1}) D_{2}$	$μ_{+ 1} (∆ e_{1}) (B_{2} + C_{2})$	$μ_{+ 1} (∆ e_{1}) A_{2}$	$μ_{- 1} (∆ e_{1}) (A_{2} + D_{2})$	$μ_{- 1} (∆ e_{1}) (B_{2} + C_{2})$
(9, 1), (9, 3)	0	$A_{1} + D_{1}$	0	0	$B_{1} + C_{1}$
(9, 2)	$A_{1}$	0	$D_{1}$	$B_{1} + C_{1}$	0
(9, 4)	$D_{1}$	0	$A_{1}$	$B_{1} + C_{1}$	0
(9, 5)	$A_{1} μ_{- 1} (e_{2})$	$(A_{1} + D_{1}) μ_{+ 1} (e_{2})$	$D_{1} μ_{- 1} (e_{2})$	$(B_{1} + C_{1}) μ_{- 1} (e_{2})$	$(A_{1} + D_{1}) μ_{+ 1} (e_{2})$
(9, 6)	$A_{1} μ_{- 1} (∆ e_{2})$	$(A_{1} + D_{1}) μ_{+ 1} (∆ e_{2})$	$D_{1} μ_{- 1} (∆ e_{2})$	$(B_{1} + C_{1}) μ_{- 1} (∆ e_{2})$	$(B_{1} + C_{1}) μ_{+ 1} (∆ e_{2})$
(9, 7)	$D_{1} μ_{+ 1} (e_{2})$	$(A_{1} + D_{1}) μ_{- 1} (e_{2})$	$A_{1} μ_{+ 1} (e_{2})$	$(B_{1} + C_{1}) μ_{+ 1} (e_{2})$	$(B_{1} + C_{1}) μ_{- 1} (e_{2})$
(9, 8)	$D_{1} μ_{+ 1} (∆ e_{2})$	$(A_{1} + D_{1}) μ_{- 1} (∆ e_{2})$	$A_{1} μ_{+ 1} (∆ e_{2})$	$(B_{1} + C_{1}) μ_{+ 1} (∆ e_{2})$	$(B_{1} + C_{1}) μ_{- 1} (∆ e_{2})$
(9, 9)	$A_{1} A_{2} + D_{1} D_{2}$	$A_{1} B_{2} + A_{1} C_{2} + D_{1} B_{2} + D_{1} C_{2}$	$A_{1} D_{2} + D_{1} A_{2}$	$B_{1} A_{2} + B_{1} D_{2} + C_{1} A_{2} + C_{1} D_{2}$	$B_{1} B_{2} + B_{1} C_{2} + C_{1} B_{2} + C_{1} C_{2}$

Cell	Expression of gain γ_ijp
(1, 1), (1, 3), (3, 1), (3, 3)	$p_{i j}^{5}$
(1, 2), (1, 4), (3, 2), (3, 4)	$p_{i j}^{4}$
(1, 5), (1, 6) (3, 5), (3, 6)	$\frac{(- p_{i j}^{4} + p_{i j}^{5}) (e_{2}; ∆ e_{2}) + (p_{i j}^{4} + p_{i j}^{5}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(1, 7), (1, 8) (3, 7), (3, 8)	$\frac{(p_{i j}^{4} - p_{i j}^{5}) (e_{2}; ∆ e_{2}) + (p_{i j}^{4} + p_{i j}^{5}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(1, 9), (3, 9)	$\frac{(p_{i j}^{4} - p_{i j}^{5}) (e_{2} ∆ e_{2}) + (p_{i j}^{4} + p_{i j}^{5}) (h_{3} h_{4})}{2 h_{3} h_{4}}$
(2, 1), (2, 3), (4, 1), (4, 3)	$p_{i j}^{2}$
(2, 2), (4, 4)	$p_{i j}^{1}$
(2, 4), (4, 2)	$p_{i j}^{3}$
(2, 5), (2, 6)	$\frac{(- p_{i j}^{1} + p_{i j}^{2}) (e_{2}; ∆ e_{2}) + (p_{i j}^{1} + p_{i j}^{2}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(2, 7), (2, 8)	$\frac{(- p_{i j}^{2} + p_{i j}^{3}) (e_{2}; ∆ e_{2}) + (p_{i j}^{2} + p_{i j}^{3}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(2, 9)	$\frac{(p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3}) (e_{2} ∆ e_{2}) + (- p_{i j}^{1} + p_{i j}^{3}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3}) h_{3} h_{4}}{4 h_{3} h_{4}}$
(4, 5), (4, 6)	$\frac{(p_{i j}^{2} - p_{i j}^{3}) (e_{2}; ∆ e_{2}) + (p_{i j}^{2} + p_{i j}^{3}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(4, 7), (4, 8)	$\frac{(p_{i j}^{1} - p_{i j}^{2}) (e_{2}; ∆ e_{2}) + (p_{i j}^{1} + p_{i j}^{2}) (h_{3}; h_{4})}{2 (h_{3}; h_{4})}$
(4, 9)	$\frac{(p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3}) (e_{2} ∆ e_{2}) + (p_{i j}^{1} - p_{i j}^{3}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3}) h_{3} h_{4}}{4 h_{3} h_{4}}$
(5, 1), (6, 1) (5, 3), (6, 3)	$\frac{(- p_{i j}^{2} + p_{i j}^{5}) (e_{1}; ∆ e_{1}) + (p_{i j}^{2} + p_{i j}^{5}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(5, 2), (6, 2)	$\frac{(- p_{i j}^{1} + p_{i j}^{4}) (e_{1}; ∆ e_{1}) + (p_{i j}^{1} + p_{i j}^{4}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(5, 4), (6, 4)	$\frac{(- p_{i j}^{3} + p_{i j}^{4}) (e_{1}; ∆ e_{1}) + (p_{i j}^{3} + p_{i j}^{4}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(5, 5), (5, 6)	$\frac{(p_{i j}^{1} - p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) (e_{1} (e_{2}; ∆ e_{2})) + (- p_{i j}^{1} - p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) (e_{1} (h_{3}; h_{4})) + (- p_{i j}^{1} + p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{1}) + (p_{i j}^{1} + p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) h_{1} (h_{3}; h_{4})}{4 h_{1} (h_{3}; h_{4})}$
(5, 7), (5, 8)	$\frac{(p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) (e_{1} (e_{2}; ∆ e_{2})) + (- p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) (e_{1} (h_{3}; h_{4})) + (- p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{1}) + (p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) h_{1} (h_{3}; h_{4})}{4 h_{1} (h_{3}; h_{4})}$
(5, 9), (6, 9)	$\frac{\begin{matrix} (- p_{i j}^{1} + 2 p_{i j}^{2} - p_{i j}^{3} + 2 p_{i j}^{4} - 2 p_{i j}^{5}) (e_{1}; ∆ e_{1}) e_{2} ∆ e_{2} + (p_{i j}^{1} - p_{i j}^{3}) (e_{1}; ∆ e_{1}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} - 2 p_{i j}^{5}) e_{2} ∆ e_{2} (h_{1}; h_{2}) + \\ (- p_{i j}^{1} - 2 p_{i j}^{2} - p_{i j}^{3} + 2 p_{i j}^{4} + 2 p_{i j}^{5}) (e_{1}; ∆ e_{1}) h_{3} h_{4} + (- p_{i j}^{1} + p_{i j}^{3}) (h_{1}; h_{2}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} + 2 p_{i j}^{5}) (h_{1}; h_{2}) h_{3} h_{4} \end{matrix}}{8 (h_{1}; h_{2}) h_{3} h_{4}}$
(6, 5), (6, 6)	$\frac{(p_{i j}^{1} - p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) (∆ e_{1} (e_{2}; ∆ e_{2})) + (- p_{i j}^{1} - p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) (∆ e_{1} (h_{3}; h_{4})) + (- p_{i j}^{1} + p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{2}) + (p_{i j}^{1} + p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) h_{2} (h_{3}; h_{4})}{4 h_{2} (h_{3}; h_{4})}$
(6, 7), (6, 8)	$\frac{(p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) (∆ e_{1} (e_{2}; ∆ e_{2})) + (- p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) (∆ e_{1} (h_{3}; h_{4})) + (- p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{2}) + (p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) h_{2} (h_{3}; h_{4})}{4 h_{2} (h_{3}; h_{4})}$
(7, 1), (8, 1) (7, 3), (8, 3)	$\frac{(p_{i j}^{2} - p_{i j}^{5}) (e_{1}; ∆ e_{1}) + (p_{i j}^{2} + p_{i j}^{5}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(7, 2), (8, 2)	$\frac{(p_{i j}^{3} - p_{i j}^{4}) (e_{1}; ∆ e_{1}) + (p_{i j}^{3} + p_{i j}^{4}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(7, 4), (8, 4)	$\frac{(p_{i j}^{1} - p_{i j}^{4}) (e_{1}; ∆ e_{1}) + (p_{i j}^{1} + p_{i j}^{4}) (h_{1}; h_{2})}{2 (h_{1}; h_{2})}$
(7, 5), (7, 6)	$\frac{(p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) (e_{1} (e_{2}; ∆ e_{2})) + (p_{i j}^{2} + p_{i j}^{3} - p_{i j}^{4} - p_{i j}^{5}) (e_{1} (h_{3}; h_{4})) + (p_{i j}^{2} - p_{i j}^{3} - p_{i j}^{4} + p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{1}) + (p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) h_{1} (h_{3}; h_{4})}{4 h_{1} (h_{3}; h_{4})}$
(7, 7), (7, 8)	$\frac{(p_{i j}^{1} - p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) (e_{1} (e_{2}; ∆ e_{2})) + (p_{i j}^{1} + p_{i j}^{2} - p_{i j}^{4} - p_{i j}^{5}) (e_{1} (h_{3}; h_{4})) + (p_{i j}^{1} - p_{i j}^{2} + p_{i j}^{4} - p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{1}) + (p_{i j}^{1} + p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) h_{1} (h_{3}; h_{4})}{4 h_{1} (h_{3}; h_{4})}$
(7, 9), (8, 9)	$\frac{\begin{matrix} (p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3} - 2 p_{i j}^{4} + 2 p_{i j}^{5}) (e_{1}; ∆ e_{1}) e_{2} ∆ e_{2} + (p_{i j}^{1} - p_{i j}^{3}) (e_{1}; ∆ e_{1}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} - 2 p_{i j}^{5}) e_{2} ∆ e_{2} (h_{1}; h_{2}) + \\ (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} - 2 p_{i j}^{4} - 2 p_{i j}^{5}) (e_{1}; ∆ e_{1}) h_{3} h_{4} + (p_{i j}^{1} - p_{i j}^{3}) (h_{1}; h_{2}) (e_{2} h_{4} + ∆ e_{2} h_{3}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} + 2 p_{i j}^{5}) (h_{1}; h_{2}) h_{3} h_{4} \end{matrix}}{8 (h_{1}; h_{2}) h_{3} h_{4}}$
(8, 5), (8, 6)	$\frac{(p_{i j}^{2} - p_{i j}^{3} + p_{i j}^{4} - p_{i j}^{5}) (∆ e_{1} (e_{2}; ∆ e_{2})) + (p_{i j}^{2} + p_{i j}^{3} - p_{i j}^{4} - p_{i j}^{5}) (∆ e_{1} (h_{3}; h_{4})) + (p_{i j}^{2} - p_{i j}^{3} - p_{i j}^{4} + p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{2}) + (p_{i j}^{2} + p_{i j}^{3} + p_{i j}^{4} + p_{i j}^{5}) h_{2} (h_{3}; h_{4})}{4 h_{2} (h_{3}; h_{4})}$
(8, 7), (8, 8)	$\frac{(p_{i j}^{1} - p_{i j}^{2} - p_{i j}^{4} + p_{i j}^{5}) (∆ e_{1} (e_{2}; ∆ e_{2})) + (p_{i j}^{1} + p_{i j}^{2} - p_{i j}^{4} - p_{i j}^{5}) (∆ e_{1} (h_{3}; h_{4})) + (p_{i j}^{1} - p_{i j}^{2} + p_{i j}^{4} - p_{i j}^{5}) ((e_{2}; ∆ e_{2}) h_{2}) + (p_{i j}^{1} + p_{i j}^{2} + p_{i j}^{4} + p_{i j}^{5}) h_{2} (h_{3}; h_{4})}{4 h_{2} (h_{3}; h_{4})}$
(9, 1), (9, 3)	$\frac{(p_{i j}^{2} - p_{i j}^{5}) (e_{1} ∆ e_{1}) + (p_{i j}^{2} + p_{i j}^{5}) (h_{1} h_{2})}{2 h_{1} h_{2}}$
(9, 2)	$\frac{(p_{i j}^{1} + p_{i j}^{3} - 2 p_{i j}^{4}) (e_{1} ∆ e_{1}) + (- p_{i j}^{1} + p_{i j}^{3}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + p_{i j}^{3} + 2 p_{i j}^{4}) h_{1} h_{2}}{4 h_{1} h_{2}}$
(9, 4)	$\frac{(p_{i j}^{1} + p_{i j}^{3} - 2 p_{i j}^{4}) (e_{1} ∆ e_{1}) + (p_{i j}^{1} - p_{i j}^{3}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + p_{i j}^{3} + 2 p_{i j}^{4}) h_{1} h_{2}}{4 h_{1} h_{2}}$
(9, 5), (9, 6)	$\frac{\begin{matrix} (- p_{i j}^{1} + 2 p_{i j}^{2} - p_{i j}^{3} + 2 p_{i j}^{4} - 2 p_{i j}^{5}) e_{1} ∆ e_{1} (e_{2}; ∆ e_{2}) + (p_{i j}^{1} - p_{i j}^{3}) (e_{2}; ∆ e_{2}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} - 2 p_{i j}^{4} - 2 p_{i j}^{5}) e_{1} ∆ e_{1} (h_{3}; h_{4}) + \\ (- p_{i j}^{1} + 2 p_{i j}^{2} - p_{i j}^{3} - 2 p_{i j}^{4} + 2 p_{i j}^{5}) (e_{2}; ∆ e_{2}) h_{1} h_{2} + (- p_{i j}^{1} + p_{i j}^{3}) (h_{3}; h_{4}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} + 2 p_{i j}^{5}) h_{1} h_{2} (h_{3}; h_{4}) \end{matrix}}{8 h_{1} h_{2} (h_{3}; h_{4})}$
(9, 7), (9, 8)	$\frac{\begin{matrix} (p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3} - 2 p_{i j}^{4} + 2 p_{i j}^{5}) e_{1} ∆ e_{1} (e_{2}; ∆ e_{2}) + (p_{i j}^{1} - p_{i j}^{3}) (e_{2}; ∆ e_{2}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} - 2 p_{i j}^{4} - 2 p_{i j}^{5}) e_{1} ∆ e_{1} (h_{3}; h_{4}) + \\ (p_{i j}^{1} - 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} - 2 p_{i j}^{5}) (e_{2}; ∆ e_{2}) h_{1} h_{2} + (p_{i j}^{1} - p_{i j}^{3}) (h_{3}; h_{4}) (e_{1} h_{2} + ∆ e_{1} h_{1}) + (p_{i j}^{1} + 2 p_{i j}^{2} + p_{i j}^{3} + 2 p_{i j}^{4} + 2 p_{i j}^{5}) h_{1} h_{2} (h_{3}; h_{4}) \end{matrix}}{8 h_{1} h_{2} (h_{3}; h_{4})}$
Note: ( $e; ∆ e$ ) implies that $e$ is for cell $C_{1}$ and $∆ e$ is for cell $C_{2}$ in the cell arrangement $C_{1}; C_{2}$ .

Controller	Parameter
Linear PD with static decoupling	$K_{P_{11}}$ = 15.03	$K_{D_{11}}$ = 1.02	$K_{P_{22}}$ = 13.01	$K_{D_{22}}$ = 1.31
Linear PD with dynamic decoupling	$K_{P_{11}}$ = 15	$K_{D_{11}}$ = 1	$K_{P_{12}}$ = 7.9	$K_{D_{12}}$ = 0.85
Linear PD with dynamic decoupling	$K_{P_{21}}$ = −19.91	$K_{D_{21}}$ = −1.61	$K_{P_{22}}$ = 13	$K_{D_{22}}$ = 1.3
Fuzzy PD with $R B_{1}$	$h_{1}$ = 0.25	$h_{2}$ = 0.006	$h_{3}$ = 0.44	$h_{4}$ = 0.009
	$a_{11}^{1}$ = 2.24	$a_{11}^{2}$ = 3.54	$a_{11}^{3}$ = 1.28	$a_{11}^{4}$ = 0.98	$a_{11}^{5}$ = 3.96
	$b_{11}^{1}$ = 171.6	$b_{11}^{2}$ = 257.4	$b_{11}^{3}$ = 101.4	$b_{11}^{4}$ = 132.6	$b_{11}^{5}$ = 117
	$a_{22}^{1}$ = 1.62	$a_{22}^{2}$ = 2.61	$a_{22}^{3}$ = 1.89	$a_{22}^{4}$ = 1.44	$a_{22}^{5}$ = 3.24
	$b_{22}^{1}$ = 289.1	$b_{22}^{2}$ = 164.3	$b_{22}^{3}$ = 72.3	$b_{22}^{4}$ = 197.1	$b_{22}^{5}$ = 131.4
	$S_{u_{1}}^{- 1}$ = 14	$S_{u_{2}}^{- 1}$ = 53
Fuzzy PD with $R B_{2}$	$h_{1}$ = 0.216	$h_{2}$ = 0.053	$h_{3}$ = 0.462	$h_{4}$ = 0.02
	$a_{11}^{1}$ = 3.21	$a_{11}^{2}$ = 4.95	$a_{11}^{3}$ = 1.74	$a_{11}^{4}$ = 1.42	$a_{11}^{5}$ = 5.50
	$b_{11}^{1}$ = 19.4	$b_{11}^{2}$ = 29.23	$b_{11}^{3}$ = 8.87	$b_{11}^{4}$ = 15.66	$b_{11}^{5}$ = 17.4
	$a_{21}^{1}$ = −3	$a_{21}^{2}$ = −3.52	$a_{21}^{3}$ = −0.34	$a_{21}^{4}$ = −1.44	$a_{21}^{5}$ = −4.02
	$b_{21}^{1}$ = −615.96	$b_{21}^{2}$ = −107.62	$b_{21}^{3}$ = −32.19	$b_{21}^{4}$ = −14.53	$b_{21}^{5}$ = −52.2
	$a_{12}^{1}$ = 1.01	$a_{12}^{2}$ = 1.44	$a_{12}^{3}$ = 3.12	$a_{12}^{4}$ = 1.95	$a_{12}^{5}$ = 2.22
	$b_{12}^{1}$ = 98.44	$b_{12}^{2}$ = 204.06	$b_{12}^{3}$ = 48.22	$b_{12}^{4}$ = 246.53	$b_{12}^{5}$ = 211.81
	$a_{22}^{1}$ = 1.29	$a_{22}^{2}$ = 2.13	$a_{22}^{3}$ = 1.62	$a_{22}^{4}$ = 1.25	$a_{22}^{5}$ = 2.76
	$b_{22}^{1}$ = 114.8	$b_{22}^{2}$ = 84.38	$b_{22}^{3}$ = 28.7	$b_{22}^{4}$ = 100.16	$b_{22}^{5}$ = 45.06
	$S_{u_{1}}^{- 1}$ = 26	$S_{u_{2}}^{- 1}$ = 53

Controller	$t_{r}$ (s)	$t_{s}$ (s)	$M_{p}$ (%)	Control energy	SSE
Note: $t_{r}$ : rise time, $t_{s}$ : settling time, $M_{p}$ : peak overshoot, SSE: steady-state error
Linear PD (static)	0.09	0.4	8	20.76	0.0061
Linear PD (dynamic)	0.095	0.4	1.65	20.59	0.0068
Fuzzy PD with $R B_{1}$	0.13	0.35	0.4	5.744	0.0022
Fuzzy PD with $R B_{2}$	0.13	0.3	0	15.44	0.001

Controller	Parameter
Linear PI with static decoupling	$K_{P_{11}}$ = 2.7	$K_{I_{11}}$ = 0.04	$K_{P_{22}}$ = 2.5	$K_{I_{22}}$ = 0.01
Linear PI with dynamic decoupling	$K_{P_{11}}$ = 2.7	$K_{I_{11}}$ = 0.04	$K_{P_{12}}$ = 2.1	$K_{I_{12}}$ = 0.008
Linear PI with dynamic decoupling	$K_{P_{21}}$ = −3.292	$K_{I_{21}}$ = −0.05	$K_{P_{22}}$ = 2.5	$K_{I_{22}}$ = 0.012
Fuzzy PI with $R B_{1}$	$h_{1}$ = 90.91	$h_{2}$ = 0.091	$h_{3}$ = 214.29	$h_{4}$ = 0.048
	$a_{11}^{1}$ = 0.1144	$a_{11}^{2}$ = 0.099	$a_{11}^{3}$ = 0.07	$a_{11}^{4}$ = 0.051	$a_{11}^{5}$ = 0.106
	$b_{11}^{1}$ = 135.3	$b_{11}^{2}$ = 202.9	$b_{11}^{3}$ = 79.95	$b_{11}^{4}$ = 104.5	$b_{11}^{5}$ = 92.25
	$a_{22}^{1}$ = 0.091	$a_{22}^{2}$ = 0.133	$a_{22}^{3}$ = 0.112	$a_{22}^{4}$ = 0.042	$a_{22}^{5}$ = 0.182
	$b_{22}^{1}$ = 351.1	$b_{22}^{2}$ = 199.5	$b_{22}^{3}$ = 87.78	$b_{22}^{4}$ = 239.4	$b_{22}^{5}$ = 159.6
	$S_{Δ u_{1}}^{- 1}$ = 0.05	$S_{Δ u_{2}}^{- 1}$ = 0.2
Fuzzy PI with $R B_{2}$	$h_{1}$ = 66.67	$h_{2}$ = 0.231	$h_{3}$ = 187.5	$h_{4}$ = 0.074
	$a_{11}^{1}$ = 0.156	$a_{11}^{2}$ = 0.135	$a_{11}^{3}$ = 0.096	$a_{11}^{4}$ = 0.069	$a_{11}^{5}$ = 0.144
	$b_{11}^{1}$ = 53.24	$b_{11}^{2}$ = 79.86	$b_{11}^{3}$ = 31.46	$b_{11}^{4}$ = 41.14	$b_{11}^{5}$ = 36.3
	$a_{21}^{1}$ = −0.046	$a_{21}^{2}$ = −0.131	$a_{21}^{3}$ = −0.1	$a_{21}^{4}$ = −0.105	$a_{21}^{5}$ = −0.252
	$b_{21}^{1}$ = −58.56	$b_{21}^{2}$ = −89.54	$b_{21}^{3}$ = −128.26	$b_{21}^{4}$ = −210.54	$b_{21}^{5}$ = −108.9
	$a_{12}^{1}$ = 0.07	$a_{12}^{2}$ = 0.1544	$a_{12}^{3}$ = 0.138	$a_{12}^{4}$ = 0.069	$a_{12}^{5}$ = 0.184
	$b_{12}^{1}$ = 380.93	$b_{12}^{2}$ = 140.8	$b_{12}^{3}$ = 312.83	$b_{12}^{4}$ = 190.98	$b_{12}^{5}$ = 317.44
	$a_{22}^{1}$ = 0.104	$a_{22}^{2}$ = 0.152	$a_{22}^{3}$ = 0.128	$a_{22}^{4}$ = 0.048	$a_{22}^{5}$ = 0.208
	$b_{22}^{1}$ = 225.28	$b_{22}^{2}$ = 128	$b_{22}^{3}$ = 56.32	$b_{22}^{4}$ = 153.6	$b_{22}^{5}$ = 102.4
	$S_{Δ u_{1}}^{- 1}$ = 0.05	$S_{Δ u_{2}}^{- 1}$ = 0.35