Design and Implementation of a Multi-Loop Type 2 Fuzzy PID Control for an Inverted Two-Input-Two-Output System

Abstract

This paper addresses a multi-loop interval Type-2 fuzzy logic proportional-integral-derivative (PID) control for a benchmark first order plus dead time system (FOPDT). In practice, Type-1 FLCs use crisp membership functions, which leads them less effective in modelling uncertainty in real-world systems where data and expert knowledge may be imprecise or noisy. In addition, this also Type-1 FLCs cannot explicitly handle uncertainties in the system dynamics, measurement noise, or modelling errors. This limits their robustness in highly dynamic or uncertain environments. Therefore, to mitigate these above mentioned draw backs of Type 1 FLC, Type 2 FLC is widely preferred. Basically, in Type 2 FLC normally incorporated an additional layer of uncertainty within the membership functions, to make more suitable for uncertain environments. In this work, the Takagi-Sugeno Fuzzy Inference System (FIS) is employed to construct the fuzzy logic system (FLS). This study comprehensively analyzes type-1 and type-2 FLS performances for a benchmark multivariable system. The simulation results reveal that the type-2 FLS outperforms the type-1 FLS and conventional PID controller. Also, to exhibit a concrete analysis, a performance analysis table has been incorporated by considering different performance indices to provide a clear visualization of output performances. The stability analysis of the system has also been described by considering the frequency domain analysis.

Keywords

multi-loop system fuzzy logic systems decoupling decentralized control

Get full access to this article

View all access options for this article.

References

Al-Fandi

Jaradat

M. A.

Sardahi

(2012). Optimal pid-fuzzy logic controller for type 1 diabetic patients. In 2012 8th International Symposium on Mechatronics and its Applications, pp. 1–7, IEEE.

Aliasghary

Eksin

Güzelkaya

Kumbasar

(2012). Design of an interval type-2˙ fuzzy logic controller based on conventional pi controller. In 2012 20th Mediterranean Conference on Control & Automation (MED), pp. 627–632, IEEE.

Cantera-Cantera

L. A.

Maya-Rodríguez

M. C.

Palomino-Resendiz

S. I.

Lozano-Hernández

Luna

(2025). Level and flow systems identification of an industrial processes module by LSOD method for PID controllers design. Results in Engineering, 25, 104347. https://doi.org/10.1016/j.rineng.2025.104347

Castillo

Amador-Angulo

Castro

J. R.

Garcia-Valdez

(2016). A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems. Information Sciences, 354, 257–274. https://doi.org/10.1016/j.ins.2016.03.026

Castillo

Melin

(2014). A review on interval type-2 fuzzy logic applications in intelligent control. Information Sciences, 279, 615–631. https://doi.org/10.1016/j.ins.2014.04.015

Castro

J. R.

Castillo

Melin

(2007). An interval type-2 fuzzy logic toolbox for control applications. In 2007 IEEE international fuzzy systems conference, pp. 1–6, IEEE.

Cervantes

Caraveo

Castillo

(2023). Performance comparison between type-1 and type-2 fuzzy logic control applied to insulin pump injection in real time for patients with diabetes. Mathematics, 11(3), 730. https://doi.org/10.3390/math11030730

Chai

Zhou

Cheng

Jia

Song

(2024, November). Industrial metaverse-based intelligent PID optimal tuning system for complex industrial processes. IEEE Transactions on Cybernetics, 54(11), 6458–6470. https://doi.org/10.1109/TCYB.2024.3386669

Dereli

Baykasoglu

Altun

Durmusoglu

Türksen

I. B.

(2011). Industrial applications of type-2 fuzzy sets and systems: A concise review. Computers in Industry, 62(2), 125–137. https://doi.org/10.1016/j.compind.2010.10.006

10.

Dey

Banerjee

Dey

(2024, September). Frequency-Domain tuning of a robust optimal 2-DOF fractional order PID controller for a maglev system. IEEE Transactions on Industrial Informatics, 20(9), 11348–11361. https://doi.org/10.1109/TII.2024.3403249

11.

Ying

(2010). Derivation and analysis of the analytical structures of the interval type-2 fuzzy-pi and pd controllers. IEEE Transactions on Fuzzy Systems, 18(4), 802–814. https://doi.org/10.1109/TFUZZ.2010.2049022

12.

Duddeti

B. B.

Naskar

A. K.

(2025). Model dominance index assisted enhanced generalized pole clustering technique for order reduction and application to PID controller design. Circuits, Systems, and Signal Processing, 44, 1–34. https://doi.org/10.1007/s00034-025-02998-x

13.

El-Bardini

El-Nagar

A. M.

(2014). Interval type-2 fuzzy PID controller: Analytical structures and stability analysis. Arabian Journal for Science and Engineering, 39, 7443–7458. https://doi.org/10.1007/s13369-014-1317-y

14.

Karnik

N. N.

Mendel

J. M.

(2001). Centroid of a type-2 fuzzy set. Information Sciences, 132(1-4), 195–220. https://doi.org/10.1016/S0020-0255(01)00069-X

15.

Kumbasar

Hagras

(2015). Interval type-2 fuzzy PID controllers. In Springer handbook of computational intelligence (pp. 285–294). Springer.

16.

Liu

Qiu

Mao

(2025). The PID high-type controller based on LQR and model-assisted LESO. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 239, 74–89. https://doi.org/10.1177/09596518241263538

17.

Mendel

J. M.

(2014). General type-2 fuzzy logic systems made simple: A tutorial. IEEE Transactions on Fuzzy Systems, 22(5), 1162–1182. https://doi.org/10.1109/TFUZZ.2013.2286414

18.

Mendel

Hagras

Tan

W.-W.

Melek

W. W.

Ying

(2014). Introduction to type-2 fuzzy logic control: theory and applications. John Wiley & Sons.

19.

Mittal

Jain

Vaisla

K. S.

Castillo

Kacprzyk

(2020). A comprehensive review on type 2 fuzzy logic applications: Past, present and future. Engineering Applications of Artificial Intelligence, 95, 103916. https://doi.org/10.1016/j.engappai.2020.103916

20.

Nie

Z. Y.

Wang

Q. G.

Gao

Luo

(2022). A unifying Ziegler–Nichols tuning method based on active disturbance rejection. International Journal of Robust and Nonlinear Control, 32(18), 9525–9541. https://doi.org/10.1002/rnc.5848

21.

Ningsih

Suharti

P. H.

Ghani

(2024). Performance analysis of IMC-PID controller on PCT-14 air pressure control system. MethodsX, 13, 102966. https://doi.org/10.1016/j.mex.2024.102966

22.

Nishanth

F. P.

Dash

S. K.

Mahapatro

S. R.

(2024). Critical study of type-2 fuzzy logic control from theory to applications: A state-of-the-art comprehensive survey. e-Prime-Advances in Electrical Engineering, Electronics and Energy 10, 100771. https://doi.org/10.1016/j.prime.2024.100771

23.

Prusty

S. B.

Pati

U. C.

Mahapatra

K. K.

(2015). A novel fuzzy based adaptive control of the four tank system. In Proceedings of the 2015 Third International Conference on Computer, Communication, Control and Information Technology (C3IT), pp. 1–6, IEEE.

24.

Raj

Mohan

(2020). General structure of interval type-2 fuzzy pi/pd controller of takagi–sugeno type. Engineering Applications of Artificial Intelligence, 87, 103273. https://doi.org/10.1016/j.engappai.2019.103273

25.

Sain

Praharaj

Mohan

B. M.

Yang

J. M.

(2025). Interval type-2 fuzzy PID controllers with interval of confidence and various types of footprints of uncertainty. Information Sciences, 699, 121795. https://doi.org/10.1016/j.ins.2024.121795

26.

Sakalli

Kumbasar

Mendel

J. M.

(2021). Towards systematic design of general type-2 fuzzy logic controllers: Analysis, interpretation, and tuning. IEEE Transactions on Fuzzy Systems, 29(2), 226–239. https://doi.org/10.1109/TFUZZ.2020.3016034

27.

Sharma

Kumar

(2022). Optimal interval type-2 fuzzy logic control based closed-loop regulation of mean arterial blood pressure using the controlled drug administration. IEEE Sensors Journal, 22(7), 7195–7207. https://doi.org/10.1109/JSEN.2022.3151831

28.

Shi

(2022). A unified general type-2 fuzzy pid controller and its comparative with type-1 and interval type-2 fuzzy pid controller. Asian Journal of Control, 24(4), 1808–1824. https://doi.org/10.1002/asjc.2578

29.

Utami

A. R.

Yuniar

R. J.

Giyantara

Saputra

A. D.

(2022). Cohen-Coon PID tuning method for self-balancing robot. In 2022 International Symposium on Electronics and Smart Devices (ISESD).

30.

(2012). On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers. IEEE Transactions on Fuzzy Systems, 20(5), 832–848. https://doi.org/10.1109/TFUZZ.2012.2186818

31.

Mendel

J. M.

(2010). On the continuity of type-1 and interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems, 19(1), 179–192. https://doi.org/10.1109/TFUZZ.2010.2091962

32.

Tan

W. W.

(2010). Interval type-2 fuzzy pi controllers: Why they are more robust. In 2010 IEEE International Conference on Granular Computing, pp. 802–807, IEEE.

33.

Zadeh

L. A.

(1975). The concept of a linguistic variable and its application to approximate reasoning-iii. Information Sciences, 9(1), 43–80. https://doi.org/10.1016/0020-0255(75)90017-1

34.

Zhou

Zhang

Tan

Dai

Duan

(2021). Design of the footprints of uncertainty for a class of typical interval type-2 fuzzy pi and pd controllers. ISA Transactions, 108, 1–9. https://doi.org/10.1016/j.isatra.2020.08.009