AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026330214
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İsmail Koray Gündoğan
Kemal Uçak
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RESEARCH ARTICLE

Adaptive MIMO fuzzy PID controller based on Runge-Kutta model

İsmail Koray Gündoğan1* ,  Kemal Uçak2,3*
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1 Department of Mechatronics Engineering, Istanbul Technical University, İstanbul , Türkiye
2 Department of Control and Automation Engineering, Istanbul Technical University, İstanbul , Türkiye
3 Network-based and Data-Driven Control Systems Laboratory, Istanbul Technical University, İstanbul , Türkiye
IJOCTA, 026330214 https://doi.org/10.36922/IJOCTA026330214
Received: 14 August 2026 | Revised: 10 October 2026 | Accepted: 15 October 2026 | Published online: 22 October 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Nonlinear multi-input multi-output (MIMO) systems are difficult to control because of input–output coupling, measurement noise, and parameter variations. This study proposes a Runge–Kutta (RK) model-based predictive adaptive MIMO fuzzy proportional–integral–derivative (PID) controller for nonlinear dynamical systems. A fourth-order Runge–Kutta (RK4) model predicts future outputs and provides the sensitivity information required to update the fuzzy PID scaling parameters online using the Levenberg–Marquardt (LM) algorithm. An additive correction mechanism provides supplementary control action during transient conditions and plant–model mismatch. Furthermore, a discrete-time Lyapunov-based stability analysis is established, guaranteeing the uniform ultimate boundedness (UUB) of the tracking errors and the boundedness of adaptive parameters under aggregate disturbances. The proposed controller is evaluated on a nonlinear three-tank system under nominal conditions, measurement noise, and time-varying parametric uncertainty. For comparative evaluation, an existing RK model-based adaptive MIMO PID controller is employed as a benchmark under the same operating scenarios. Both controllers maintain bounded tracking responses, control inputs, and adaptive parameters. The composite performance-index values obtained with the proposed fuzzy PID controller are 0.2660, 0.2701, and 1.0739, corresponding to improvements of 6.11%, 6.12%, and 3.59%, respectively, over the benchmark controller. In the parametric-uncertainty case, the RK-based parameter estimator follows the variations in the uncertain inter-tank flow coefficient and reduces the resulting plant–model mismatch. Although the benchmark adaptive PID controller has lower computational requirements, the average and maximum cycle times of both controllers remain substantially below the sampling period. The results demonstrate that the proposed adaptive MIMO fuzzy PID controller improves the combined tracking and control-variation performance while retaining computational feasibility.

Keywords
Adaptive control
Levenberg-Marquardt
Lyapunov stability
MIMO fuzzy PID controller
Parameter estimation
Runge–Kutta model
Funding
This study was supported by Scientific and Technological Research Council of Türkiye(TUBITAK) under the Grant Number 125E336. The author thanks to TUBITAK for their supports.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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