A Backstepping Method for Controlling A Nonlinear System of Fuzzy Delay Differential Equations Saja Faeq Noaman (1) , Mustafa Wassef Hasan (1) , Shaimaa Shukri Abd.Alhalim (1,2) and Rusul Khalid Abdulsattar (1) saja.f.noaman@uotechnology.edu.iq, mustafa.w.hasan@uotechnology.edu.iq, shaimaa.s.abdalhalim@uotechnology.edu.iq, rusul.k.abdulsattar@uotechnology.edu.iq 1: Department of Electrical Engineering, University of Technology- Iraq, Baghdad 2: University of Sfax, ENIS, Laboratory of Control & Energy Management (CEM-Lab), Sfax, Tunisia Abstract— This paper presents a control strategy for a nonlinear system of fuzzy delay differential equations (FDDEs) using the generalizable backstepping method (GBSM). Despite the inherent challenges posed by uncertain parameters and time delays, this method successfully achieves system stability at specific time intervals using the general functions of control which have been created based on the Lyapunov function. Additionally, the method of successive integrations is employed to transform the original FDDEs system into an equivalent system of fuzzy differential equations (FDEs), facilitating the application of the GBSM. Finally, substitution the control functions in the transformed system (FDEs) in final step that leads to a new system is an exponentially stable. As a result, the proposed control strategy for stabilizing FDDEs has been effectively demonstrated by numerical simulation (MATLAB and Mathcad platforms). Keywords— control problems, backstepping method, fuzzy system, method of successive integrations I. INTRODUCTION In the field of science and technology, most mathematical models can be used to express a lot of dynamic real-world problems. Some of them may be expressed as ordinary differential equations or partial differential equations. A particular type of ordinary differential equation is delay differential equations where the derivative of an unknown function at a specific time is in terms of the function's values at previous times. This form of equation represents a significant and crucial category of dynamical systems that frequently arise in either natural or artificial control problems [1]. It is well known that delay differential equations (DDEs) have been extensively covered in the literature, focusing on functional equations and their practical applications. At the same time, stability is considered one of the applications of differential equations. It has a significant effect in both theoretical and practical contexts. Modelling real-life problems normally includes variations or fuzziness in some factors. In this context, the evolvement of fuzzy theory has played an important role in addressing this issue by facilitating the modelling of uncertain systems [2]. Recently, many authors have worked to control the stability of fuzzy differential equations. For instance, In 2005 Le introduced essential conditions for stability and asymptotically stable of FDDEs [3]. While Mizukoshi developed the fuzzy problem with initial values characterized by fuzzy sets [4]. Furthermore, the concept of fuzzy equilibrium stability was also discussed. Wang, Qiu, Gao and Wang [5] worked on developing a fuzzy control for nonlinear industrial processes based on network technology and distributed fuzzy H-infinity filtering problems non-linear multirate networked with double-layer discovered by Wang, Qiu and Fu [6] via fuzzy modelling technique [7]. Lupulescu addressed results on existence and uniqueness [8]. The studies are guided by Liu process. Subsequently, numerous works have explored fuzzy delay differential equations in various domains [9]. A backstepping method is considered one of the most commonly used methods in the control problems, BSM is a systematic and iterative design approach for nonlinear systems, that provides the option to accommodate unmodelled nonlinear effects and parameter uncertainties. The concept of backstepping design involves recursively suitable functions of state variables in the form of pseudo- control inputs for low-dimensional subsystems inside the overarching system. Hence, each stage of backstepping results in a new pseudo-control design, stated by using the pseudo-control design from previous design phases. The procedure finishes a feedback design for the real control input which realizes the original design goal by a final Lyapunov function which is formed by summing the Lyapunov functions related to each design stage. 979-8-3315-4272-6/25/$31.00 ©2025 IEEE 2025 IEEE 22nd International Multi-Conference on Systems, Signals & Devices (SSD) 794 2025 IEEE 22nd International Multi-Conference on Systems, Signals & Devices (SSD) | 979-8-3315-4272-6/25/$31.00 ©2025 IEEE | DOI: 10.1109/SSD64182.2025.10989857 Authorized licensed use limited to: Infineon Technologies AG. Downloaded on June 02,2025 at 16:30:58 UTC from IEEE Xplore. Restrictions apply.