RELIABILITY ANALYSIS OF FRACTIONAL - ORDER FUZZY PID CONTROLLERS

JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 18, issue 01, 2025 ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431 www.wordlyknowledge.uz ILMIY METODIK JURNAL Sodiqov Baxtiyor Qahhor o‘g‘li, Phd Student of Navoiy state university of mining and technologies Phone number:+998913365510 Email: [email protected] RELIABILITY ANALYSIS OF FRACTIONAL - ORDER FUZZY PID CONTROLLERS ABSTRACT: This paper presents a comprehensive reliability analysis of Fractional-Order Fuzzy PID (FOFPID) controllers applied to dynamic and nonlinear industrial systems. The FOFPID controller combines the advantages of fractional calculus and fuzzy logic to improve control precision, adaptability, and robustness under uncertain operating conditions. In this study, the reliability indicators of the controller—such as mean time between failures (MTBF), fault tolerance, stability margin, and sensitivity to parameter variations—are analyzed both theoretically and through simulation. The proposed reliability model is verified using MATLAB/Simulink environments for different types of dynamic plants, including first-order and higher-order nonlinear processes. The obtained results demonstrate that the FOFPID controller maintains high performance stability even under component degradation and stochastic disturbances, outperforming classical PID and FOPID structures. KEYWORDS: Fractional-Order Fuzzy PID, reliability analysis, fault tolerance, robustness, intelligent control, nonlinear systems. INTRODUCTION The reliability of control systems has become a critical factor in modern industrial automation, where continuous operation and safety are essential. Traditional PID controllers often fail to provide high reliability when exposed to nonlinear dynamics, parameter uncertainty, or sensor noise. To overcome these challenges, the Fractional-Order Fuzzy PID (FOFPID) controller has emerged as a promising alternative. This controller integrates fractional calculus operators with fuzzy logic inference, allowing for better adaptability, robustness, and long-term stability. Fractional differentiation offers a more accurate dynamic memory effect, while fuzzy logic ensures smooth control under uncertain or noisy environments. However, the reliability performance of FOFPID controllers has not been comprehensively analyzed in many studies. This research focuses on evaluating reliability indicators such as stability margins, mean time between failures, and fault tolerance of FOFPID systems. Using simulation in MATLAB/Simulink, comparative analyses with classical PID and FOPID controllers are carried out. The results demonstrate that FOFPID controllers provide enhanced operational reliability and reduced failure probability, making them suitable for critical industrial applications like cryogenic air separation, milling processes, and energy-efficient automation systems. METHODOLOGY The reliability of the Fractional-Order Fuzzy PID (FOFPID) controller was analyzed using MATLAB/Simulink simulations on a nonlinear process model. The controller integrated fractional calculus operators and fuzzy logic rules for adaptive parameter tuning. Reliability indicators such as stability margin, JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 18, issue 01, 2025 ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431 www.wordlyknowledge.uz ILMIY METODIK JURNAL mean time between failures (MTBF), and fault tolerance were evaluated. Comparative results with classical PID and FOPID controllers showed that the FOFPID achieved higher reliability, reduced overshoot, and improved robustness under noise and disturbances. The results confirm that FOFPID controllers provide superior reliability and stability for industrial automation applications. RESULTS To quantitatively assess the performance and reliability of the Fractional-Order Fuzzy PID (FOFPID) controller, two major statistical indicators were used: the Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE). These indices provide a clear measure of how closely the controller’s output follows the desired reference signal and how effectively it handles dynamic disturbances. The RMSE evaluates the magnitude of deviation between the desired output ( ) and the actual system output ( ). It gives greater weight to large errors, which is useful for identifying instability or poor tuning. The mathematical expression is: The MAE, on the other hand, measures the average of absolute differences between the desired and actual outputs, providing a more balanced assessment of general control accuracy: In the conducted simulations, the FOFPID controller achieved lower RMSE and MAE values compared to classical PID and FOPID controllers. Specifically, RMSE decreased by approximately 20%, and MAE by 15%, indicating improved precision and smoother control actions. These reductions confirm the enhanced reliability and adaptability of the FOFPID design under noise and parameter uncertainties. The fuzzy inference mechanism minimized transient errors, while fractional calculus provided finer dynamic adjustment. Overall, the results validate that FOFPID controllers ensure higher stability, accuracy, and operational reliability in complex industrial control systems. Initially, we open MATLAB 2022 program because, this application is very useful for simulations on sphere automation, modelling and simulation on chemical or mining technological processes. There SIMULINK package for advanced simulation. For FOFPID controller we had created specialized model and we used transfer function, step, scope, summator and this kind of functional blocks. Results of model is given in pictures below. JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 18, issue 01, 2025 ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431 www.wordlyknowledge.uz ILMIY METODIK JURNAL Fig.1. Simulation results of FOFPID controller Figure 1 illustrates the dynamic responses of PID, FOPID, and FOFPID controllers compared to the reference signal. The FOFPID output closely follows the reference curve with minimal overshoot and faster settling time, confirming its superior accuracy and robustness. Figure 2 presents a comparative analysis of RMSE and MAE values for the three controllers. The FOFPID controller achieved the lowest RMSE and MAE, indicating significantly reduced steady-state and transient errors. These results demonstrate that combining fractional-order calculus with fuzzy logic improves reliability, enhances stability margins, and increases tolerance to disturbances and parameter variations. Consequently, the FOFPID controller provides better overall performance and reliability than conventional PID and FOPID controllers in industrial control applications. CONCLUSION This study presented a comprehensive reliability analysis of the Fractional-Order Fuzzy PID (FOFPID) controller in comparison with classical PID and FOPID controllers. Simulation results demonstrated that the FOFPID controller achieved the lowest RMSE and MAE values, indicating superior accuracy and reduced error dynamics. The integration of fractional calculus and fuzzy logic significantly improved system robustness, stability margins, and fault tolerance under varying operational conditions. Furthermore, the FOFPID controller maintained consistent performance despite noise, parameter uncertainty, and nonlinear disturbances. These findings confirm that the proposed FOFPID design ensures higher reliability and adaptability, making it a promising solution for safety-critical industrial processes such as cryogenic air separation and ore grinding systems. Future work will focus on implementing realtime experimental validation and hardware-in-the-loop testing to further evaluate its reliability in practical environments. REFERENCES 1. Padula, F., & Visioli, A. (2011). Tuning rules for optimal PID and fractional-order PID controllers. Journal of process control, 21(1), 69-81. 2. Lachhab, N., Svaricek, F., Wobbe, F., & Rabba, H. (2013, July). Fractional order PID controller (FOPID)-toolbox. In 2013 European control conference (ECC) (pp. 3694-3699). IEEE. 3. Zamani, M., Karimi-Ghartemani, M., Sadati, N., & Parniani, M. (2009). Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Engineering Practice,17(12), 1380-1387.. 4. Hamamci, S. E. (2007). An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Transactions on Automatic Control,52(10), 1964-1969. JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 18, issue 01, 2025 ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431 www.wordlyknowledge.uz ILMIY METODIK JURNAL 5. Soukkou, A., Belhour, M. C., & Leulmi, S. (2016). Review, design, optimization and stability analysis of fractional-order PID controller. International Journal of Intelligent Systems and Applications, 8(7), 73.