Fractional-order Control Research Articles

Fractional-order self-tuning high-gain feedback for a class of linear systems

This paper presents a new fractional-order self-tuning high-gain control approach designed to stabilize a class of linear time-invariant (LTI) systems with minimum phase and relative degree one. The key contribution of this work lies in the incorporation of a fractional-order integrator into the adaptation law, which ensures that the adaptation gain ϕ(t) converges to a desired stabilizing value. This method addresses a critical limitation of classical high-gain adaptive controllers, where the gain increases monotonically, potentially leading to actuator saturation. The stability of the proposed control strategy is rigorously established by extending the classical high-gain lemma, accompanied by a detailed mathematical analysis. To validate the proposed method, we conduct numerical simulations that demonstrate its ability to achieve stable behavior. The results provide strong evidence of the efficacy of the proposed fractional-order self-tuning high-gain controller in ensuring stability for this class of systems while avoiding the risks associated with conventional high-gain strategies.

Promoted load frequency control system using optimal multi-objective design of fractional-order interval type-II fuzzy controller

Promoted load frequency control system using optimal multi-objective design of fractional-order interval type-II fuzzy controller

Adaptive fractional order sliding mode control for dive plane manoeuvring of supercavitating vehicles

ABSTRACT In this paper, the vehicle's dynamical behaviour is first described through a bifurcation analysis to give some insights for its robust control synthesis. Then a novel adaptive fractional-order sliding mode controller (AFOSMC) is designed to effectively control the uncertain supercavitating vehicle against payload changes, nonlinear planing force and external disturbances. The control scheme utilises the fractional calculus and the adaptive law for the sliding mode control (SMC) to provide active control actions. The fractional calculus can guarantee more flexibility and more freedom for tuning active control synthesis than the integer-order counterpart. The adaptation law has been presented to directly handle the payload change effects. The dynamic performance of the proposed controller is verified through extensive simulation results, which demonstrate faster responses than existing integer order controllers. Furthermore, the fractional order controller shows its effectiveness in dealing with external disturbance, when eliminating most of the uncertainty effect.

Fixed-Time Adaptive Synchronization of Fractional-Order Memristive Fuzzy Neural Networks with Time-Varying Leakage and Transmission Delays

Finite-time synchronization depends on the initial conditions of the system in question. However, the initial conditions of an actual system are often difficult to estimate or even unknown. Therefore, a more valuable and pressing problem is fixed-time synchronization (FTS). This paper addresses the issue of FTS for a specific class of fractional-order memristive fuzzy neural networks (FOMFNNs) that include both leakage and transmission delays. We have designed two distinct discontinuous control methodologies that account for these delays: a state feedback control scheme and a fractional-order adaptive control strategy. Leveraging differential inclusion theory and fractional-order differential inequalities, we derive several novel algebraic conditions that are independent of delay. These conditions ensure the FTS of drive–response FOMFNNs in the presence of leakage and transmission delays. Additionally, we provide an estimate for the upper bound of the settling time required to achieve FTS. Finally, to validate the feasibility and applicability of our theoretical findings, we present two numerical examples which are accompanied by simulations.

Reducing power ripple for multi-rotor wind energy systems using FOPDPI controllers

Energy systems based on wind energy have now become a necessity due to their many advantages. However, using traditional turbines has several disadvantages, the most notable of which is the low performance and quality of the resulting energy. This work proposes an energy system based on the use of a multi-rotor wind turbine (MRWT) with a double-fed induction generator. In addition, to obtain high-quality power, a new control is used based on the development and modification of direct power control (DPC) based on proportional-integral (PI) controllers. This designed approach has many advantages, such as fast dynamic response, easy adjustment, high robustness, and ease of implementation. In this designed technique, the PI controller is dispensed with, and they are replaced by the designed fractional-order proportional-integral proportional derivative controller. This designed technique is based on power estimation, where the same estimation equations used in the DPC-PI technique are used. Robustness, high competence, and effectiveness in improving power quality are among the most prominent features of the designed technique compared to the DPC-PI approach and some research works. The designed technique was verified using MATLAB, comparing the results obtained in the case of different wind speed profiles with those using the DPC-PI approach. Numerical results demonstrated the efficacy of the designed technique over the DPC-PI approach, as the steady-state error and active power ripples were minimized in the first test by rates estimated at 44.72% and 46.67%, respectively. Also, the reactive power overshoot and ripples were reduced in the fifth test by rates calculated at 81.40% and 42.18%, respectively. On the other hand, the total harmonic distortion of the current was minimized by rates estimated in the five suggested tests at 33.80%, 34.88%, 11.52%, 37.58%, and 33.33%. These percentages show the strength of the performance and effectiveness of the designed technique in enhancing the robustness and efficiency of the MRWT system, making it a promising solution.

Design, Tuning, and Experimental Validation of Switched Fractional-Order PID Controllers for an Inverted Pendulum System

Stabilizing inverted pendulum systems remains a challenging and open control problem due to their inherent instability and relevance in a wide range of real-world applications, including robotics and aerospace systems. While PID and fractional-order PID (FOPID) controllers offer distinct advantages, they individually suffer from trade-offs between performance and control energy. This paper presents the design, implementation, and experimental validation of a switched SW FOPID-PID controller for the stabilization of an inverted pendulum (InvP) system, aiming to achieve an improved balance between system performance and control energy used. The controller was tuned offline using particle swarm optimization (PSO) and a mathematical model of the system for simulation. Additional PID and FOPID controllers were also designed, tuned and validated for comparison purposes. Their performance was assessed through key indicators, including ITAE, ISI, settling time, peak values, and variance and compared against a manufacturer-provided PID controller. The experimental results demonstrated that all three designed controllers outperformed the manufacturer’s PID under nominal conditions. The SW FOPID-PID controller achieved the best overall performance, balancing control energy efficiency and response quality. Under external disturbances, the FOPID and SW FOPID-PID controllers exhibited superior robustness, with the switched controller being the most effective, responding quickly to disturbances while minimizing positional and angular errors. Still, this research is limited to a specific plant and switching strategy; thus, further validation on other systems and switching criteria is necessary to generalize these findings.

Enhanced frequency control of a hybrid microgrid integrated with EV aggregator using [FOI-(PDN+1)] controller optimized by osprey optimization algorithm

Abstract Progressively increasing power demand and increasing penetration of renewable energy resources (RESs) have led to decreased system inertia that in turn increased frequency fluctuation in power system and resulted into increased focus on load frequency control (LFC). Different controllers are proposed in recent time to improve the LFC of microgrid. This paper proposes a novel fractional order integral minus proportional derivative with filter plus one [FOI-(PDN+1)] controller to minimize frequency deviation of the microgrid containing hydro-thermal power plant with photovoltaic and wind power generation. This controller provides more flexibility due to having fractional order control and also provides improved dynamics. To compensate the frequency deviation caused by load disturbance and RES variation, fast acting energy storage system capacitive energy storage (CES) and electric vehicle aggregators are employed. Multiple nonlinearities such as generation rate constraints, governor dead band, communication time delay, boiler dynamic are also taken into consideration. The proposed controller parameters are tuned by a bio-inspired metaheuristic algorithm namely osprey optimization algorithm. Sensitivity of the proposed controller is inspected at different scenarios considering load perturbations, wide variation of system parameters, as well as different weather conditions. Robustness of the proposed controller is validated on Typhoon-HIL based real time emulator and the resilience of the proposed controller is investigated for LFC during denial-of-service attack. The proposed controller is effective in controlling the frequency deviation under distinct load change by stabilizing it 2.5 to 4 time faster compared to conventional, CES unit and 1.5 to 2.5 time faster compared to 2DOF-FOPID, 3DOF-FOPID and FOPTID+1. The result establishes the superiority of the studied approach.

Heading control of USV based on fractional-order model predictive control

Heading control of USV based on fractional-order model predictive control

Optimized fractional-order PID controller for temperature control of CSTH: a comparative study with PID cascade controller

Optimized fractional-order PID controller for temperature control of CSTH: a comparative study with PID cascade controller

Enhanced Stability and Performance of Islanded DC Microgrid Systems Using Optimized Fractional Order Controller and Advanced Energy Management

ABSTRACTThe increasing demand for electrical power is placing heavy pressure on the power system transmission network. To reduce this burden and conversion losses, a distributed generation‐based DC microgrid system is favorable due to its flexibility and reliability. However, traditional control approaches for the power grid are impractical for islanded microgrids, making stability a primary concern. Using a conventional PI controller results in significant deviations in DC bus voltage and prolonged settling times. Additionally, conventional techniques fail to ensure adequate current sharing between the battery and supercapacitor, which are employed to control the DC bus voltage. To address these issues, this study proposes the use of an optimized fractional order PI (FOPI) controller and an efficient energy management algorithm. The FOPI controller regulates processes while meeting constraints, with the control rule generated from minimizing a cost function. A complete mathematical model of the DC microgrid is developed along with the proposed controller to facilitate detailed analyses. The developed energy management system (EMS) ensures energy balance within the microgrid with regulated DC bus voltage and state‐of‐charge (SoC) limits. Comparing the performance of the FOPI controller with that of the conventional PI controller reveals significant improvements: the DC bus voltage deviation is reduced by 63.63%, and the settling time is 16.67% faster during PV variation. During load variation, the maximum DC bus voltage deviation is 2.25 V, and the settling time is 202 ms, which are within acceptable ranges.

Is it Curbing-spread of SARS-CoV-2 Variants by Considering Non-linear Predictive Control?

Although SARS-COV-2 started in 2019, its losses are still significant, and it takes victims. In the present study, the epidemic patterns of SARS-COV-2 disease have been investigated from the point of view of mathematical modeling. Also, the effect of quarantine has been considered. This mathematical model is designed in the form of fractional calculations along with a model predictive control (MPC) to monitor this model. The fractional-order model has the memory and hereditary properties of the system, which can provide more adjustable parameters to the designer. Because the MPC can predict future outputs, it can overcome the conditions and events that occur in the future. The results of the simulations show that the proposed nonlinear model predictive controller (NMPC) of fractional-order has a lower mean squared error in susceptible people compared to the optimal control of fractional-order (~3.6e-04 vs. 47.4). This proposed NMPC of fractional-order can be used for other models of epidemics.

Fractional-Order Modeling and Control of HBCS-MG in Off-Grid State

Half-bridge converter series microgrid (HBCS-MG) is susceptible to a variety of uncertainties and disturbances during operation, and therefore, the use of the traditional integer-order models cannot accurately reflect the effects of environmental variations on internal components of the off-grid system, such as converters, filters, and loads, including factors like time delays, memory effects, and multi-scale coupling. The fractional-order control method is better equipped to deal with these disturbances, thereby enhancing the robustness and stability of the system. In the off-grid state, a fractional-order PI (FOPI) controller is employed for double-closed-loop control, and the load voltage feedforward control is utilized to offset the impact of load voltage fluctuations on the system. A new simplified equivalent circuit calculation method for the fractional-order inductor is proposed, and a complete fractional mathematical model of the system in the dq rotating coordinate system is established to obtain the transfer function between the load voltage and the input voltage. Furthermore, the impact of the fractional-order variation of the FOPI controllers and the fractional elements on system performance in the frequency domain and time domain is described in detail. The simulation results are compared with the theoretical analysis to demonstrate the accuracy of the mathematical model. The overshoot of the load voltage at the switching instant of 0.7 s is reduced by 4.2% compared with the integer-order PI controller, which proves that the fractional-order controller can improve the system control accuracy.

A relative analysis to cascaded fractional-order controllers in microgrid non-minimum phase converters using EHO

Microgrids integrate various distributed energy resources to enhance energy reliability and sustainability. Power electronic converters are vital in microgrids since they provide efficient, reliable, and flexible operation. There are numerous controllers available that can be applied to these converters, and lately, fractional-order controllers (FOC) have gathered huge recognition. These controllers provide enhanced flexibility and superior performance in managing dynamic behavior. There are various structures of FOCs, and this article predominantly focuses on comparing different cascaded fractional order controllers (C-FOC). Four distinct topologies of cascaded fractional order proportional integral (C-FOPI) controllers are selected for comparison with one another and with the cascaded proportional integral controller used in a non-minimum phase converter, such as the boost converter employed in a microgrid system. The controllers are optimized using the Elephant Herd Optimization (EHO) algorithm with the Integral of Time-weighted Absolute Error (ITAE) serving as the performance metric. Each controller is subject to variation in system changes, and the outcomes are documented and correlated to ascertain the optimum structure. The simulation outcomes endorsed notable advancements in terms of transient and steady-state performance, featuring improved resilience to parameter changes, a reduction of 36.6% in settling time, 15% in overshoot, 20.1% in rise time, an improved phase margin of more than 51% and more than 50% reduction in performance indices compared to traditional cascaded proportional integral controllers (PI-PI).

Robust nonlinear fractional-order state feedback control for variable-speed wind turbines using PSO algorithm

Abstract This paper designs a robust nonlinear fractional-order state feedback control (NFOSFC) method for variable-speed wind turbines. This method’s control law is constructed from the fractional-order two-mass model. The use of the fractional-order two-mass model is justified by its improved flexibility in modeling the complex dynamics of the wind turbine, especially for handling flexible modes induced by low shaft stiffness at low speeds. To ensure the best performance of the NFOSFC, the Particle Swarm Optimization (PSO) algorithm is used to extract the optimal control parameters. Comparative studies with some existing control strategies demonstrate that the proposed NFOSFC scheme significantly improves the system performance, offering smoother transient responses and improved power capture and load mitigation efficiency.

REVISITING BODE’S IDEAL LOOPS: INTEGRAL SQUARE ERROR OPTIMALITY OF BODE’S IDEAL LOOPS AND BODE’S IDEAL LOOP INVERSE CONTROLLER DESIGN

Bode’s ideal loop models have been utilized in control system designs to obtain performance robustness for the Direct Current (DC) gain variations. However, effects of crossover frequency and fractional order on control optimality of Bode's ideal loop reference models have not been sufficiently discussed, and it may raise a question of whether a control system design based on Bode’s ideal loop reference model is optimal. In this regard, this study revisits Bode’s ideal loops to investigate Integral Square Error (ISE) optimality of Bode’s ideal loops. For this purpose, the ISE optimality of Bode’s ideal loop is theoretically investigated by using Parseval’s theorem, and an ISE optimality condition is suggested in terms of crossover frequency and fractional order. Then, a generic controller family, which achieves an exact matching with the characteristics of the ISE optimal Bode’s ideal loop, was implemented for inverse fractional-order control of a class of minimum phase fractional order plant models. The authors present a numerical experiment on an ISE optimal Bode’s ideal loop inverse (OBILI) controller, and the ISE optimality and performance robustness of the designed control system were investigated for closed-loop control of a heat furnace system model via Monte Carlo simulations.

Barrier Lyapunov-based fractional-order sliding-mode direct yaw-moment constrained control design for four-wheel-independent-drive EVs

Barrier Lyapunov-based fractional-order sliding-mode direct yaw-moment constrained control design for four-wheel-independent-drive EVs

Design tools to stabilize and to synchronize fractional-order energy resources system based on fractional-order control approaches: a review

Design tools to stabilize and to synchronize fractional-order energy resources system based on fractional-order control approaches: a review

Trajectory tracking of 2DOF robotic manipulator system based on adaptive exponential forgetting recursive least squares fractional-order PID control

Trajectory tracking of 2DOF robotic manipulator system based on adaptive exponential forgetting recursive least squares fractional-order PID control

Empirical modelling, simulation and control of coffee brewing

Coffee brewing is a complex process governed by various parameters, such as water temperature, extraction time, and particle size of ground coffee. These parameters affect the quality of brewed coffee in terms of flavour and intensity of aromatic compounds. This research focused on studying the effect of water temperature on caffeine concentration in brewed coffee. This was done by varying the water temperature (80°C, 85°C and 90°C) in infusion coffee brewing while keeping other parameters such as coffee type (Arabica), coffee particle size (500 µm), and coffee-to-water ratio constant. The results showed that the concentration of caffeine in the brewed coffee increased as the water temperature increased, with the highest concentration observed at 90°C with 210.58 g/L. An empirical model, represented as a first-order model with time delay (FOPTD), was developed to study the effect of temperature in the brewing process. The model was then simulated and controlled using a Proportional Integral (PI) controller tuned using different methods. Performance and robustness analysis were conducted, and the best results were obtained using the fractional-order controller with an Integral of Absolute Error (IAE) of 64.67. In conclusion, the effect of water temperature on caffeine concentration during coffee brewing was determined, and a brewing model at different water temperatures was developed and simulated with different tuning rules for the PI controller.

Investigation of fractional order model for glucose-insulin monitoring with PID and controllability

The global prevalence of diabetes, a chronic condition that disrupts glucose homeostasis, is rapidly increasing. Patients with diabetes face heightened challenges due to the COVID-19 pandemic, which exacerbates symptoms associated with the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection. In this study, we developed a mathematical model utilizing the Mittag–Leffler kernel in conjunction with a generalized fractal fractional operator to explore the complex dynamics of diabetes progression and control. This model effectively captures the disease’s inherent memory effects and delayed responses, demonstrating improved accuracy over traditional integer-order models. We identified a single equilibrium point that represents the stable glucose level in healthy individuals. To establish the existence and uniqueness of the model, we employed fixed point theory alongside the Lipschitz condition. The Ulam–Hyers stability of the proposed model was also examined. Subsequently, we analyzed the chaotic behavior of the diabetic model using feedback control approaches, focusing on controllability and PID techniques. The application of chaos theory revealed that glucose-insulin dynamics are highly sensitive to initial conditions, leading to complex oscillatory behavior that can result in unstable glucose levels. By implementing fractional-order PID controllers, we effectively stabilized chaotic glucose dynamics, achieving more reliable blood sugar regulation compared to conventional methods, with a notable reduction in oscillation amplitude. We conducted numerical simulations to validate our findings, employing the Newton polynomial method across various fractal and fractional order values to assess the robustness of the results. A discussion of the graphical outcomes from the numerical simulations, conducted using MATLAB version 18, is provided, illustrating the dynamics of glucose regulation under different fractal-fractional orders. This comprehensive approach enhances our understanding of the underlying mechanics driving chaotic behavior in glucose-insulin dynamics.