Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

In this paper, a self-tuning hierarchical controller in which a Fuzzy logic controller supervises the control actions of a conventional PID has been proposed, implemented and presented. The controller has been applied to a control study of Fluid Catalytic Cracking (FCC) unit riser temperature, and regenerator temperature respectively. Performance comparison of the proposed Fuzzy-PID controller and the conventional PID was made in simulation studies of regulatory and servo performances of the two controller types. Six performance measures: Percent overshoot (OS), settling time (ST), integral absolute error (IAE), integral square error (ISE), integral time absolute error (ITAE) and integral time square error (ITSE) were employed as the tools for performance comparison between the conventional PID and the Fuzzy-PID controller. For the tracking of riser temperature with a set point at 524oC, the performance indicators under PID control gave the following results overshoot (14.5%); settling time (40 seconds) Integral absolute error (8.246), integral square error (3.3); integral time absolute error(1762);integral time square error (43.77) while for the same indicators under Fuzzy-PID control the following values: overshoot (3.3%); settling time (40 seconds) ;Integral absolute error (8.811); integral square error (14.5); integral time absolute error(280),;integral time square error (31.11) .The results allude to the superiority of the fuzzy-PID scheme over the PID scheme in tracking the optimal set point of riser temperature. More so, for tracking the regenerator set point temperature of 746oC, comparative study of step response under the two schemes gave the following results in six performance indicators: overshoot (PID(12.6%)/Fuzzy-PID(6%)); settling time (PID(80 seconds)/Fuzzy-PID(20seconds));Integral absolute error(PID(14.29)/Fuzzy-PID(8.63)); integral square error(PID(6.713).Fuzzy-PID(4.506)); integral time absolute error(PID(2858)/Fuzzy-PID(305.9)), integral time square error (PID(77.55)/Fuzzy-PID(33.05)).Moreover, the fuzzy-PID controller also showed superior performance over the conventional PID controller in terms disturbance rejection (regulatory response) of both riser and regenerator temperature. The results from this study suggest that the application of fuzzy-PID scheme to temperature control offers good promise of improved fluid catalytic cracking unit (FCCU) operations.

Background: As a high-performance functional material, stacked piezoelectric actuator can produce a displacement under the effect of changing voltage. Its advantages of fast response and easy operation make it to be widely applied in the precision structure field. However, its small displacement stroke and hysteresis nonlinearity affect the accuracy of the output. : In the next step, some experiments were undertaken based on the constructed platform. Methods: In order to enlarge the displacement of piezoelectric actuator and reduce the influence of hysteresis, this study designs a diamond-shaped amplifying mechanism to amplify the output of the piezoelectric actuator, and then develops a self-tuning fuzzy fractional-order PIλDμ controller for the high precision displacement control of the proposed amplifying mechanism. After analyzing the working principle and modeling the amplifying mechanism, the fractional-order PIλDμ control model of the proposed mechanism was built and discretized according to the theoretical base of the fractional calculus in the time domain. Moreover, the fuzzy control algorithm was also introduced to achieve self-turning of parameters. Besides, the amplifying mechanism was also adopted for a microdroplet jetting dispenser to verify the practicability of the mechanism and control strategy. In the next step, some experiments were undertaken based on the constructed platform. Results: Experiments show that the displacement overshoots, the times of reaching a steady state of the traditional integer-order controller and the fractional-order controller are 5.08%, 1.17% and 17.25 s, 12.00 s, respectively. However, the fuzzy PIλDμ controller lowers the overshoot and the time of reaching a steady state to 0.95% and 9.00 s, respectively. The control algorithm can not only improve the follow-ability of the output displacement of the proposed mechanism, but also maintain the deviation within the range of 0.4% after the displacement stroke is stable and reduce the entering time of the mechanism up to 47.8%. In actual application, the droplet volume of micro-droplet jetting dispenser under fuzzy fractional-order PID control method is more stable, and its repeatability accuracy can reach up to 1.6475%. Conclusion: Experimental results indicate that the self-tuning fuzzy fractional-order PIλDμ controller can significantly improve the tracking performances of the PID and the integer-order PID with regard to the amplifying mechanism with the advantages of good dynamic character and regulation precision. Furthermore, the diamond-shaped amplification mechanism and control strategy can be applied for some micro-droplet jetting dispensers used in microelectronic packaging, life science and 3D printing fields.

In this paper, a new fractional-order (FO) PIλDµ controller is designed with the desired gain and phase margin for the automatic rudder of underactuated surface vessels (USVs). The integral order λ and the differential order μ are introduced in the controller, and the two additional adjustable factors make the FO PIλDµ controller have better accuracy and robustness. Simulations are carried out for comparison with a ship’s digital PID autopilot. The results show that the FO PIλDµ controller has the advantages of a small overshoot, short adjustment time, and precise control. Due to the uncertainty of the model parameters of USVs and two extra parameters, it is difficult to compute the parameters of an FO PIλDµ controller. Secondly, this paper proposes a novel particle swarm optimization (PSO) algorithm for dynamic adjustment of the FO PIλDµ controller parameters. By dynamically changing the learning factor, the particles carefully search in their own neighborhoods at the early stage of the algorithm to prevent them from missing the global optimum and converging on the local optimum, while at the later stage of evolution, the particles converge on the global optimal solution quickly and accurately to speed up PSO convergence. Finally, comparative experiments of four different controllers under different sailing conditions are carried out, and the results show that the FO PIλDµ controller based on the IPSO algorithm has the advantages of a small overshoot, short adjustment time, precise control, and strong anti-disturbance control.

In this paper, a fractional order controller is designed for a class of nonlinear systems by using the high-order fully actuated (HOFA) system approach. First, the original system in an affine state-space form is transformed into a HOFA system model. Second, a state-feedback controller involving the Caputo fractional derivatives is derived based on the HOFA system model. Then, in the framework of close-loop fractional order linear time invariant (LTI) systems, the gain of the fractional order controller is designed via parametric eigenstructure assignment approach. Finally, numerical simulations verify the efficiency of the designed fractional order controller.

There are several parameters required for viscoelastic material modeling to describe the material behavior. On the other hand, the parameter identification of viscoelastic materials often leads to difficult experimental procedures. Since the fractional controllers are robust to parameter uncertainties in the plant model, in this paper, this type of controller is proposed to avoid the difficulties of parameters identification in viscoelastic materials. As a prototype, a viscoelastic beam covered by piezoelectric patches is modeled for suppressing vibrations arising from transient excitations of viscoelastic materials. Different fractional-order PIλDμ controllers are used to illustrate the effect of fractional integral and fractional derivative orders separately. The results show that the fractional-order controllers are successful and effective for vibration suppression of viscoelastic beams.

The artificial gorilla troops optimization (GTO) is a recently-introduced metaheuristic optimization technique for non-linear, non-convex and non-smooth problems with multiple types of variables. This algorithm, which is a nature-based technique, has been inspired by the social intelligence of gorilla troops. This study delves into how optimal can a power system stabilizer (PSS) unit be tuned incorporating GTO algorithm. The Integral Time Square Error (ITSE) was utilized as a fitness function, which desirably should be minimized. Four different controllers are employed to investigate the model of a single-machine scheme as a model for the infinite bus. The tilt-integral-derivative (TID) shows a better performance response in comparison with the PID controller, lead-lag controller, as well as fractional-order PID controller (FOPID) investigated. The results have implied that the GTO algorithm has obtained faster convergence over the other compared optimization techniques, while it has superiority in not trapping in local optimum points. It has shown also a high level of precision.

O NE of the critical issues for space tether missions is the stable deployment of the tether [1,2]. Many effective control strategies and devices have been proposed for the deployment [3–7]. Among them, the tension feedback control has been proven the most effective because of its simplicity, implementability, real-time capability, and effectiveness [8]. Although effective, the existing tension control laws either take a relatively long time to deploy the tetherwith an overshoot in tether length or employ complex nonlinear feedback control strategies that require precise dynamic models of the space tether system. Recently, the fractional-order control theory has emerged as an effective generalization of the classic integer-order control theory [9– 11], such as Commande Robusted' Ordre Non Entier (CRONE) strategy, fractional PIλDμ controller, and fractional-order lead–lag compensators [12]. Compared with its integer-order counterpart, the fractional-order control has three unique advantages that specifically appealing to the space tether deployment. First, it is more suitable for modeling flexible structures because of its unique historical memory effect. Second, it provides an effective and clear-cut robust control strategy, especially for distributed parameter systems. Third, it responds to control input faster, with smaller overshoots than its integer counterpart. To our best knowledge, no attempt has been made to control the deployment of space tethers with a fractionalorder control law. In this Note, a new fractional-order tension control law is developed for the fast and stable deployment of space tethers.

Although a considerable amount of research has been carried out in the field of fractional-order controllers, a simplified tuning routine has yet to be established. Most of the tuning techniques for fractional-order controllers deal with complex computations and optimization routines. This paper proposes a simple yet efficient methodology based on a vector representation of the fractional-order controllers. This simplifies considerably the computations and derivation of the fractional-order controller parameters. The tuning procedure is exemplified first for a fractional-order PI controller designed for a simple first-order process, as well as for a fractional-order PD controller for a servoing system. In this case, the experimental results are also included, showing that this novel tuning approach is a viable replacement for the more complex tuning procedures currently employed in the design of different fractional-order controllers.

This paper presents an algorithm for load frequency control of multi-area power system. A more realistic models is used to represent the elements of power system. The frequency bias factor (B) is set optimally. The effect of generation rate constraints (GRC) and the dead band characteristics of the governor is taken into consideration. A simultaneous optimization of the fractional order (PIλ Dµ) control parameters, the speed regulation (R) of the governor and the frequency bias factor (B) is proposed for the first time. The objective function to be minimizes is a modified integral time absolute error (MITAE). The tuning of the control parameters is based on using cuckoo search algorithm (CSA) to minimize the objective function. The proposed algorithm is applied to a typical interconnected three-area power systems with a different levels of load step change. Results of simulation of the presented method are compared with those obtained by applying particle swarm optimization technique. The results obtained reveal the robustness of the suggested algorithm in terms of settling time and peak over shoot.

This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.

Performance comparison of wind turbine based doubly fed induction generator system using fault tolerant fractional and integer order controllers

Coot optimization algorithm-tuned neural network-enhanced PID controllers for robust trajectory tracking of three-link rigid robot manipulator

Abstract: This study presents a comparative analysis of five metaheuristic algorithms for optimizing Proportional Integral Derivative (PID) controller parameters in a vehicle cruise control system. The selected algorithms include the Rain Optimization Algorithm (ROA), Student Psychology-Based Optimization (SPBO), Sine Cosine Algorithm (SCA), Gradient-Based Optimization (GBO), and Grey Wolf Optimization (GWO). The PID parameters are optimized using four common performance criteria: Integral Time Absolute Error (ITAE), Integral Absolute Error (IAE), Integral SquareError (ISE), and Integral Time Square Error (ITSE). Unlike conventional tuning methods, metaheuristic approaches provide more flexible and adaptive parameter selection, enhancing the control system's robustness. The study analyses the algorithms under several internal and external disturbances, including road incline, passenger mass, and measurement noise. The findings demonstrate that no singular algorithm is universally superior; instead, efficacy is contingent upon the particular system conditions and control objectives. The Sine Conine Algorithm (SCA) has superior overall performance under typical settings, while GWO attains the quickest response time for slope fluctuations. Conversely, ROA reduces overshoot, rendering it appropriate for applications that necessitate stability. Additionally, the proposed methods are compared with the classical Ziegler-Nichols tuning approach, demonstrating the advantages of metaheuristic algorithms in terms of reduced overshoot, faster settling time, and improved robustness. The findings indicate that the SCA with the ITAE criterion yields superior performance under varying disturbance conditions. However, the fastest system response under different slope conditions isachieved with the parameter values determined by the GWO employing the ITSE criterion. The system model is designed in a MATLAB environment and the detailed comparative results are provided in figures and tables.Cite this article as: F. Heybetli, Y. Danayiyen, A. Begüm Taşdemir and Ş. Şenyiğit, "Comparative analysis of metaheuristic algorithms in PID-based vehicle cruisecontrol systems," Electrica, 25, 0051, 2025. doi: 10.5152/electrica.2025.24051.

Fractional-order modeling and controller design by a simplified way is the demanding research area and is gearing more and more momentum. This paper is the attempt of application of fractional-order modeling and controller design for the power plant gas turbine. The Gas Turbine is most important equipment in power, aviation and automotive industry. It converts the thermal energy of fuel into the mechanical power. Therefore, important requirement of gas turbine system is to control the flow of input fuel. The existing identified model of the gas turbine between the input fuel flow and the output speed, is of high-order and integer type, which is reduced to the simple and compact integer-order (IO) and fractional-order (FO) models using local optimization technique. The fractional-order internal model controller (FO-IMC) is designed and to show the performance efficacy it is compared with integer-order internal model controller (IO-IMC), which is also designed using the same methodology and specification. Simulation results show that FO-IMC based controller gives better performance for the set point tracking, plant uncertainty and disturbance rejection than the IO-IMC. FO-IMC controller also satisfy the robust stability condition.

Various controller design strategies have been studied in the past for under-actuated and fully actuated robotic systems to control a specific task in the best possible manner. This chapter proposes an adaptive fractional order (FO) controller design for under-actuated and fully actuated robotic manipulators. Adaptive controllers can adapt to a particular uncertainty inside the plant and give the best possible controlled response [1]. The change may due to variation of an individual parameter or may be due to disturbances. A suitable controller is supposed to adapt to these uncertainties and give the best results.