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This paper aims to design a modified fractional order proportional integral derivative (PID) (FO[PI]^λD^µ) controller based on the principle of fractional calculus and investigate its performance for a class of a second-order plant model under different operating conditions. The effectiveness of the proposed controller is compared with the classical controllers.

The fractional factor related to the integral term of the standard FO[PI]^λD^µ controller is applied as a common fractional factor term for the proportional plus integral coefficients in the proposed controller structure. The controller design is developed using the regular closed-loop system design specifications such as gain crossover frequency, phase margin, robustness to gain change and two more specifications, namely, noise reduction and disturbance elimination functions.

The study results of the designed controller using matrix laboratory software are analyzed and compared with an integer order PID and a classical FOPI^λD^µ controller, the proposed FO[PI]^λD^µ controller exhibit a high degree of performance in terms of settling time, fast response and no overshoot.

This paper proposes a methodology for the FO[PI]^λD^µ controller design for a second-order plant model using the closed-loop system design specifications. The effectiveness of the proposed control scheme is demonstrated under different operating conditions such as external load disturbances and input parameter change.

The purpose of this manuscript, a state feedback gain depends on the optimal design of fractional order PID controller to time-delay system is established. In established optimal design known as advanced cuttlefish optimizer and random decision forest that is combined performance of random decision forest algorithm (RDFA) and advanced cuttlefish optimizer (ACFO).

The proposed ACFO uses the concept of crossover and mutation operator depend on position upgrading to enhance its search behavior, calculational speed as well as convergence profile at basic cuttlefish optimizer.

Fractional order proportional-integrator-derivative (FOPID) controller, apart from as tuning parameters (kp, ki and kd) it consists of two extra tuning parameters λ and µ. In established technology, the increase of FOPID controller is adjusted to reach needed responses that demonstrated using RDFA theory as well as RDF weight matrices is probable to the help of the ACFO method. The uniqueness of the established method is to decrease the failure of the FOPID controller at greater order time delay method with the help of controller maximize restrictions. The objective of the established method is selected to consider parameters set point as well as achieved parameters of time-delay system.

In the established technique used to evade large order delays as well as reliability restrictions such as small excesses, time resolution, as well as fixed condition defect. These methods is implemented at MATLAB/Simulink platform as well as outcomes compared to various existing methods such as Ziegler-Nichols fit, curve fit, Wang method, regression and invasive weed optimization and linear-quadratic regression method.

The purpose of this paper is to detail the design of a fractional controller which was developed for the suppression of the flexural vibrations of the first mode of a smart beam.

During the design of the fractional controller, in addition to the classical control parameters such as the controller gain and the bandwidth; the order of the derivative effect was also included as another design parameter. The controller was then designed by considering the closed loop frequency responses of different fractional orders of Continued Fraction Expansion (CFE) method.

The first, second, third and fourth order approximations of CFE method were studied for the performance analysis of the controller. It was determined that the increase in the order resulted in better vibration level suppression at the resonance. The robustness analysis of the developed controllers was also conducted.

The experimentally obtained free and forced vibration results indicated that the increase in the order of the approximations yielded better performance around the first flexural resonance region of the smart beam and proved to yield better performance than the classical integer order controllers.

Evaluation of the performance of a developed fractional controller was realized by using different approach orders of the CFE method for the suppression of the flexural vibrations of a smart beam.

The purpose of this paper is to address a fractional order fuzzy PID (FOFPID) control approach for solving the problem of enhancing high precision tracking performance and robustness against to different reference trajectories of a 6-DOF Stewart Platform (SP) in joint space.

For the optimal design of the proposed control approach, tuning of the controller parameters including membership functions and input-output scaling factors along with the fractional order rate of error and fractional order integral of control signal is tuned with off-line by using particle swarm optimization (PSO) algorithm. For achieving this off-line optimization in the simulation environment, very accurate dynamic model of SP which has more complicated dynamical characteristics is required. Therefore, the coupling dynamic model of multi-rigid-body system is developed by Lagrange-Euler approach. For completeness, the mathematical model of the actuators is established and integrated with the dynamic model of SP mechanical system to state electromechanical coupling dynamic model. To study the validness of the proposed FOFPID controller, using this accurate dynamic model of the SP, other published control approaches such as the PID control, FOPID control and fuzzy PID control are also optimized with PSO in simulation environment. To compare trajectory tracking performance and effectiveness of the tuned controllers, the real time validation trajectory tracking experiments are conducted using the experimental setup of the SP by applying the optimum parameters of the controllers. The credibility of the results obtained with the controllers tuned in simulation environment is examined using statistical analysis.

The experimental results clearly demonstrate that the proposed optimal FOFPID controller can improve the control performance and reduce reference trajectory tracking errors of the SP. Also, the proposed PSO optimized FOFPID control strategy outperforms other control schemes in terms of the different difficulty levels of the given trajectories.

To the best of the authors’ knowledge, such a motion controller incorporating the fractional order approach to the fuzzy is first time applied in trajectory tracking control of SP.

The purpose of this paper is to present a two-loop approach for velocity control of a permanent magnet synchronous motor (PMSM) under mechanical uncertainties.

The inner loop calculates the two-axis stator reference voltages through a feedback linearization method. The outer loop employs an RST control structure to compute the q-axis stator reference current. To increase the robustness of the proposed method, the RST controller parameters are adapted through a fractional order model reference adaptive system (FO-MRAS). The fractional order gradient and Lyapunov methods are utilized as adaptation mechanisms.

The effect of the fractional order derivative in the load disturbance rejection, transient response speed and the robustness is verified through computer simulations. The simulation results show the effectiveness of the proposed method against the external torque and mechanical parameters uncertainties.

The proposed FO-MRAS based on Lyapunov adaptation mechanism is proposed for the first time. Moreover, application of the FO-MRAS for velocity control of PMSM is presented for the first time.

In recent years, use of sensorless control methods for electrical motor-based variable speed drive systems has been increasing rapidly to compensate the increasing costs in industrial systems. Also, use of induction motors is popular for a long time to decrease the cost of these industrial systems. This study aims to design an effective controller to improve the sensorless speed control performance of induction motor. To achieve this, a conformable fractional order proportional integral (CFOPI) controller is designed.

The system is modeled based on small signal analysis by using the input–output data, experimentally. To do this, system identification toolbox of Matlab is used. The proposed controller is established on conformable fractional integral approach proposed by Khalil et al. (2014). CFOPI controller coefficients are optimized using particle swarm optimization method on the created small signal-based simulation model of the system to minimize the integral time absolute error. To prove the success of the proposed method, a traditional fractional order proportional integral (TFOPI) controller is tested under the same experimental system with the CFOPI controller.

TFOPI and CFOPI controllers are tested with the optimum parameters. Reference and actual speed trends are obtained for both methods. In induction motor start-up test, settling-times are measured as 8.73 and 8.44 s and steady-state oscillations are 2.66% and 0% (almost) for TFOPI and CFOPI controllers, respectively. In variable referenced speed tracking test, CFOPI performs well at all speed levels, while TFOPI fails to reach the reference speed at most speed levels.

Proposed CFOPI control method can be easily implemented in industrial systems, thanks to its simple algorithm. digital signal peripheral interface controller (dsPIC) based driver circuit with designed CFOPI controller used in this study can be applied directly to industrial systems such as elevators, conveyors, cranes and drills. Moreover, it can improve the performance of induction motor-based variable speed drive systems.

The proposed method provides robust performance for induction motor used in control systems. Additionally, it does this by using less complex algorithm written on the processors according to the traditional fractional order controllers.

With the increasing demands of industrial applications, it is imperative for robots to accomplish good contact-interaction with dynamic environments. Hence, the purpose of this research is to propose an adaptive fractional-order admittance control scheme to realize a robot–environment contact with high accuracy, small overshoot and fast response.

Fractional calculus is introduced to reconstruct the classical admittance model in this control scheme, which can more accurately describe the complex physical relationship between position and force in the interaction process of the robot–environment. In this control scheme, the pre-PID controller and fuzzy controller are adopted to improve the system force tracking performance in highly dynamic unknown environments, and the fuzzy controller is used to improve the trajectory, transient and steady-state response by adjusting the pre-PID integration gain online. Furthermore, the stability and robustness of this control algorithm are theoretically and experimentally demonstrated.

The excellent force tracking performance of the proposed control algorithm is verified by constructing highly dynamic unstructured environments through simulations and experiments. In simulations and experiments, the proposed control algorithm shows satisfactory force tracking performance with the advantages of fast response speed, little overshoot and strong robustness.

The control scheme is practical and simple in the actual industrial and medical scenarios, which requires accurate force control by the robot.

A new fractional-order admittance controller is proposed and verified by experiments in this research, which achieves excellent force tracking performance in dynamic unknown environments.

The purpose of this paper is to improve transient response and dynamic performance of automatic voltage regulator (AVR).

This paper proposes a novel fractional order proportional–integral–derivative plus derivative (PI^λD^µD^µ2) controller called FOPIDD for AVR system. The FOPIDD controller has seven optimization parameters and the equilibrium optimizer algorithm is used for tuning of controller parameters. The utilized objective function is widely preferred in AVR systems and consists of transient response characteristics.

In this study, results of AVR system controlled by FOPIDD is compared with results of proportional–integral–derivative (PID), proportional–integral–derivative acceleration, PID plus second order derivative and fractional order PID controllers. FOPIDD outperforms compared controllers in terms of transient response criteria such as settling time, rise time and overshoot. Then, the frequency domain analysis is performed for the AVR system with FOPIDD controller, and the results are found satisfactory. In addition, robustness test is realized for evaluating performance of FOPIDD controller in perturbed system parameters. In robustness test, FOPIDD controller shows superior control performance.

The FOPIDD controller is introduced for the first time to improve the control performance of the AVR system. The proposed FOPIDD controller has shown superior performance on AVR systems because of having seven optimization parameters and being fractional order based.

This study aims to compare the performance of the conventional and fractional order proportional-integral-derivative (PID and FOPID) controllers tuned with a particle swarm optimization (PSO) and genetic algorithm (GA) for quadrotor control.

In this study, the gains of the controllers were tuned using PSO and GA, which are included in the heuristic optimization methods. The tuning processes of the controller’s gains were formulated as optimization problems. While generating the objective functions (cost functions), four different decision criteria were considered separately: integrated summation error (ISE), integrated absolute error, integrated time absolute error and integrated time summation error (ITSE).

According to the simulation results and comparison tables that were created, FOPID controllers tuned with PSO performed better performances than PID controllers. In addition, the ITSE criterion returned better results in control of all axes except for altitude control when compared to the other cost functions. In the control of altitude with the PID controller, the ISE criterion showed better performance.

While a conventional PID controller has three parameters (Kp, Ki, Kd) that need to be tuned, FOPID controllers have two additional parameters (µ). The inclusion of these two extra parameters means more flexibility in the controller design but much more complexity for parameter tuning. This study reveals the potential and effectiveness of PSO and GA in tuning the controller despite the increased number of parameters and complexity.

The puspose of this paper, a novel systematic design of fractional order proportional integral derivative (FOPID) controller-based speed control of sensorless brushless DC (BLDC) motor using multi-objective enhanced genetic algorithm (EGA). This scheme provides an excellent dynamic and static response, low computational burden, the robust speed control.

The EGA is a meta-heuristic-inspired algorithm for solving non-linearity problems such as sudden load disturbances, modeling errors, power fluctuations, poor stability, the maximum time of transient processes, static and dynamic errors. The conventional genetic algorithm (CGA) and modified genetic algorithm (MGA) are not very effective in solving the above-mentioned problems. Hence, a multi-objective EGA optimized FOPID (EGA-FOPID) controller is proposed for speed control of sensorless BLDC motor under various conditions such as constant load conditions, varying load conditions, varying set speed (Ns) conditions, integrated conditions and controller parameters uncertainty.

This systematic design of the multi-objective EGA-FOPID controller is implemented in MATLAB 2020a with Simulink models for optimal speed control of the BLDC motor. The overall performance of the EGA-FOPID controller is observed and evaluated for computational burden, time integral performance indexes, transient and steady-state characteristics. The hardware experiment results confirm that the proposed EGA-FOPID controller can precisely change the BLDC motor speed is desired range with minimal effort.

The conventional real time issues such as nonlinearity characteristics, poor controllability and stability.

It is clearly evident that out of these three intelligent controllers, the EGA optimized FOPID controller gives enhanced performance by minimizing the time domain parameters, performance Indices error and convergence time. Also, the hardware experimental setup and the results of the proposed EGA-FOPID controller are presented.

It shows the effectiveness of the proposed controllers is completely verified by comparing the above three intelligent optimization algorithms. It is clearly evident that out of these three intelligent controllers, the EGA optimized FOPID controller gives enhanced performance by minimizing the time domain parameters, performance Indices error and convergence time. Also, the hardware experimental setup and the results of the proposed EGA-FOPID controller are presented.