Fractional Order PIλ Research Articles

Decentralized fractional-order PIλ control applied to a TITO distillation column: a comparative study with PID and MPC

This paper investigates the use of fractional-order PIλ (FO-PI) controllers and compares their performance with classical integer-order PID (IO-PID) and fractional-order PID (FO-PID) controllers for a two-input, two-output (TITO) distillation column. Additionally, a comparative study between FO-PI and model predictive control (MPC) is conducted on the same multivariable system. While fractional control in multivariable systems has been explored, evaluations of global performance metrics—such as integral square error (ISE), integral absolute value of error (IAE), integral time weighted absolute error (ITAE), integral time weighted square error (ITSE), and integral of manipulated variables (IU)—compared to IO-PID and MPC are still scarce. This work addresses this gap by examining these metrics alongside the steps necessary for fractional control design, including Laplace numerical inversion and controller tuning. The results demonstrate that the FO-PI controller shows superior performance compared to FO/IO-PID and MPC controllers. The global relative gain indexes show improvements, with FO-PI achieving gains of 36.9% to 59.9% compared to FO/IO-PID, and a reduction of 19% to 44.1% in global index compared to MPC. The distillation column dynamics were modeled using transfer functions, allowing for a detailed comparison of control strategies. The studies conducted in this work indicate that fractional-order controllers (FO-PI) show promise in controlling the dynamics of the TITO distillation column, as evidenced by simulations and comparison with integer-order (IO-PID) and fractional-order (FO-PID) control strategies, as well as with MPC control. The study highlights the potential benefits of using fractional controllers for a classical multivariable system, allowing to advance the state of the art of this type of control approach.

Dual-Loop Voltage–Current Control of a Fractional-Order Buck-Boost Converter Using a Fractional-Order PIλ Controller

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current is established according to the fractional-order calculus theory. The fractional-order PIλ control system of the fractional-order Buck-Boost converter is designed to compare the performance of the integer-order PI controller with the fractional-order controller. Secondly, the sparrow search algorithm is applied to the optimal design of the fractional-order PIλ control system of the fractional-order Buck-Boost converter to improve the system’s phase margin, stability, and robustness. Finally, the simulation is verified on the Matlab/Simulink simulation platform and compared with the integer-order PI controller.

Design of PIλ-PDμ controller for high-order systems based on model order reduction using BB-BC and time moment matching

In this paper, a practical fractional proportional–integral and proportional–derivative (PIλ-PDμ) controller tuning method and a model reduction method are proposed for high-order systems with or without time delay. First, the Big Bang–Big Crunch (BB-BC) algorithm and the time moment matching method are used to reduce the order of the high-order system function, and the approximate low-order system function is obtained. Second, the parameter region of fractional-order PIλ-PDμ controller is obtained according to the D partition method, and the improved weighted geometric center (WGC) method is introduced to select the special points in the stable region as the control parameters. Then, the fractional-order PIλ controller is discretized by Muir method, and the fractional-order PDμ controller is discretized by continued fraction expansion (CFE). Finally, the BB-BC algorithm is used to optimize the control parameters, and the parameters with better control performance are obtained. The simulation results show that the fractional PIλ-PDμ parameter setting method based on BB-BC algorithm can improve the control performance of the system, and the model reduction scheme based on the combination of moment matching method and BB-BC algorithm is effective in both time domain and frequency domain analysis.

An enhanced whale optimized fractional order piλ controller based power quality improvement for sensorless brushless dc motor under dynamic operating conditions

This paper presents an Enhanced Whale Optimization Algorithm (EWO) approach for tuning to perfection of Fractional Order Proportional Integral and integral order Controller (FOPI λ) is used to sensorless speed control of permanent magnet Brushless DC (PMBLDC) motor under the operating dynamic condition such as (i) speed change by set speed command signal (ii) varying load conditions, (iii) integrated conditions and (iv) controller parameters uncertainty. On the other hand, it deals with a reduced THD (Total Harmonic Distortion) under dynamic operating conditions to improve the power quality for the above control system. Here present are three optimization techniques, namely (i) Enhanced Whale Optimization (EWO), (ii) Invasive Weed Optimization (IWO), and (iii) Social Spider Optimization (SSO) for fine-tuning of the FOPI λ controller parameters with reduction of THD. The proposed optimization algorithm optimized FOPI λ controller are compared under various BLDC motor operating conditions. Based on the results of MATLAB/Simulink models, the proposed algorithms are evaluated. Here, both the simulation and the results of the experiments are validated for the proposed controller technique. It demonstrates that the effectiveness of the proposed controllers is completely validated by comparing the three intelligent optimization techniques mentioned above. The EWO optimized FOPI λ controller for speed control of sensorless PMBLDC motor clearly outperforms the other two intelligent controllers by minimizing the time domain parameters, THD, performance Indices error, convergence time, control efforts, cost function, mean and standard deviation.

Research on Control Strategy of Three-Phase Inverter Based on Fractional Calculus

Three-level inverters has widely used in Medium pressure due to its advantages of low total harmonics, low voltage resistance of power switch tubes and low switching losses. In order to further improve the output voltage waveform quality and optimal control effect of the Three-level inverter, this paper introduces fractional calculus in the control method of the three-level inverter circuit. First, designed a control strategy for the fractional voltage outer loop. Secondly, the use of vector representation simplifies the parameter tuning of fractional-order controller, the optimization process of the parameters of the fractional order PIλ and integer-order PI is completed by utilizing the ITAE rules. At the same time, the Oustaloup approximation method realizes the fractional integral operator in the controller, thereby achieving the purpose of improving the output voltage of the system. For three-level inverter circuit model, comparative analysis of the control effects of the fractional-order PIλ and integer-order PI. Finally, simulation results show that fractional-order PIλ controllers have excellent flexibility, robustness.

SBW System Control Strategy for Vehicle Based on Fractional-order PIλ Dμ Controller

SBW System Control Strategy for Vehicle Based on Fractional-order PIλ Dμ Controller

Ractional PI Stabilization of Delay Systems: Application to a Thermal System

In this paper, an application of fractional-order PIλ controller is given as an alternative to solve some control problems that can arise. It aims to apply the analytical tuning procedure to control the heat flow systems. This system, modeled by first-order system involving time delay, is one with open loop characteristic equations are fractional order quasi-polynomials. Using the proposed method, the entire stability region of PI λ controllers is obtained and visualized in the plane (Kp,Ki, λ ). The simulation was carried out on thermal systems and the results demonstrate the effectiveness of the proposed type of controllers and the tuning rule.

Fractional Order PIλDµ Control Design for a Class of Cyber-Physical Systems with Fractional Order Time-Delay models

The control of cyber-physical systems (CPS) is a great challenge for researchers in control theory and engineering mainly because of delays induced by merging computation, communication, and control of physical processes. Consequently, control solutions for time-delay systems can be applied efficiently for many CPS system configurations. In this article, a fractional order PIλ and PIλDµ control design is investigated for a class of fractional order time-delay systems. The proposed control design approach is simple and efficient. The controller parameter's adjustment is achieved in two steps: first, the relay approach is used to compute satisfactory classical PID coefficients, namely kp, Ti and Td. Then, the fractional orders λ and µ are optimized using performance criteria. Simulation results show the efficiency of the proposed design technique and its ability to enhance the PID control performance.

Fractional-order composite control for the transient creep of a piezoelectric actuator

In this study, a fractional-order composite controller is proposed to decrease the settling time and narrow the range of the transient creep of a piezoelectric actuator. The controller is composed of backlash modules and a fractional-order PIλ controller. The backlash modules reduce the effect of the transient creep and the hysteresis, while the fractional-order controller decreases the settling time. Experiments show that the settling time is decreased from 11 ms to 6 ms. The controller exhibits good performance on step creep and hysteresis when the input signal is discretised into steps, whether 10, 20, 40 or 80 steps. All standard deviations are suppressed to levels 39.3% lower than that of a standard PI controller and the range of the transient creep is reduced to 19.46%.

A Fractional-Order Noise-Shaping Integrator for Sigma-Delta Modulator

A novel fractional-order Sigma-Delta modulator, which can improve the signal to noise ratio (SNR) and is of robust stability of a high order Sigma-Delta modulator, is presented in the paper. Traditional Sigma-Delta modulator mainly focused on forming an integer high order integrator to improve the SNR, whereas the paper presents a fractional-order integrator instead to obtain higher SNR and better stability. Specifically, a 4th order digital integrator and a fractional-order PIλ controller are inserted between the AFE and the quantizer. The 4th order digital integrator increases overall loop order to produce much higher SNR for Sigma-Delta modulator and the fractional-order PIλ controller allows fine tuning accuracy in seeking the coefficients for Sigma-Delta modulator and provides stability in the placement of poles and zeros for compensating the phase delays caused by system nonlinearity. The parameters of PIλ controller are optimized by using swarm intelligent optimization algorithm. The stability of the proposed fractional-order Sigma-Delta modulator is analyzed and the robustness of the proposed fractional-order Sigma-Delta modulator is demonstrated. Further simulation from the fractional-order Sigma-Delta modulator, results of SNR>145dB and the noise floor under -190dB are obtained in frequency range of [5-150Hz].

Fractional order [formula omitted] controller synthesis for steam turbine speed governing systems

The current state of the art of fractional order stability theory is hardly to build connection between the time domain analysis and frequency domain synthesis. The existing tuning methodologies for fractional order PIλDμ are not always satisfy the given gain crossover frequency and phase margin simultaneously. To overcome the drawbacks in the existing synthesis of fractional order controller, the synthesis of optimal fractional order PIλ controller for higher-order process is proposed. According to the specified phase margin, the corresponding upper boundary of gain crossover frequency and stability surface in parameter space are obtained. Sweeping the order parameter over λ∈(0,2), the complete set of stabilizing controller which guarantees both pre-specifying phase frequency characteristic can be collected. Whereafter, the optimal fractional order PIλ controller is applied to the speed governing systems of steam turbine generation units. The numerical simulation and hardware-in-the-loop simulation demonstrate the effectiveness and satisfactory closed-loop performance of obtained fractional order PIλ controller.

Fractional Order PIλ Controller Application for Limited Memory System

Bu çalışmada, asenkron motorun hız kontrolünde kesirli PI λ yöntemi, sınırlı hafızaya sahip olan bir dsPIC mikrodenetleyicisi kullanılarak gerçekleştirilmiştir. Kesirli integralin hesaplanması işleminde kullanılan hata fonksiyonu ve ağırlık fonksiyonu veri setleri sadece son 50 veriyi kullanacak şekilde sınırlandırılmıştır. Bu sınırlandırma işlemi kesirli kontrolörün dsPIC mikrodenetleyicisinde kullanılabilmesini sağlamıştır. Farklı k p , k i ve λ değerleri için hız kontrolü deneyleri gerçekleştirilmiştir. Sonuçlar tablolarda ve şekillerde sunulmuştur.

Optimal tuning of fractional order Pλ D µA controller using Particle Swarm Optimization algorithm

In this paper an optimal design method for the tuning of the fractional order proportional-integral-derivative-acceleration (PI λ DµA) controller parameters is presented. The fractional order PIλ DµA controller is a conventional proportional-integral-derivative-acceleration (PIDA) controller whose integration action order λ and differentiation action order μ are real rather than integer. The two more parameters λ and μ will provide a better flexibility in controller design for a wide range of dynamical systems. The six parameters of the fractional order PI λ DµA controller are tuned through the optimization of the Integral Time Absolute Error (ITAE) performance criteria using the Particle Swarm Optimization (PSO) algorithm. The fractional order PIλ DµA controller design for two real world system models of a feedback control system have been proposed to illustrate its benefits and effectiveness. The simulation results show that the proposed fractional PIλ DµA controller exhibits dynamical performances improvements compared to the conventional PIDA controller of the feedback control system.

Fractional order PIλDμ controller design for satisfying time and frequency domain specifications simultaneously

In order to achieve a desired control performance characterized by satisfying specifications in both frequency-domain and time-domain simultaneously, an optimal fractional order proportional integral derivative (PIλDμ) controller design strategy is proposed based on analytical calculation and Differential Evolution algorithm for a permanent magnet synchronous motor (PMSM) servo system in this paper. In this controller design, the frequency-domain specifications can guarantee the system stability with both gain margin and phase margin, and also the system robustness to loop gain variations. The time-domain specifications can ensure the desired step response performance with rapid rising curve, constrained overshoot, and proper power consuming. Compared with the PIλ controller and the traditional PID controller, PIλDμ controller can get obvious benefits from two more degrees of freedom of the fractional orders λ and μ on satisfying multiple constraints simultaneously and achieving better servo tracking performance for the PMSM servo system. PMSM speed tracking simulations and experiments are demonstrated to show the significant advantages of using the proposed optimal PIλDμ controller over the optimal fractional order PIλ controller and traditional integer order PID controller.

A comparative study of fractional order PIλ/PIλDμ tuning rules for stable first order plus time delay processes

Conventional PID tuning methods may not be sufficient to deal with complex processes of modern industry. For better control, fractional orderPIλDμ controller was introduced as the generalization of classical PID controller with the help of non-integer order (fractional order) calculus. Thefractional calculus uses integration and differentiation with a fractional order or complex order. The major advantage of fractional derivative is theability to inherit the nature of the processes. In general, the control loop includes both fractional order process model and fractional order controller.However, the processes to be controlled are usually modeled as integer order models and controlled using fractional order controllers. But if theplant model is obtained as fractional model, it is converted into integer order model by approximating the fractional terms using differentapproximations proposed in the literature. With all the above mentioned advantages, several fractional order PIλ/PIλDμ tuning rules are proposedin the literature for integer order systems and researchers are still proposing the new rules. The main aim of this paper is to compare fractional orderPI/PID tuning methods based on Integral of Absolute Error (IAE), Total Variation (TV) and Maximum Sensitivity (Ms). The main reason forchoosing fractional order PIλ/PIλDμ controllers is their additional degrees of freedom that result in better control performance. These tuning ruleswere applied on several first order plus time delay processes subjected to step change in setpoint and disturbance.Six recent tuning methods, three for fractional order PIλ and the remaining for fractional order PIλDμ, were considered. Finally, from thesimulation results the optimal tuning method is recommended based on the control objective of the process and the process dead time (L) to timeconstant (T) ratio. It is observed that the performance of tuning methods vary with the nature of the process like lag dominant, balanced and delaysignificant processes. The FOPTD processes were checked for robustness with increasing L/T ratio with respect to IAE, TV and Ms.

A comparative study of fractional order PIλ/PIλDµ tuning rules for stable first order plus time delay processes

Conventional PID tuning methods may not be sufficient to deal with complex processes of modern industry. For better control, fractional order PIλDµ controller was introduced as the generalization of classical PID controller with the help of non-integer order (fractional order) calculus. The fractional calculus uses integration and differentiation with a fractional order or complex order. The major advantage of fractional derivative is the ability to inherit the nature of the processes. In general, the control loop includes both fractional order process model and fractional order controller. However, the processes to be controlled are usually modeled as integer order models and controlled using fractional order controllers. But if the plant model is obtained as fractional model, it is converted into integer order model by approximating the fractional terms using different approximations proposed in the literature. With all the above mentioned advantages, several fractional order PIλ/PIλDµ tuning rules are proposed in the literature for integer order systems and researchers are still proposing the new rules. The main aim of this paper is to compare fractional order PI/PID tuning methods based on Integral of Absolute Error (IAE), Total Variation (TV) and Maximum Sensitivity (Ms). The main reason for choosing fractional order PIλ/PIλDµ controllers is their additional degrees of freedom that result in better control performance. These tuning rules were applied on several first order plus time delay processes subjected to step change in setpoint and disturbance. Six recent tuning methods, three for fractional order PIλ and the remaining for fractional order PIλDµ, were considered. Finally, from the simulation results the optimal tuning method is recommended based on the control objective of the process and the process dead time (L) to time constant (T) ratio. It is observed that the performance of tuning methods vary with the nature of the process like lag dominant, balanced and delay significant processes. The FOPTD processes were checked for robustness with increasing L/T ratio with respect to IAE, TV and Ms.

Robust stabilizing regions of fractional-order PIλ controllers for fractional-order systems with time-delays

This study focuses on a graphical approach to determine the stabilizing regions of fractional-order PIź(proportional integration) controllers for fractional-order systems with time-delays. By D-decomposition technique, the existence conditions and calculating methods of the real root boundary (RRB) curves, complex root boundary (CRB) curves and infinite root boundary (IRB) lines are investigated for a given stability degree. The stabilizing regions in terms of the RRB curves, CRB curves and IRB lines are identified by the proposed criteria in this paper. Finally, two illustrative examples are given to verify the effectiveness of this graphical approach for different stability degrees.

A Study on Parameter Tuning of Fractional Order PI<sup>λ</sup>-PD<sup>μ</sup> Controller

Aiming at a kind of special object with fractional characteristics,a new kind of fractional order PIλ-PDμ controller is put forward in this paper,In this control system, the forward channel of system contains a fractional order PIλ controller,while the feedback loop adopts PDμ controller.The control structure in this fractional order system can achieve a good performance of the closed loop control.Moreover,the impacts on system dynamic performance and static performance are compared when the parameters in the controller are changed.The results show that when these parameters are in the reasonable range,the system can achieve better control performance.

Application of Fractional-Order PI<sup>λ</sup> Controller for PMSM Speed Control

Permanent Magnet Synchronous Motor(PMSM) is a multivariable, nonlinear system with strong couple. It’s sensitive to the parameters and load variations. To improve the system static, dynamic performance and robustness, the fractional-order PIλ control strategy is proposed. Compared with integer-order PI controller, the fractional-order PIλ controller achieves better control performance due to an extra adjustable knob λ. The simulation results show that the fractional-order PIλ controller can track the given speed quickly and accurately, and has better anti-interference with load variation.

Real Time Implementation of Third Generation CRONE Control Strategy for DC Motor Speed Control System

This work focuses on real time implementation of third generation CRONE control strategy for the speed control of DC motor. The dynamic second order transfer function model is analytically derived for the speed control system based on real time data. An attempt has been made to compare third generation CRONE controller with Fractional order (FO) PIλ controller. The real time run of both the controller results are recorded and reported in time domain and integral time performance indices. The real time result shows that the third generation CRONE controller may be more effective and gives superior performance over the FO PIλ controller