Fractional Order Controller Design Research Articles

Optimal design of fractional-order proportional integral derivative controllers for structural vibration suppression

In designing control systems, it is known that fractional-order proportional integral derivative (FOPID) controllers often provide greater flexibility than conventional proportional integral derivative (PID) controllers. This higher level of flexibility has proven to be extremely valuable for various applications such as vibration suppression in structural engineering. In this paper, we study the optimization of FOPID controllers using twelve well-established algorithms to minimize structural responses under seismic excitations. The algorithms include crystal structure algorithm (CryStAl), stochastic paint optimizer, particle swarm optimization, krill herd, harmony search, ant colony optimization, genetic algorithm, grey wolf optimizer, Harris hawks optimization, sparrow search algorithm, hippopotamus optimization algorithm, and duck swarm algorithm. In addition to highlighting the benefits of fractional calculus in structural control, this study provides a detailed analysis of FOPID controllers as well as a brief description of the algorithms used to optimize them. To evaluate the efficiency of the proposed techniques, two building models with different numbers of stories are examined. FOPID controllers are designed based on oustaloup’s approximation and the El Centro earthquake data. Using five well-known metrics, the performances of the developed methods are evaluated against five earthquake scenarios, including the recent earthquake in Turkey. A non-parametric (Friedman) test is also employed to compare the algorithms based on their corresponding vibration reduction. The findings of this analysis show that CryStAl consistently performs better than the other algorithms for both building models, thus resulting in superior vibration suppression.

Optimal Fractional Order Controller Design for a DC Buck Converter

Optimal Fractional Order Controller Design for a DC Buck Converter

Smith predictor based fractional order controller design for improved performance and robustness of unstable FOPTD processes

Abstract Performance and robustness are essential characteristics for the application of unstable time-delayed systems. As tasks become more complex, traditional control methods cannot meet such demands for performance and robustness. The present work aims to develop fractional order-based controllers for enhanced Smith predictor-based unstable first-order plus time-delayed systems (FOPTD) with improved performance and robustness. In the current work, fractional order controllers using a Genetic Algorithm (GA) are designed with enhanced SP (Smith Predictor) structure to control unstable first-order time-delayed processes to improve performance. Furthermore, in the feedback path a fractional order (FO) filter is used to further improve robustness and performance. A systematic methodology is proposed for obtaining the optimum fractional order filter parameters based on the minimization of Integral Absolute Error (IAE). The recommended approach is beneficial to balance the necessary tradeoff between performance and robustness. Also, the proposed method provides flexibility in tuning the degree of freedom by adding a fractional order integrator, thus leading to robust performance. The efficacy of the recommended controller is analyzed by simulating numerical examples from the literature. The proposed controller provides enhanced performance and robustness compared to the literature.

Non-parametric fractional-order controller design for stable process based on modified relay

The relay feedback approach is an effective approach for real-time process control. However, it requires accurate limit cycle data. This paper proposes a novel relay feedback approach for fractional-order (FO) controller design to regulate the stable process performance. The proposed methodology does not require prior knowledge of the process and assures accurate limit cycle. Accurate limit cycle parameters are first estimated using the proposed relay. Then, the FO controller is designed based on the output of the limit cycle obtained through the proposed relay, phase, and Nyquist stability criteria. Further, the accuracy of the proposed relay is verified analytically through harmonic analysis. Various performance parameters and indices are estimated and compared with recently published techniques to assess the effectiveness. Additionally, disturbance and parametric uncertainty analyses are conducted to examine the disturbance rejection capacity and robustness. The efficacy of the proposed methodology is further validated on a real-time liquid-level control four-tank process.

An adaptive fractional order controller design: A realization for liquid level regulation in liquid level plant

In this paper, an adaptive FO − IλD1−λ controller design is proposed using a frequency domain approach. The proposed adaptive controller parameters are tuned based on the frequency domain specifications. To realize the velocity error constant for the proposed controller, Oustaloup Recursive Approximation (ORA) method is employed. The proposed controller is realized for liquid level regulation in the Liquid Level Plant (LLP). An identification technique is proposed to identify a linear adaptable parameter transfer function (LAPTF) model of LLP. The performance of the proposed controller is compared with the existing works. It is observed that the proposed controller outperforms the existing controllers in the literature.

Novel applications of oustaloup recursive approximation and modified oustaloup recursive approximation methods in linear fractional order control design

Novel applications of oustaloup recursive approximation and modified oustaloup recursive approximation methods in linear fractional order control design

Observer-Based Decentralized Control for Non-Strict-Feedback Fractional-Order Nonlinear Large-Scale Systems With Unknown Dead Zones.

This article addresses the output-feedback decentralized control issue for the fractional-order nonlinear large-scale nonstrict-feedback systems with states immeasurable and unknown dead zones. The unknown nonlinear functions are identified by neural networks (NNs), and immeasurable states are estimated by establishing an NNs' decentralized state observer. The algebraic loop issue is solved by using the property of NN basis functions and designing the fractional-order adaptation laws. In addition, the fractional-order dynamic surface control (FODSC) design technique is introduced in the adaptive backstepping control algorithm to avoid the issue of ``explosion of complexity.'' Then, by treating the nonsymmetric dead zones as the time-varying uncertain systems, an adaptive NNs' output-feedback decentralized control scheme is developed via the fractional-order Lyapunov stability criterion. It is proven that the controlled fractional-order systems are stable, and the tracking and observer errors can converge to a small neighborhood of zero. Two simulation examples are given to confirm the validity of the put forward control scheme.

Fractional order controller design via gazelle optimizer for efficient speed regulation of micromotors

In this paper, the design of an efficient fractional-order proportional-integral-derivative (FOPID) controller, tailored specifically for the regulation of micro direct current (DC) motors, is explored. A fresh approach is introduced using the gazelle optimization algorithm (GOA), a cutting-edge method set to manage the speed control of these small but vital motors. With the adoption of GOA, fine-tuning the parameters of the FOPID controller is aimed. This is achieved by employing a time-based performance metrics-based cost function as the guiding compass. The GOA is implemented to discover the optimal controller settings for the attainment of peak performance. Through thorough simulations and careful statistical analysis, the worth of GOA-based FOPID controller is demonstrated. Not only are top-notch cost function values achieved, but excellence across various time domain-based performance metrics is also excelled in. The results suggest that the GOA proves to be efficient in fine-tuning of FOPID controller parameters. A comprehensive analysis in the time domain further solidifies the superiority of GOA-based FOPID-controlled micro-DC motor system. Outperformance is observed when compared to alternative algorithms employing different controllers, such as an advanced hybrid stochastic fractal search-based FOPID controller, a grey wolf optimizer-based FOPID controller, a slime mold algorithm-based PID controller, an enhanced arithmetic Harris hawks optimization-based PID controller, and a hybrid atom search optimization and simulated annealing-based PID controller. In essence, the results highlight the potential of the GOA as a powerful tool poised to reshape the optimization of FOPID controllers for micro-DC motor speed regulation.

Bi-fractional-order reference model parameters-based control system design

It is known that analytical manipulation of the time-domain representation of fractional order transfer function is, hitherto, a challenging task. In this study, a controller design methodology based on the direct synthesis design method is proposed using bi-fractional order transfer function as a reference model and certain time-domain criteria. First, an analysis examines the effects of bi-fractional order transfer function parameters, that is, commensurate fractional order, damping ratio and natural frequency, on the system time-domain criteria. This examination has allowed us to set up a relation between damping ratio and commensurate fractional order. Next, a polynomial function fitting is established in order to express the damping ratio in terms of this commensurate order. Bi-fractional-order reference model is regenerated using the above-mentioned relationship. Furthermore, a control design algorithm that utilizes the newly derived bi-fractional order reference model is developed by considering control signal limitations. The proposed fractional order control design method is then compared with globally optimized fractional order proportional-integral-derivative (PID) controllers under the same circumstances and performance index. Finally, the implementation of the proposed controller design algorithm is done on a real-time active suspension system. The results are very satisfactory and incoherent with the simulations.

Design and implementation of fractional-order controller in delta domain

In this work, a fractional-order controller (FOC) is designed in a discrete domain using delta operator parameterization. FOC gets rationally approximated using continued fraction expansion (CFE) in the delta domain. Whenever discretization of any continuous-time system takes place, the choice of sampling time becomes the most critical parameter to get most accurate results. Obtaining a higher sampling rate using conventional shift operator parameterization is not possible and delta operator parameterized discretize time system takes the advantages to circumvent the problem associated with the shift operator parameterization at a high sampling limit. In this work, a first-order plant with delay is considered to be controlled with FOC, and is implemented in discrete delta domain. The plant model is designed using MATLas well as in hardware. The fractional-order controller is tuned in the continuous domain and discretized in delta domain to make the discrete delta FOC. Continuous time fractional order operator (s??) is directly discretized in delta domain to get the overall FOC in discrete domain. The designed controller in implemented using MATLABSimulink and dSPACE board such that dSPACEboard acts as the hardware implemented FOC. The step response characteristics of the closed-loop system using delta domain FOC resembles to that of the results obtained by continuous time controller. It proves that at a high sampling rate, the continuous-time result and discrete-time result are obtained hand to hand rather than the two individual cases. Therefore, the analysis and design of FOC parameterized with delta operator opens up a new area in the design and implementation of discrete FOC, which unifies both continuous and discrete-time results. The discrete model performance characteristics are evaluated in software simulation using MATLAB, and results are validated through the hardware implementation using dSPACE.

STOCHASTIC STABILITY AND PARAMETRIC CONTROL IN A GENERALIZED AND TRI-STABLE VAN DER POL SYSTEM WITH FRACTIONAL ELEMENT DRIVEN BY MULTIPLICATIVE NOISE

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic [Formula: see text]-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.

Stochastic stability of the fractional and tri-stable Van der vol oscillator with time-delay feedback driven by Gaussian white noise

The stochastic P-bifurcation behavior of tri-stability in a fractional-order van der Pol system with time-delay feedback under additive Gaussian white noise excitation is investigated. Firstly, according to the equivalent principle, the fractional derivative and the time-delay term can be equivalent to a linear combination of damping and restoring forces, so the original system can be simplified into an equivalent integer-order system. Secondly, the stationary probability density function of the system amplitude is obtained by the stochastic averaging, and based on the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the properties of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing corresponding parameters in each sub-region divided by the transition set curves. The consistence between numerical results obtained by Monte-Carlo simulation and analytical solutions has verified the accuracy of the theoretical analysis. The method used in this paper has a direct guidance in the design of fractional-order controller to adjust the dynamic behavior of the system.

Zaman gecikmeli sistemler için Padé yaklaşımı ve kesirli dereceli PI kontrolör için yeni bir tasarım metodu

Kesirli dereceli PI kontrolörler, günden güne getirdiği kolaylıklar sayesinde birçok makalenin araştırma konusu olmuştur. Bu tür bir kontrolör tam sayı dereceli PI ile karşılaştırıldığında çok daha faydalı sonuçlar sunar. Fakat işlemleri daha karmaşık bir hale getirir. Bu çalışmada kararlılık sınır eğrisi metodu kullanılarak geliştirilmiş olan dışbükey kararlılık bölgesinin ağırlık merkezine dayalı basit bir kesirli dereceli PI kontrolör tasarım yöntemi önerilmiştir. Bunun için önce kesirli dereceli PI kontrolörün parametre düzleminde tüm kararlı kontrolörleri sağlayan kararlılık bölgesi elde edilir. Daha sonra bu kararlılık bölgesinin köşe ve tepe noktaları belirlenerek dışbükey kararlılık bölgesi elde edilir. Son olarak elde edilen dışbükey kararlılık bölgesinin ağırlık merkezi bulunarak kontrolör parametreleri hesaplanır. Kullanılan yöntem, kontrolör parametrelerinin karmaşık grafiksel yöntemler kullanılmadan hesaplanması, kapalı döngü sisteminin kararlılığını sağlaması ve hesaplama yükünü oldukça azaltması bakımından önemli avantajlar sağlamaktadır. Ayrıca, bu çalışmada kararlılık bölgesinin elde edilmesinde kullanılan denklemler için literatürdeki çalışmalardan farklı olarak zaman gecikmesi terimi yerine Padé yaklaşımlarının kullanılması tercih edilmiştir. Özellikle ikinci dereceden Padé yaklaşımı kullanılarak elde edilen sonuçların neredeyse orijinal sistemle birebir örtüştüğü görülmektedir. Önerilen yöntemin etkinliğini ve basitliğini göstermek için bazı sayısal örnekler verilmiştir.

Fractional-Order Linear Active Disturbance Rejection Control Design and Optimization Based Improved Sparrow Search Algorithm for Quadrotor UAV with System Uncertainties and External Disturbance

This paper presents a generic robust controller that allows applications in various quadrotor unmanned aerial vehicle (UAV) systems effectively even when facing severe system uncertainties and unknown external disturbances. The fractional-order linear active disturbance rejection control (FOLADRC) scheme has combined the advantages of the fractional-order PID (FOPID) with the linear active disturbance rejection control (LADRC). Firstly, the structure of the FOLADRC-based quadrotor UAV is designed. Then, considering the difficulty of parameter tuning of FOLADRC and the demand for accuracy and rapidity of the controller, the improved sparrow search algorithm is applied. Finally, to illustrate the robustness and effectiveness of FOLADRC, the FOLADRC-based quadrotor UAV is firstly compared with PID and LADRC. The simulation and experiment results show that the FOLADRC method can suppress the influence of system uncertainties and external disturbance effectively, where the superiority compared to PID and LADRC has been demonstrated clearly.

Fractional Order Controller Design for Wind Turbines

According to recent studies, it has been concluded that renewable electricity generation is being requested to replace all other fuels more often. In China and the USA, among renewable electricity sources, wind usage has increased significantly compared to 2020. Given these circumstances, the aim of this study was to develop a suitable speed control method for wind power systems in order to achieve maximum power generation while reducing mechanical loads. Several control strategies have been proposed in the literature, all of which offer a compromise between performance and robustness. The present research developed fractional order PID (FOPID) controllers and proved which would be the most suitable controller to address the challenges that wind turbine systems face. The parameters of the FOPID controllers (KP, KI, KD, λ and µ) were tuned with the help of the following optimization algorithms: a genetic algorithm (GA), a multi-objective genetic algorithm (MOGA) and particle swarm optimization (PSO). The results from these three turning methods were then compared to find the method that offered the best performance and system robustness.

Conformable Fractional Order Controller Design and Implementation for Per-Phase Voltage Regulation of Three-Phase SEIG Under Unbalanced Load

Self-excited induction generators (SEIGs) usage is an effective way to generate electrical power in off-grid locations. This study aims to performance improvement for voltage regulation of the SEIG in wind turbine systems. To achieve this, a fixed-capacitor with thyristor controlled-reactor based per-phase voltage controller is designed with a conformable fractional order proportional integral (CFOPI) method. The SEIG 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. in 2014. Proposed CFOPI controller parameters Kp , Ki and γ are optimized using particle swarm optimization (PSO) according to minimization of integral time absolute error (ITAE). To validate the success of proposed scheme, CFOPI, traditional fractional order PI (TFOPI) and integer order PI (IOPI) controllers are tested under same experimental setup and results are compared. Three experiments, after voltage build-up process, under balanced and unbalanced load are done for each controller. Experimental results demonstrate that proposed voltage regulator improves the performance of SEIG systems. Moreover, robust voltage regulation is ensured thanks to independent control loops created for each phase.

Design of fractional-order NPID controller for the NPK model of advanced nuclear reactor

Advanced heavy water reactor (AHWR) is a type of advanced nuclear reactor which is the most efficient nuclear fission reactor for power generation due to the large thorium reserve. Its safe, stable, and efficient operations are critical for the revival of the fission energy sector. Incorporation of a robust control scheme into the nuclear reactor model is required for proper trajectory tracking of demand power. The AHWRs are coupled, significantly nonlinear, and multi-input multi-output (MIMO) systems. External disruptions and time-varying characteristics harm the systems' performance. As a result, the controller built for these systems must be able to deal with the complexity, which is most challenging for control engineers. A fractional-order nonlinear proportional, integral, and derivative (FONPID) control method is presented in this study for normalized power distribution management of the AHWR using normalized point kinetic equations (NPKEs) for trajectory tracking, disturbance rejection, and noise suppression tasks to improve the output power. All controller settings are fine-tuned using a genetic algorithm (GA) with the sum integral of time and square error (ITSE). The suggested FONPID controllers' performance is compared to its integer-order control structure, i.e., NPID and classical PID control structure. External disturbances at controller output and random noise at the sensor output are tested for resilience to establish the usefulness of the suggested control methods. The simulation results showcased that the proposed FONPID controller outperforms its integer-order (IO) counterpart as well as the traditional PID controller.

A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function

In this study, a robust fractional-order controller design methodology for a type of fractional-order or integer-order model with dead time is proposed using phase and gain margin specifications. The delayed Bode’s ideal transfer function is used as a reference model to design the controller analytically. The delay term in delayed Bode’s ideal transfer function provides the exact determination of these frequency domain specifications when the system owns a dead time. The analytical robust controller design problem is transformed to solving four nonlinear equations with four unknown variables, two of which are the desired specifications; namely, phase and gain margins. The remaining two are the phase and gain cross-over frequencies. Next, some conditions are set based on the desired specifications so that nonlinear equations provide a unique solution. The proposed method is compared with the other existing robust controller methods based on the same frequency domain specifications. The simulation results reveal that the proposed method outperforms the other methods and also gives closer outcomes to the desired specifications.

Operator & Fractional Order Based Nonlinear Robust Control for a Spiral Counter-Flow Heat Exchanger with Uncertainties and Disturbances

This paper studies operator and fractional order nonlinear robust control for a spiral counter-flow heat exchanger with uncertainties and disturbances. First, preliminary concepts are presented concerning fractional order derivative and calculus, fractional order operator theory. Then, the problem statement about nonlinear fractional order derivative equation with uncertainties is described. Third, the design of an operator fractional order controller and fractional order PID controller and determination of several related parameters is described. Simulations were performed to verify tracking and anti-disturbance performance by comparison to different control cases; verification is described and concluding remarks provided.

Fractional-Order PI Controller Design Based on Reference–to–Disturbance Ratio

The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order controllers have now been researched intensively in terms of improving the closed-loop results and robustness of the control system, compared to the standard integer-order controllers. In this study, a novel tuning method for fractional-order controllers is developed. The tuning is based on improving the disturbance attenuation of periodic disturbances with an estimated frequency. For this, the reference–to–disturbance ratio is used as a quantitative measure of the control system’s ability to reject disturbances. Numerical examples are included to justify the approach, quantify the advantages and demonstrate the robustness. The simulation results provide for a validation of the proposed tuning method.