Fractional Order Processes Research Articles

Experimentally validated analytical single parametric FOTID control for time-delayed fractional order processes

In current studies, non-integer order (NIO) plants with dead time have gained the significant attention of many researchers in the area of control theory. However, the design methods for fractional order controller tuning suffer from a lack of direct systematic tuning strategies due to a higher number of tuneable parameters. Most of the works on such design are optimisation-based whereas the analytical solutions are achieved with assumptions and approximations leading to compromised closed-loop robustness. This paper proposes a simple analytical design of a fractional-order tilt-integral-derivative (FOTID) controller for time-delayed NIO processes with a single tuneable parameter. Explicit tuning rules in terms of plant parameters are derived with user-defined stability margins. To prove the efficiency of the proposed control law, simulation comparisons with recently developed tuning rules are included. Lastly, the proposed control law is tested through experiments in a two-tank level loop for real-time validation.

A frequency domain-based loop shaping procedure for the parameter estimation of the fractional-order tilt integral derivative controller

<p>This paper demonstrates a frequency domain-based loop shaping method for the parameter estimation of a fractional order tilt integral derivative (FOTID) controller for the interval integer and fractional order time-delay systems. Along with the five nonlinear constraints usually considered for the design of the fractional order proportional integral derivative (FOPID) controller, a more flat phase concept signifying an enhanced robustness of the system towards gain variations is adopted as the sixth constraint for the tuning of a six variable tunable FOTID controller. The optimization toolbox fmincon in MATLAB is utilized for the solution process of the above constraint minimization problem. A certain class of fractional order plus time delay process is considered for the implementation and validation of the above procedure. The robustness of the FOTID controller optimized by the proposed method is tested against variations of the system parameters. By considering different numerical examples, the technical superiority of the FOTID controller over the FOPID controller is demonstrated through suitable comparisons in this current work.</p>

IMC-based fractional order TID controller design for different time-delayed chemical processes: case studies on a reactor model

Abstract To achieve good servo and regulatory responses, a generalized fractional-order tilt-integral-derivative (FOTID) control is developed in this study for time-delayed fractional-order processes. To enhance the closed-loop robustness, the controller parameters are calculated based on maximum sensitivity (M s ). To check the performance and robustness of the suggested control law, several case studies on industrial processes like DC servo systems, Level-loop, Bio-reactors, Fuel cells and CSTR are performed and compared with existing designs. The robustness of the proposed controller is analysed by employing 20 percent perturbation in plant parameters. Lastly, a comparison of the closed-loop response on different error indices is included.

Remote Monitoring the Parameters of Interest in the 18O Isotope Separation Technological Process

This manuscript presents the remote monitoring of the main parameters in the 18O isotope separation technological process. It proposes to monitor the operation of the five cracking reactors in the isotope production system, respectively, the temperature in the preheating furnaces, the converter reactors and the cracking reactors. In addition, it performs the monitoring of the two separation columns from the separation cascade structure, respectively, the concentrations of the produced 18O isotope and the input nitric oxides flows. Even if the production process is continuously monitored by teams of operators, the professionals who designed the technical process and those who can monitor it remotely have the possibility to intervene with the view of making the necessary adjustments. Based on the processing of experimental data, which was gathered from the actual plant, the proposed original model of the separation cascade functioning was developed. The process computer from the monitoring system structure runs the proposed mathematical model in parallel with the real plant and estimates several signal values, which are essential to be known by the operators in order to make the appropriate decisions regarding the plant operation. The separation process associated with the final separation column from the separation cascade structure is modeled as a fractional-order process with variable and adjustable differentiation order, which represents another original aspect. Neural networks have been employed in order to implement the proposed mathematical model. The accuracy, validity and efficiency in the operation of the proposed mathematical model is demonstrated through the simulation results presented in the final part of the manuscript.

Proposal of a General Identification Method for Fractional-Order Processes Based on the Process Reaction Curve

This paper aims to present a general identification procedure for fractional first-order plus dead-time (FFOPDT) models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting three arbitrary points on the process reaction curve. In order to validate this procedure and check its effectiveness for the identification of fractional-order models from the process reaction curve, analytical expressions of the FFOPDT model parameters have been obtained for both situations: as a function of any three points and three points symmetrically located on the reaction curve, respectively. Some numerical examples are provided to show the simplicity and effectiveness of the proposed procedure. Good results have been obtained in comparison with other well-recognized identification methods, especially when simplicity is emphasized. This identification procedure has also been applied to a thermal-based experimental setup in order to test its applicability and to obtain insight into the practical issues related to its implementation in a microprocessor-based control hardware. Finally, some comments and reflections about practical issues relating to industrial practice are offered in this context.

FOPI/FOPID Tuning Rule Based on a Fractional Order Model for the Process

This paper deals with the design of a control system based on fractional order models and fractional order proportional-integral-derivative (FOPID) controllers and fractional-order proportional-integral (FOPI) controllers. The controller design takes into account the trade-off between robustness and performance as well as the trade-off between the load disturbance rejection and set-point tracking tasks. The fractional order process model is able to represent an extensive range of dynamics, including over-damped and oscillatory behaviors and this simplifies the process modelling. The tuning of the FOPID and FOPI controllers is achieved by using an optimization, as a first step, and in a second step, several fitting functions were used to capture the behavior of the optimal parameters of the controllers. In this way, a new set of tuning rules called FOMCoRoT (Fractional Order Model and Controllers Robust Tuning) is obtained for both FOPID and FOPI controllers. Simulation examples show the effectiveness of the proposed control strategy based on fractional calculus.

Dynamic analysis of mean-field and fractional-order epidemic vaccination strategies by evolutionary game approach

Individual's perception of the participant in a vaccine program reflects their intrinsic appreciation of the trade-off between vaccine behavior, risk of infection, and memory effect. Here we present a mean-field approximation and fractional-order model embedded with the evolutionary game theory (EGT) process for susceptible-vaccinated-infected-recovered (SVIR) epidemic dynamics, where the two types of immunity, artificial and natural immunity, are considered. Besides the well-known vaccination game models, we successfully establish a theoretical approach of fractional-order dynamics for vaccination games in which epidemic spread and individual decisions are supposed to evaluate social behavior. The EGT governs the strategy adoption, while the fractional-order process directs the memory effect state. Our analytical forecasts are validated by numerical simulation of the finite difference method and Adams-Bashforth-Moulton algorithm for the mean-field and Caputo fractional-order derivative that assesses various graphs at varying parameters. Our results show that an effective vaccine may meaningfully minimize the risk of infection. However, despite vaccines' obvious impacts and advantages on a community, many individuals choose not to vaccinate for various reasons, leading to anti-vaccination groups and vaccine apprehension. In this aspect, people are interested in gaining natural immunity. Notably, the individual's risk perception is fundamental for controlling the infection, while the fractional-order dynamics mainly dictate the degree of freedom with memory effect. Higher fractional order with EGT leads to an improved vaccination intake, while a cheaper and reliable vaccine ensures to lessening of contagious disease. The proposed model captures relevant disease behavior and the memory effect of a synchronized pandemic event, emphasizing the underlying role of social schemes.

Artificial Intelligence in Fractional-Order Systems Approximation with High Performances: Application in Modelling of an Isotopic Separation Process

This paper presents a solution for the modelling, implementation and simulation of the fractional-order process of producing the enriched 13C isotope, through the chemical exchange between carbamate and carbon dioxide. To achieve the goal of implementation and simulation of the considered process, an original solution for the approximation of fractional-order systems at the variation of the system’s differentiation order is proposed, based on artificial intelligence methods. The separation process has the property of being strongly non-linear and also having fractional-order behaviour. Consequently, in the implementation of the mathematical model of the process, the theory associated with the fractional-order system’s domain has to be considered and applied. For learning the dynamics of the structure parameters of the fractional-order part of the model, neural networks, which are associated with the artificial intelligence domain, are used. Using these types of approximations, the simulation and the prediction of the produced 13C isotope concentration dynamics are made with high accuracy. In order to prove the efficiency of the proposed solutions, a comparation between the responses of the determined model and the experimental responses is made. The proposed model implementation is made based on using four trained neural networks. Moreover, in the final part of the paper, an original method for the online identification of the separation process model is proposed. This original method can identify the process of fractional differentiation order variation in relation to time, a phenomenon which is quite frequent in the operation of the real separation plant. In the last section of the paper, it is proven that artificial intelligence methods can successfully sustain the system model in all the scenarios, resulting in the feasible premise of designing an automatic control system for the 13C isotope concentration, a method which can be applied in the case of other industrial applications too.

Robust design of fractional order IMC controller for fractional order processes with time delay

AbstractThis article presents a novel approach to the robust design of fractional order IMC controller for the fractional‐order (FO) plus time delay processes. The proposed methodology provides the flexibility to adjust the controller performance based on the application requirement using a single parameter. Also, it does not require any external filter. Thus, it ensures a better response over existing methods. A mathematical analysis is provided to select the tuning parameter to obtain the optimal solution. The effectiveness of the proposed methodology is measured for different FO processes with time delay using time‐domain specifications, performance indices and frequency domain parameters. Load disturbance analysis under various disturbances and robust analysis under uncertainty in process gain in the presence of the disturbances is carried out to study the sensitivity and robustness of the controller. The performance parameters are estimated in both analyses and compared with the different approaches available in the literature.

Controlled Fractional Order Processes with Distributed Parameters

This work is devoted to the study of the problem of pursuit in systems controlled with distributed parameters of fractional order. Derivatives with respect to spatial variables are ordinary, integer, and even of arbitrary order. Designed and studied sampling schemes. A numerical method is constructed for finding strategies in suitable classes and for constructing the corresponding control laws. Theorems are proved for the possibility of completing the pursuit by the finite difference method, and they can be used to solve finite-difference optimal control problems or numerical solutions of differential games of fractional order. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.

An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment

The present work tackles the modeling of the motion dynamics of an object submerged in a non-Newtonian environment. The mathematical model is developed starting from already known Newtonian interactions between the submersible and the fluid. The obtained model is therefore altered through optimization techniques to describe non-Newtonian interactions on the motion of the vehicle by using real-life data regarding non-Newtonian influences on submerged thrusting. For the obtained non-Newtonian fractional order process model, a fractional order control approach is employed to sway the submerged object’s position inside the viscoelastic environment. The presented modeling and control methodologies are solidified by real-life experimental data used to validate the veracity of the presented concepts. The robustness of the control strategy is experimentally validated on both Newtonian and non-Newtonian environments.

Fractional Order Processing of Satellite Images

Nowadays, satellite images are used in many applications, and their automatic processing is vital. Conventional integer grey-scale edge detection algorithms are often used for this. This study shows that the use of color-based, fractional order edge detection may enhance the results obtained using conventional techniques in satellite images. It also shows that it is possible to find a fixed set of parameters, allowing automatic detection while maintaining high performance.

Design of Indirect Fractional Order IMC Controller for Fractional Order Processes

In this brief, indirect design estimation of fractional order systems is proposed. In indirect fractional order approach, fractional order plant is shifted in the frequency domain and the equivalent plant is modeled by employing binomial approximation. The equivalent fractional order plant obtained is used for the design of the controller. While designing fractional order controllers, robustness and handling sensitivity to parametric variations are of key importance. Therefore, indirect fractional order approach is used, which gives the flexibility to adjust the maximum sensitivity according to the system requirements and eliminates the need for external IMC filter. IMC filter added externally results in an additional phase lag of the system. The proposed design method excludes external IMC filter and the binomial expansion used helps in achieving the internal filter which results in better transient response. This is termed as Indirect Fractional order IMC based fractional order PID controller.

A Robust Internal Model-Based Fractional Order Controller for Fractional Order Plus Time Delay Processes

This letter proposes a fractional order internal model controller (FO-IMC) for a fractional order plus time delay (FrOPTD) process model to satisfy desired gain margin ( $A_{m}$ ) and phase margin ( $\phi _{m}$ ) specifications. A method is proposed to solve four equations arising from the $A_{m}$ and $\phi _{m}$ specifications, which also generates solution of the system gain-cross-over frequency ( $\omega _{g}$ ) and phase-cross-over frequency ( $\omega _{p}$ ) as intermediate variables. The designed FO-IMC is implemented on laboratory DC servo-system and compared with other integer order controller and fractional order controller.

Characteristic analysis of a simple fractional-order chaotic system with infinitely many coexisting attractors and Its DSP implementation

A new simple third-order chaotic system is proposed. The numerical solution of the proposed chaotic system is calculated by using the Adomian decomposition method. The phenomena of infinitely many coexisting attractors is found in this new chaotic system. This interesting physical phenomenon don’t disappear after fractional-order processing. Conversely, with the order of fractional-order system changes, it shows more complex dynamical characteristic than the original system. In particular, the dynamical behavior of the new fractional-order system is analyzed by using the methods of bifurcation diagram, complexity. Finally, the chaotic attractors are physically implemented by DSP experiment. An extremely simple chaotic system as the demonstration of the fractional-order characteristic, it is of great significance to the application of the special chaotic system in related fields.

Performance Assessment of FO-PID Temperature Control System Using a Fractional Order LQG Benchmark

In this paper, a fractional order LQG benchmark is proposed for the control performance assessment of fractional order control systems. Similar to the conventional LQG benchmark, the fractional order LQG performance benchmark curve is determined by the numerical calculation method, which avoids the calculation of the complex interaction matrix. The fractional order process model is discretized via fractional order calculus. Meanwhile, the fractional order integral is introduced into the conventional LQG cost function. Then solving the linear quadratic Gaussian problem under the fractional order model and fractional control, the optimal input and output variances are determined for different weighting factors and the performance curves can be achieved. The comparison between fractional order LQG and the conventional LQG shows the improvement of the proposed benchmark under the same condition. The proposed benchmark can provide a more direct and superior reference standard to evaluate the performance of fractional order control system. Finally, a case study of fractional order PID(FO-PID) controller in industrial heating furnace temperature control experiment with model matching and model mismatch conditions is used to verify the effectiveness of the proposed benchmark.

A Novel Design of Fractional PI/PID Controllers for Two-Input-Two-Output Processes

In this paper, a new formulation of fractional order proportional integral (PI)/ proportional integral derivative (PID) controller is proposed. The proposed controller will be justified for some well-known two-input-two-output (TITO) processes. In order to deal with interactions between process variables in a multivariable system, as well as multiple delay times in process transfer functions, the simplified decoupling Smith predictor (SDSP) structure is also used. The issue of decoupling realizability is solved by the PSO algorithm and fractional order processes are also suggested for model reduction. The tuning rules of the controller are derived in analytical terms based on the internal model control (IMC) structure. The effectiveness and robust stability of the proposed approach are illustrated by comparing it with other methods. To have a fair comparison, the robustness criterion using the M-Δ structure with μ-synthesis is adopted and the μ value of the proposed method is always kept smaller than the value of the others.

Extracting soil salinization information with a fractional-order filtering algorithm and grid-search support vector machine (GS-SVM) model

ABSTRACTThe remote sensing information on the extraction method is of great importance to improve the accuracy and efficiency of soil salinization information. The objective of this study is to develop remote sensing extraction techniques to improve soil salinization maps. The following procedures were used in this study: (1) developed a fractional-order algorithm-based methodology of filter from high-resolution remote sensing imagery (Sentinel-2 MSI); (2) investigated the changing trend of image under different order filters; and (3) used a grid-search algorithm-support vector machines (GS-SVM) classification to employ extraction information of soil salinization. The results showed that the Fractional-order filter method outperformed the integer derivative in extracted information of soil salinization. In comparison of the classification accuracy between fractional-order processing algorithm and integer-order image processing algorithm, the fractional order has improved remarkably. The optimal classification model was 0.6 order, 0.8 order, 1.4 order, 1.6 order, and 1.8 order models. The overall accuracy and kappa coefficient (κ) of these models are 91.90% and 0.90, respectively. Analysing and comparing between soil salt index and filtering algorithm (1.2 order), the researchers found that the classification results of the two methods are similar. In general, this method can successfully extract soil salinization information in dry regions.

Extended Kalman filters for nonlinear fractional-order systems perturbed by colored noises

The fractional-order extended Kalman filter (FEKF) algorithm for a nonlinear fractional-order system perturbed by the colored noise is presented. Firstly, the first-order Taylor expansion is employed to linearize the nonlinear functions in the estimated system. Then, Grünwald–Letnikov difference (GLD) and the concept of fractional-order average derivative (FOAD) are employed to discretize nonlinear fractional-order systems perturbed by colored fractional-order process or measurement noise. An augmented system determined by the state and colored noises is presented to treat colored noises. Hence, the FEKFs using GLD and FOAD are carried out, respectively. By comparing two kinds of Kalman filters, FEKFs using FODA can gain the better effect of filtering for colored process or measurement noise to raise the estimation precision. Finally, we discuss three examples to show the validity of investigated FEKFs.

Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is to demonstrate a simple and effective method for the treatment of the output of fractional processes in the form of time series. The proposed method is based on fractional-order differentiation/integration using the Grünwald–Letnikov definition of the fractional-order operators. With this simple approach, we observe important properties in the time series and make decisions in real process control. Finally, an illustrative example for a real data set from a steelmaking process is presented.