Fractional Order Plants Research Articles

State Space Model Reference Adaptive Control for a Class of Second-Degree Fractional Order Systems with Different Eigenvalues

This study proposes an adaptive control synthesis for a class of second-degree fractional order systems with different eigenvalues in the state-space domain. The proposed fractional order adaptive controller is a generalization of the MRAC controller for the class of scalar fractional order systems. In order to control the fractional order plant, an adaptive state space feedback controller is applied based on the error between the system output and a chosen reference model using a fractional adaptation law to make the fractional order plant track the fractional order reference model. We show that the resulting adaptive regulator is able to stabilize the fractional order second degree system with a satisfying performance. A simulation example illustrating these performance properties is provided along with a comparison with a fractional order sliding mode control (FOSMC) to demonstrate the superiority of the proposed control scheme.

A graphical method to determine robust stabilizing region of FOPID controllers for stable/unstable fractional-order plants with interval uncertainties of a fractional order and model coefficients

This paper focuses on robustly stabilizing stable and unstable fractional-order plants with one uncertain fractional-order term and interval uncertainties using fractional order P I μ D λ controllers. Two necessary and sufficient conditions are provided to check the robust stability of the closed-loop control system. Moreover, the D-decomposition technique is utilized to determine the robust stability region of the system. Subsequently, evolutionary algorithms, such as the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE), can be utilized to discover a fractional-order controller within the region of robust stability. This work introduces three primary contributions, outlined as follows: (1) Utilizing a graphical approach, a set of stabilizing controller is obtained. (2) Rather than employing just a single stabilizing fractional-order controller, a collection of controllers is provided for the control system. (3) Employing evolutionary algorithms to find an optimal fractional-order controller. Finally, four numerical examples are presented to validate the results.

Analyzing complex fractional order systems physical phenomena in IOPI controller design

AbstractThe Fractional Complex Order Plant model, which has lately gained popularity in applied physics and control systems, is the main subject of this study. The major contribution of this study to the literature is the discussion of the physical phenomena of complex plant models and how they affect the stability and robustness of the systems. Because the Fractional Complex Order Plant model is the most general mathematical form, other plant models covering the Integer Order Plant and the Fractional Order Plant can be easily created with this benefit. The proposed approach using the classical Proportional Integral controller which is recalled as the Integer‐Order PI controller in this paper gives the calculation equations of the physical alterations of plants having integer, fractional, and complex orders. Along with the visuals and with the aid of simulations, the consequences of the parameters on the system are described. Additionally, the advantages and disadvantages of the proposed controller designs for each of the three plant species are discussed.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

An examination of complex fractional order physical phenomena in IOPD controller design

This research focuses on the fractional complex order plant (FCOP). The significant contribution is the role of complex plant models in system stability and robustness and associated physical phenomena. A general transfer function is studied in the paper. Other plant models may be built with this structure since the FCOP is a general mathematical form covering integer order plant (IOP) and fractional order plant (FOP). Using the equations produced with the proposed technique and the recommended integer order proportional derivative (IOPD controller, physical changes in integer, fractional and complex coefficients, and orders are observed within this paper. Analysis of the plant controlled with an IOPD controller is done by applying an integrator to reveal the differences. The effects of the parameters are discussed together with the visuals, supported by simulations. The aim is to tune the controller parameters to achieve the phase and specifications as the researcher desired. It is observed that the integrator greatly takes part in reducing the steady‐state error. The IOP with the integrator showed the lowest steady‐state error, and also, the settling and overshoot time were enhanced. Increase in the phase margin also caused an increase in the phase crossover frequency. It is also observed that the fractional order affected the phase crossover frequency comparing with the IOP, and the complex order modification also had an effect comparing to the fractional order version. The complex order of the system is considered with its conjugate components in the imaginary part thus, the results are found separately for each case.

Modeling and analysis of sustainable approach for dynamics of infections in plant virus with fractal fractional operator

In this paper, studying a sustainable approach to see the dynamics of infection in the plant by using a fractal fractional derivative. To depict a time-fractional order plant virus model including disease consequences, we suggested a set of fractional differential equations. For the fractional order system, both qualitative and quantitative studies are carried out. To prove the existence and uniqueness of the proposed model with the effect of global derivative, Linear growth and Lipschitz conditions are used. Positiveness and boundedness of solutions of the fractional order model are verified. Local stability analysis and global stability is verified by using the Lyapunov function with the first and second derivative tests. Sensitivity analysis is performed to see the influence of parameters on the fractional order model. To study the effect of the fractional operator, which demonstrates the impact of the illness on plants, solutions are generated with a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel. To observe the behavior of the fractional order plant virus model, numerical simulation is used. Such an inquiry will help to comprehend how the virus behaves and to create defences against infected plants.

Robust Stability Analysis of Smith Predictor Based Interval Fractional-Order Control Systems: A Case Study in Level Control Process

The robust stability study of the classic Smith predictor-based control system for uncertain fractional-order plants with interval time delays and interval coefficients is the emphasis of this work. Interval uncertainties are a type of parametric uncertainties that cannot be avoided when modeling real-world plants. Also, in the considered Smith predictor control structure it is supposed that the controller is a fractional-order proportional integral derivative (FOPID) controller. To the best of the authors' knowledge, no method has been developed until now to analyze the robust stability of a Smith predictor based fractional-order control system in the presence of the simultaneous uncertainties in gain, time-constants, and time delay. The three primary contributions of this study are as follows: i) a set of necessary and sufficient conditions is constructed using a graphical method to examine the robust stability of a Smith predictor-based fractional-order control system—the proposed method explicitly determines whether or not the FOPID controller can robustly stabilize the Smith predictor-based fractional-order control system; ii) an auxiliary function as a robust stability testing function is presented to reduce the computational complexity of the robust stability analysis; and iii) two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction. Finally, four numerical examples and an experimental verification are presented in this study to demonstrate the efficacy and significance of the suggested technique.

Analysis of fractional order systems using newton iteration-based approximation technique

<span lang="EN-US">Fractional differential equations play a major role in expressing mathematically the real-world problems as they help attain good fit to the experimental data. It is also known that fractional order controllers are more flexible than integer order controllers. But when it comes to the numerical approximation of fractional order functions inaccuracies arise if the conversion technique is not chosen properly. So, when a fractional order plant model is approximated to an integer order system, it is required that the approximated model be accurate, as the overall system performance is based on the estimated integer order model. Nitisha-Pragya-Carlson (NPC) is a recent approximation technique proposed in 2018 to derive the rational approximation of fractional order differ-integrators. In this paper, three fractional order plant models having fractional powers 3.1, 1.25 and 1.3 is analyzed in frequency domain in terms of magnitude and phase response. The performance of approximated third and second order NPC based integer model is studied and compared with the integer models developed using other existing technique. The approximation error is calculated by comparing the frequency response of the developed models with the ideal response. It has been found that in all the three examples NPC based models are very much close to the ideal values. Hence proving the efficacy of NPC technique in approximation of fractional order systems.</span>

Stabilizing region of fractional-order proportional integral derivative controllers for interval fractional-order plants

This paper focuses on computing the stabilizing region of fractional-order proportional integral derivative (FOPID) controllers for interval fractional-order plants. In this study, first, a theorem is presented to investigate the robust stability of the given closed-loop system. Then, an algorithm is proposed to solve the problem of robustly stabilizing the interval fractional-order plant using FOPID controllers. Also, two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction. Finally, four numerical examples are presented to confirm the obtained results.

A simple modelling strategy for integer order and fractional order interval type-2 fuzzy PID controllers with their simulation and real-time implementation

This paper presents a simple approach to the mathematical modelling of the Integer Order (IO) and Fractional Order (FO) Interval Type-2 Fuzzy Proportional Integral Derivative (IT2FPID) controllers. In this approach, the control effort produced by the IT2FPID controller is obtained by combining the individual control efforts generated by the Interval Type-2 Fuzzy Proportional (IT2FP), Interval Type-2 Fuzzy Integral (IT2FI), and Interval Type-2 Fuzzy Derivative (IT2FD) controllers. The analytical structures of each of the IT2FPID components are unveiled using one-dimensional (1D) input space and without using any AND or OR operator. To the best of the authors’ knowledge, such a strategy was never used and is completely new for the modelling of the IT2FPID controllers. Properties, computational issues, and design guidelines of the newly obtained controllers are discussed, and their applicability is delineated using five simulation examples and one experimental case study. A chemical reactor plant, two fractional order plants, a nonlinear plant, and a Twin Rotor Multi Input Multi Output System (TRMS) are considered in the simulation study. The experimental study is conducted on an unstable nonlinear Magnetic Levitation System (MLS). Moreover, to exhibit the usefulness of the newly derived controllers, simulation and real-time results are also compared with the existing results available in the literature. As the proposed controllers are plant model-free, they can be used for other control applications also.

Robust stabilization criteria of a general form of fractional-order controllers for interval fractional-order plants with complex uncertain parameters

This paper studies the problem of robust stabilization of interval fractional-order plants with complex uncertain parameters by using fractional-order controllers. An interval fractional-order plant with complex uncertain parameters means that the coefficients of the numerator and denominator of the plant are all uncertain and may be complex numbers and lie in specified intervals. At first, by using a graphical approach, necessary and sufficient conditions are presented for the stabilization of the fractional-order plant containing complex coefficients. Then, by using some interesting geometric features of convex polygons, a robust stability checking function is presented for the stabilization. Also, an upper frequency bound is introduced to reduce the computational burden. Finally, six examples are provided to illustrate the results.

Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system

<p style='text-indent:20px;'>The approximation of the fractional-order controller (FOC) has already been recognized as a distinguished field of research in the literature of system and control. In this paper, a two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractional-order plant model. The design goals are based on some frequency domain specifications. The first stage of the work is focused on developing the pure continuous-time FOC, while the second stage actually realizes the FOPI controller in discrete-time representation. The presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain. In the process of realizing the FOC, the delta operator has been involved as a generating function due to its exclusive competency to unify the discrete-time system and its continuous-time counterpart at low sampling time limit. The well-known continued fraction expansion (CFE) method has been employed to approximate the FOPI controller in delta-domain. Simulation outcomes exhibit that the discrete-time FOPI controller merges to its continuous-time counterpart at the low sampling time limit. The robustness of the overall system is also investigated in delta-domain.</p>

Robust Stability Analysis of Interval Fractional-Order Plants With Interval Time Delay and General Form of Fractional-Order Controllers

This letter aims to investigate the robust stability of an interval fractional-order plant with an interval time delay by a general form of fractional-order controllers. Based on a simple graphical procedure, necessary and sufficient criteria are proposed to analyze the robust stability of an interval fractional-order plant. Then, the idea of “robust stability testing function” is extended to analyze the robust stability of the considered type of systems. Besides, an upper frequency bound is provided which is useful for the simulation purposes and it helps to reduce the computational cost. Finally, two examples are presented to verify the usefulness of the obtained results.

Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllers

In this paper, the robust stability of interval fractional order plants with one time delay controlled by fractional order controllers is investigated in a general form. For robust stability analysis of the closed loop system by the zero exclusion principle, the distance between the origin and the value set of the characteristic function needs to be checked. It is known that the outer vertices of this value set may change at some switching frequencies and the repetitive calculation of these vertices at switching frequencies leads to additional calculations. In this study initially, new necessary and sufficient conditions are proposed to check the robust stability of a delayed fractional order closed loop system. Then, a novel robust stability testing function is presented based on some vertices, which are fixed for all positive frequencies. Therefore, no additional calculation is needed to obtain the outer vertices of the characteristic function value set for any pair of the switching frequencies. Also, a finite frequency range is presented to reduce the computational cost noticeably. Eventually, three numerical examples are given to verify the efficiency of the results of this paper.

Robust stability analysis of interval fractional-order plants by fractional-order controllers: an approach to reduce additional calculation

This paper investigates the robust stability of fractional-order plants suffering from interval uncertainties by using fractional-order controllers. Based on the zero-exclusion principle, at first, new necessary and sufficient criteria are proposed to analyze the robust stability of the corresponding characteristic polynomial. Then, an auxiliary function is presented to investigate the robust stability criteria. The results of this paper also help to reduce additional calculations noticeably since the vertices used in the auxiliary function are fixed for all positive frequencies. Finally, three numerical examples are given to indicate the effectiveness of the paper results.

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.

Tuning of complex coefficient PI/PD/PID controllers for a universal plant structure

Controllers are tuned either analytically or through optimisation techniques for integer, fractional and fractional complex order plants with dead time by considering only the positive frequency information of the plant. This is applicable only for integer / fractional order plants containing real coefficients which have an even symmetrical magnitude and odd symmetrical phase behaviour in frequency responses. The frequency response for (i) integer / fractional order plants containing complex coefficients and (ii) fractional complex order plants containing real / complex coefficients have unsymmetrical magnitude and phase behaviour. Tuning of controllers for such plants by considering only its positive frequency response produces reduced stability margins and in turn deteriorates its time response. Hence, tuning of controllers for such plants requires both positive and negative frequency information which in turn demands complex coefficient controllers. To address the problem, complex coefficient PI/PD/PID controllers are proposed in this paper. Unified expressions are derived for complex coefficient PI/PD/PID controllers by considering both positive and negative frequency information of plant with complex coefficients and complex order derivatives plus dead time. Numerical simulations are performed for three different case studies with the proposed complex coefficient integer order controllers and results are compared with real coefficient PI/PD/PID controllers under various operating conditions.

Robust stabilizability of fractional order proportional integral controllers for fractional order plants with uncertain parameters: A new value set based approach

In this paper, a new value set based approach is presented to study the robust stability of a fractional order plant with parametric uncertainty of the interval type by using a fractional order proportional integral controller. The concept of interval uncertainty means that the parameters of the system transfer functions are uncertain parameters that each adopts a value in a real interval. Based on the zero exclusion principle, it is necessary to check whether the value set of the characteristic polynomial for the infinite frequency range includes the origin or not. To this end, in this paper, an auxiliary function is defined. The sign of this auxiliary function within a finite frequency range verifies whether the origin is included in the value set of the characteristic polynomial or not. The upper bound and the lower bound of this finite frequency range are obtained using the triangle inequality. Finally, the results of the numerical examples are provided, which confirm the effectiveness of the paper results.

Fractional-order partial pole assignment for time-delay systems based on resonance and time response criteria analysis

In this paper, a novel fractional-order partial pole assignment (FPPA) control algorithm is proposed for systems with time-delay. The FPPA control algorithm is essentially an extension of the original pole assignment, which could change undesired pole locations into desired pole locations. The presented control scheme can be used on open loop poorly damped or unstable systems, which is superior to most other time-delay compensation schemes. The discussion on choosing desirable pole locations is presented based on stability and resonance conditions in the frequency domain. The controlled system is also studied in the time domain based on different transient performance indicators, namely overshoot, settling time, and rising time. In addition, the parameters of the proposed FPPA control algorithm are tunable, thus the control scheme can be used to satisfy different control requirements. Simulation results of stable and unstable fractional-order plants with time-delay are shown to verify the effectiveness and practicability of the FPPA control algorithm.

Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay

This study investigates the robust stability of the retarded type of interval fractional order plants with an interval time delay. To this end, the characteristic quasi-polynomial is divided into two terms. The first term is simply the denominator interval polynomial of the open loop system and the second term is the multiplication of the interval delay term in the numerator of the open loop system which is an interval polynomial. Each of these two terms of the characteristic quasi-polynomial makes their own value sets in the complex plane for a given frequency. In this paper, based on these two value sets and by using the zero exclusion principle, the robust stability of the closed loop system by applying a FOPID controller is analyzed. Finally, two numerical examples and an experimental verification are provided to demonstrate the effectiveness of the proposed method in the robust stabilization of fractional order plants with interval uncertainties and interval time delay.