Non-integer Order Systems Research Articles

A deterministic design approach of tilt integral derivative controller for integer and fractional-order system with time delay

A tilt integral derivative (TID) controller modifies the proportional integral derivative (PID) controller in the fractional domain. It converts the proportional gain as a function of frequency and is thereby capable of achieving optimal system response. The usual practice for the parameter estimation of the TID controller is by minimization of the error-based objective functions using optimization techniques. Although precise results can be achieved, these nature-inspired algorithms are stochastic and hence produce different solutions during different iterations. Therefore, a comparative statistical study is usually necessary to validate the best possible result. This study shows a deterministic analytical procedure for the paramssseter estimation of TID controllers. The magnitude and phase angle criteria, along with the frequency-domain loop shaping specifications, are used for the explicit evaluation of the TID parameters. Because of its model-independent nature, this tuning strategy can be used for a variety of integral and nonintegral order systems with different plant structures. In this article, the authenticity of the applied procedure is demonstrated through suitable numerical examples. The complexity of the design problem is enhanced by using it for both integer and non-integer (fractional) order plus time-delay systems. Further, the robustness of the control system in the presence of a TID controller was examined under the influence of external parameters and input reference changes. Simulation studies validate the supremacy of TID controllers over PID controllers in terms of reference tracking and disturbance rejection capabilities.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Finite-time sliding mode control methods for a class of non-integer-order systems with input saturations and its application

Control and synchronization of chaotic dynamical systems is a key issue in engineering that has numerous applications in the applied sciences. In this research, single input finite-time sliding mode (FTSMC) control algorithms are developed to synchronize and stabilize a class of three-dimensional non-integer order systems where input saturation is present. Using the non-integer version of the Lyapunov stability theory (LST) and the dynamic-free idea, techniques are devised to suppress the improper behavior of the aforementioned fractional-order (FO) chaotic systems without unpleasant chattering phenomena. The proposed FTSMC approach can be utilized to stabilize and synchronize systems that include model uncertainty, external disturbances, and input saturation. The developed single input techniques have the benefits of being model-free, robust to uncertainty, user-friendly, and establishing equilibrium in a finite amount of time. In addition, the efficacy and applicability of the FTSMC approaches are shown by synchronizing two different industrial FO chaotic systems and chaos suppressing of the PMSM chaotic system utilizing these methods.

Computational analysis of different Pseudoplatystoma species patterns the Caputo-Fabrizio derivative

Abstract Pattern formation processes in non-integer order systems are increasingly becoming the subject of activity considered by many scientists and engineers for scenarios associated with spatial heterogeneity or anomalous diffusion. The major drawback encountered is the computation of the Caputo and Fabrizio fractional operator which leads to non-locality issues or memory problems in time. We formulate efficient numerical schemes which are based on the novel finite difference method and the spectral algorithms are developed for solving time-fractional reaction-diffusion equations. The proposed technique is showcased by solving the fractional BVAM model in one-, two- or three-dimensional spaces, to explore the dynamic richness of pattern formation with the concept of fractional derivatives. Experimental results obtained mimicked various Pseudoplatystoma catfish structures involving the tiger sorubim in Pseudoplatystoma tigrinum, spotted sorubim in Pseudoplatystoma corruscans, the barred sorubim in Pseudoplatystoma reticulatum, spotted and vertical bar structures in Pseudoplatystoma punctifer species, many other unrecognized patterns are classified as Pseudoplatystoma fasciatum species. Experimental results are given for different values of fractional power to address any points or queries that may occur.

On stability of delayed differential systems of arbitrary non-integer order

On stability of delayed differential systems of arbitrary non-integer order

Direct Synthesis Based Fractional Order PID Controller Design for Non - Integer Order Systems with Time Delay

Present study focuses on design and implementation of PI D controller to obtain the closed loop response of for non-integer order systems (NIOS). Controller is designed using direct synthesis (DS) method. Performance analysis in terms of IAE, ISE and ITAE is made and compared with that of literature reported methods. Robustness in terms of Maximum Sensitivity (Ms) is also analyzed. Tuning parameters q and , are selected with an arbitrary value for set-point trajectory and disturbance rejection. The closed loop response is studied for various non-integer order systems. Tuning parameters with respect to q (adjustable tuning parameter in assumed closed loop transfer function) and are arrived at for different case studies, q varying from 0.05 to 0.5 and varying from 0.5 to 6. FOMCON tool box of Simulink in MATLAB is employed for the simulation study.

A Novel Non-integer Indirect Adaptive Control for Non-integer Order Systems with Non-prior Knowledge

In this study, a new non-integer indirect adaptive control method with reference model is suggested for the class of non-integer order systems. The objective of model reference control is to include the output of the given reference fractional model in tracking the output of a controlled plant by using the concept of on-line goal adaptation. The stability of the closed-loop system is analyzed via the Lyapunov method. Finally, Matlab simulation results are presented to illustrate the effectiveness of the proposed method of indirect fractional model reference adaptive control.

Model Order Reduction: A Comparison between Integer and Non-Integer Order Systems Approaches

In this paper, classical and non-integer model order reduction methodologies are compared. Non integer order calculus has been used to generalize many classical control strategies. The property of compressing information in modelling systems, distributed in time and space, and the capability of describing long-term memory effects in dynamical systems are two features suggesting also the application of fractional calculus in model order reduction. In the paper, an open loop balanced realization is compared with three approaches based on a non-integer representation of the reduced system. Several case studies are considered and compared. The results confirm the capability of fractional order systems to capture and compress the dynamics of high order systems.

Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

Nowadays, the dynamics of non-integer order system or fractional modelling has become a widely studied topic due to the belief that the fractional system has hereditary properties. Hence, as part of understanding the dynamic behaviour, in this paper, we will perform the computation of stability criterion for a fractional Shimizu–Morioka system. Different from the existing stability analysis for a fractional dynamical system in literature, we apply the optimal Routh–Hurwitz conditions for this fractional Shimizu–Morioka system. Furthermore, we introduce the way to calculate the range of adjustable control parameter β to obtain the stability criterion for fractional Shimizu–Morioka system. The result will be verified by using the predictor-corrector scheme to obtain the time series solution for the fractional Shimizu–Morioka system. The findings of this study can provide a better understanding of how adjustable control parameter β influences the stability criterion for fractional Shimizu–Morioka system.

Neuro-adaptive tracking control of non-integer order systems with input nonlinearities and time-varying output constraints

This paper studies the design of neuro-adaptive tracking control schemes for non-integer order non-square systems subject to time-varying output constraints and input nonlinearities. It should first be stated that by employing the mean-value theorem, the original non-affine non-square system with actuator nonlinearities is converted into an equivalent affine square form. Neural networks, Barrier Lyapunov Functions and Nussbaum functions are then incorporated to overcome the difficulties raised by the uncertain nonlinear dynamics, output constraints and unknown control directions, respectively. In a further step, in order to systematically derive the control signals and updating laws, the Backstepping technique is applied. It is shown that by using the proposed adaptive controller, the semi-global asymptotic tracking and the boundedness of all variables in the closed-loop system are guaranteed without transgression of the constraints. The foremost contributions of this paper include: (1) by means of new lemmas and corollaries based on Caputo fractional derivative definitions, techniques and approaches related to the stability analysis and controllability of integer-order plants are extended to fractional-order non-square ones, and (2) the ‘explosion of complexity’ issue is fixed. Finally, simulation results are provided to reveal the effectiveness of the proposed control scheme.

Global Stabilization for a class of nonlinear fractional-order systems

The problem of Mittag–Leffler stabilization (MLS) is studied for a class of nonlinear non-integer order systems. The stabilizer is constructed by using the Lyapunov function and backstepping algorithm. The continuous controller is designed to ensure that the state of the nonlinear fractional-order closed-loop system converges to the equilibrium. Two simulation examples are given to illustrate the effectiveness of the method.

Computational study of noninteger order system of predation.

In this paper, we analyze the stability of the equilibrium point and Hopf bifurcation point in the three-component time-fractional differential equation, which describes the predator-prey interaction between different species. In the dynamics, the classical first-order derivative in time is modelled by either the Caputo or the Atangana-Baleanu fractional derivative of order α,0<α<1. We utilized a fractional version of the Adams-Bashforth formula to discretize these fractional derivatives in time. The results of the linear stability analysis presented are confirmed by computer simulation results.

A novel approach of fractional-order time delay system modeling based on Haar wavelet

In this paper, fractional-order time delay system modeling is presented using Haar wavelet operational matrix of integration. Therefore, it does not require any prior knowledge of transfer function structure or partial information about fractional differentiation order. It allows the estimation of the implicit time delay parameter together with other model parameters by utilizing new delay operational matrix of Haar wavelet based on Riemann-Liouville definition. The proposed technique reduces the complexity of identification by converting the complex fractional calculus equations into simple algebra. The efficacy of the approach is verified on various integer and non-integer (fractional) order systems in simulation. For realistic condition, proposed method is verified in the presence of noise in simulation and also demonstrated on the real-time process control temperature system.

Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios

In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise.

On Digital Realizations of Non-integer Order Filters

Typical approach to non-integer order filtering consists of analogue design and implementation. Digital realization of non-integer order systems is susceptible to problems such as infinite memory requirement and sensitivity to numerical errors. The aim of this paper is to present two efficient methods for digital realization of non-integer order filters: discrete time-domain Oustaloup approximation and Laguerre impulse response approximation. Properties of both methods are investigated with use of non-integer low-pass filter. Filters realized with presented methods are then used for filtering of EEG signal. Paper concludes with discussion of merits and flaws of both methods.

Convergence of Laguerre Impulse Response Approximation for Noninteger Order Systems

One of the most important issues in application of noninteger order systems concerns their implementation. One of the possible approaches is the approximation of convolution operation with the impulse response of noninteger system. In this paper, new results on the Laguerre Impulse Response Approximation method are presented. Among the others, a new proof ofL1convergence of approximation is given, allowing less strict assumptions. Additionally, more general results are given including one regarding functions that are in the joint part ofL1andL2spaces. The method was also illustrated with examples of use: analysis of “fractional order lag” system, application to noninteger order filters design, and parametric optimization of fractional controllers.

The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters

The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters

Maximum sensitivity based fractional IMC–PID controller design for non-integer order system with time delay

A simple approach with a small number of tuning parameters is a key goal in fractional order controller design. Recently there have been a number of limited attempts to bring about improvements in these areas. In this paper, a new design method for a fractional order PID controller based on internal model control (IMC) is proposed to handle non-integer order systems with time delay. In order to reduce the number of tuning parameters and mitigate the impact of time delay, the fractional order internal model control scheme is used. Considering the robustness of the control system with respect to process variations and model uncertainty, maximum sensitivity is applied to the tuning of the parameters. The resulting controller has the structure of a fractional order PID which is cascaded with a filter. This is named a fractional IMC–PID controller. Numerical results are given to show the efficiency of the proposed controller.

Applications of Direct Lyapunov Method in Caputo Non-Integer Order Systems

Paper presents certain properties of Lyapunov direct method for non-integer order systems. Mittag-Leffler stability is defined and its relationship with Lyapunov stability is investigated. General results for Lyapunov functions are presented and a new result allowing constructive stability analysis is proved. Results are illustrated with the examples of stability analysis for problem of cooling, chemical reaction evolution and parametric stability analysis.DOI: http://dx.doi.org/10.5755/j01.eee.21.2.11505

Time domain simulation of Li-ion batteries using non-integer order equivalent electrical circuit

For electric vehicle (EV) or hybrid EV (HEV) development and integration of renewables in electrical networks, battery monitoring systems have to be more and more precise to take into account the state-of-charge and the dynamic behavior of the battery. Some non-integer order models of electrochemical batteries have been proposed in literacy with a good accuracy and a low number of parameters in the frequential domain. Nevertheless, time simulation of such models required to approximate this non-integer order system by an equivalent high integer order model. An adapted algorithm is then proposed in this article to simulate the non-integer order model without any approximation, thanks to the construction of a 3-order generalized state-space system. This algorithm is applied and validated on a 2.3 A.h Li-ion battery.