Fractional Time Delay Research Articles

Exponential Stability of a Kirchhoff Plate Equation with Structural Damping and Internal Time Delay

In this manuscript, we investigate the exponential stability of a Kirchhoff plate equation with free boundary conditions in a bounded domain of R2. First, we consider the case where the model is subjected only to structural damping and, using the energy method, we prove the exponential decay of the energy associated with this model. Second, we consider the case where structural damping with a linear internal time delay term is applied, and we use the same technique to derive the exponential stability of the new model. Finally, we examine the scenario where the model is governed by structural damping with an added fractional time delay term and, by constructing an appropriate Lyapunov function, we show that the energy associated with the system is exponentially stable. This work improves upon prior results concerning the Kirchhoff plate equation with frictional damping, where exponential stability could not be achieved.

A frequency domain fractional order tilted integral derivative controller design for fractional order time delay processes

AbstractIn real‐time processes, time delays are inherent due to various factors, such as delays in volume, mass, and information transfer, leading to a deterioration in process performance and stability. Therefore, this paper proposes a fractional‐order tilted integral derivative (TID) controller design for stable and unstable fractional‐order processes with time delay, where the optimal controller parameters are designed using a deterministic approach. An inner loop controller is designed using stability analysis and a graphical approach for unstable fractional‐order time delay processes. The proposed approach offers flexibility and fine‐tuning in designing the controller for a specific application. A systematic reference to the disturbance ratio is carried out to assess the disturbance handling capacity of the proposed controller. It shows the sensitivity of the designed controller against disturbances in the frequency spectrum graphically. Numerous performance indices and time‐domain parameters are computed and compared with state‐of‐the‐art techniques to analyze the efficacy. Furthermore, it is validated on the experimental setup of a four‐tank system. Simulation and experimental results demonstrate that the proposed approach outperforms recent techniques in terms of process control and disturbance rejection.

Identification of fractional order time delay system with measurement noise using variable period integration operational matrix

This paper proposes a parameter identification method for fractional-order time-delay systems with measurement noise, based on the Legendre wavelet variable-period integration operational matrix. A variable-period integration operational matrix and a variable-period time-delay integration operational matrix are constructed by variable-period sampling principle. For the impulse noise, global Gaussian noise and local Gaussian noise present in the identification data, the variable-period sampling, operational matrix global coefficient decomposition and matrix local coefficient decomposition methods are used to reconstruct the variable-period integration operational matrix to separate the noise from the data; the fractional-order time-delay system is expressed as an integral equation by using the reconstructed variable-period operational matrix, this reduces the influence of measurement noise on system identification accuracy. To further improve the parameter identification accuracy, the integral equation is solved by the multi-innovation least squares algorithm. The proposed method’s effectiveness is verified through several simulations and an experimental study of a constant-temperature system.

Simulating time delays and space–time memory interactions: An analytical approach

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear (1+2)-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

An analytical criterion for stability testing of delayed Bode’s transfer function with unity feedback

The unity feedback where delayed Bode’s transfer function is placed in forward path is used as reference transfer function in controller design for fractional time delay systems. In this study, bounded input-bounded output stability of this reference transfer function is investigated. This investigation is based on a theorem for fractional order time delay systems using Hassard’s techniques. Sufficient and necessary conditions are obtained for testing the stability of this reference transfer function. Moreover, an explicit formula is given for calculating the number of unstable roots in case of instability. The illustrative examples show that this stability criterion works efficiently.

Effects of Fractional Time Delay as a Low-Power True Time Delay Digital Beamforming Architecture

Effects of Fractional Time Delay as a Low-Power True Time Delay Digital Beamforming Architecture

Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method.

In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented.

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system’s solutions, we investigate stochasticity’s influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

A Variable Fractional Delay Filter Based True-Time-Delay Array Architecture for Wideband Beamforming

True-time delay (TTD) units are critical in Wideband Large-Scale Antenna Systems (LSASs) for resolving beam squint and frequency selective fading. In this paper, we propose a TTD array architecture for wideband beamforming that addresses the challenges posed by LSASs by utilizing variable fractional delay (VFD) filters and time delay estimations (TDEs). This study introduces a convex optimization problem to develop a novel method for designing VFD filters that can achieve high precision time delay resolution while maintaining an acceptable frequency response peak error. Using these VFD filters, we present a TDE approach with Newton-Raphson iteration update that corrects the difference in arrival time between sensors. Due to the implementation of VFD filters and fast TDE modules, the new architecture can complete beam steering and tracking in 10 ms with MATLAB simulation. Furthermore, an example was designed to demonstrate that the proposed architecture is capable of producing precise beam pointing while maintaining the bit error rate (BER) that is extremely close to the theoretical curve over the entire bandwidth.

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>

A Non-Local wave equation with General fractional derivatives and time delay

We analyze, spatially one dimensional, non-local elastic body with deformation measure given in the form of General fractional derivative of Riesz type with truncated power-law kernel. It is assumed that stress at the point x and time t depends on the strain at point ξ at time t−|x−ξ|/c, where c is a constant having dimension of time. We formulate spatially one dimensional wave equation for such a body and show the influence of the parameter c on the solution.

An improved repetitive controller with fractional time-delay for discrete-time linear systems: Synthesis and comparison study

This paper develops a discrete-time repetitive control (RC) system with a fractional-delay internal model. Unlike the conventional RC, the time delay for constructing the internal model is not necessarily an integer, implying that the time delay is allowed to be fractional. In this work, a fractional delay-based repetitive control system is presented. Firstly, the fractional time delay is calculated based on the information of the reference or disturbance along with the associated sampling period. Secondly, the fractional time delay is divided into two parts, i.e., the large integer and small fractional delays. Finally, the fractional part is realized by a stable and causal infinite impulse response (IIR) filter, whose coefficients are calculated using the Thiran formula. The controller structure and stability analysis are developed for three different RCs, namely a fractional-delay general RC (FD-GRC), a fractional delay-odd harmonics RC (FD-OHRC), and a fractional delay-(6l±1) RC (FD-(6l±1)RC). Experiments and comparison studies of the integer delay-based RCs are conducted to verify the effectiveness and improved tracking accuracy of the proposed method.

A Class of Parameter Interpolation Iteration Algorithm based on Clem- Equations

The Newdon Iterative method and its deformation mostly require calculating the derivative of nonlinear equations. In practice, many functions are complex, and there are limitations in finding high-order derivatives. This paper fits an existing iterative format by establishing n times interpolated polynomials, n conditions are used to construct a system of linear equations with n equations and n + 1 unknowns, in which the previous n unknowns can be expressed by the n + 1 th unknown by Clem’s rule, to achieve the purpose of reducing the number of derivative calculations. Convergence analysis and numerical examples verify that the algorithm is better than Newdon iteration, and can be better used in path trajectory optimization and fractional time delay control.

Forecasting China’s total renewable energy capacity using a novel dynamic fractional order discrete grey model

Accurately forecasting renewable energy capacity is crucial for informing national energy policies and promoting sustainable socio-economic growth. This study proposes a novel dynamic fractional-order discrete grey model (DFDGM (1,1)) for forecasting China's total renewable energy capacity. The model constructs a dynamic fractional time delay polynomial, which enhances the adaptability of the model to different sample sizes. The time response function of model is directly derived from the discrete equation, preventing errors caused by the transition from discreteness to continuity. Furthermore, the proposed model represents the unified form of six grey models, effectively expanding the model's application domain and scope. The method introduces a double error approach to facilitate optimal parameter selection, successfully addressing overfitting and randomness challenges in parameter selection. Empirical studies show that the proposed model outperforms eleven other methods, with fitting and forecasting errors of 0.39% and 1.13%, respectively. Finally, the proposed model forecasts the future development trend of China’s total renewable energy capacity. In general, this research not only improves the precision of predictions but also progresses the dynamic theory in prediction studies.

Finite-Time Stability to Fractional Delay Cauchy Problem of Hilfer Type

The purpose of this work is to look at the solution’s representation for the non-homogeneous fractional timedelay Cauchy problem of Hilfer type in terms of the cosine and sine fractional delayed matrices of two parameters. The solutions are determined using the constant variation approach. Following that, finite-time stability under moderate circumstances is examined. At last, an example is offered to show how the theoretical results may be used.

Dynamical Analysis of a Local Lengley-Epstein System Coupled with Fractional Delayed Differential Equations

We consider a system of fractional delayed differential equations. The ordinary differential version of the system without delay is introduced in the Lengyel-Epstein reaction-diffusion system. We evaluate the system with and without delay and explore the stability of the unique positive equilibrium. We also prove the existence of Hopf bifurcation for both cases. Furthermore, the impacts of Caputo fractional order parameter and time delay parameter on the dynamics of the system are investigated with numerical simulations. It is also concluded that for different values of time delay parameter, the decreament of the Caputo fractional order parameter has opposite effects on the system in terms of stability.

Stability and bifurcation analysis for a fractional-order cancer model with two delays

Considering that healthy tissue cells and tumor cells restrain each other’s the average growth rate with a corresponding delay, a fractional-order cancer model with time delays is established. Firstly, by selecting different delays as bifurcation parameters, the delay-independent stability and Hopf bifurcation conditions of such system are derived. Then, some interesting dynamical phenomena of the system under different initial values can be observed, such as a pair of coexisted stable periodic solution, equilibrium point transition, and even chaos. Finally, some numerical simulations are exhibited to substantiate the obtained results. The work reveals that the fractional order and delays play an important role in the stability behavior and bifurcation characteristics of the system, and that a suitable control strategy of the system can be designed by selecting reasonable system parameters (fractional orders and time delays).

Control Design for Fractional Order Leader and Follower Systems with Mixed Time Delays: A Resilience-Based Approach

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into the delayed and non-delayed controller part to make it more practical. Our method involves proposing algebraic criteria by utilizing non-delayed and delayed protocols, applying the Razumikhin technique and graph theory respectively. The presented method can well overcome the difficulty resulting from fractional calculus, time delays and fractional derivatives. To demonstrate the validity and effectiveness of our findings, we provide an example at the end of our study.

Memory-dependent and fractional order analysis of the initially stressed piezo-thermoelastic medium

In the context of the triple-phase lag model, comparison studies are conducted between the memory-dependent derivative and fractional order derivative on the propagation of plane waves in a half-space of initially stressed transversely isotropic piezothermoelastic material. The coupled governing equations of the considered model involving relaxation time, time delay, fractional order parameter, and kernel function are selected according to specific problems and solved using the normal mode analysis technique and obtaining the analytical expression. The non-dimensional temperature, displacement, electrical displacements, and stresses at different values of the fractional order parameter and time delay factor are obtained graphically.

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.