Fractional Time Delay Research Articles

Chaos in the fractional order nonlinear Bloch equation with delay

The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ=0, we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α=0.8548 with subsequent period doubling that leads to chaos at α=0.9436. A periodic window is observed for the range 0.962<α<0.9858, with chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10ms delay is similar: two cycles appear at the value α=0.8532, and the transition from two to four cycles at α=0.9259. With further increases in the fractional order, period doubling continues until at α=0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α=0.8441, and α=0.8635, respectively. However, the system exhibits chaos at much lower values of α (α=0.8635). A periodic window is observed in the interval 0.897<α<0.9341, with chaos again appearing for larger values of α. In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a delay-dependent model that predicts instability in this non-linear fractional order system consistent with the experimental observations of spin turbulence.

Fractional-Order Controller Design for Oscillatory Fractional Time-Delay Systems Based on the Numerical Inverse Laplace Transform Algorithms

Fractional-order time-delay system is thought to be a kind of oscillatory complex system which could not be controlled efficaciously so far because it does not have an analytical solution when using inverse Laplace transform. In this paper, a type of fractional-order controller based on numerical inverse Laplace transform algorithm INVLAP was proposed for the mentioned systems by searching for the optimal controller parameters with the objective function of ITAE index due to the verified nature that fractional-order controllers were the best means of controlling fractional-order systems. Simulations of step unit tracking and load-disturbance responses of the proposed fractional-order optimalPIλDμcontroller (FOPID) and corresponding conventional optimal PID (OPID) controller have been done on three typical kinds of fractional time-delay system with different ratio between time delay (L) and time constant (T) and a complex high-order fractional time delay system to verify the availability of the presented control method.

Fractional time delay estimation algorithm based on the maximum correntropy criterion and the Lagrange FDF

In this paper, a novel algorithm for fractional time delay estimation is proposed. It is based on the maximum correntropy criterion and the Lagrange fractional delay filter (FDF). The instantaneous correntropy is introduced to measure the similarity between received signals and estimated ones in the proposed algorithm. It leads to an effective performance under both Gaussian and impulsive noises. The performances, including the convergence of the algorithm and the variance of the time delay estimation, are theoretically analyzed. Simulations demonstrate that the time delay estimation precision of the proposed algorithm is higher than that utilizing the mixed modulated Lagrange explicit time delay estimation (MMLETDE) especially under low generalized signal-to-noise ratio (GSNR).

Regression model for tuning the PID controller with fractional order time delay system

In this paper a regression model based for tuning proportional integral derivative (PID) controller with fractional order time delay system is proposed. The novelty of this paper is that tuning parameters of the fractional order time delay system are optimally predicted using the regression model. In the proposed method, the output parameters of the fractional order system are used to derive the regression function. Here, the regression model depends on the weights of the exponential function. By using the iterative algorithm, the best weight of the regression model is evaluated. Using the regression technique, fractional order time delay systems are tuned and the stability parameters of the system are maintained. The effectiveness and feasibility of the proposed technique is demonstrated through the MATLAB/Simulink platform, as well as testing and comparison using the classical PID controller, Ziegler–Nichols tuning method, Wang tuning method and curve fitting technique base tuning method.

Nonlinear dynamics of a class of digital tan-lock loops with non-ideal phase detector

The consequences of unequal propagation delay of the sampled control signals along the two arms of a class of digital tan-lock loops (DTLs) have been studied in this paper. The said delay is considered as a fraction of the basic sampling period and it is modeled by including a discrete time version of a unit-amplitude pure phase shift network in one of the arms of the tan-lock phase detector (PD). The DTL with such unbalanced fractional delay (named as FD-DTL) is studied through bifurcation analysis and time domain numerical simulation of the system equation. Obtained results indicate that for a FD-DTL loop gain values could be taken higher and frequency acquisition ranges (FARs) are wider compared to those for a conventional DTL (CDTL). Thus a deliberate inclusion of a fractional time delay network in one of the arms of a CDTL is a novel technique to improve its dynamical responses.

Preprocessed Fractional Lower-order Covariance Time Delay Estimation under Symmetric α-stable Conditions

Preprocessed Fractional Lower-order Covariance Time Delay Estimation under Symmetric α-stable Conditions

Towards a Non-linear Theory for Induced Seismicity in Shales

We analyze the pore transmission of fluid pressure p and solute density ρ in porous rocks. To investigate shale deformation, fundamental for deep oil drilling and for gas extraction (“fracking”), a non-linear model of mechanic and chemo-poroelastic interactions among fluid, solute and the solid matrix is discussed. The solutions of the model are quick non-linear Burger's solitary waves, potentially destructive for deep operations. Following Civan [2], diffusive and shock waves are applied to fine particles filtration. Then the delaying effects of fine particles filtration is compared with fractional model time delay and the fractional order parameter can be realistically estimate.

PID auto-tuning using new model reduction method and explicit PID tuning rule for a fractional order plus time delay model

In this paper, a new model reduction method and an explicit PID tuning rule for the purpose of PID auto-tuning on the basis of a fractional order plus time delay model are proposed. The model reduction method directly fits the fractional order plus time delay model to frequency response data by solving a simple single-variable optimization problem. In addition, the optimal tuning parameters of the PID controller are obtained by solving the Integral of the Time weighted Absolute Error (ITAE) minimization problem and then, the proposed PID tuning rule in the form of an explicit formula is developed by fitting the parameters of the formula to the obtained optimal tuning parameters. The proposed tuning method provides almost the same performance as the optimal tuning parameters. Simulation study confirms that the auto-tuning strategy based on the proposed model reduction method and the PID tuning rule can successfully incorporate various types of process dynamics.

Legendre multiwavelet collocation method for solving the linear fractional time delay systems

Abstract In this article the Legendre multiwavelet basis with aid of collocation method has been applied to give approximate solution for fractional delay systems. The properties of Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of algebraic system. Numerical results and comparison with exact solutions in the cases when we have exact solution are given in test examples in order to demonstrate the applicability and efficiency of the method.

Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay

In this paper, a Duffing-van der Pol oscillator having fractional derivatives and time delays is investigated by the residue harmonic method. The angular frequencies and limit cycles of periodic motions are expanded into a power series of an order-tracking parameter and the unbalanced residues resulting from the truncated Fourier series are considered iteratively to improve the accuracy. The periodic bifurcations are examined using the fractional order, feedback gain and time delay as continuation parameters. It is shown that jumps and hysteresis phenomena can be delayed or removed. Transition from discontinuous bifurcation to continuous bifurcation is observed. The approximations are verified by numerical integration. We find that the proposed method can easily be programmed and can predict accurate periodic approximations while the system parameters being unfolded.

Comments on “Closed Form Variable Fractional Time Delay Using FFT”

In this letter drawbacks of the aforementioned paper are pointed out. The proposed approach is improved with minor modifications of the discrete frequency response. This allows for design of fractional delay filters which are close to optimal and can be efficiently implemented in the frequency domain using the sliding DFT based structure. Alternatively, the derived equivalent closed form formulae for offset windows can be used to design filters in the time domain.

Subsample time delay estimation of chirp signals using FrFT

A new subsample time delay estimator is proposed in this paper. It is evaluated based on the time delay property of the fractional Fourier transform and is suitable for processing chirp signals. The theoretic derivation of the algorithm is also presented. The validity and superiority to the fractional time delay estimator (FRTDE), the conventional direct correlator (DC) and the average magnitude squared fractional difference function (AMSFDF) estimator of this new time delay estimator are demonstrated via computer simulations.

BIBO stability analysis of fractional order time delay systems

In this paper, an analytical method is introduced for determining the stability map of a general class of fractional order systems with single and multiple commensurate time delays, against delay uncertainties. The complexity arises due to the exponential type transcendental terms and fractional order in their characteristic equation. It is shown that this method analytically reveals all possible stability windows exclusively in the parametric space of the time delay. Using the approach presented in this study, first, without using any approximation or substitution, the transcendental characteristic equation is converted to an algebraic one with some specific crossing points. The resulting algebraic equation also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin. As an added strength, it does not require the delay-free system under consideration to be stable. The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Finally, the concept of stability is expressed as a function of delay. Five illustrative examples are presented to confirm the proposed method results.

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.

Further Results on Finite Time Partial Stability of Fractional Order Time Delay Systems

This paper proposes sufficient conditions for finite time partial stability for the (non)homogeneous fractional order systems with time varying delay. New stability criteria for this class of fractional order systems are derived using “classical” Bellman-Gronwall inequality as well as a suitable “fractional” inequality instead of generalized Gronwall inequality. Finally, a numerical example is given to illustrate the application of the proposed stability procedure.

Closed Form Variable Fractional Time Delay Using FFT

In this letter, a novel tunable frequency response is proposed for variable fractional delay filter with careful consideration on conjugate symmetry. This proposed fractional delay method can be efficiently realized by FFT using the tunable fractional delay parameter. Its equivalent closed form to windowing method is derived and presented. Several comparisons to other windowing methods are also demonstrated, such as the magnitude response, group delay, and approximation error between ideal frequency response.

Adaptive fuzzy H ∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems

In this paper, a fuzzy logic controller equipped with training algorithms is developed such that the H∞ tracking performance should be satisfied for a model-free nonlinear fractional order time delay system which is infinite dimensional in nature and time delay is a source of instability. In order to deal with the linguistic uncertainties caused from delay terms, the adaptive time delay fuzzy logic system is constructed to approximate the unknown time delay system functions. By incorporating Lyapunov stability criterion with H∞ tracking design technique, the free parameters of the adaptive fuzzy controller can be tuned on line by output feedback control law and adaptive law. Moreover, the tracking error and external disturbance can be attenuated to arbitrary desired level. The numerical results show the effectiveness of the proposed adaptive H∞ tracking scheme.

Fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces

This paper concerns the fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces. A suitable mild solution of the fractional finite time delay evolution systems is introduced. Using the singular version of the Gronwall inequality with finite time delay, we obtain some sufficient conditions for the existence, uniqueness and continuous dependence of mild solutions of these control systems. A formulation for the fractional Lagrange problem is introduced. The existence of optimal pairs of fractional-time-delay evolution systems is also presented. Finally, an example is given for demonstration.

Existence of mild solutions for fractional delay evolution systems

Abstract In this paper, we prove some sufficient conditions for the local and global existence of fractional nonlinear finite time delay evolution equations whose linear part is the infinitesimal generators of analytic semigroups. The results are obtained by applying the generalized singular versions of integral inequalities under some different conditions on nonlinear term. At last, two examples are given for demonstration.

A new correntropy based TDE method under α-stable distribution noise environment

This paper presents a robust time delay estimation algorithm for the α-stable noise based on correntropy. Many time delay estimation algorithms derived for impulsive stable noise are based on the theory of Fractional Lower Order Statistics (FLOS). Unlike previously introduced FLOS-type algorithms, the new algorithm is proposed to estimate the time delay by maximizing the generalized correlation function of two observed signals needing neither prior information nor estimation of the numerical value of the stable noise’s characteristic exponent. An interval for kernel selection is found for a wide range of characteristic exponent values of α-stable distribution. Simulations show the proposed algorithm offers superior performance over the existing covariation time delay estimation, least mean p-norm time delay estimation and achieves slightly improved performance than fractional lower order covariance time delay estimation at lower signal to noise ratio when the noise is highly impulsive