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This paper investigates the global Mittag-Leffler synchronization problem of Caputo fractional-order inertial memristive neural networks with time-varying delays. First, the model of fractional-order inertial memristive neural network with time-varying delays is proposed, which is more universal and less conservative than the integer-order inertial memristive neural network with time-varying delays. Second, one lemma about the compound property of Caputo fractional derivative and integral is given. According to the compound property of Caputo fractional derivative and appropriate variable substitution, the original inertial memristive system translates a routine inertial memristive system. And, using the Filippov discontinuous theory, Lyapunov stability theory, and Mittag-Leffler convergence, several sufficient conditions are derived to ensure the global Mittag-Leffler synchronization of Caputo fractional-order inertial memristive neural networks. An effective feedback controller is proposed for fractional-order inertial memristive neural networks with time-varying delays, such that the global Mittag-Leffler synchronization between the slave system and the master system can be achieved. Finally, a simulation example is given to verify the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

• A novel fractional order gradient descent method based on quadratic loss function is proposed. • A novel two-stage fractional order gradient descent method is proposed with random weight particle swarm optimization algorithm. • The scope of application of the fractional order gradient descent is expanded. This article introduces a novel fractional order gradient descent method for the quadratic loss function. Based on Riemann-Liouville definition, a more practical fractional order gradient descent method with variable initial value is proposed to ensure convergence to the actual extremum. On this basis, the random weight particle swarm optimization algorithm is introduced to select the appropriate initial value, which not only accelerates the convergence speed, but also enhances the global convergence ability of the algorithm. To avoid complicated problems of the chain rule in fractional calculus, the parameters of output layers is trained by the new designed method, while the parameters of hidden layers still use the conventional method. By selecting proper hyper-parameters, the proposed method shows faster convergence speed than others. Finally, numerical examples are given to verify that the proposed algorithm has fast convergence speed and high accuracy under a adequate large number of independent runs. [ABSTRACT FROM AUTHOR]

This paper investigates the time–domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag–Leffler function. In particular, we find the limitation of discrete Mittag–Leffler when adopted in describing the response. Then we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time–domain response; and ii) the dynamic behavior of the zero input response. Finally, a simulation study is provided to show the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

The paper focuses on the numerical approximation of discrete fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed model of discrete fractional order system is introduced. Then, resorting the nabla discrete Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology. [ABSTRACT FROM AUTHOR]

Summary: This article mainly studies the fractional‐order active disturbance rejection control (FOADRC) schemes for the underactuated commensurate fractional‐order systems (FOSs). The FOADRC framework for linear FOSs‐based fractional proportion integration differentiation is constructed by using the fractional‐order tracking differentiator and the fractional‐order extended state observer, and the necessary conditions for the system to have stable controllers are provided. The FOADRC scheme for underactuated FOSs based on differential flatness is proposed. For underactuated FOSs, a set of flat output expressions with a fixed format is given under the controllable condition of the system. Moreover, making the flat output as the equivalent of the system output is simple and easy to analyze and calculate. Subsequently, the FOADRC scheme is designed by using the flat output. Finally, the scheme proposed in this article is verified by a simulation example. [ABSTRACT FROM AUTHOR]

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results. [ABSTRACT FROM AUTHOR]

This paper comes up with a stable bias-compensated fractional order normalized least mean square (BC-FONLMS) algorithm with noisy inputs. This kind of bias-compensated algorithm needs the estimation of input noise variance to avoid the bias caused by noisy inputs. Yet, existing algorithms either cause instability because of the method used to estimate input noise variance, or surmount the instability problems at the price of performance diminishment. This paper introduces fractional order calculus into LMS algorithm to be a new BC-FONLMS algorithm. Then, analyze the stability of the BC-FONLMS algorithm through probing the recursive equations of mean deviation (MD) and mean square deviation (MSD). On the basis of the stability analysis, methods to estimate input noise variance and to adjust step size are suggested to stabilize the algorithm and likewise to enhance the performance such as convergence speed and steady-state error. Numerical simulations are given at last, whose results show that the proposed BC-FONLMS algorithm performs well. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on the fractional difference of Lyapunov functions related to Riemann–Liouville, Caputo and Grünwald–Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results. Highlights • New method to build Lyapunov functions for discrete fractional order systems. • Five inequalities on the fractional difference of Lyapunov functions. • The applicability of the proposed results for three different definitions. [ABSTRACT FROM AUTHOR]

Motivated essentially by the success of the applications of the discrete Mittag-Leffler functions (DMLF) in many areas of science and engineering, the authors present, in a unified manner, a detailed numerical implementation method of the Mittag-Leffler function. With the proposed method, the overflow problem can be well solved. To further improve the practicability, the state transition matrix described by discrete Mittag-Leffler functions are investigated. Some illustrative examples are provided to verify the effectiveness of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

Abstract This paper mainly investigates the numerical implementation issue of fractional order systems. First, a pattern of fixed pole schemes are developed to approximate fractional integrator/differentiator, whose common is that the poles keep constant for different α. Then, two solutions are proposed to improve the approximation performance around α = 0. Afterwards, the simulation schemes are introduced for two kinds of fractional order systems. In those schemes, the configuration problem of nonzero initial value is considered. Finally, a fair and solid comparison to the classical approximation methods is presented, demonstrating the effectiveness and efficiency of the elaborated algorithms. Highlights • A universal fixed pole approximation algorithm is derived for fractional integrator/differentiator. • The improvement is made to achieve a better performance with smaller order. • Two approximation schemes are presented for implementing fractional transfer function model. • Three approximation schemes are proposed for implementing fractional state space model. • The configuration of nonzero initial system state is discussed for fractional order system. [ABSTRACT FROM AUTHOR]

This paper deals with the time-optimal control problem for a class of fractional order systems. An analytic solution of the time-optimal problem is proposed, and the optimal transfer route is provided. Considering it is usually adopted in the discrete situation for actual control system, the sampling date may induce chattering phenomenon, an alternative sub-optimal solution is constructed. Additionally, the special and meaningful application of fractional order tracking differentiator is introduced to explain our main results. The effectiveness and advantages of the proposed method have been illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, steady-state error of fractional order control systems (FOCSs) is analyzed based on the concept of type number. Some practical definitions of the reference and the error for FOCSs with non-unity feedback are first given. Based on these definitions, the type number concept is first introduced to FOCSs with non-unity feedback. Then steady-state error of various types of FOCSs are analyzed respectively via type number concept. It is also shown that FOCSs with type number more than 1 are not able to track particular reference without changing the sign of the tracking error. [ABSTRACT FROM AUTHOR]

Energy production and conversion have a significant impact on the economic development of all countries in the world. China's energy production and conversion are large. Therefore, accurate mid-to-long term China's energy production and conversion forecasting is becoming more and more important for integrating energy systems and energy strategic planning. For this purpose, a novel fractional grey sequence is proposed based on Grunwald–Letnikov fractional calculus. Furthermore, a novel self-adaptive fractional multivariable grey model is proposed based on the novel sequence. In this article, we compare several classical optimization algorithms and finally choose Particle Swarm Optimization (PSO) to compute the parameters. In addition, Monte-Carlo simulation and probability density analysis (PDA) are presented in this article to verify the model's performance. Monte-Carlo simulation reduces the randomness of the results of the model runs to a certain extent. Probability density analysis visualizes this randomness through kernel density estimation (KDE). This paper compares the new model with the existing seven grey models and predicts the total energy consumption per capita, energy conversion efficiency and total renewable energy in China, respectively. The experimental results show that the new model is superior to the other seven models in terms of stability and prediction accuracy. [ABSTRACT FROM AUTHOR]

To overcome the imprecise shortcomings of keeping low-level outline signal in denoise method based on fractional calculus filtering and losing high-level detail signal in BM3D (block-matching 3D) filtering, this paper proposed a fractional block-matching 3D filtering method. It discussed the basic mathmatic theory of fractional calculus to apply it to the image processing, constructured the fractional integral denoising templates, and discussed the impact of fractional order selection for de-noising results. And it used two standard methods to evaluate result: subjective visual perception and objective peak signal to noise ratio (PSNR). The experiment results show that this method gets better results in reservations high-level details compared with wavelet filtering and nonlocal mean filtering. Through numerical realizing the method and analysing it with some advanced denoise methods today, it proves the correctness and reasonability of the theory and it gets the conclution that this method is better. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability of fractional-order neural networks (FNNs) without and with delay by the fractional Lyapunov direct method and the fractional Razumikhin-type theorem, respectively. S-procedure is applied to handle the nonlinear constraints to obtain a wider parameter selection of the systems. For FNNs without delay, the improved less conservative conditions of the existence and uniqueness of the equilibrium point and the global Mittag-Leffler stability are all derived in the form of linear matrix inequalities (LMIs). Moreover, an LMI-based uniform stability condition of FNNs with time delay is established, which simplifies and extends some previous work. Finally, the validity of the presented results is indicated by some numerical examples. [ABSTRACT FROM AUTHOR]

Highlights • The Fx-NLMS algorithm for feedforward vibration rejection is improved. • Fx-NLMS algorithm using fractional order descent method is proposed. • Orth-FONLMS algorithm using orthogonal transform method is formed. Abstract This paper mainly discusses a pattern of vibration rejection by using a feedforward controller. Conventional filtered-x normalized least mean squares (Fx-NLMS) algorithm converges slow especially when signals are not independent on each other. Therefore, a novel orthogonalized fractional order filtered-x normalized least mean squares (Orth-FONLMS) algorithm is proposed to obtain the precise parameters of the feedforward controller. The proposed algorithm is an improved Fx-NLMS algorithm by using fractional calculus and orthogonal transform. To implement the algorithm, fractional order gradient method and adaptive lattice filter are introduced, respectively. Numerical simulation proves that the proposed algorithm outperforms over the Fx-NLMS and fractional order filtered-x normalized least mean squares (Fx-FONLMS) algorithm in terms of convergence rate and accuracy. [ABSTRACT FROM AUTHOR]

Abstract The online calculation of fractional derivative is an important technique used in the control engineering area. To achieve satisfactory computing performance, this paper proposes a useful and effective tool named fractional order tracking differentiator (FOTD), which is easy to be implemented. A constructive method for designing the FOTD is developed using a class of asymptotically stable fractional order systems. It is shown that FOTD is the pioneering one to achieve it with simple structure, convenient construction and predominant performance. Finally, several examples are conducted to illustrate the effectiveness and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Highlights • A generalized fractional central difference Kalman filter is proposed. • An adaptive fractional central difference Kalman filter is proposed to estimate the system state and noise parameters simultaneously. • The unbiasedness of the proposed adaptive algorithm is analyzed and then an unbiased recursive algorithm is developed. • The approximate accuracy and numerical complexity of the proposed algorithm are analyzed. Abstract In this paper, a generalized fractional central difference Kalman filter for nonlinear discrete fractional dynamic systems is proposed. Based on the Stirling interpolation formula, the presented algorithm can be implemented as no derivatives are needed. Besides, in order to estimate the state with unknown prior information, a maximum a posteriori principle based adaptive fractional central difference Kalman filter is derived. The adaptive algorithm can estimate the noise statistics and system state simultaneously. The unbiasedness of the proposed algorithm is analyzed. Several numerical examples demonstrate the accuracy and effectiveness of the two Kalman filters. [ABSTRACT FROM AUTHOR]

This article presents a novel fractional order LMS (FOLMS) algorithm, which involves a variable gradient order scheme. The fractional order gradient descent method is revisited firstly. A variable initial value scheme is proposed to attenuate the non–locality of fractional order calculus and to ensure the convergence of the proposed FOLMS algorithm. Furthermore, it is noticed that a contradiction between rapidity and accuracy always appears together with the advancement of FOLMS algorithm; namely, a larger value of the gradient order can not only give a faster convergence speed, but also correspond to a larger estimation error. For the purpose of removing the contradiction between rapidity and accuracy, a variable gradient order scheme is designed for the FOLMS algorithm. Based on a sufficient large number of independent runs, the efficiency and superiority of the proposed algorithm are demonstrated in numerical examples finally. [ABSTRACT FROM AUTHOR]