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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]

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]

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]