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- Fractional-order derivative approximations in discrete-time control systemsPublication . Machado, J. A. TenreiroThe theory of fractional calculus goes back to he beginning of the theory of differential calculus but its inherent complexity postponed the application of the associated concepts. in the last decade the progress in the areas of chaos and fractals revealed subtle relationships with the fractional calculus leading to an increasing interest in the development of the new paradigm. in the area of automatic control preliminay work has already been carried out but the proposed algorithms are restricted to the frequency domain. The paper discusses the design of fractional-order discrete-time controllers. the algorithms and discrete-time implementation.
- Fractional Control of Legged RobotsPublication . Silva, Manuel S.; Machado, J. A. TenreiroFractional calculus (FC) is being used in several distinct areas of science and engineering, being recognized its ability to yield a superior modelling and control in many dynamical systems. This article illustrates the application of FC in the area of robot control. A Fractional order PDu controller is proposed for the control of an hexapod robot with 3 dof legs. It is demonstrated the superior performance of the system by using the FC concepts.
- Dynamics of the Dow Jones and the NASDAQ stock indexesPublication . Duarte, Fernando B.; Machado, J. A. Tenreiro; Duarte, Gonçalo MonteiroThe goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indices. We analyze the Dow Jones Industrial Average ( ∧ DJI) and the NASDAQ Composite ( ∧ IXIC) indexes at a daily time horizon. The methods and algorithms that have been explored for description of physical phenomena become an effective background, and even inspiration, for very productive methods used in the analysis of economical data. We start by applying the classical concepts of signal analysis, Fourier transform, and methods of fractional calculus. In a second phase we adopt a pseudo phase plane approach.
- Fractional Order Generalized InformationPublication . Machado, J. A. TenreiroThis paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.
- A piecewise spectral-collocation method for solving fractional Riccati differential equation in large domainsPublication . Azin, H.; Mohammadi, F.; Tenreiro Machado, J. A.This paper addresses the approximate solution of the fractional Riccati differential equation (FRDE) in large domains. First, the solution interval is divided into a finite number of subintervals. Then, the Legendre–Gauss–Radau points along with the Lagrange interpolation method are employed to approximate the FRDE solution in each subinterval. The method has the advantage of providing the approximate solutions in large intervals. Additionally, the convergence analysis of the numerical algorithm is also provided. Three illustrative examples are given to illustrate the efficiency and applicability of the proposed method.
- Fractional generalization of memristor and higher order elementsPublication . Machado, J. A. TenreiroFractional calculus generalizes integer order derivatives and integrals. Memristor systems generalize the notion of electrical elements. Both concepts were shown to model important classes of phenomena. This paper goes a step further by embedding both tools in a generalization considering complex-order objects. Two complex operators leading to real-valued results are proposed. The proposed class of models generate a broad universe of elements. Several combinations of values are tested and the corresponding dynamical behavior is analyzed.
- Entropy Analysis of a Railway Network’s ComplexityPublication . Valério, Duarte; Lopes, António M.; Machado, J.A.TenreiroRailway networks are among the many physical systems that reveal a fractal structure. This paper studies the Portuguese railway system, and analyzes how it evolved over time, namely what concerns the structure of its different levels, and its distribution over the territory. Different mathematical tools are adopted, such as fractal dimension, entropy and state space portrait. The results are consistent with the historical evolution of the network.
- A fractional perspective on the trajectory control of redundant and hyper-redundant robot manipulatorsPublication . Machado, J.A.Tenreiro; Lopes, António M.The manuscript develops a new perspective for studying the trajectory control of planar manipulators using the Moore–Penrose pseudoinverse. Different mechanical structures are compared, namely redundant and hyper-redundant robots. The proposed method is based on fractional calculus and fractional matrix powers. The signals can be interpreted as time- space waves propagating along the trajectory planing system. Several simulations demon- strate the performance of the novel scheme in the analysis of pseudoinverse-based closed- loop systems.
- Particle swarm optimization algorithm using complex-order derivative concept: A comprehensive studyPublication . Abedi Pahnehkolaei, Seyed Mehdi; Alfi, Alireza; Machado, J. A. TenreiroThis paper presents a comprehensive study of the Particle Swarm Optimization (PSO) algorithm, called complex-order PSO (CPSO). In the core of new set of algorithms, we employ the complex-order derivative and the conjugate order differential concepts in the position and velocity adaption mechanisms. To determine the influence of the control parameters on the quality of the results, a sensitivity analysis is conducted. A number of value- and rank-based tests assesses the algorithms’ performance. For a suite of benchmark functions, the standard deviation and the mean best of the results are reported. Additionally, the Friedman test specifies the average ranking from the obtained results. The effect of the complex-order operation and the population size are analyzed using the Taguchi test. An application example illustrates the performance of the CPSO.
- Entropy analysis of systems exhibiting negative probabilitiesPublication . Machado, J. A. TenreiroThis paper addresses the concept of negative probability and its impact upon entropy. An anal-ogy between the probability generating functions, in the scope of quasiprobability distribu-tions, and the Grünwald–Letnikov definition of fractional derivatives, is explored. Two distinct cases producing negative probabilities are formulated and their distinct meaning clarified. Nu-merical calculations using the Shannon entropy characterize further the characteristics of the two limit cases.