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- Integer and Fractional Order Entropy Analysis of Earthquake Data-seriesPublication . Lopes, António M.; Machado, J.A.TenreiroThis paper studies the statistical distributions of worldwide earthquakes from year 1963 up to year 2012. A Cartesian grid, dividing Earth into geographic regions, is considered. Entropy and the Jensen–Shannon divergence are used to analyze and compare real-world data. Hierarchical clustering and multi-dimensional scaling techniques are adopted for data visualization. Entropy-based indices have the advantage of leading to a single parameter expressing the relationships between the seismic data. Classical and generalized (fractional) entropy and Jensen–Shannon divergence are tested. The generalized measures lead to a clear identification of patterns embedded in the data and contribute to better understand earthquake distributions.
- Performance analysis of multi-legged systemsPublication . Silva, Manuel; Tenreiro Machado, J. A.; Lopes, António M.This paper studies periodic gaits of multi-legged robot locomotion systems based on dynamic models. The purpose is to determine the system performance during walking and the best set of locomotion variables. For that objective the prescribed motion of the robot is completely characterized in terms of several locomotion variables such as gait, duty factor, body height, step lenght, stroke pitch, foot clearance, legs link lengths, foot-hip offset, body and legs mass and cycle time. In this perspective, we formulate four performance measures of the walking robot namely, the locomobility of the foot, the mean absolute power, the mean power dispersion and the mean power lost in the joint actuators per walking distance. A set of model-based experiments reveals the influence of the locomotion variables in the proposed indices.
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional CalculusPublication . Machado, J. A. Tenreiro; Lopes, António M.In this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.
- Root-locus practical sketching rules for fractional-order systemPublication . Lopes, António M.; Machado, J. A. TenreiroFor integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.
- Entropy analysis of human death uncertaintyPublication . Machado, J. A. Tenreiro; Lopes, António M.Uncertainty about the time of death is part of one’s life, and plays an important role in demographic and actuarial sciences. Entropy is a measure useful for characterizing complex systems. This paper analyses death uncertainty through the concept of entropy. For that purpose, the Shannon and the cumulative residual entropies are adopted. The first may be interpreted as an average information. The second was proposed more recently and is related to reliability measures such as the mean residual lifetime. Data collected from the Human Mortality Database and describing the evolution of 40 countries during several decades are studied using entropy measures. The emerging country and inter-country entropy patterns are used to characterize the dynamics of mortality. The locus of the two entropies gives a deeper insight into the dynamical evolution of the human mortality data series.
- Fractional order models of leavesPublication . Lopes, António M.; Machado, J. A. TenreiroLeaves are mainly responsible for food production in vascular plants. Studying individual leaves can reveal important characteristics of the whole plant, namely its health condition, nutrient status, the presence of viruses and rooting ability. One technique that has been used for this purpose is Electrical Impedance Spectroscopy, which consists of determining the electrical impedance spectrum of the leaf. In this paper we use EIS and apply the tools of Fractional Calculus to model and characterize six species. Two modeling approaches are proposed: firstly, Resistance, Inductance, Capacitance electrical networks are used to approximate the leaves’ impedance spectra; afterwards, fractional-order transfer functions are considered. In both cases the model parameters can be correlated with physical characteristics of the leaves.
- Multidimensional Scaling Visualization using Parametric EntropyPublication . Lopes, António M.; Machado, J. A. Tenreiro; Galhano, Alexandra M.This paper studies complex systems using a generalized multidimensional scaling (MDS) technique. Complex systems are characterized by time-series responses, interpreted as a manifestation of their dynamics. Two types of time-series are analyzed, namely 18 stock markets and the gross domestic product per capita of 18 countries. For constructing the MDS charts, indices based on parametric entropies are adopted. Multiparameter entropies allow the variation of the parameters leading to alternative sets of charts. The final MDS maps are then assembled by means of Procrustes’ method that maximizes the fit between the individual charts. Therefore, the proposed method can be interpreted as a generalization to higher dimensions of the standard technique that represents (and discretizes) items by means of single “points” (i.e. zero-dimensional “objects”). The MDS plots, involving one-, two- and three-dimensional “objects”, reveal a good performance in capturing the correlations between data.
- 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.
- Integer vs. fractional order control of a hexapod robotPublication . Silva, Manuel F.; Machado, J. A. Tenreiro; Lopes, António M.This paper studies the performance of integer and fractional order controllers in a hexapod robot with joints at the legs having viscous friction and flexibility. For that objective the robot prescribed motion is characterized in terms of several locomotion variables. The controller performance is analised through the Nyquist stability criterion. A set of model-based experiments reveals the influence of the different controller implementations upon the proposed metrics.