Fractional dynamics in liquid manipulation

This paper presents a fractional calculus perspective in the study of signals captured during the movement of a mechanical manipulator carrying a liquid container. In order to study the signals an experimental setup is implemented. The system acquires data from the sensors, in real time, and, in a second phase, processes them through an analysis package. The analysis package runs off-line and handles the recorded data. The results show that the Fourier spectrum of several signals presents a fractional behavior. The experimental study provides useful information that can assist in the design of a control system and the trajectory planning to be used in reducing or eliminating the effect of vibrations.

Fractional order calculus: basic concepts and engineering applications

The fractional order calculus (FOC) is as old as the integer one although up to recently its application was exclusively in mathematics. Many real systems are better described with FOC differential equations as it is a well-suited tool to analyze problems of fractal dimension, with long-term “memory” and chaotic behavior. Those characteristics have attracted the engineers' interest in the latter years, and now it is a tool used in almost every area of science. This paper introduces the fundamentals of the FOC and some applications in systems' identification, control, mechatronics, and robotics, where it is a promissory research field.

Fractional order models of leaves

Leaves 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.

And I say to myself: “What a fractional world!”

This paper discusses several complex systems in the perspective of fractional dynamics. For prototype systems are considered the cases of deoxyribonucleic acid decoding, financial evolution, earthquakes events, global warming trend, and musical rhythms. The application of the Fourier transform and of the power law trendlines leads to an assertive representation of the dynamics and to a simple comparison of their characteristics. Moreover, the gallery of different systems, both natural and man made, demonstrates the richness of phenomena that can be described and studied with the tools of fractional calculus.

Fractional-order impulse response of the respiratory system

This paper presents the measurement, frequency-response modeling and identification, and the corresponding impulse time response of the human respiratory impedance and admittance. The investigated adult patient groups were healthy, diagnosed with chronic obstructive pulmonary disease and kyphoscoliosis, respectively. The investigated children patient groups were healthy, diagnosed with asthma and cystic fibrosis, respectively. Fractional order (FO) models are identified on the measured impedance to quantify the respiratory mechanical properties. Two methods are presented for obtaining and simulating the time-domain impulse response from FO models of the respiratory admittance: (i) the classical pole-zero interpolation proposed by Oustaloup in the early 90s, and (ii) the inverse discrete Fourier Transform (DFT). The results of the identified FO models for the respiratory admittance are presented by means of their average values for each group of patients. Consequently, the impulse time response calculated from the frequency response of the averaged FO models is given by means of the two methods mentioned above. Our results indicate that both methods provide similar impulse response data. However, we suggest that the inverse DFT is a more suitable alternative to the high order transfer functions obtained using the classical Oustaloup filter. Additionally, a power law model is fitted on the impulse response data, emphasizing the intrinsic fractal dynamics of the respiratory system.

Root locus of fractional linear systems

In this paper an algorithm for the calculation of the root locus of fractional linear systems is presented. The proposed algorithm takes advantage of present day computational resources and processes directly the characteristic equation, avoiding the limitations revealed by standard methods. The results demonstrate the good performance for different types of expressions.

Fractional dynamics of a system with particles subjected to impacts

This paper studies the dynamics of a system composed of a collection of particles that exhibit collisions between them. Several entropy measures and different impact conditions of the particles are tested. The results reveal a Power Law evolution both of the system energy and the entropy measures, typical in systems having fractional dynamics.

Fractional order control on a wind turbine benchmark

This paper is about a hierarchical structure with an event-based supervisor in a higher level and a fractional-order proportional integral (FOPI) in a lower level applied to a wind turbine. The event-based supervisor analyzes the operation conditions to determine the state of the wind turbine. This controller operate in the full load region and the main objective is to capture maximum power generation while ensuring the performance and reliability required for a wind turbine to be integrated into an electric grid. The main contribution focus on the use of fractional-order proportional integral controller which benefits from the introduction of one more tuning parameter, the integral fractional-order, taking advantage over integer order proportional integral (PI) controller. Comparisons between fractional-order pitch control and a default proportional integral pitch controller applied to a wind turbine benchmark are given and simulation results by Matlab/Simulink are shown in order to prove the effectiveness of the proposed approach.

Fuzzy, integer and fractional-order control: application on a wind turbine benchmark model

This paper presents a comparison between proportional integral control approaches for variable speed wind turbines. Integer and fractional-order controllers are designed using linearized wind turbine model whilst fuzzy controller also takes into account system nonlinearities. These controllers operate in the full load region and the main objective is to extract maximum power from the wind turbine while ensuring the performance and reliability required to be integrated into an electric grid. The main contribution focuses on the use of fractional-order proportional integral (FOPI) controller which benefits from the introduction of one more tuning parameter, the integral fractional-order, taking advantage over integer order proportional integral (PI) controller. A comparison between proposed control approaches for the variable speed wind turbines is presented using a wind turbine benchmark model in the Matlab/Simulink environment. Results show that FOPI has improved system performance when compared with classical PI and fuzzy PI controller outperforms the integer and fractional-order control due to its capability to deal with system nonlinearities and uncertainties. © 2014 IEEE.

Modeling of the lung impedance using a fractional order ladder network with constant phase elements

The self similar branching arrangement of the airways makes the respiratory system an ideal candidate for the application of fractional calculus theory. The fractal geometry is typically characterized by a recurrent structure. This study investigates the identification of a model for the respiratory tree by means of its electrical equivalent based on intrinsic morphology. Measurements were obtained from seven volunteers, in terms of their respiratory impedance by means of its complex representation for frequencies below 5 Hz. A parametric modeling is then applied to the complex valued data points. Since at low-frequency range the inertance is negligible, each airway branch is modeled by using gamma cell resistance and capacitance, the latter having a fractional-order constant phase element (CPE), which is identified from measurements. In addition, the complex impedance is also approximated by means of a model consisting of a lumped series resistance and a lumped fractional-order capacitance. The results reveal that both models characterize the data well, whereas the averaged CPE values are supraunitary and subunitary for the ladder network and the lumped model, respectively.

A fractional approach for the motion planning of redundant and hyper-redundant manipulators

The trajectory planning of redundant robots through the pseudoinverse control leads to undesirable drift in the joint space. This paper presents a new technique to solve the inverse kinematics problem of redundant manipulators, which uses a fractional differential of order α to control the joint positions. Two performance measures are defined to examine the strength and weakness of the proposed method. The positional error index measures the precision of the manipulator's end-effector at the target position. The repeatability performance index is adopted to evaluate if the joint positions are repetitive when the manipulator execute repetitive trajectories in the operational workspace. Redundant and hyper-redundant planar manipulators reveal that it is possible to choose in a large range of possible values of α in order to get repetitive trajectories in the joint space.

Recent history of fractional calculus

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date.

Offshore wind turbine simulation: multibody drive train. Bacl-to-back NPC (Neutral Point Clamped) converters: Fractional-order control

This paper is on offshore wind energy conversion systems installed on the deep water and equipped with back-to-back neutral point clamped full-power converter, permanent magnet synchronous generator with an AC link. The model for the drive train is a five-mass model which incorporates the dynamic of the structure and the tower in order to emulate the effect of the moving surface. A three-level converter and a four-level converter are the two options with a fractional-order control strategy considered to equip the conversion system. Simulation studies are carried out to assess the quality of the energy injected into the electric grid. Finally, conclusions are presented. (C) 2014 Elsevier Ltd. All rights reserved.

Particle swarm optimization: dynamical analysis through fractional calculus

This chapter considers the particle swarm optimization algorithm as a system, whose dynamics is studied from the point of view of fractional calculus. In this study some initial swarm particles are randomly changed, for the system stimulation, and its response is compared with a non-perturbed reference response. The perturbation effect in the PSO evolution is observed in the perspective of the fitness time behaviour of the best particle. The dynamics is represented through the median of a sample of experiments, while adopting the Fourier analysis for describing the phenomena. The influence upon the global dynamics is also analyzed. Two main issues are reported: the PSO dynamics when the system is subjected to random perturbations, and its modelling with fractional order transfer functions.

Optimal location of supports in beam structures using genetic algorithms

Real structures can be thought as an assembly of components, as for instances plates, shells and beams. This later type of component is very commonly found in structures like frames which can involve a significant degree of complexity or as a reinforcement element of plates or shells. To obtain the desired mechanical behavior of these components or to improve their operating conditions when rehabilitating structures, one of the eventual parameters to consider for that purpose, when possible, is the location of the supports. In the present work, a beam-type structure is considered, and for a set of cases concerning different number and types of supports, as well as different load cases, the authors optimize the location of the supports in order to obtain minimum values of the maximum transverse deflection. The optimization processes are carried out using genetic algorithms. The results obtained, clearly show a good performance of the approach proposed. © 2014 IEEE.