Fractional order modelling of zero length column desorption response for adsorbents with variable particle sizes

This manuscript analyses the data generated by a Zero Length Column (ZLC) diffusion experimental set-up, for 1,3 Di-isopropyl benzene in a 100% alumina matrix with variable particle size. The time evolution of the phenomena resembles those of fractional order systems, namely those with a fast initial transient followed by long and slow tails. The experimental measurements are best fitted with the Harris model revealing a power law behavior.

Application of continuous wavelet transform to the analysis of the modulus of the fractional Fourier transform bands for resolving two component mixture

In this paper, the fractional Fourier transform (FrFT) is applied to the spectral bands of two component mixture containing oxfendazole and oxyclozanide to provide the multicomponent quantitative prediction of the related substances. With this aim in mind, the modulus of FrFT spectral bands are processed by the continuous Mexican Hat family of wavelets, being denoted by MEXH-CWT-MOFrFT. Four modulus sets are obtained for the parameter a of the FrFT going from 0.6 up to 0.9 in order to compare their effects upon the spectral and quantitative resolutions. Four linear regression plots for each substance were obtained by measuring the MEXH-CWT-MOFrFT amplitudes in the application of the MEXH family to the modulus of the FrFT. This new combined powerful tool is validated by analyzing the artificial samples of the related drugs, and it is applied to the quality control of the commercial veterinary samples.

Fractional generalization of memristor and higher order elements

Fractional 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.

Fractional order modelling of fractional-order holds

Discrete time control systems require sample- and-hold circuits to perform the conversion from digital to analog. Fractional-Order Holds (FROHs) are an interpolation between the classical zero and first order holds and can be tuned to produce better system performance. However, the model of the FROH is somewhat hermetic and the design of the system becomes unnecessarily complicated. This paper addresses the modelling of the FROHs using the concepts of Fractional Calculus (FC). For this purpose, two simple fractional-order approximations are proposed whose parameters are estimated by a genetic algorithm. The results are simple to interpret, demonstrating that FC is a useful tool for the analysis of these devices.

Multidimensional scaling analysis of fractional systems

This paper investigates the use of multidimensional scaling in the evaluation of fractional system. Several algorithms are analysed based on the time response of the closed loop system under the action of a reference step input signal. Two alternative performance indices, based on the time and frequency domains, are tested. The numerical experiments demonstrate the feasibility of the proposed visualization method.

The effect of fractional order in variable structure control

This paper studies fractional variable structure controllers. Two cases are considered namely, the sliding reference model and the control action, that are generalized from integer into fractional orders. The test bed consists in a mechanical manipulator and the effect of the fractional approach upon the system performance is evaluated. The results show that fractional dynamics, both in the switching surface and the control law are important design algorithms in variable structure controllers.

Introducing the fractional-order Darwinian PSO

One of the most well-known bio-inspired algorithms used in optimization problems is the particle swarm optimization (PSO), which basically consists on a machinelearning technique loosely inspired by birds flocking in search of food. More specifically, it consists of a number of particles that collectively move on the search space in search of the global optimum. The Darwinian particle swarm optimization (DPSO) is an evolutionary algorithm that extends the PSO using natural selection, or survival of the fittest, to enhance the ability to escape from local optima. This paper firstly presents a survey on PSO algorithms mainly focusing on the DPSO. Afterward, a method for controlling the convergence rate of the DPSO using fractional calculus (FC) concepts is proposed. The fractional-order optimization algorithm, denoted as FO-DPSO, is tested using several well-known functions, and the relationship between the fractional-order velocity and the convergence of the algorithm is observed. Moreover, experimental results show that the FO-DPSO significantly outperforms the previously presented FO-PSO.

Memória de longo prazo nos retornos acionistas dos índices de referência da euronext, implicações para a hipótese de mercados eficientes e contributo fractal para aperfeiçoamento do capital asset pricing model

Não existe uma definição única de processo de memória de longo prazo. Esse processo é geralmente definido como uma série que possui um correlograma decaindo lentamente ou um espectro infinito de frequência zero. Também se refere que uma série com tal propriedade é caracterizada pela dependência a longo prazo e por não periódicos ciclos longos, ou que essa característica descreve a estrutura de correlação de uma série de longos desfasamentos ou que é convencionalmente expressa em termos do declínio da lei-potência da função auto-covariância. O interesse crescente da investigação internacional no aprofundamento do tema é justificado pela procura de um melhor entendimento da natureza dinâmica das séries temporais dos preços dos ativos financeiros. Em primeiro lugar, a falta de consistência entre os resultados reclama novos estudos e a utilização de várias metodologias complementares. Em segundo lugar, a confirmação de processos de memória longa tem implicações relevantes ao nível da (1) modelação teórica e econométrica (i.e., dos modelos martingale de preços e das regras técnicas de negociação), (2) dos testes estatísticos aos modelos de equilíbrio e avaliação, (3) das decisões ótimas de consumo / poupança e de portefólio e (4) da medição de eficiência e racionalidade. Em terceiro lugar, ainda permanecem questões científicas empíricas sobre a identificação do modelo geral teórico de mercado mais adequado para modelar a difusão das séries. Em quarto lugar, aos reguladores e gestores de risco importa saber se existem mercados persistentes e, por isso, ineficientes, que, portanto, possam produzir retornos anormais. O objetivo do trabalho de investigação da dissertação é duplo. Por um lado, pretende proporcionar conhecimento adicional para o debate da memória de longo prazo, debruçando-se sobre o comportamento das séries diárias de retornos dos principais índices acionistas da EURONEXT. Por outro lado, pretende contribuir para o aperfeiçoamento do capital asset pricing model CAPM, considerando uma medida de risco alternativa capaz de ultrapassar os constrangimentos da hipótese de mercado eficiente EMH na presença de séries financeiras com processos sem incrementos independentes e identicamente distribuídos (i.i.d.). O estudo empírico indica a possibilidade de utilização alternativa das obrigações do tesouro (OT’s) com maturidade de longo prazo no cálculo dos retornos do mercado, dado que o seu comportamento nos mercados de dívida soberana reflete a confiança dos investidores nas condições financeiras dos Estados e mede a forma como avaliam as respetiva economias com base no desempenho da generalidade dos seus ativos. Embora o modelo de difusão de preços definido pelo movimento Browniano geométrico gBm alegue proporcionar um bom ajustamento das séries temporais financeiras, os seus pressupostos de normalidade, estacionariedade e independência das inovações residuais são adulterados pelos dados empíricos analisados. Por isso, na procura de evidências sobre a propriedade de memória longa nos mercados recorre-se à rescaled-range analysis R/S e à detrended fluctuation analysis DFA, sob abordagem do movimento Browniano fracionário fBm, para estimar o expoente Hurst H em relação às séries de dados completas e para calcular o expoente Hurst “local” H t em janelas móveis. Complementarmente, são realizados testes estatísticos de hipóteses através do rescaled-range tests R/S , do modified rescaled-range test M - R/S e do fractional differencing test GPH. Em termos de uma conclusão única a partir de todos os métodos sobre a natureza da dependência para o mercado acionista em geral, os resultados empíricos são inconclusivos. Isso quer dizer que o grau de memória de longo prazo e, assim, qualquer classificação, depende de cada mercado particular. No entanto, os resultados gerais maioritariamente positivos suportam a presença de memória longa, sob a forma de persistência, nos retornos acionistas da Bélgica, Holanda e Portugal. Isto sugere que estes mercados estão mais sujeitos a maior previsibilidade (“efeito José”), mas também a tendências que podem ser inesperadamente interrompidas por descontinuidades (“efeito Noé”), e, por isso, tendem a ser mais arriscados para negociar. Apesar da evidência de dinâmica fractal ter suporte estatístico fraco, em sintonia com a maior parte dos estudos internacionais, refuta a hipótese de passeio aleatório com incrementos i.i.d., que é a base da EMH na sua forma fraca. Atendendo a isso, propõem-se contributos para aperfeiçoamento do CAPM, através da proposta de uma nova fractal capital market line FCML e de uma nova fractal security market line FSML. A nova proposta sugere que o elemento de risco (para o mercado e para um ativo) seja dado pelo expoente H de Hurst para desfasamentos de longo prazo dos retornos acionistas. O expoente H mede o grau de memória de longo prazo nos índices acionistas, quer quando as séries de retornos seguem um processo i.i.d. não correlacionado, descrito pelo gBm(em que H = 0,5 , confirmando- se a EMH e adequando-se o CAPM), quer quando seguem um processo com dependência estatística, descrito pelo fBm(em que H é diferente de 0,5, rejeitando-se a EMH e desadequando-se o CAPM). A vantagem da FCML e da FSML é que a medida de memória de longo prazo, definida por H, é a referência adequada para traduzir o risco em modelos que possam ser aplicados a séries de dados que sigam processos i.i.d. e processos com dependência não linear. Então, estas formulações contemplam a EMH como um caso particular possível.

An introduction to the fractional continuous-time linear systems

A brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of the fractional derivatives. We will show that the computation of the impulse and step responses is very similar to the classic. The main difference lies in the substitution of the exponential by the Mittag-Leffler function. We will present also the main formulae defining the fractional derivatives.

On the fractional central differences and derivatives

Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied.

Time-delay and fractional derivatives

This paper proposes the calculation of fractional algorithms based on time-delay systems. The study starts by analyzing the memory properties of fractional operators and their relation with time delay. Based on the Fourier analysis an approximation of fractional derivatives through timedelayed samples is developed. Furthermore, the parameters of the proposed approximation are estimated by means of genetic algorithms. The results demonstrate the feasibility of the new perspective.

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.

Fractional dynamics in financial indexes

The goal of this study is the analysis of the dynamical properties of financial data series from 32 worldwide stock market indices during the period 2000–2009 at a daily time horizon. Stock market indices are examples of complex interacting systems for which a huge amount of data exists. The methods and algorithms that have been explored for the description of physical phenomena become an effective background in the analysis of economical data. In this perspective are applied the classical concepts of signal analysis, Fourier transform and methods of fractional calculus. The results reveal classification patterns typical of fractional dynamical systems.