Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

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.

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.

Modelização de equivalente eléctrico para rede de nervuras de folhas vegetais

A tese tem como objectivo principal a criação de um modelo equivalente eléctrico da rede de nervuras de algumas folhas vegetais e analisar o seu comportamento a estímulos eléctricos, analisando-se também a respectiva resposta em frequência. A motivação desta tese passa pela observação dos sistemas existentes na natureza. Neste caso, as folhas vegetais e analisar se são sistemas de ordem fraccionária ou não. Para a sua elaboração, fez-se uma breve abordagem à estrutura das plantas, sob o ponto de vista da botânica e elaborou-se um método de fotografia das amostras, mapeamento da rede de nervuras e medição dos segmentos que compõem essa mesma rede. A tese termina com um capítulo de resultados experimentais.

Development of fractional order capacitors based on electrolyte processes

In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

On the analysis of data emerging in non-linear and complex systemscomparison of two X-Ray prony spectra

New arguments proving that successive (repeated) measurements have a memory and actually remember each other are presented. The recognition of this peculiarity can change essentially the existing paradigm associated with conventional observation in behavior of different complex systems and lead towards the application of an intermediate model (IM). This IM can provide a very accurate fit of the measured data in terms of the Prony's decomposition. This decomposition, in turn, contains a small set of the fitting parameters relatively to the number of initial data points and allows comparing the measured data in cases where the “best fit” model based on some specific physical principles is absent. As an example, we consider two X-ray diffractometers (defined in paper as A- (“cheap”) and B- (“expensive”) that are used after their proper calibration for the measuring of the same substance (corundum a-Al2O3). The amplitude-frequency response (AFR) obtained in the frame of the Prony's decomposition can be used for comparison of the spectra recorded from (A) and (B) - X-ray diffractometers (XRDs) for calibration and other practical purposes. We prove also that the Fourier decomposition can be adapted to “ideal” experiment without memory while the Prony's decomposition corresponds to real measurement and can be fitted in the frame of the IM in this case. New statistical parameters describing the properties of experimental equipment (irrespective to their internal “filling”) are found. The suggested approach is rather general and can be used for calibration and comparison of different complex dynamical systems in practical purposes.

Robot Manipulator Systems: Analysis and Control

Manipulator systems are rather complex and highly nonlinear which makes difficult their analysis and control. Classic system theory is veil known, however it is inadequate in the presence of strong nonlinear dynamics. Nonlinear controllers produce good results [1] and work has been done e. g. relating the manipulator nonlinear dynamics with frequency response [2–5]. Nevertheless, given the complexity of the problem, systematic methods which permit to draw conclusions about stability, imperfect modelling effects, compensation requirements, etc. are still lacking. In section 2 we start by analysing the variation of the poles and zeros of the descriptive transfer functions of a robot manipulator in order to motivate the development of more robust (and computationally efficient) control algorithms. Based on this analysis a new multirate controller which is an improvement of the well known “computed torque controller” [6] is announced in section 3. Some research in this area was done by Neuman [7,8] showing tbat better robustness is possible if the basic controller structure is modified. The present study stems from those ideas, and attempts to give a systematic treatment, which results in easy to use standard engineering tools. Finally, in section 4 conclusions are presented.

A multi-objective model for scheduling of short-term incentive-based demand response programs offered by electricity retailers

In this paper, we formulate the electricity retailers’ short-term decision-making problem in a liberalized retail market as a multi-objective optimization model. Retailers with light physical assets, such as generation and storage units in the distribution network, are considered. Following advances in smart grid technologies, electricity retailers are becoming able to employ incentive-based demand response (DR) programs in addition to their physical assets to effectively manage the risks of market price and load variations. In this model, the DR scheduling is performed simultaneously with the dispatch of generation and storage units. The ultimate goal is to find the optimal values of the hourly financial incentives offered to the end-users. The proposed model considers the capacity obligations imposed on retailers by the grid operator. The profit seeking retailer also has the objective to minimize the peak demand to avoid the high capacity charges in form of grid tariffs or penalties. The non-dominated sorting genetic algorithm II (NSGA-II) is used to solve the multi-objective problem. It is a fast and elitist multi-objective evolutionary algorithm. A case study is solved to illustrate the efficient performance of the proposed methodology. Simulation results show the effectiveness of the model for designing the incentive-based DR programs and indicate the efficiency of NSGA-II in solving the retailers’ multi-objective problem.

Using demand response to deal with unexpected low wind power generation in the context of smart grid

Demand response is assumed an essential resource to fully achieve the smart grids operating benefits, namely in the context of competitive markets. Some advantages of Demand Response (DR) programs and of smart grids can only be achieved through the implementation of Real Time Pricing (RTP). The integration of the expected increasing amounts of distributed energy resources, as well as new players, requires new approaches for the changing operation of power systems. The methodology proposed aims the minimization of the operation costs in a smart grid operated by a virtual power player. It is especially useful when actual and day ahead wind forecast differ significantly. When facing lower wind power generation than expected, RTP is used in order to minimize the impacts of such wind availability change. The proposed model application is here illustrated using the scenario of a special wind availability reduction day in the Portuguese power system (8th February 2012).

The role of demand response in future power systems

Power systems are planed and operated according to the optimization of the available resources. Traditionally these tasks were mostly undertaken in a centralized way which is no longer adequate in a competitive environment. Demand response can play a very relevant role in this context but adequate tools to negotiate this kind of resources are required. This paper presents an approach to deal with these issues, by using a multi-agent simulator able to model demand side players and simulate their strategic behavior. The paper includes an illustrative case study that considers an incident situation. The distribution company is able to reduce load curtailment due to load flexibility contracts previously established with demand side players.

Demand response in future power systems management – a conceptual framework and simulation tool

Mestrado em Engenharia Electrotécnica – Sistemas Eléctricos de Energia

Differences in Mismatch Negativity (MMN) response to Pure-tone and Speech sounds in normal subjects: an additional explanation

The relation of automatic auditory discrimination, measured with MMN, with the type of stimuli has not been well established in the literature, despite its importance as an electrophysiological measure of central sound representation. In this study, MMN response was elicited by pure-tone and speech binaurally passive auditory oddball paradigm in a group of 8 normal young adult subjects at the same intensity level (75 dB SPL). The frequency difference in pure-tone oddball was 100 Hz (standard = 1 000 Hz; deviant = 1 100 Hz; same duration = 100 ms), in speech oddball (standard /ba/; deviant /pa/; same duration = 175 ms) the Portuguese phonemes are both plosive bi-labial in order to maintain a narrow frequency band. Differences were found across electrode location between speech and pure-tone stimuli. Larger MMN amplitude, duration and higher latency to speech were verified compared to pure-tone in Cz and Fz as well as significance differences in latency and amplitude between mastoids. Results suggest that speech may be processed differently than non-speech; also it may occur in a later stage due to overlapping processes since more neural resources are required to speech processing.

Evaluating P-NET message's response time with fixed priority queuing at application process level

P-NET is a multi-master fieldbus standard based on a virtual token passing scheme. In P-NET each master is allowed to transmit only one message per token visit. In the worst-case, the communication response time can be derived considering that, in each token cycle, all stations use the token to transmit a message. In this paper, we define a more sophisticated P-NET model, which considers the actual token utilisation. We then analyse the possibility of implementing a local priority-based scheduling policy to improve the real-time behaviour of P-NET.

Communication response time in P-NET networks: worst-case analysis considering the actual token utilisation

Fieldbus networks aim at the interconnection of field devices such as sensors, actuators and small controllers. Therefore, they are an effective technology upon which Distributed Computer Controlled Systems (DCCS) can be built. DCCS impose strict timeliness requirements to the communication network. In essence, by timeliness requirements we mean that traffic must be sent and received within a bounded interval, otherwise a timing fault is said to occur. P-NET is a multi-master fieldbus standard based on a virtual token passing scheme. In P-NET each master is allowed to transmit only one message per token visit, which means that in the worst-case the communication response time could be derived considering that the token is fully utilised by all stations. However, such analysis can be proved to be quite pessimistic. In this paper we propose a more sophisticated P-NET timing analysis model, which considers the actual token utilisation by different masters. The major contribution of this model is to provide a less pessimistic, and thus more accurate, analysis for the evaluation of the worst-case communication response time in P-NET fieldbus networks.

Response time analysis of COTS-Based multicores considering the contention on the shared memory bus

The current industry trend is towards using Commercially available Off-The-Shelf (COTS) based multicores for developing real time embedded systems, as opposed to the usage of custom-made hardware. In typical implementation of such COTS-based multicores, multiple cores access the main memory via a shared bus. This often leads to contention on this shared channel, which results in an increase of the response time of the tasks. Analyzing this increased response time, considering the contention on the shared bus, is challenging on COTS-based systems mainly because bus arbitration protocols are often undocumented and the exact instants at which the shared bus is accessed by tasks are not explicitly controlled by the operating system scheduler; they are instead a result of cache misses. This paper makes three contributions towards analyzing tasks scheduled on COTS-based multicores. Firstly, we describe a method to model the memory access patterns of a task. Secondly, we apply this model to analyze the worst case response time for a set of tasks. Although the required parameters to obtain the request profile can be obtained by static analysis, we provide an alternative method to experimentally obtain them by using performance monitoring counters (PMCs). We also compare our work against an existing approach and show that our approach outperforms it by providing tighter upper-bound on the number of bus requests generated by a task.