Design Templates for Some Fractional Order Control Systems

Time domain characteristics of first and second order systems are well known. But the same simplicity and explicitness do not exist for low order fractional order systems (FOSs). Considering the step response, the templates are developed for designing the behavior of simple FOSs with a 2-term denominator polynomial (one is unity and the other involves fractional power). Although the explicit relations between design parameters and the performance parameters such as time constant, rise time, overshoot, settling time for fractional order control systems (FOCSs) do not exist and can't be obtainable as in the ordinary integer order control systems, the obtained templates in this paper can be used for designing low order FOCSs. Hence, the drawback of non-existence of similar explicit formulas for FOCSs is eliminated by using these templates

Various analytical models for supercapacitors: a mathematical study

Supercapacitors (SCs) are used extensively in high-power potential energy applications like renewable energy systems, electric vehicles, power electronics, and many other industrial applications. This is due to SCs containing high-power density and the ability to respond spontaneously with fast charging and discharging demands. Advancements in material and fabrication techniques have induced a scope for research to improve the application of SCs. Many researchers have studied various SC properties and their effects on energy storage and management performance. In this paper, various fractional calculus-based SC models are summarized, with emphasis on analytical studies from derived classical SC models. Study prevails such parameterized resistor- capacitor networks have simplified the representation of electrical behavior of SCs to deal with the complicated internal structure. Fractional calculus has been used to develop SC models with the aim of understanding their complicated structure. Finally, the properties of different SC models utilized by various researchers to understand the behavior of SCs are listed using an equivalent circuit

Analysis of the Band-Pass and Notch Filter with Dynamic Damping of Fractional Order Including Discrete Models

The paper presents analysis of the second order band-pass and notch filter with a dynamic damping factor βd of fractional order. Factor βd is given in the form of fractional differentiator of order a, i.e. βd=β/sa, where β and a are adjustable parameters. The aim of the paper is to exploit an extra degree of freedom of presented filters to achieve the desired filter specifications and obtain a desired response in the frequency and time domain. Shaping of the frequency response enables achieving a better phase response compared to the integer-order counterparts which is of great concern in many applications. For the implementation purpose, the paper presents a comparison of four discretization techniques: the Osutaloup’s Recursive Algorithm (ORA+Tustin), Continued Fractional Expansion (CFE+Tustin), Interpolation of Frequency Characteristic (IFC+Tustin) and recently proposed AutoRegressive with eXogenous input (ARX)-based direct discretization method

CFOA-Based Fractional Order PIλDδ Controller

Conventional Current Feedback Operational Amplifier (CFOA) is not current controllable or not electronically controllable. It is thus of interest to add a current mirror into the CFOA in order to make it current controllable. This modification can be achieved by using Diamond Transistor (DT) instead of going through complicated IC fabrication process. This work applies the modified CFOA in fractional order proportional integral derivative (PIλDδ) controller. Both simulation and experimental results confirm that the modified CFOA is electronically controllable

Measurement and control of emergent phenomena emulated by resistive-capacitive networks, using fractionalorder internal model control and external adaptive control

A fractional-order internal model control technique is applied to a three-dimensional resistive-capacitive network to enforce desired closed loop dynamics of first order. In order to handle model mismatch issues resulting from the random allocation of the components within the network, the control law is augmented with a model-reference adaptive strategy in an external loop. By imposing a control law on the system to obey first order dynamics, a calibrated transient response is ensured. The methodology enables feedback control of complex systems with emergent responses and is robust in the presence of measurement noise or under conditions of poor model identification. Furthermore, it is also applicable to systems that exhibit higher order fractional dynamics. Examples of feedback-controlled transduction include cantilever positioning in atomic force microscopy or the control of complex de-excitation lifetimes encountered in many types of spectroscopies, e.g., nuclear magnetic, electron-spin, microwave, multiphoton fluorescence, Förster resonance, etc. The proposed solution should also find important applications in more complex electronic, microwave, and photonic lock-in problems. Finally, there are further applications across the broader measurement science and instrumentation community when designing complex feedback systems at the system level, e.g., ensuring the adaptive control of distributed physiological processes through the use of biomedical implants

Modelagem de circuitos elétricos com efeito de memória através do cálculo fracionário

Orientador: Prof. Dr. Cesar Augusto DartoraCoorientador: Prof. Ph.D. Eduardo Gonçalves de LimaDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Eletrica. Defesa : Curitiba, 21/02/2019Inclui referências: p.75-79Área de concentração: Sistemas EletrônicosResumo: Nos últimos anos houve um grande aumento de pesquisas referente ao cálculo fracionário e nos mais diversos ramos da ciencia, como a física, controle de sistemas, economia, biomedicina, areas da ciencia aplicada e engenharias. Atualmente a modelagem dos circuitos sao realizadas com o calculo de ordem inteira, porem sabe-se que os circuitos eletricos possuem efeito de memoria e nao-linearidade no qual atraves da modelagem usual nao se observa todo esse comportamento, e a modelagem com o calculo fracionario e uma das possíveis formas de analisar esse efeito. Neste trabalho e exemplificado a modelagem de um circuito usual, no caso um circuito RC, para diferentes valores da derivada, e, tambem, proposto um modelo matematico adaptado da definicao de Grunwald-Letnikov, para ser inserido na modelagem com estrutura de blocos em serie e para a estrutura de blocos em paralelo, e assim realizar a modelagem comportamental em amplificadores de potencia de radio frequencia. Foram utilizados três conjuntos de dados, contendo amostras de entrada e saída, de dois amplificadores de potencia para testar o modelo proposto e comparar as respostas com outros modelos ja utilizados na literatura, como por exemplo o Polinomio de Memoria e a modelagem de blocos em serie, atraves do erro quadrático medio normalizado. Realizadas as simulacoes para esses conjuntos de dados, nota-se que o modelo proposto, implementado nas estruturas em serie e paralelo, obteve respostas iguais aos modelos comparativos, isso para mesmas ordens de truncamentos e quantidade de coeficientes. Observa-se tambem, que o modelo proposto e capaz de reduzir a quantidade de coeficientes sem reduzir a precisao da modelagem, onde simulacoes foram realizadas com reducao de ate 50% dos coeficientes e nao houve perda considerável, em relacao ao modelo em comparacao. Palavras Chave: Calculo Fracionario, Derivadas Fracionarias, Efeito de Memoria, Modelagem de Circuitos.Abstract: In the recent years, an increase in research in the area of fractional calculus and in the most diverse branches of science has been recorded, such as physics, control systems, economics, biomedicine, applied science and engineering. Current circuit modeling techniques are performed with calculations of integer order, but it is known that electrical circuits have memory effect and non linearity, which cannot be observed by means of the usual modeling methods, thus the fractional calculation approach is one of the possible ways of analyzing this effect. In this work, an usual circu it, which is a RC circuit, is modeled and exemplified for different values of the derivative. Besides, a mathematical model adapted from the Grunwald-Letnikov definition is proposed to perform behavioral modeling in radiofrequency power amplifiers, one for serial block structure modeling and another for parallel block structure modeling. Input and output samples from three different data sets obtained for two power amplifiers were used to test the proposed model and to compare the responses with other models already used in the literature, such as the Memory Polynomial model and the serial block modeling, using the normalized mean square error. Once the simulations were performed for the data sets, the proposed models, implemented in series and parallel structures, obtained similar answers of the comparative models, for the same truncation orders and the same number of coefficients. It is also observed that the proposed models are able to reduce the number of coefficients without reducing the precision of the modeling, in which simulations were performed with reduction of up to 50% of the coefficients and it did not present considerable loss, when compared to other models. Keywords: Fractional Calculus, Fractional Derivatives, Memory Effect, Circuit Modeling

Analysis of transient processes, energy balance and frequency characteristics of electrical circuits with fractional elements

Električni elementi, napravljeni od materijala kod kojih se procesi polarizacije i magnetizacije ne dešavaju trenutno, već zavise i od istorije procesa, mogu se konstitutivno modelirati uopštenjem klasične konstitutivne jednačine dodavanjem člana koji uključuje istoriju fizičkog procesa. Matematički model frakcionog kondenzatora, pored člana koji odgovara klasičnom kondenzatoru, uključuje i član koji sadrži frakcioni integral, te uvažava da količina naelektrisanja na frakcionom kondenzatoru ne zavisi samo od trenutne vrednosti napona, već i od njegove istorije. U slučaju frakcionog uopštenja modela kalema, odnosno u slučaju kada se proces magnetizacije materijala ne dešava trenutno, već je uključena i istorija njene promene, fluks vektora magnetske indukcije se može izraziti superpozicijom člana koji u obzir uzima trenutnu vrednost struje i člana kojim se, korišćenjem frakcionog integrala, modelira uticaj istorije promene struje. Navedeni konstitutivni modeli su korišćeni za formulisanje jednačina frakcionih rednih RC, RL i RLC kola, koje daju struju kao odziv kola na pobudu u vidu elektromotorne sile.Takođe je proučavan prelazni i kvazistacionarni režim kola, energetski bilans i frekvencijske karakteristike.Electrical elements made of materials in which the processes of polarization and magnetization do not occur instantaneously, but also depend on the history of the process, can be constitutively modeled by generalizing the classical constitutive equation by adding a hereditary type term that includes the history of the physical process. The mathematical model of the fractional capacitor, in addition to the term corresponding to the classical capacitor, also includes the term containing the fractional integral, taking into account that the amount of charge on the fractional capacitor depends not only on the current voltage value, but also on its history. In the case of fractional generalization of the inductor model, ie in the case when the process of magnetization of the material does not occur instantaneously, but the history of its change is included, the total magnetic flux can be expressed in terms of current by superposition of the instantaneous and hereditary term, that is expressed trough the fractional integral. These constitutive models are further used to formulate the governing equations of fractional series RC, RL and RLC circuits, which provide circuit’s current as a response to the excitation in the form of electromotive force. The transient and steady state regime, energy balance and frequency characteristics of the circuits are also studied

Fully-Differential Frequency Filters with Modern Active Elements

Tato disertační práce se zaměřuje na výzkum v oblasti frekvenčních filtrů. Hlavním cílem je navrhnout a analyzovat plně diferenční kmitočtové filtry pracující v proudovém módu a využívající moderní aktivní prvky. Prezentované filtry jsou navrženy za použití proudových sledovačů, operačních transkonduktančních zesilovačů, plně diferenčních proudových zesilovačů a transrezistančních zesilovačů. Návrh se zaměřuje na možnost řídit některý z typických parametrů filtru pomocí řiditelných aktivních prvků, které jsou vhodně umístněny do obvodové struktury. Jednotlivé prezentované filtry jsou navrženy v nediferenční a diferenční verzi. Velký důraz je věnován srovnání plně diferenčních struktur s jejich odpovídajícími nediferenčními formami. Funkčnost jednotlivých návrhů je ověřena simulacemi a v některých případech i experimentálním měřením.This doctoral thesis focuses on research in the field of frequency filters. The main goal is to propose and analyze fully-differential current-mode frequency filters employing modern active elements. Presented filters are proposed using current followers, operational transconductance amplifiers, digitally adjustable current amplifiers and transresistance amplifiers. The proposal is focusing on ability to control some of the typical filter parameter or parameters using controllable active elements suitably placed in the circuit structure. Individual presented filters are proposed in their single-ended and fully-differential forms. Great emphasis is paid to a comparison of the fully-differential structures and their corresponding single-ended forms. The functionality of each proposal is verified by simulations and in some cases also by experimental measurements.

Analog Implementation of Fractional-Order Elements and Their Applications

With advancements in the theory of fractional calculus and also with widespread engineering application of fractional-order systems, analog implementation of fractional-order integrators and differentiators have received considerable attention. This is due to the fact that this powerful mathematical tool allows us to describe and model a real-world phenomenon more accurately than via classical “integer” methods. Moreover, their additional degree of freedom allows researchers to design accurate and more robust systems that would be impractical or impossible to implement with conventional capacitors. Throughout this thesis, a wide range of problems associated with analog circuit design of fractional-order systems are covered: passive component optimization of resistive-capacitive and resistive-inductive type fractional-order elements, realization of active fractional-order capacitors (FOCs), analog implementation of fractional-order integrators, robust fractional-order proportional-integral control design, investigation of different materials for FOC fabrication having ultra-wide frequency band, low phase error, possible low- and high-frequency realization of fractional-order oscillators in analog domain, mathematical and experimental study of solid-state FOCs in series-, parallel- and interconnected circuit networks. Consequently, the proposed approaches in this thesis are important considerations in beyond the future studies of fractional dynamic systems