Dissertations / Theses on the topic 'Fractional Order'

This research thesis within the XXVI Ph.D. Course in Systems Engineering is centered on Fractional, or also noticed as Non-Integer Order, Systems. The Ph.D. Thesis' purpose is to present the auto-tuning procedure of Fractional Order PID Controllers PID proposing also a future implementation via fractional order elements. The Thesis is organized in four chapters to outline the different research activity phases developed during the Ph.D. course. In the Chapter 1 an overview on Fractional Order Systems is presented. The Chapter 2 focuses on the auto-tuning procedure proposed for Non-Integer Order Proportional-Integral-Derivative (PID) controllers and some examples are presented. The Chapter 3 is divided in two parts: the first one is dedicated to the modeling and control of Ionic Polymer Metal Composite (IPMC) and than a study on Fractional Order Elements modeling as possible building block for controller implementation is proposed. Finally the conclusions are presented in the Chapter 4.

This thesis considers the development of novel parameter estimation and system identification methods for fractional-order continuous-time nonlinear systems from sampled input-output signals. It is recognised that there is no universal model parameter estimation method which would be suitable for all types of nonlinear system models. In this work, two parameter estimation methods, targeting specific nonlinear model structures, have been developed. The first proposed parameter estimation method is based on an extension of the simplified and refined instrumental variable method for identification of integer-order continuous-time linear transfer function models. The proposed extended method is able to estimate parameters of fractional-order continuous-time Hammerstein–Wiener (HWFC) models, where the case of estimation of linear, Hammerstein (HFC), and Wiener (WFC) models is considered as a special case. It is also possible to estimate the classical integerorder model counterparts as a special case. Subsequently, the proposed extension to the simplified and refined instrumental variable methods for HWFC model estimation is abbreviated to HWSRIVCF and HWRIVCF, respectively. The refined version, HWRIVCF, considers the noise model to be of Box-Jenkins type, while the simplified version, HWRIVCF, assumes an output error measurement noise scenario. The advantage of this novel extension, compared to published methods, is that the output static nonlinearity of the Wiener model part does not need to be invertible. The second proposed fractional-order parameter estimation method is based on an existing delayed integer-order state variable filtering technique. In general, the developed method is able to estimate parameters of continuous-time fractional-order nonlinear models, when formulated in input-output form. The individual elements of the inputoutput equation (regression model) comprises of higher order time derivatives, signal powers, and products between the input and output signals and their powers. In this thesis, the focus is on a special model subclass, namely, a class of bilinear system models due mainly to its previous use in control engineering applications. A comprehensive case study, presenting the full system identification cycle, is also given. In this study, fractional-order continuous-time transfer function model of a 1D linear solid a diffusion process has been identified from sampled input-output data. The data was generated from a governing diffusion equation solved by the finite volume method.

Dans cette thèse, nous nous intéressons à résoudre certains problèmes inverses pour des équations différentielles aux dérivées fractionnaires. Un problème inverse est généralement mal posé. Un problème mal posé est un problème qui ne répond pas à l’un des trois critères de Hadamard pour être bien posé, c’est-à-dire, soit l’existence, l’unicité ou une dépendance continue aux données n'est plus vraie, à savoir, des petits changements dans les données de mesure entraînent des changements indéfiniment importants dans la solution. La plupart des difficultés à résoudre des problèmes mal posés sont causées par l’instabilité de la solution. D’autre part, les équations différentielles fractionnaires deviennent un outil important dans la modélisation de nombreux problèmes de la vie réelle et il y a eu donc un intérêt croissant pour l’étude des problèmes inverses avec des équations différentielles fractionnaires. Le calcul fractionnaire est une branche des mathématiques qui fait référence à l’extension du concept de dérivation classique à la dérivation d’ordre non entier. Calculer une dérivée fractionnaire à un certain moment exige tous les processus précédents avec des propriétés de mémoire. C’est l’avantage principal du calcul fractionnaire d’expliquer les processus associés aux systèmes physiques complexes qui ont une mémoire à long terme et / ou des interactions spatiales à longue distance. De plus, les équations différentielles fractionnaires peuvent nous aider à réduire les erreurs découlant de paramètres négligés dans la modélisation des phénomènes physiques
In this thesis, we are interested in solving some inverse problems for fractional differential equations. An inverse problem is usually ill-posed. The concept of an ill-posed problem is not new. While there is no universal formal definition for inverse problems, Hadamard [1923] defined a problem as being ill-posed if it violates the criteria of a well-posed problem, that is, either existence, uniqueness or continuous dependence on data is no longer true, i.e., arbitrarily small changes in the measurement data lead to indefinitely large changes in the solution. Most difficulties in solving ill-posed problems are caused by solution instability. Inverse problems come into various types, for example, inverse initial problems where initial data are unknown and inverse source problems where the source term is unknown. These unknown terms are to be determined using extra boundary data. Fractional differential equations, on the other hand, become an important tool in modeling many real-life problems and hence there has been growing interest in studying inverse problems of time fractional differential equations. The Non-Integer Order Calculus, traditionally known as Fractional Calculus is the branch of mathematics that tries to interpolate the classical derivatives and integrals and generalizes them for any orders, not necessarily integer order. The advantages of fractional derivatives are that they have a greater degree of flexibility in the model and provide an excellent instrument for the description of the reality. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time, i.e., calculating timefractional derivative at some time requires all the previous processes with memory and hereditary properties

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 231-237).
This thesis is focused on the theoretical characterization of topological order in non-Abelian fractional quantum Hall (FQH) states. The first part of the thesis is concerned with the ideal wave function approach to FQH states, where the idea is to try to obtain model wave functions and model Hamiltonians for all possible FQH states and to have a physical way of characterizing their topological order. I will explain recent attempts to do this through the so-called pattern of zeros framework and its relation to conformal field theory. The first chapter about the pattern of zeros introduces the basic concepts for single-component FQH states, how it relates to the conformal field theory approach to FQH wave functions, and how it can be used to derive various topological properties of FQH states. The second chapter extends the pattern of zeros framework to multi-component non-Abelian FQH states; this is an attempt at a full classification of possible topological orders in FQH states. Aside from the ideal wave function methods. the other known general method of constructing non-Abelian FQH states is through the parton construction. Here the idea is to break apart the electron into other fermions, called partons. and assume that they form integer quantum Hall states. This method allows us to describe all known FQH states. After reviewing the parton construction, I will demonstrate how it can be used to derive the low energy effective field theories for some of the most well-known non-Abelian FQH states, the Zk parafermion (Laughlin/Moore-Read/Read-Rezayi) states. The parton construction will motivate yet another topological field theory, the U(1) x U(1) x Z2 Chern-Simons (CS) theory. I will demonstrate how to calculate many highly non-trivial topological properties of the U(1) x U(1) x Z2 CS theory, such as ground state degeneracy on genus g surfaces and various fusion properties of the quasiparticles. Using the U(1) x U(1) x Z2 CS theory, we will study phase transitions between bilayer Abelian states and non-Abelian states. The non-Abelian ones contain a series of new states, which we call the orbifold FQH states. These orbifold FQH states turn out to be important for the conceptual foundations of the pattern of zeros/vertex algebra approach to ideal FQH wave functions. We also find a series of non-Abelian topological phases - which are not FQH states and do not have protected gapless edge modes - that are separated from the deconfined phase of ZN gauge theories by a continuous phase transition. We give a preliminary analysis of these Z2 "twisted" ZN topological phases.
by Maissam Barkeshli.
Ph.D.

Although the concept of fractional calculus was known for centuries, it was not considered in engineering due to the lack of implementation tools and acceptable performance of integer order models and control. However, recently, engineers and researchers started to investigate the potentially high performance of fractional calculus in various fields among which are acoustics, conservation of mass, diffusion equation and specifically in this thesis control theory. The intention of this thesis is to analyze the relative performance of fractional versus integer order PID controller for a quadcopter. Initially, the dynamics of the quadcopter is presented with additional consideration of the ground effect and torque saturation. Then, are introduced the concept of fractional calculus and the mathematical tools to be used for modeling fractional order controller. Finally, the performance of the fractional order controller is evaluated by comparing it to an integer order controller.

Fractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include non-integer orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Single-term and Multi-term FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Single-term and Multi-term methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a non-linear Multi-term Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.

Surface temperature control of a thin plate is investigated. Temperature is controlled on one side of the plate using the other side temperature measurements. This is a decades-old problem, reactivated more recently by the awareness that this is a fractional-order problem that justifies the investigation of the use of fractional order calculus. The approach is based on a transfer function obtained from the one-dimensional heat conduction equation solution that results in a fractional-order s-domain representation. Both the inverse problem approach and the fractional controller approach are studied here to control the surface temperature, the first one using inverse problem plus a Proportional only controller, and the second one using only the fractional controller. The direct problem defined as the ratio of the output to the input, while the inverse problem defined as the ratio of the input to the output. Both transfer functions are obtained, and the resulting fractional-order transfer functions were approximated using Taylor expansion and Zero-Pole expansion. The finite number of terms transfer functions were used to form an open-loop control scheme and a closed-loop control scheme. Simulation studies were done for both control schemes and experiments were carried out for closed-loop control schemes. For the fractional controller approach, the fractional controller was designed and used in a closed-loop scheme. Simulations were done for fractional-order-integral, fractional-order-derivative and fractional-integral-derivative controller designs. The experimental study focussed on the fractional-order-integral-derivative controller design. The Fractional-order controller results are compared to integer-order controller’s results. The advantages of using fractional order controllers were evaluated. Both Zero-Pole and Taylor expansions are used to approximate the plant transfer functions and both expansions results are compared. The results show that the use of fractional order controller performs better, in particular concerning the overshoot.

Cette thèse vise à concevoir des estimateurs non-asymptotiques et robustes pour les systèmes linéaires d’ordre fractionnaire dans un environnement bruité. Elle traite une classe des systèmes linéaires d’ordre fractionnaire modélisée par la dite pseudo représentation d’état avec des conditions initiales inconnues. Elle suppose également que les systèmes étudiés ici peuvent être transformés sous la forme canonique de Brunovsky. Pour estimer le pseudo-état, la forme précédente est transformée en une équation différentielle linéaire d’ordre fractionnaire en prenant en compte les valeurs initiales des dérivées fractionnaires séquentielles de la sortie. Ensuite, en utilisant la méthode des fonctions modulatrices, les valeurs initiales précédentes et les dérivées fractionnaires avec des ordres commensurables de la sortie sont données par des formules algébriques avec des intégrales à l’aide d’une méthode récursive. Ainsi, ces formules sont utilisés pour calculer le pseudo-état dans le cas continu sans bruit. En outre, elle fournit un algorithme pour construire les fonctions modulatrices requises à l’accomplissement de l’estimation. Deuxièmement, inspiré par la méthode des fonctions modulatrices développée pour l’estimation de pseudo-état, cette méthode algébrique basée sur un opérateur est introduite pour estimer la dérivée fractionnée avec un ordre arbitraire fractionnaire de la sortie pour les systèmes considérés. Cet opérateur sert à annuler les valeurs initiales non désirées, puis permet d’estimer la dérivée fractionnaire souhaitée par une nouvelle formule algébrique à l’aide d’une méthode récursive. Troisièmement, l’estimateur du pseudo-état et le différenciateur d’ordre fractionnaire obtenus précédemment sont étudiés respectivement dans le cas discret et bruité. Chacun d’entre eux contient une erreur numérique due à la méthode d’intégration numérique utilisée et au bruit. En particulier, elle fournit une analyse pour diminuer la contribution du bruit au moyen d’une d’erreur bornée qui permet de sélectionner les degrés optimaux des fonctions de modulation à chaque instant. Ensuite, des exemples numériques sont donnés pour mettre en évidence la précision, la robustesse et la propriété non-asymptotique des estimateurs proposés. En outre, les comparaisons avec certaines méthodes existantes et avec un nouvel observateur d’ordre fractionnaire de typeH1sont montrées. Enfin, elle donne des conclusions
This thesis aims to design non-asymptotic and robust estimators for a class of fractional order linear systems in noisy environment. It deals with a class of commensurate fractional order linear systems modeled by the so-called pseudo-state space representation with unknown initial conditions. It also assumed that linear systems under study can be transformed into the Brunovsky’s observable canonical form. Firstly, the pseudo-state of the considered systems is estimated. For this purpose, the Brunovsky’s observable canonical form is transformed into a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output. Then, using the modulating functions method, the former initial values and the fractional derivatives with commensurate orders of the output are given by algebraic integral formulae in a recursive way. Thereby, they are used to calculate the pseudo-state in the continuous noise-free case. Moreover, to perform this estimation, it provides an algorithm to build the required modulating functions. Secondly, inspired by the modulating functions method developed for pseudo-state estimation, an operator based algebraic method is introduced to estimate the fractional derivative with an arbitrary fractional order of the output. This operator is applied to cancel the former initial values and then enables to estimate the desired fractional derivative by a new algebraic formula using a recursive way. Thirdly, the pseudo-state estimator and the fractional order differentiator are studied in discrete noisy case. Each of them contains a numerical error due to the used numerical integration method, and the noise error contribution due to a class of stochastic processes. In particular, it provides ananalysis to decrease noise contribution by means of an error bound that enables to select the optimal degrees of the modulating functions at each instant. Then, several numerical examples are given to highlight the accuracy, the robustness and the non-asymptotic property of the proposed estimators. Moreover, the comparisons to some existing methods and a new fractional orderH1-like observer are shown. Finally, conclusions are outlined with some perspectives

This work presents a new control strategy using fractional order operators in threephase grid-connected photovoltaic generation systems with unity power factor for any situation of solar radiation. The modeling of the space vector pulse width modulation inverter and fractional order control strategy using Park’s transformation are proposed. The system is able to compensate harmonic components and reactive power generated by the loads connected to the system. A fractional order extremum seeking control and “Bode’s ideal cut-off extremum seeking control” are proposed to control the power between the grid and photovoltaic system, to achieve the maximum power point operation. Simulation results are presented to validate the proposed methodology for grid-connected photovoltaic generation systems. The simulation results and theoretical analysis indicate that the proposed control strategy improves the efficiency of the system by reducing the total harmonic distortion of the injected current to the grid and increases the robustness of the system against uncertainties. Additionally, the proposed maximum power point tracking algorithms provide more robustness and faster convergence under environmental variations than other maximum power point trackers.

This thesis is primarily concerned with developing new models and numerical methods based on the fractional generalisation of the Bloch and Bloch-Torrey equations to account for anomalous MRI signal attenuation. The two main contributions of the research are to investigate the anomalous evolution of MRI signals via the fractionalised Bloch equations, and to develop new effective numerical methods with supporting analysis to solve the time-space fractional Bloch-Torrey equations.

S pokroky v teorii počtu neceločíselného řádu a také s rozšířením inženýrských aplikací systémů neceločíselného řádu byla značná pozornost věnována analogové implementaci integrátorů a derivátorů neceločíselného řádu. Je to dáno tím, že tento mocný matematický nástroj nám umožňuje přesněji popsat a modelovat fenomén reálného světa ve srovnání s klasickými „celočíselnými“ metodami. Navíc nám jejich dodatečný stupeň volnosti umožňuje navrhovat přesnější a robustnější systémy, které by s konvenčními kondenzátory bylo nepraktické nebo nemožné realizovat. V předložené disertační práci je věnována pozornost širokému spektru problémů spojených s návrhem analogových obvodů systémů neceločíselného řádu: optimalizace rezistivně-kapacitních a rezistivně-induktivních typů prvků neceločíselného řádu, realizace aktivních kapacitorů neceločíselného řádu, analogová implementace integrátoru neceločíselného řádů, robustní návrh proporcionálně-integračního regulátoru neceločíselného řádu, výzkum různých materiálů pro výrobu kapacitorů neceločíselného řádu s ultraširokým kmitočtovým pásmem a malou fázovou chybou, možná realizace nízkofrekvenčních a vysokofrekvenčních oscilátorů neceločíselného řádu v analogové oblasti, matematická a experimentální studie kapacitorů s pevným dielektrikem neceločíselného řádu v sériových, paralelních a složených obvodech. Navrhované přístupy v této práci jsou důležitými faktory v rámci budoucích studií dynamických systémů neceločíselného řádu.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

This work revolves around the use of fractional order calculus in control science. Techniques such as fractional order universal adaptive stabilization (FO-UAS), and the fascinating results of their application to real-world systems, are presented initially. A major portion of this work deals with fractional order modeling and control of real-life systems like heat flow, fan and plate, and coupled tank systems. The fractional order models and controllers are not only simulated, they are also emulated using analog hardware. The main aim of all the above experimentation is to develop a fractional order controller for plasma position control of the Saskatchewan torus-1, modified (STOR-1M) tokamak at the Utah State University (USU) campus. A new method for plasma position estimation has been formulated. The results of hardware emulation of plasma position and its control are also presented. This work performs a small scale test measuring controller performance, so that it serves as a platform for future research efforts leading to real-life implementation of a plasma position controller for the tokamak.

This thesis developed a new practical tuning method for fractional order proportional and integral controllers (FO-PI) for varying time-delay systems like networked control systems (NCS), sensor networks, etc. Based on previously proposed FO-PI controller tuning rules using fractional Ms constrained integral gain optimization (F-MIGO), simultaneous maximization of the jitter margin and integrated time weighted absolute error (ITAE) performance for a set of hundred gain delay time-constant (KLT) systems having different time-constants and time-delay values are achieved. A multi-objective optimization algorithm is used to simultaneously maximize the ITAE factor and jitter margin of the plants at initial F-MIGO gain parameters. The new values of controller gain parameters are generalized to give a new set of optimal fractional order proportional integral (OFOPI) tuning rules such that the jitter margin and system performance of closed-loop KLT systems are maximized and yet the closed-loop feedback system is stable. This is further tested and verified by simulation techniques. Comparisons are made with other existing proportional integral derivative (PID) and fractional order proportional integral (PI) tuning rules to prove the efficiency of the new technique. It is further shown that OFOPI tuning rules perform better than traditional tuning methods for lag-dominated FOPDT systems, because it can take the varying time-delay better into account. The tuning method is modified to work with discrete-time controllers in the context of NCSs. Furthermore, experimental results in a NCS platform, Stand-alone Smart Wheel (omnidirectional networked control robot wheel), are reported using the tuning rules developed in this thesis. The optimization tuning method performed almost equally well in practice as in simulations. The thesis also shows that the tuning rule development procedure for OFOPI is not only valid for FOPDT systems but is also applicable for other general classes of plants which could be reduced to first order plant systems. Temperature control in heat flow apparatus and water-level control in a coupled tank system using FO-PI tuning rules are other major contributions of this thesis work.

Disertační práce se zabývá syntézou a analýzou nových obvodových struktur neceločíselného (fraktálního) řádu s řiditelnými parametry. Hlavní cíl této práce je návrh nových řešení filtračních struktur fraktálního řádu v proudovém módu, emulátorů prvků fraktálního řádu a také oscilátorů. Práce obsahuje návrh tří emulátorů pasivního prvku fraktálního řádu, tři filtrační struktury a dva oscilátory navržené na základě využití pasivního prvku fraktálního řádu v jejich obvodové struktuře a dvě obecné koncepce filtrů fraktálního řádu založené na využití aproximace přenosové funkce fraktálního řádu. Na základě obecných koncepcí jsou v práci navrženy filtry fraktálního řádu typu dolní a horní propust. Díky aktivním prvkům s přeladitelnými parametry, které jsou užity v obvodových strukturách je zajištěna řiditelnost řádu filtru, jeho pólového kmitočtu a některých případech i činitele jakosti. Vlastnosti všech zapojení jsou ověřeny počítačovými simulacemi za pomoci behavioralních simulačních modelů aktivních prvků. Některé z uvedených obvodů byly realizovány na DPS a jejich vlastnosti ověřeny experimentálním měřením.

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In this paper, a new fractional order chaotic system without equilibrium is proposed, analyti-cally and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigation were used to describe the system dynamical behaviors including, the system equilibria, the chaotic attractors, the bifurcation diagrams and the Lyapunov expo-nents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attrac-tors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive con-trol theory has been developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state varia-bles for the master and slave. Consequently, the update laws of the slave parameters are ob-tained, where the slave parameters are assumed to be uncertain and estimate corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results are obtained by MATLAB and the Arduino Due boards respectively, where a good consistent between the simulation results and the ex-perimental results. indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

Fractional calculus is ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. The idea of generalizing differential operators to a non-integer order, in particular to the order 1/2, first appears in the correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contributed to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand and Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only after the First Conference on Fractional Calculus and its applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Recently, many mathematicians and applied researchers have tried to model real processes using the fractional calculus. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can be successfully achieved by using fractional calculus. In other words, the nature of the definition of the fractional derivatives have provided an excellent instrument for the modeling of memory and hereditary properties of various materials and processes.
Il calcolo frazionario e` ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. L’ idea di generalizzare operatori differenziali ad un ordine non intero, in particolare di ordine 1/2, compare per la prima volta in una corrispondenza di Leibniz con L’Hopital (1695), Johann Bernoulli (1695), e John Wallis (1697), come una semplice domanda o forse un gioco di pensieri. Nei successive trecento anni molti matematici hanno contribuito al calcolo frazionario: Laplace (1812), Lacroix (1812), di Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand e Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), e molti altri. Eppure, è solo dopo la prima conferenza sul calcolo frazionario e le sue applicazioni che questo tema diventa una delle le aree più intensamente studiate dell’analisi matematica. Recentemente, molti matematici e ingegneri hanno cercato di modellare i processi reali utilizzando il calcolo frazionario. Questo a causa del fatto che spesso, la modellazione realistica di un fenomeno fisico non è locale nel tempo, ma dipende anche dalla storia, e questo comportamento può essere ben rappresentato attraverso modelli basati sul calcolo frazionario. In altre parole, la definizione dei derivata frazionaria fornisce un eccellente strumento per la modellazione della memoria e delle proprietà ereditarie di vari materiali e processi.

In this dissertation, a broad spectrum of research in fractional-order (FO) proportional-integral-derivative (PID) controllers is directed to fundamental control problems such as stability, performance, and robustness. First, nominal stability was considered by finding all the possible FO PID controllers that stabilize a closed-loop system with respect to arbitrary values of the fractional orders λ and μ of the FO PID controller. The findings are presented on the (Kp, Ki), (Kp, Kd), and (Ki, Kd) planes. In order to meet nominal performance specifications, a sensitivity function weight was introduced and FO PID controllers were sought to meet the weighted sensitivity constraint. This led to a complete set of possible values of FO PID parameters that satisfy the given performance specifications. Following the nominal stability and performance, robust stability and performance were investigated. For a robust stability requirement, a multiplicative weight was selected to bound all multiplicative errors of a closed-loop system. Such FO PID controllers allow the closed-loop to remain stable for all the sets of perturbed plants. Nominal performance and robust stability are the prerequisite conditions for the robust performance of a closed-loop system. Though, in robust stability analysis, the closed-loop system was designed only to remain stable, it was required not only to remain stable for all the uncertain plants but also to satisfy given performance specifications in the robust performance analysis. A substantial contribution of this research is the establishment of a complete set of solutions for FO PID controllers, with respect to nominal stability and performance and robust stability and performance. The use of frequency response of a system makes it possible to apply the results presented in this dissertation even when a system transfer function is not known or unavailable, as long as the experimental frequency data of a system can be obtained.
Thesis (Ph.D.)--Wichita State University, College of Engineering, Dept. of Electrical Engineering and Computer Science

碩士
輔仁大學
電機工程學系碩士班
103
The passive inductors are employed broadly in communication circuits whereas due to the large chip area requirement, the applications of the inductor is resisted in some special circuits such as power amplifiers. On the other hand, active inductors are prevalent due to the tunablity of their quality factor and inductance. In this thesis, we introduce the fundamentals of fractional-order calculus. To realize the fractional-order active inductors, the conventional gyrator architecture and a bipolar junction transistor (BJT) are employed. The proposed fractional-order active inductor is realized by configuring with a bipolar junction transistor and the architecture of the conventional gyrator. According to the mathematical derivation and Hspice simulation results, the feasibility of the proposed active inductors is validated. The measured S-parameter of the circuit configured with discrete components can also further verify the spectrum of the proposed fractional-order active inductor.

In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives has found wide application in the control of dynamical systems, when the controlled system or/and the controller is described by a set of fractional order differential equations. In the present work a fractional order system has been represented by a higher integer order system, which is further approximated by second order plus time delay (SOPTD) model. The approximation to a SOPTD model is carried out by the minimization of the two norm of the actual and approximated system. Further, the effectiveness of a fractional order controller in meeting a set of frequency domain specifications is determined based on the frequency response of an integer order PID and a fractional order PID (FOPID) controller, designed for the approximated SOPTD model. The advent of fuzzy logic has led to greater flexibility in designing controllers for systems with time varying and nonlinear characteristics by exploiting the system observations in a linguistic manner. In this regard, a fractional order fuzzy PID controller has been developed based on the minimization different optimal control based integral performance indices. The indices have been minimized using genetic algorithms. Simulation results show that the fuzzy fractional order PID controller is able to outperform the classical PID, fuzzy PID and FOPID controllers.

博士
國立高雄第一科技大學
電腦與通訊工程所
98
For the past decade, the design of variable digital filters became one of the most important branches in digital signal processing because of the self-adjustable ability of a variable digital filter online. The variable digital filters are generally classified into two categories. One is the filters with adjustable magnitude response such as the filters with variable cut-off frequencies and the variable fractional-order differentiators/integrators. The other is the filters with variable fractional-delay response. In this dissertation, the weighted least-squares method will be proposed to design variable digital filters. Generally, a general weighted least-squares method can be applied directly to find the optimal solution when the objective error can be formulated in a linear function. On the contrary, when the problem concerns a nonlinear optimization, an iterative quadratic method is applied. Furthermore, if it is desirable to minimize a specified maximum error, the technique of iterative weighted least-squares method will be used which constitutes the inner loop of the overall procedures while the iterative method stated above makes up the outer loop. In this dissertation, the stated method will be applied to the following topics: ? Minimax design of variable fractional-delay FIR digital filters by iterative weighted least- squares approach (Chapter 2). ? A new criterion for the design of variable fractional-delay FIR digital filters (Chapter 3). ? A new structure for the design of variable fractional-delay FIR digital filters (Chapter 4). ? Minimax phase error design of allpass variable fractional-delay digital filters by iterative weighted least-squares method (Chapter 5). ? A new method for least-squares and minimax group-delay error design of allpass variable fractional-delay digital filters (Chapter 6). ? A new method for the design of variable fractional-delay IIR digital filters (Chapter 7). ? An iterative method for the design of variable fractional-order FIR differintegrators (Chapter 8). ? A new structure for the design of wideband variable fractional-order FIR differentiators (Chapter 9). ? Iterative design of variable fractional-order IIR differintegrators (Chapter 10).

碩士
國立高雄應用科技大學
電機工程系博碩士班
103
A optimal Fractional Order controller design problem is studied in this thesis, the particle swarm optimization method is used to search the best parameters of the FO controllers. Among different particle swarm optimization methods, this study shows the one with time-varying acceleration coefficients and time-varying intertia weight could result in a smallest performance index, The result show that the modified PSO method is highly suitable for the searching for optimal controller parameters of the FO systems.

This thesis deals with developing Galerkin based solution strategies for several important classes of differential equations involving derivatives and integrals of various fractional orders. Fractional order calculus finds use in several areas of science and engineering. The use of fractional derivatives may arise purely from the mathematical viewpoint, as in controller design, or it may arise from the underlying physics of the material, as in the damping behavior of viscoelastic materials. The physical origins of the fractional damping motivated us to study viscoelastic behavior of disordered materials at three levels. At the first level, we review two first principles models of rubber viscoelasticity. This leads us to study, at the next two levels, two simple disordered systems. The study of these two simplified systems prompted us towards an infinite dimensional system which is mathematically equivalent to a fractional order derivative or integral. This infinite dimensional system forms the starting point for our Galerkin projection based approximation scheme. In a simplified study of disordered viscoelastic materials, we show that the networks of springs and dash-pots can lead to fractional power law relaxation if the damping coefficients of the dash-pots follow a certain type of random distribution. Similar results are obtained when we consider a more simplified model, which involves a random system coefficient matrix. Fractional order derivatives and integrals are infinite dimensional operators and non-local in time: the history of the state variable is needed to evaluate such operators. This non-local nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Following this, we identify eight important classes of fractional differential equations (FDEs) and fractional integrodifferential equations (FIEs), and develop separate Galerkin based solution strategies for each of them. Distinction between these classes arises from the fact that both Riemann-Liouville as well as Caputo type derivatives used in this work do not, in general, follow either the law of exponents or the commutative property. Criteria used to identify these classes include; the initial conditions used, order of the highest derivative, integer or fractional order highest derivative, single or multiterm fractional derivatives and integrals. A key feature of our approximation scheme is the development of differential algebraic equations (DAEs) when the highest order derivative is fractional or the equation involves fractional integrals only. To demonstrate the effectiveness of our approximation scheme, we compare the numerical results with analytical solutions, when available, or with suitably developed series solutions. Our approximation scheme matches analytical/series solutions very well for all classes considered.

This thesis deals with developing Galerkin based solution strategies for several important classes of differential equations involving derivatives and integrals of various fractional orders. Fractional order calculus finds use in several areas of science and engineering. The use of fractional derivatives may arise purely from the mathematical viewpoint, as in controller design, or it may arise from the underlying physics of the material, as in the damping behavior of viscoelastic materials. The physical origins of the fractional damping motivated us to study viscoelastic behavior of disordered materials at three levels. At the first level, we review two first principles models of rubber viscoelasticity. This leads us to study, at the next two levels, two simple disordered systems. The study of these two simplified systems prompted us towards an infinite dimensional system which is mathematically equivalent to a fractional order derivative or integral. This infinite dimensional system forms the starting point for our Galerkin projection based approximation scheme. In a simplified study of disordered viscoelastic materials, we show that the networks of springs and dash-pots can lead to fractional power law relaxation if the damping coefficients of the dash-pots follow a certain type of random distribution. Similar results are obtained when we consider a more simplified model, which involves a random system coefficient matrix. Fractional order derivatives and integrals are infinite dimensional operators and non-local in time: the history of the state variable is needed to evaluate such operators. This non-local nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Following this, we identify eight important classes of fractional differential equations (FDEs) and fractional integrodifferential equations (FIEs), and develop separate Galerkin based solution strategies for each of them. Distinction between these classes arises from the fact that both Riemann-Liouville as well as Caputo type derivatives used in this work do not, in general, follow either the law of exponents or the commutative property. Criteria used to identify these classes include; the initial conditions used, order of the highest derivative, integer or fractional order highest derivative, single or multiterm fractional derivatives and integrals. A key feature of our approximation scheme is the development of differential algebraic equations (DAEs) when the highest order derivative is fractional or the equation involves fractional integrals only. To demonstrate the effectiveness of our approximation scheme, we compare the numerical results with analytical solutions, when available, or with suitably developed series solutions. Our approximation scheme matches analytical/series solutions very well for all classes considered.

博士
國立中正大學
化學工程所
94
Fractional-order systems are characterized by differential operator of non-integer orders. They are more adequate to describe real-world system than those of integer-order models. Moreover, fractional-order controllers have been introduced for improving the robustness and performance of feedback control systems. However, the lack of efficient stability analysis and controller design methods for fractional-order systems have hindered the development of fractional-order feedback control systems. This is only particularly true for those systems having time delays or distributed delays. This dissertation is focused on the stability analysis and controller design of fractional-order systems. The subjects studied include: stability analysis of fractional delay systems with the Lambert W function; numerical stability test for a general fractional order systems; the computation of integral-squared-error performance index of fractional order feedback systems; the stabilization of unstable first-order plus dead time process using fractional-order proportional derivative controllers; and optimal digital redesign for fractional order control systems. The Lambert W function is defined to be the multivalued inverse of the function $w ightarrow we^w=z$. This function has been used in an extremely wide variety of applications, including the stability analysis of fractional-order as well as integer-order time-delay systems . The latter application is based on taking the $m$th power and/or $n$th root of the transcendental characteristic equation (TCE) and representing the roots of the derived TCE(s) in terms of W functions. We re-examine such an application of using the Lambert W function through actually computing the root distributions of the derived TCEs of some chosen orders. It is found that the rightmost root of the original TCE is not necessarily a principal branch Lambert W function solution, and that a derived TCE obtained by taking the $m$th power of the original TCE introduces superfluous roots to the system. With these observations, some deficiencies displayed in the literature (Chen and Moore 2002a, 2002b) are pointed out. Moreover, we clarify the correct use of Lambert W function to stability analysis of a class of time-delay systems. This will actually enlarge the application scope of the Lambert W function, which is becoming a standard library function for various commercial symbolic software packages, to time-delay systems. We also extend the approach of using Lambert W function to time-domain analysis of a class of feedback fractional-order time-delay systems. It is pointed out that, due to the multivaluedness of a transfer function of fractional order, the approach has two pitfalls which must be circumvented with care. Since remodeling the TCE of a feedback fractional delay system to allow for the Lambert W function representation of roots introduces superfluous poles to the original TCE, we make a clarification of the relationship between the roots of the remodeled TCEs and the poles of the system. As a result, the time response function of the system can be approximated by a finite series of eigenmodes written in terms of Lambert W functions. As the singularities of a fractional-order system include both the poles and the branch cut(s) of the transfer function, the neglect of the response portion contributed by the branch cut(s) incurs a significant transient response error. In order to compensate such a transient response error, three schemes of optimal approximation with specified poles are developed. Simulation results show that the proposed approaches to time domain analysis of feedback fractional delay systems can indeed enlarge the application scope of the emerging Lambert W function. Further, we presents an effective numerical algorithm for testing the BIBO stability of fractional delay systems described by fractional-order delay-differential equations. It is based on using Cauchy's integral theorem and solving an initial-value problem. The algorithm has a reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of feedback control for both integer- and fractional-order systems having time delays. The quadratic cost functional or integral-square-error (ISE) defined as $I=int_0^infty e^2(t)dt$ has been widely used in the analytical design of optimal control systems. In most control literature the integral $I$, by virtue of Parseval's theorem, is represented by the complex integral $(1/i 2pi)int_{-iinfty}^{iinfty} E(s)E(-s)ds, i=sqrt{-1}$. The concept of product-to-sum decomposition $E(s)E(-s)= F(s)+F(-s)$, where $E(s)$ is a Hurwitz stable transfer function, has been extensively applied to derive parametric expressions for quadratic cost functionals of linear time-invariant systems, including a class of commensurate pure delay and distributed delay systems. We show that due to the multivaluedness of the transfer function and the existence of non-removable branch-cut singularity, the extension of such a concept to obtain closed-form expression for the integral-squared-error of fractional-order systems is generally not possible. Hence, the calculation of the quadratic functionals of fractional-order systems has to resort a numerical integration scheme. In this aspect, a reliable and efficient numerical approach based on solving a differential equation is suggested for obtaining accurate solutions. Moreover, we especially consider the evaluation of ISE for linear feedback control of systems involving a distributed delay $exp(- au s/sqrt{s^2+b^2})$. It is shown that due to multivalued square root function ${(s^2+b^2)}^{1/2}$ has a non-removable branch-cut singularity on the imaginary axis, the product-to-sum decomposition approach fails to generate a parametric expression for the evaluation of $I$. Also shown is that pitfall exists with the use of the above-mentioned representation of Parseval identity when value of $I$ is computed by a numerical integration of the complex integral in a computer. The findings gained from numerical results indeed clarify the correct use of a useful numerical approach. The problem of stabilizing unstable first-order time-delay (FOTD) processes using fractional-order proportional derivative (PD) controllers is considered. It investigates how the fractional derivative order $mu$ in the range $(0,2)$ affects the stabilizability of unstable FOTD processes. The D-partition technique is used to characterize the boundary of the stability domain in the space of process and controller parameters. The characterization of stability boundary allows one to describe and compute the maximum stabilizable time delay as a function of derivative gain and/or proportional gain. It is shown that for the the same derivative gain, a fractional-order PD controller with derivative order less than unity has greater ability to stabilize unstable FOTD processes than an integer-order PD controller. Such a fractional-order PD controller can allow the use of higher derivative gain than an integer-order PD controller. However, the setting of derivative gain greater than unity makes the maximum stabilizable time delay decrease drastically. When the derivative order $mu$ is greater than unity, the allowable derivative gain is restricted to be less than unity, as in the case of using an integer-order PD controller, and, for a fixed derivative gain, the maximum stabilizable time delay decreases as the derivative order $mu$ is increased. Finally, we present a digital redesign technique which takes into account both closed loop and intersample behaviour. This is to replace an existing well-designed fractional-order analog feedback controller by a digital one along with a ideal sampler and a fractional order hold. More precisely, we give a numerical procedure to optimally discrete an analog controller such that the integral of squared error between the closed-loop step response of the analog and digital controlled systems is minimized.

碩士
國立雲林科技大學
電機工程系
106
In this thesis, the self-made fan system is used as an experimental plant to simulate a wind speed system. By controlling the wind speed, we can replace the method of changing the size of air conditioner vent to reduce the energy consumption. We use the designed controllers with MATLAB for Model 626 control card to control the fan system that is a convenient method. We designed a fractional-order PID controller in this thesis. It is a classical controller that derived from the integer-order PID controller. However, the fractional-order PID controller has more two parameters than the integer-order PID controller to diversify adjustment. Compared with the integer-order PID controller, the fractional-order PID controller is compatible with transient and steady-state response for system robustness. In order to verify the robustness of the fractional-order PID controller, we also used the sliding mode controller for comparison. We designed a stable sliding line which allowed the system trajectory to approach the sliding line. Hence, the system trajectory would converge to origin along with the sliding line. Sliding mode control used the sign function to deal with the uncertainties. It has the robustness and stability with chatting phenomenon.

碩士
輔仁大學
電機工程學系
101
The resister, inductor and capacitor are the most important circuit elements although some coupling parasitic effects may lead to non-ideality. Conventionally, we treat these elements as linear in much research to develop different kind of oscillators, filters and other digital and analog circuits whereas they exhibit some non-linear characteristics. In the past, we were desperately trying to avoid non-linearity by employing some circuit design techniques. Since these nonlinearities always exist, why we do not use these non-linear features to develop even high-performance circuits or systems? Fractional-order elements (FOEs) are non-linear elements. In theory, this kind of elements can be used to improve several different kinds of circuits, oscillators, filters , etc. Although FOEs have many advantages, the most difficult application on circuits is, in fact the real FOEs does not exist. There are many numerical approaches in certain frequencies whereas they cannot be integrated and used in high frequency domain. In this thesis, we investigate a new method to develop the FOEs, especially in fractional-order capacitors, on which we focus in the research to develop the device in integrated circuits realized with MOSFETs. The feasibility is validated by Hspice simulation .

碩士
國立臺灣海洋大學
電機工程學系
101
Fractional-order calculus is a generalization of the classical calculus. It allows the order of the differentiation operation to be a fractional number. It has been used in many applications with more accurate descriptions for system dynamic characteristics. The main purpose of this thesis is to investigate the application of the fractional-order theory to the design of power system stabilizer which provides a supplementary control signal to generator excitation system and has been widely adopted in the enhancement of power system stability. This study utilizes the Proportional-Integral controller in the design of power system stabilizer and examine whether the system can resume a new stable operating point in a shorter time period after being perturbed by small disturbance. The results show that the proposed power system stabilizer can provide positive damping for the system and thus can suppress low frequency oscillation to improve power system stability.

Fractional order circuits incorporating fractional calculus concept have various applications in many fields namely, bio-medical engineering, control system, analog signal processing/generation, etc. In the present dissertation, along with a brief review of different methods of approximations used for the fractional order differentiator and integrator operator, fractional order analog universal filter circuits using operational transconductance amplifier and LT 1228 ICs have been presented. The workability of all the fractional order filter circuits along with fractional operator have been verified through PSPICE simulation and MATLAB simulation. Also, stability of all the designed fractional order filters have been discussed briefly.

Fractional order calculus (FOC) has wide applications in modeling natural behavior of systems related to different areas of engineering including bioengineering, viscoelasticity, electronics, robotics, control theory and signal processing. This thesis aims at modeling a lossy transmission line using fractional order calculus and identifying its parameters. A lossy transmission line is considered where its behavior is modeled by a fractional order transfer function. A semi-infinite lossy transmission line is presented with its distributed parameters R, L, C and ordinary AC circuit theory is applied to find the partial differential equations. Furthermore, applying boundary conditions and the Laplace transformation a generalized fractional order transfer function of the lossy transmission line is obtained. A finite length lossy transmission line terminated with arbitrary load is also considered and its fractional order transfer function has been derived. Next, the frequency responses of lossy transmission lines from their fractional order transfer functions are also derived. Simulation results are presented to validate the frequency responses. Based on the simulation results it can be concluded that the derived fractional order transmission line model is capable of capturing the phenomenon of a distributed parameter transmission line. The achievement of modeling a highly accurate transmission line requires that a realistic account needs to be taken of its parameters. Therefore, a parameter identification technique to identify the parameters of the fractional order lossy transmission line is introduced. Finally, a few open problems are listed as the future research directions.
Controls

Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.

The fractional–order capacitors add an additional degree of freedom over conventional capacitors in circuit design and facilitate circuit configurations that would be impractical or impossible to implement with conventional capacitors. We propose a generic strategy for fractional-order capacitor fabrication that integrates layers of conductive, semiconductor and ferroelectric polymer materials to create a composite with significantly improved constant phase angle, constant phase zone, and phase angle variation performance. Our approach involves a combination of dissolving the polymer powders, mixing distinct phases and making a film and capacitor of it. The resulting stack consisting of ferroelectric polymer-based composites shows constant phase angle over a broad range of frequencies. To prove the viability of this method, we have successfully fabricated fractional-order capacitors with the following: nanoparticles such as multiwall carbon nanotube (MWCNT), Molybdenum sulfide (MoS2) inserted ferroelectric polymers and PVDF based ferroelectric polymer blends. They show better performance in terms of fabrication cost and dynamic range of constant phase angle compared to fractional order capacitor from graphene percolated polymer composites. These results can be explained by a universal percolation model, where the combination of electron transport in fillers and the dielectric relaxation time distribution of the permanent dipoles of ferroelectric polymers increase the constant phase angle level and constant phase zone of fractional-order capacitors. This approach opens up a new avenue in fabricating fractional capacitors involving a variety of heterostructures combining the different fillers and different matrixes.

碩士
國立臺灣大學
電信工程學研究所
95
In this thesis, we introduce a few designs of digital integrator, and a few designs of fractional-order differintegrator. We apply some numerical integration rules and fractional delay filters to obtain the closed form design of IIR digital integrators. There are three types of numerical integration rules to be investigated: Newton-Cotes quadrature rule, Gauss-Legendre quadrature rule and Clenshaw-Curtis quadrature rule. The fractional delay involved in the design will be implemented by FIR Lagrange and IIR allpass fractional delay filters. Also, a combined version is proposed. Several digital filter design examples are illustrated to demonstrate the effectiveness. Chapter 5 is to show the designs of the fractional-order differintegrator. We find a suitable generating function to fit the ideal fractional-order differintegrator. Then discretize the fractional-order with a power series expansion or continued fraction expansion. Last, we discuss the different methods to decrease the absolute magnitude error. Moreover, the filter properties will also be presented at the end of the chapter. Finally, we make a conclusion of this thesis and suggest the future work in chapter 6.

碩士
中華大學
機械工程學系碩士班
98
The dynamics of the fractionally-order systems have attracted increasing attentions in recent years. Chaos and synchronization of the fractional-order Ueda oscillator is researched in this study. The fractional-order Ueda oscillator is solved by a Adams-Bashforth-Moulton predictor-corrector method. The phase portraits, the Poincaré map technique and bifurcation are used to study the effect of frequency of external force on the dynamic behaviors of the motion. Then design an electronic circuit to realizate of the fractional-order Ueda oscillator. Further the controll laws are also designed to suppress the chaotic behaviors. Finally, the one-way coupling and two-way coupling method are also designed to make two identical Ueda oscillators to achieve chaos synchronization.

Fractional order PID controller is gaining popularity because the presence of two extra degrees of freedom, which have the potential to meet up the extra degrees in terms of uncertainty, robustness, output controllability .In other words, the fractional order PID controller is the generalization of the conventional PID controller. In the current study, the fractional order PID controller is designed and implemented for the complex and plant-wide processes. Distillation is the most effective separation process in the chemical and petroleum industries but with a drawback of energy intensivity To reduce the energy consumption two distillation columns can be combined into one column, which is known as dividing wall distillation column (DWC).Though the control of DWC has been addressed but it requires further R&D efforts considering the complexity in control of this process In this work the DWC is controlled by the advanced control strategy like fractional order PID controller. One of the challenging field in the process control is to design control system for the entire chemical plant. We have presented the control system for the HDA plant by implementing the fractional order PID controller. Both the discussed processes are multi-input-multi-output (MIMO) system and these processes are difficult to tune because of the presence of the interaction between the control loops. For the DWC process, the traditional simplified decoupler is used, while for the HDA plant process the equivalent transfer function model is used to handle the MIMO system. For tuning of the fractional-order PID controllers the optimization techniques have been used. The DWC controllers have been tuned by the ev-MOGA multi objective algorithm and the HDA plant controllers are tuned by the cuckoo search method.

碩士
國立高雄應用科技大學
電機工程系博碩士班
103
Fractional order PID controller design for Fractional order systems using internal model method is studied in this thesis. A fuzzy inference system is added into the controller to adjust the parameters increasing the performances. Compared to the traditional PID controllers, the obtained performance using the modified fuzzy time-varying PID controller is improved. The analysis of numerical simulations demonstrates the effectiveness.

碩士
國立高雄應用科技大學
電機工程系博碩士班
106
The problems of robust stability and stabilization for continuous time uncertain fractional order systems are solved in this thesis. The parametric uncertainty considered is of polytopic type. The purpose of the robust stability problem is to give condition such that the uncertain fractional order system is stable for all admissible uncertainties, while the purpose of robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. A strict linear matrix inequality (LMI) design approach is derived, and an explicit expression for the desired robust state feedback control law is also given. Numerical examples are provided to demonstrate the application of the proposed method, and finally the controller design method is applied to the fault-tolerant control to show the applicability of the design method. Keywords: Fractional Order PID Controller, Linear Matrix Inequality (LMI), Robust Stability, Fault-Tolerant Control.

Process control in industry is improving gradually with the innovations and implementation of new technology. Different control techniques are being used for process control. Proportional Integral and Derivative (PID) controller is employed in every facet of industrial automation. In any of control application, controller design is the most important part. There are different types of controller architectures available in control literature. The applications of PID controller span from small industries to high technology industries. Designing a PID controller to meet gain and phase margin specifications is a well-known design technique. If the gain and phase margin are not specified carefully, then the design may not be optimum in the sense that a large phase margin (more robust) that could give better performance. This research outlines the development and design of an infrared radiation heating profile controller. An attempt has been made to theoretically analyze the system, design of the Controller, their simulation, and real-time implementation of an infrared ceramic heating profile controller. The Controller has been subjected to comparative testing with a proportional control model to observe its performance and validate its effectiveness. PID controllers of this nature that are commercially available either lack the functionality of this unit or are too expensive to implement for research purposes. This unit has been designed with cost-effectiveness in mind but still meets the standards required for an industrial controller. Heating profiles are necessary and useful tools for the proper processing of a host of materials. The Controller developed in this research is able to meet a level of a fair degree of accuracy and track a heating profile. The results confirm that this programmable control model will be advantageous and a valuable tool in temperature regulation. This means that intensive studies into the effects of infrared radiation on materials are now feasible. Research of this nature could possibly expand the application of infrared as a heating mechanism. Although tests were conducted on this Controller, they are not meant to serve as an exhaustive analysis. The conclusions of these simulations do reveal the benefit of such a v controller. More rigorous investigation is suggested as a subject for further study. System identification of this nonlinear process is done using black box model, which is identified to be nonlinear and approximated to be a First Order plus Dead Time (FOPDT) model. In order to obtain an accurate mathematical expression of the IR heater used in this research, a step response test of the IR heater has been completed. This method of testing has been done in accordance with the Ziegler-Nichols, Astrom Hügglund, and Cohen-Coon methods. Simulation of the obtained transfer function, using Mat Lab software, showed good agreement. Although the transfer function represented a first-order model with transportation lag, the simulated results reflected an acceptable accuracy. An exhaustive study has been done on different PID controller tuning techniques. The PID controller of the model has been designed using the classical method, and the results have been analyzed. A compromise has been made between robustness and tracking performance of the system in the presence of time delay. The results of the simulation indicate the validity of the study. Integer order PID controller (IOPID) based on Bode plot and Nyquist plot has been designed. The results illustrate that the IOPID controllers have the capability of minimizing the control objectives better than the previously designed controllers (Ziegler-Nichols, Astrom-Hügglund, and Cohen-Coon method-based Controller).With the change in temperature occurs, the oscillations of the controlled system outputs are eliminated and the output steady state errors become very small. The results demonstrate that the IOPID controller is stable and it suppresses the cost function (Maximum overshoot, Rise Time, Settling time and Peak time) even in case of significant disturbances. IOPID controller has also been designed using Bode plot and Nyquist plot for high gain system. The results have shown that the responses of Ceramic IR heater temperature profile have been reduced to very small value and prove that the IOPID controller is still stable and it suppresses the cost function even if significant disturbances have occurred. Fuzzy Logic controller-based model reference has been designed. Its implementation indicates that the proposed Controller suppresses the output of the controlled system. The results illustrate that the proposed Controller only slightly vi improves the performance of the cost function. The various AI techniques (GA) and Soft Computing (bio inspired) based algorithm (BFO, ACO) for PID controller offers several advantages. These methods can be used for higher order process models in complex problems. Approximations that are typical to classical tuning rules are not needed. Compared to conventionally tuned system, GA, PSO, BFO and ACO tuned system provides good steady state response and performance indices. The genetic approaches can achieve better temperature control with smaller settling time, overshoot and undershoot, and zero steady error. The control signal changes more frequently and with larger magnitude as the genetic algorithms are stochastic in nature. The PSO has an additional unique advantage that it adapts any change in system conditions, and obtains different system dynamics accurately in a short time period. It is a random search method but if combined with an artificial intelligence features, it tracks required system dynamics accurately in short time (small number of iterations). The BFO based Controller has the advantage of a better closed loop time constant, which enables the Controller to act faster with a balanced overshoot and settling time. The response of the conventional Controller is more sluggish than the BFO based Controller. Compared to conventionally tuned system, BFO tuned system has better steady state response and performance indices. Ant- Colony algorithm (ACO) has no special requirements on the characteristics of optimal designing problems, which has a fairly good universal adaptability and a reliable operation of program with ability of global convergence. Simulation results show the controlled system has satisfactory response and the proposed method has an effective tuning strategy. ACO shows better performance for PID controller parameter tuning of the considered control system. The simulation results show that the proposed method achieve minimum tracking error and estimate the parameter values with high accuracy. The work presents tuning method for fractional order proportional Integral and derivative controllers (FOPID) for the first order plus time delay (FOPTD) class of systems based on gain and phase margin. Techniques such as fractional order PID controller design and the results of their application to real-world system vii have been presented. A comparative study has been done using different control techniques to analyze the performance of different controllers. First, the conventional PID controller is implemented as primary Controller. The performance of PID, IOPID, Fuzzy Logic Controller, and Artificial Intelligence based PID, Bio inspired based PID controller and FOPID controllers have been examined. It has been concluded that the overall performance of the FOPID-based Controller is better than other controllers. In real-time implementation, the performance of the process control includes the time required by the heater to be settled on the initial set-up temperature. The rising of temperature is slow due to the resistance heating element used in ceramic infrared heaters. So the settling time is very high. The results obtained by simulation and real-time implementation with fractional order PID controller show overall better performance( rise time , settling time, peak time and peak overshoot) in comparison with other designed and implemented Controllers embedded with ceramic infrared systems. Further stability problems of fractional order system with leakage delay and distributed delay with hybrid feedback controller have been solved (with examples) using the Mittag-Leffler function and Lyapunov direct method and proved Global Mittag-Leffler stability of fractional order system of the proposed model which implies faster convergence rate of the network model which represents the stability of the system. This work performs a small-scale test measuring controller performance so that it serves as a platform for future research efforts leading to the real-life implementation of a Ceramic Infrared Heater Temperature control system.