Fractional Differential Equations

The science and technology of ultracapacitors are reviewed for a number of electrode materials, including carbon, mixed metal oxides, and conducting polymers. More work has been done using microporous carbons than with the other materials and most of the commercially available devices use carbon electrodes and an organic electrolytes. The energy density of these devices is 3¯5 Wh/kg with a power density of 300¯500 W/kg for high efficiency (90¯95%) charge/discharges. Projections of future developments using carbon indicate that energy densities of 10 Wh/kg or higher are likely with power densities of 1¯2 kW/kg. A key problem in the fabrication of these advanced devices is the bonding of the thin electrodes to a current collector such the contact resistance is less than 0.1 cm2. Special attention is given in the paper to comparing the power density characteristics of ultracapacitors and batteries. The comparisons should be made at the same charge/discharge efficiency.

Ultracapacitors: Why, How, and Where is the Technology

http://ieeexplore.ieee.org/iel5/5610941/5624686/05624745.pdf

Power electronics

System Identification I

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ordinary differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ordinary calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.This XXI century has thus started to think-exactly for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.

Functional Fractional Calculus

A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modelling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.

On the selection of tuning methodology of FOPID controllers for the control of higher order processes.

Background: Subcortical band heterotopia (SBH) is a neuronal migration disorder. DCX mutations are responsible for almost all familial cases, 80% of sporadic female cases, and 25% of sporadic male cases of SBH, and are associated with more severe gyral and migration abnormality over the anterior brain regions. Somatic mosaicism has previously been hypothesized in a patient with posteriorly predominant SBH and a mutation of the LIS1 gene, which is usually mutated in patients with severe lissencephaly. The authors identified mosaic mutations of LIS1 in two patients (Patients 1 and 2) with predominantly posterior SBH. Methods: After ruling out DCX mutations, the authors performed sequencing of the LIS1 gene in lymphocyte DNA. Because sequence peaks in both patients were suggestive of mosaic mutations, they followed up with denaturing high-pressure liquid chromatography analysis on blood and hair root DNA and compared the areas of heteroduplex and homoduplex peaks. A third patient showing the same mutation as Patient 2 but with no evidence of mosaicism was used for comparing the phenotype of mosaic vs full mutation. Results: The two patients with posterior SBH harbored a missense (Arg241Pro) and a nonsense (R8X) mosaic mutation of LIS1 . The rate of mosaicism in Patient 1 was 18% in the blood and 21% in the hair roots, whereas in Patient 2 it was 24% and 31% in the same tissues. The patient with a full R8X mutation of LIS1 had severe lissencephaly. Conclusions: Subcortical band heterotopia can occur with mosaic mutations of the LIS1 gene. Mutation analysis of LIS1 , using highly sensitive techniques such as denaturing high-pressure liquid chromatography, should be considered for patients with posteriorly predominant subcortical band heterotopia and pachygyria.

Shantanu Das

Papers

Functional Fractional Calculus

A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices

On the selection of tuning methodology of FOPID controllers for the control of higher order processes.

Mosaic mutations of the LIS1 gene cause subcortical band heterotopia.

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index