26 research outputs found
Towards the Formalization of Fractional Calculus in Higher-Order Logic
Fractional calculus is a generalization of classical theories of integration
and differentiation to arbitrary order (i.e., real or complex numbers). In the
last two decades, this new mathematical modeling approach has been widely used
to analyze a wide class of physical systems in various fields of science and
engineering. In this paper, we describe an ongoing project which aims at
formalizing the basic theories of fractional calculus in the HOL Light theorem
prover. Mainly, we present the motivation and application of such formalization
efforts, a roadmap to achieve our goals, current status of the project and
future milestones.Comment: 9 page
On algebraic time-derivative estimation and deadbeat state reconstruction
This note places into perspective the so-called algebraic time-derivative
estimation method recently introduced by Fliess and co-authors with standard
results from linear state-space theory for control systems. In particular, it
is shown that the algebraic method can in a sense be seen as a special case of
deadbeat state estimation based on the reconstructibility Gramian of the
considered system.Comment: Maple-supplements available at
https://www.tu-ilmenau.de/regelungstechnik/mitarbeiter/johann-reger
Wavelet Theory Demystified
In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theoryâincluding some new extensions for fractional ordersâin a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the -sense and a sharper theorem stating that smoothness implies order
Guest Editorial Introduction to the Special Section on Nonlinear Fractional-Order Circuits and Systems: Advanced Analysis and Effective Implementation
Nowadays, fractional order differential operators, as a generalization for classical differential operators, have established their key-role in modeling, analysis, and implementation of specific circuits and systems in which one typically faces nonlinear behaviors. It enforces to exploit analysis and implementation methods covering simultaneously "fractionality" and "nonlinearity" aspects. This special section, entitled "Nonlinear Fractional Order Circuits and Systems: Advanced Analysis and Effective Implementation," aims at introducing some of these methods
An Updated Vision of Continuous-Time Fractional Models
A few days before the end of the revision procedure, my friend J. Tenreiro Machado had a sudden cardio-respiratory arrest and died. Here I want to express my gratitude and tribute to a great man and scientist. He was a very friendly and helpful person, with an unusual work capacity that allowed him to publish interesting articles on a wide range of topics.This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is shown that such systems can have integer and fractional components. From the integer component we deduce the stability. The fractional order component is always stable. The initial-condition problem is analyzed and it is verified that it depends on the structure of the system. For a correct definition and backward compatibility with classic systems, suitable fractional derivatives are also introduced. The GrĂŒnwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated.authorsversionpublishe
Self-Similarity: Part IâSplines and Operators
The central theme of this pair of papers (Parts I and II in this issue) is self-similarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of L-splines; these are defined in the following terms: s(t) is a cardinal L-spline iff L\{s(t)\} = \sum _{ k \in Z } a\[k\] \delta (t-k), where L is a suitable pseudodifferential operator. Our starting point for the construction of "self-similar" splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a two-parameter family of generalized fractional derivatives, , where Îł is the order of the derivative and Ï is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator is used to define a scale-invariant energy measureâthe squared -norm of the Îłth derivative of the signalâwhich provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order 2Îł, which admits a stable representation in a B-spline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order 2Îł. We also establish a formal link between the regularization parameter λ and the cutoff frequency of the smoothing spline filter: . Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions
Cardinal Exponential Splines: Part IIâThink Analog, Act Digital
By interpreting the Green-function reproduction property of exponential splines in signal processing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the B-spline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we introduce an extended family of cardinal splinesâthe generalized E-splinesâto generalize the concept for all convolution operators with rational transfer functions. We construct the corresponding compactly-supported B-spline basis functions, which are characterized by their poles and zeros, thereby establishing an interesting connection with analog filter design techniques. We investigate the properties of these new B-splines and present the corresponding signal processing calculus, which allows us to perform continuous-time operations, such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the B-spline domain. In particular, we show how the formalism can be used to obtain exact, discrete implementations of analog filters. Finally, we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real-world analog-to-digital (A-to-D) and D-to-A conversion systems
Fractional Calculus and the Future of Science
Newton foresaw the limitations of geometryâs description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newtonâs laws. Mandelbrotâs mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newtonâs macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newtonâs laws to describe the many guises of complexity, most of which lay beyond Newtonâs experience, and many had even eluded Mandelbrotâs powerful intuition. The bookâs authors look behind the mathematics and examine what must be true about a phenomenonâs behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding