Igor Podlubny

Technical communique: Mittag-Leffler stability of fractional order nonlinear dynamic systems

In this paper, we propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability

Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. With the definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.

Analogue Realizations of Fractional-Order Controllers

An approach to the design of analogue circuits, implementingfractional-order controllers, is presented. The suggestedapproach is based on the use of continued fraction expansions;in the case of negative coefficients in a continued fractionexpansion, the use of negative impedance converters is proposed.Several possible methods for obtaining suitable rational appromixationsand continued fraction expansions are discussed. An exampleof realization of a fractional-order Iλ controlleris presented and illustrated by obtained measurements.The suggested approach can be used for the control of veryfast processes, where the use of digital controllers isdifficult or impossible.

Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubnys matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

Using Fractional Order Adjustment Rules and Fractional Order Reference Models in Model-Reference Adaptive Control

This paper investigates the use of Fractional Order Calculus (FOC) inconventional Model Reference Adaptive Control (MRAC) systems. Twomodifications to the conventional MRAC are presented, i.e., the use offractional order parameter adjustment rule and the employment offractional order reference model. Through examples, benefits from theuse of FOC are illustrated together with some remarks for furtherresearch.

Robust Stability Test of A Class of Linear Time-Invariant Interval Fractional-Order System Using Lyapunov Inequality

This paper provides a new analytical robust stability checking method of fractional-order linear time invariant interval uncertain system. This paper continues the authors’ previous work [YangQuan Chen, Hyo-Sung Ahn, I. Podlubny, Robust stability check of fractional-order linear time invariant systems with interval uncertainties, in: Proceedings of the IEEE Conference on Mechatronics and Automation, Niagara Falls, Canada, July, 2005, pp. 210–215] where matrix perturbation theory was used. For the new robust stability checking, Lyapunov inequality is utilized for finding the maximum eigenvalue of a Hermitian matrix. Through numerical examples, the usefulness and the effectiveness of the newly proposed method are verified.

Review: On the fractional signals and systems

A look into fractional calculus and its applications from the signal processing point of view is done in this paper. A coherent approach to the fractional derivative is presented, leading to notions that are not only compatible with the classic but also constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequency responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the input-ouput computations. Some biomedical applications are described.

On Fractional PID Controllers: A Frequency Domain Approach

Abstract A more general structure for the classical PID controller is proposed in this paper by using fractional integral and differential operators. A frequency domain approach is used to show the advantages of using these fractional PID controllers, which can be sumarized in the possibility of dealing with a more general class of control problems, in which the fractional nature of the controller can be imposed by the fractional nature of the system to be controlled, or by the special nature of the required time or frequency responses. Some illustrative examples and comments on controller tuning and realizations are given.

Modelling heat transfer in heterogeneous media using fractional calculus

This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub- or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in a new form. This leads to the transfer function that describes the dependency between the heat flux at the beginning of the beam and the temperature at a given distance. This article also presents the experimental results of modelling real plant in the frequency domain based on the obtained transfer function.

Comparison of scientific impact expressed by the number of citations in different fields of science

SummaryCitation distributions for 1992, 1994, 1996, 1997, 1999, and 2001, which were published in the 2004 report of the National Science Foundation, USA, are analyzed. It is shown that the ratio of the total number of citations of any two broad fields of science remains close to constant over the analyzed years. Based on this observation, normalization of total numbers of citations with respect to the number of citations in mathematics is suggested as a tool for comparing scientific impact expressed by the number of citations in different fields of science.