A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and dieren tiation (i.e., integration and dieren tiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and dieren tiation, the Caputo fractional dieren tiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.

http://people.tuke.sk/igor.podlubny/pspdf/0110241.pdf

Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

/pdf/fractional-order-control-a-tutorial-4zjusf99pg.pdf

Fractional order control - A tutorial

Studies on fractional order differentiators and integrators: A survey

What is a fractional derivative

This paper presents a generalization of the Kalman filter for linear and nonlinear fractional order discrete state-space systems. Linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified kalman filter for the linear case is called the fractional Kalman filter and its nonlinear extension is named the extended fractional Kalman filter. The background and motivations for using such techniques are given, and some algorithms are discussed. The paper also shows a simple numerical example of linear state estimation. Finally, as an example of nonlinear estimation, the paper discusses the possibility of using these algorithms for parameters and fractional order estimation for fractional order systems. Numerical examples of the use of these algorithms in a general nonlinear case are presented.

/pdf/fractional-kalman-filter-algorithm-for-the-states-parameters-1jx4fuwyfc.pdf

Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation

In this paper an interpretation of fractional operators in the time domain is given. The interpretation is based on the four concepts of fractal geometry, linear filters, construction of a Cantor set and physical realisation of fractional operators. It is concluded here that fractional operators may be grouped as filters with partial memory that fall between two extreme types of filters with complete memory and those with no memory. Fractional operators are capable of modelling systems with partial loss or partial dissipation. The fractional order of a fractional integral is an indication of the remaining or preserved energy of a signal passing through such system. Similarly, the fractional order of a differentiator reflects the rate at which a portion of the energy has been lost.

J.K. Hammond

Papers

Physical and geometrical interpretation of fractional operators