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Dynamic systems of an arbitrary real order (fractionalorder systems) are considered. A concept of a fractional-order PI Dcontroller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller both for the open and closed loop. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI Dcontrollers is given.

a b s t r a c tStability 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.Published by Elsevier Ltd

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On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.

a b s t r a c tA 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 Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) . 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.

This paper attempts to present an expository review of continued fraction expansion (CFE) based discretization schemes for fractional order differentiators defined in continuous time domain. The schemes reviewed are limited to infinite impulse response (IIR) type generating functions of first and second orders, although high-order IIR type generating functions are possible. For the first-order IIR case, the widely used Tustin operator and Al-Alaoui operator are considered. For the second order IIR case, the generating function is obtained by the stable inversion of the weighted sum of Simpson integration formula and the trapezoidal integration formula, which includes many previous discretization schemes as special cases. Numerical examples and sample codes are included for illustrations.

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