On fractional derivatives, fractional-order dynamic systems and PI/sup λ/D/sup μ/-controllers

On Fractional-Order PID Design

“…In order to address this problem, different methods for the design of a FOPID controller have been proposed in the literature. The concept of FOPID controllers was proposed by Podlubny in 1997 (Podlubny et al, 1997;Podlubny, 1999a). He also demonstrated the better response of this type of controller, in comparison with the classical PID controller, when used for the control of fractional order systems.…”

“…There are some definitions for fractional derivatives. The commonly used definitions are Grunwald-Letnikov, Riemann-Liouville, and Caputo definitions (Podlubny, 1999b). The Grunwald-Letnikov definition is given by…”

Tuning fractional order proportional integral controllers for fractional order systems

“…As explained in Kostial et al [52], the idea of using fractional order controllers for the dynamic system control belongs to Oustaloup et al [46,53] and generalized fractional order proportional integral derivative (PID) controller was proposed by Podlubny [54]. Advantages of using fractional order PID controller have been introduced in a number of publications.…”

“…Vinagre et al provided frequency domain analysis to illustrate the superiority of the fractional order PID controller applied to both the fractional dynamic system and the integer dynamic system [56]. In Podlubny et al [52], it was claimed that fractional order PID controller is an adequate controller for the fractional order mathematical models and it is less sensitive to shifts of parameters of a controlled system and to variations of parameters of the controller. Particularly, in work of Podlubny [45], it was illustrated that the fractional order PID controller is a suitable way for the control of the fractional system.…”

Reachability of Cone Fractional Continuous-Time Linear Systems

“…Mathematical fundamentals of fractional calculus are given in the monographs (Miller and Ross, 1993;Nishimoto, 1984;Oldham and Spanier, 1974;Oustaloup, 1995;Podlubny, 1999). Fractional order controllers were developed in (Oustaloup, 1993;Podlubny et al, 1997). A generalization of the Kalman filter for fractional order systems was proposed in (Sierociuk and Dzieliński, 2007).…”