There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

A new collection of real world applications of fractional calculus in science and engineering

This chapter contains sections titled: Historical Review Supervised Multilayer Networks Unsupervised Neural Networks: Kohonen Network Unsupervised Networks: Adaptive Resonance Theory Network Model Validation Summary References Recommended Exercises

Nonlinear System Identification

Review of fractional PID controller

FOMCON is a new fractional-order modeling and control toolbox for MATLAB. It offers a set of tools for researchers in the field of fractional-order control. In this paper we present all the major modules comprising the toolbox and discuss the corresponding mathematical concepts. Fractional-order system analysis, identification and fractional PID controller design, tuning and optimization in the context of the toolbox are presented and discussed.

FOMCON: Fractional-order modeling and control toolbox for MATLAB

http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-LOD7-0029-0056/c/Tepljakov_Petlenkov_Belikov_FOMCON_2_2011.pdf

FOMCOM: a MATLAB toolbox for fractional-order system identification and control

Fractional-order Modeling and Control of Dynamic Systems

FOPID Controllers and Their Industrial Applications: A Survey of Recent Results

The interest in fractional-order (FO) control can be traced back to the late nineteenth century. The growing tendency towards using fractional-order proportional-integral-derivative (FOPID) control has been fueled mainly by the fact that these controllers have additional “tuning knobs” that allow coherent adjustment of the dynamics of control systems. For instance, in certain cases, the capacity for additional frequency response shaping gives rise to the generation of control laws that lead to superior performance of control loops. These fractional-order control laws may allow fulfilling intricate control performance requirements that are otherwise not in the span of conventional integer-order control systems. However, there are underpinning points that are rarely addressed in the literature: (1) What are the particular advantages (in concrete figures) of FOPID controllers versus conventional, integer-order (IO) PID controllers in light of the complexities arising in the implementation of the former? (2) For real-time implementation of FOPID controllers, approximations are used that are indeed equivalent to high-order linear controllers. What, then, is the benefit of using FOPID controllers? Finally, (3) What advantages are to be had from having a near-ideal fractional-order behavior in control practice? In the present paper, we attempt to address these issues by reviewing a large portion of relevant publications in the fast-growing FO control literature, outline the milestones and drawbacks, and present future perspectives for industrialization of fractional-order control. Furthermore, we comment on FOPID controller tuning methods from the perspective of seeking globally optimal tuning parameter sets and how this approach can benefit designers of industrial FOPID control. We also review some CACSD (computer-aided control system design) software toolboxes used for the design and implementation of FOPID controllers. Finally, we draw conclusions and formulate suggestions for future research.

/pdf/towards-industrialization-of-fopid-controllers-a-survey-on-16lzyj4qif.pdf

Aleksei Tepljakov

Papers

FOMCON: Fractional-order modeling and control toolbox for MATLAB

FOMCOM: a MATLAB toolbox for fractional-order system identification and control

Fractional-order Modeling and Control of Dynamic Systems

FOPID Controllers and Their Industrial Applications: A Survey of Recent Results

Towards Industrialization of FOPID Controllers: A Survey on Milestones of Fractional-Order Control and Pathways for Future Developments