# Fractional calculus in control applications

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The beginning of fractional calculus dates back to the early days of classical differential calculus, although its inherent complexity postponed its use and application to the engineering world. Nowadays, its use in control engineering has been gaining more and more popularity in terms of controller tuning. Generally speaking, fractional calculus may be defined as a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The seminar focuses on the basics of fractional calculus in control applications, with a focus upon the advantages that a controller employing fractional order differentiators or integrators might bring to the overall performance of the closed loop system.

The seminar includes a brief historical review of the early steps of fractional order controllers and how Bode’s ideal loop transfer function has been used to shape the idea of robust fractional order controllers. The stability of fractional order systems is also presented, along with a suggestive example used to indicate how the analysis of stability regions might represent a useful tool in designing fractional order control strategies. The main topic of the seminar is centered around the fractional order PIμDλ controllers. The design problem of fractional order controllers has been the interest of many authors, with some valuable works, in which the fractional order controllers have been applied to a variety of processes to enhance the robustness and performance of the control systems. The seminar includes a general tuning procedure for fractional order PIμ, PDλ and PIμDλ controllers, as well as an alternative approach based on vector representation. The tuning procedure is also extended to time delay processes. Finally, ways of implementing these fractional order controllers are addressed, including both analog and digital realizations. Apart from the fractional order PID , several attempts have been made to design advanced fractional order control strategies, by combining fractional calculus with optimal control, predictive control, robust or adaptive control. The presentation will end with such a tuning procedure for fractional order Internal Model Control strategies.

This talk is part of the CUED Control Group Seminars series.

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