Sonin operator via the orthonormal polynomials: properties and applications

In this report we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. We study the Sonin operator as a generalisation of the Riemann-Liouville operator. In its own turn, the achieved generalization allows us to spread the obtained technique on practical problems that connected with various physical-chemical processes. Thanking the great features of fractional order differential operators, in the past decades they have been successfully applied in more precise modeling of different real-world phenomena and enhancing the performance of traditional control systems [A]. By generalizing these differential operators (e.g., generalizing the Riemann-Liouville operator to reach the Sonin operator), the opportunity of more progress in increasing the accuracy of the dynamic models describing real-world processes and improving the performance of control system loops can be provided. In order to move in this direction, as a primary step, finding the properties of the dynamic models constructed on the basis of the generalized differential operators is of great importance. For instance, the following issues can be considered as future relevant research directions: • Finding the conditions guaranteeing the monotonicity of the responses in the linear class of the considered generalized dynamic models (such as that previously obtained for fractional order dynamic models [B, C]) • Study on the complex behaviors in the nonlinear class of the considered generalized dynamic models (for example, study on chaotic behavior [D]) The achievements on the above-mentioned issues can helpful in revealing the properties (and consequently the potentials) of the considered generalized dynamic models.
References: [A] M. S. Tavazoei, "From Traditional PI Control to Fractional PI Control: A Key for Generalization" IEEE Industrial Electronics Magazine, Volume 6, Issue 3, 2012, Pages 41-51. [B] M. S. Tavazoei, "On Monotonic and Non-Monotonic Step Responses in Fractional Order Systems," IEEE Transactions on Circuits and Systems II, Volume 58, Issue 7, August 2011, Pages 447-451. [C] M. S. Tavazoei, "Fractional/Distributed Order Systems and Irrational Transfer Functions with Monotonic Step Responses," Journal of Vibration and Control, Volume 20, Issue 11, 2014, Pages 1697-1706. [D] M. S. Tavazoei, "Fractional order chaotic systems: history, achievements, applications, and future challenges," The European Physical Journal Special Topics, Volume 229, 2020, Pages 887-904.