Time response optimal rational approximation: Improvement of time responses of MSBL based approximate fractional order derivative operators by using gradient descent optimization

Abstract

Fractional calculus has become a popular topic in recent years. Its applications have been seen frequently in the analysis and solution of many engineering problems. Since, the exact realization of fractional order (FO) operators is not computationally feasible, several integer order approximations methods such as Oustaloup, Matsuda, Curve Fitting, modified stability boundary locus (MSBL) are utilized for frequency domain approximation of FO operators. These methods present advantages in frequency response approximation. However, real-world engineering applications require more accurate time responses to implement FO operators. Thus, any improvement in the approximate transfer function of FO systems will positively contribute to realization performance of FO system models in real-world applications. In this work, time responses of MSBL based approximate FO derivative models are improved by using gradient descent optimization (GDO) technique. Frequency domain approximation methods do not guarantee optimal approximation performance for the step response of FO elements. To deal with this weakness, author proposes a new GDO based rational approximation scheme that can considerably enhance the step response performance of MSBL method without deteriorating frequency response approximation performance. Results for the improved approximate FO model obtained by GDO (GDO-MSBL) are compared with the results of approximate FO models for popular FO approximation methods. The analog circuit realization is performed for the derivative operator, and the performance improvements for the proposed method are shown on the FO-PID controller and FO filter.

Keywords

• Gradient descent optimization
• MSBL approximation method
• Fractional order operator
• Analog realization