Control of non-minimum phase systems with dead time: a fractional system viewpoint

Abstract

In this paper, a unified theory is proposed for the fractional control of the non-minimum phase plus dead time systems via the introduction of the fractional filter-PID control and stability inequality. More precisely, three Theorems from a fractional system-theoretic viewpoint are developed. The first Theorem deals with the controller design through the notion of generalised internal model control. That embeds the fractional filter-PID controller coupled with the unified form of the Smith predictor and the inverse compensator. The second Theorem designs a bound on the fractional filter parameter that leads to a stable feedback system. The third Theorem deals with the stability of a fractional quasi-characteristic polynomial of the commensurate order arising from the control of non-minimum phase systems with dead time. Two illustrative second-order plus dead time non-minimum phase systems are presented to test the efficacy of the resulting fractional filter-PID controller that obeys the stability bound. Controller and sensitivity performance indices for the nominal as well as mismatch cases reveal the effectiveness and robustness of the proposed method for enhanced closed-loop performance. Thus, the generalised inverse compensator, the fractional filter-PID controller and the stability inequality in corroboration with numerical simulations demonstrate the better control of non-minimum phase systems with dead time.

KEYWORDS:

• Non-minimum phase systems with dead time
• fractional filter-PID controller
• fractional quasi-characteristic polynomial
• stability bound

Acknowledgements

The Authors express gratefulness to anonymous Reviewers and the Associate Editor for their constructive comments, suggestions and ideas. That has led to an improvement in the content of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Shaival Hemant Nagarsheth

Shaival Hemant Nagarsheth received the B.E. degree in Instrumentation & Control from Sarvajanik College of Engineering & Technology, Surat, Gujarat, India in 2014, with a gold medal. In 2016 he received his M. Tech degree in Instrumentation & Control with specialisation in Control & Automation from Nirma University, Ahmedabad, Gujarat, India, with a gold medal. He is currently pursuing his doctoral degree in control theory from the Electrical Engineering Department at Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, India. He was a recipient of EECI (European Embedded Control Institute) Overseas Grant amongst the ten research scholars all around the world. He was awarded an amount of 500 Euros for attending 2019-IGSC (International Graduate School on Control) held at Automatic Control Laboratory, ETH Zürich, Switzerland. His research includes matrix calculus, controller design for multivariable systems, sensitivity integrals, fractional control design. He is a member of the IEEE Young Professionals, IEEE student member, IEEE Control System Society, IEAE student member, ISA student member, IFAC Affiliate, IFAC-ACDOS (NMO) member.

Shambhu Nath Sharma

Shambhu Nath Sharma received the B.E. degree from Government College of Engineering, Rewa (M.P.), India in 1994, the M Tech degree from Banaras Hindu University (Now IIT BHU), UP, India in 2000, and the PhD degree from Delhi University in 2007. Currently, he is working as a Professor in the Electrical Engineering Department of the National Institute of Technology, Surat, India. He specialises in the areas of stochastic systems, control theory, stochastic filtering, and stochastic differential equations with applications to electrical and electronic networks. One of his works is known as a pioneering work in stochastic systems. In addition to these, he is also interested in multivariable system theory as well as non-linear system theory. He had a visiting academic appointment at the Department of Systems and Control of Jožef Stefan Institute, Ljubljana, the Republic of Slovenia under the joint programme of the Indian National Science Academy (INSA, New Delhi) and the Slovenian Academy of Sciences and Arts (SASA, Ljubljana).