Normalized Robust FOPID Controller Regulation Based on Small Gain Theorem

Shuo Zhang,Lu Liu

doi:10.1155/2018/5690630

Abstract

In this paper, a normalized robust FOPID controller regulation algorithm is proposed. Only one parameter k is necessary to be tuned in the controller regulation process, so the proposed control algorithm is convenient to be applied on both fractional-order systems and integer-order systems. A robustness evaluation function is constructed based on the small gain theorem. Larger robustness evaluation function value will help the system achieve better robustness performance. Another parameter, β, is also available to serve as a tuning knob when larger robust evaluation function value is needed. Therefore, the controlled systems can be stabilized and can achieve quite satisfactory robust control performance using the proposed algorithm. The corresponding robust analysis results are obtained according to different conditions in the discussion. For a special case of widely used fractional-order systems, the FOPI and FOID controllers are presented based on the same tuning scheme together with their robustness discussion. Some examples are shown to verify the robustness of systems controlled by the proposed algorithm.

Highlights

In 1695, fractional calculus was first mentioned in Leibniz’s letter to L’Hospital, as a generalization of classical integerorder calculus
Fractional-order PID (FOPID or PIλ Dμ) control is an important application of fractional calculus which was presented in [5]
We investigate a normalized regulation algorithm for robust fractionalorder PID controller (FOPID) controller design

Summary

Introduction

In 1695, fractional calculus was first mentioned in Leibniz’s letter to L’Hospital, as a generalization of classical integerorder calculus. References [4, 14,15,16] point out that fractional calculus could provide one more dimension in control process It owns more potential in enhancing system robustness as well as transient control performance. Fractional-order PID (FOPID or PIλ Dμ) control is an important application of fractional calculus which was presented in [5]. Some FOPID tuning methods based on flat phase property for servo control and disturbance compensation for different motion systems were demonstrated in [9]. The proposed controller is convenient and simple to be applied on different kinds of systems, including fractional-order and integer-order ones. The robustness of a special kind of fractional-order systems which is commonly used in practice is considered under FOPI and FOID controllers.

Preliminaries

Robust Analysis for Fractional-Order Systems under FOPID Controllers

Robust Analysis for Some Special Cases

Robustness Analysis Examples

Conclusion