# A universal framework of GKYP lemma for singular fractional order systems

## TL;DR

This paper develops a universal finite frequency band GKYP lemma for singular fractional order systems, enabling improved control design and performance analysis in limited frequency ranges.

## Contribution

It introduces a novel framework for the GKYP lemma applicable to singular fractional order systems, filling a theoretical gap in control analysis.

## Key findings

- Established a universal finite frequency band GKYP lemma for singular fractional order systems.
- Derived bounded real lemma for various frequency ranges.
- Designed controllers that enhance L performance indices for these systems.

## Abstract

The well-known GKYP is widely used in system analysis, but for singular systems, especially singular fractional order systems, there is no corresponding theory, for which many control problems for this type of system can not be optimized in the limited frequency ranges. In this paper, a universal framework of finite frequency band GKYP lemma for singular fractional order systems is established. Then the bounded real lemma in the sense of L is derived for different frequency ranges. Furthermore, the corresponding controller is designed to improve the L performance index of singular fractional order systems. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.