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A universal framework of GKYP lemma for singular fractional order systems

Yuman Li, Yiheng Wei, Yuquan Chen, Yong Wang

TL;DR

This paper addresses the lack of a universal finite-frequency GKYP framework for singular fractional-order systems (SFOS) and their performance analysis. It develops a universal GKYP lemma for SFOS using the S-procedure with an admissible, rank-one separable set ${\mathbb M}$ and derives ${L_ fty}$ bounded real lemmas across low, middle, high, and full frequency bands expressed as LMIs. An ${L_ fty}$ controller synthesis method via state feedback is proposed, yielding a real gain ${K = YX^{-1}}$ under LMIs that guarantee the bound on the closed-loop transfer. Three numerical examples validate the theory, demonstrating finite-frequency control of SFOS and suggesting practical impact for networks, power, and aerospace applications.

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.

A universal framework of GKYP lemma for singular fractional order systems

TL;DR

This paper addresses the lack of a universal finite-frequency GKYP framework for singular fractional-order systems (SFOS) and their performance analysis. It develops a universal GKYP lemma for SFOS using the S-procedure with an admissible, rank-one separable set and derives bounded real lemmas across low, middle, high, and full frequency bands expressed as LMIs. An controller synthesis method via state feedback is proposed, yielding a real gain under LMIs that guarantee the bound on the closed-loop transfer. Three numerical examples validate the theory, demonstrating finite-frequency control of SFOS and suggesting practical impact for networks, power, and aerospace applications.

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.

Paper Structure

This paper contains 10 sections, 14 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular fractional order system model
  4. S-procedure
  5. Frequency range
  6. GKYP lemma for SFOS
  7. ${L_\infty }$ bounded real lemmas for the SFOS
  8. ${L_\infty }$ controller synthesis for the SFOS
  9. Simulation Study
  10. Conclusions

Key Result

Lemma 1

Iwasaki2005GKYPlemma Let $\mathbb{M} \subset {\mathbb{H}_q}$ be defined by (3) and (4). Then for any matrix $\Theta \in {\mathbb{H}_q}$, if and only if the set $\mathbb{M}$ is admissible and rank-one separable, the strict S-procedure is lossless.

Figures (5)

  • Figure 1: Maximum singular values of Example 1.
  • Figure 2: The circuit structure diagram of Example 2.
  • Figure 3: Maximum singular values of the SFOS in Example 2.
  • Figure 4: Maximum singular values of the open-loop system in Example 3.
  • Figure 5: Maximum singular values of the closed-loop system in Example 3.

Theorems & Definitions (22)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Remark 2
  • Lemma 4
  • Lemma 5
  • Theorem 1
  • ...and 12 more