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H∞ controller design for Nabla discrete fractional-order systems.

Jin-Xi Zhang1, Yuanda Lv2, Xuefeng Zhang2

  • 1State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, China.

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|February 19, 2025
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Summary

This study introduces criteria for Nabla discrete fractional-order systems (FOSs) to achieve H∞ performance. It develops an H∞ state feedback controller using linear matrix inequality (LMI) methods for system stabilization and performance optimization.

Keywords:
Discrete fractional-order systemsH(∞) performance indexLinear matrix inequality (LMI)State feedback control

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Area of Science:

  • Control Systems Engineering
  • Fractional Calculus
  • Systems Theory

Background:

  • Fractional-order systems (FOSs) offer enhanced modeling capabilities over integer-order systems.
  • Ensuring stability and performance (H∞) in discrete FOSs remains a challenge.
  • Nabla calculus provides a framework for discrete-time fractional analysis.

Purpose of the Study:

  • To establish sufficient conditions for achieving a specific H∞ performance index in Nabla discrete FOSs.
  • To develop a linear matrix inequality (LMI)-based H∞ state feedback controller for these systems.
  • To verify the proposed methodology through numerical simulations.

Main Methods:

  • Approximation of the instability region using fan-shaped domains.
  • Application of the generalized Kalman-Yakubovič-Popov lemma.
  • Utilization of the projection lemma for controller design.
  • Formulation of criteria using linear matrix inequalities (LMIs).

Main Results:

  • Derivation of LMI-based criteria for H∞ norm boundedness of Nabla discrete FOSs.
  • Design of an H∞ state feedback controller that simultaneously stabilizes the system and optimizes the H∞ performance index.
  • Validation of the controller's effectiveness through two simulation examples.

Conclusions:

  • The proposed LMI-based approach effectively addresses H∞ performance for Nabla discrete fractional-order systems.
  • The developed controller ensures system stability and optimizes the H∞ performance index.
  • The methodology is validated by accurate numerical solutions and system responses.