H_∞ controller design for Nabla discrete fractional-order systems

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.