The convergence analysis of P-type iterative learning control with initial state error for some fractional system

Area of Science:

• Control Theory
• Fractional Calculus
• Applied Mathematics

Background:

• Iterative learning control (ILC) is crucial for repetitive tasks.
• Fractional equations present unique challenges in control system analysis.
• Initial state errors can significantly impact system convergence.

Purpose of the Study:

• To investigate the convergence of iterative learning control for fractional equations with initial state errors.
• To introduce and analyze the concept of mild solutions in this context.
• To derive sufficient conditions for the convergence of P-type ILC.

Main Methods:

• Utilizing Laplace transforms to analyze the fractional dynamics.
• Employing the Mittag-Leffler (M-L) function to define and analyze mild solutions.
• Developing theoretical frameworks to establish convergence criteria for open and closed-loop P-type ILC.

Main Results:

• The concept of mild solutions for the studied fractional equation is established.
• Sufficient conditions for the convergence of both open-loop and closed-loop P-type iterative learning controllers are derived.
• The theoretical findings are validated through illustrative examples.

Conclusions:

• The paper provides a rigorous analysis of iterative learning control for fractional systems with initial errors.
• The derived convergence conditions offer practical guidelines for designing effective P-type ILC systems.
• This work contributes to the advancement of control theory for fractional-order systems.