Controllability and observability of fractional linear systems with two different orders

Area of Science:

• Control Theory
• Dynamical Systems
• Fractional Calculus

Background:

• Fractional-order systems are increasingly utilized in various engineering fields.
• Controllability and observability are fundamental concepts for analyzing system behavior.
• Existing research on fractional systems often focuses on single orders.

Purpose of the Study:

• To investigate controllability and observability for fractional linear systems with two different orders.
• To establish rigorous mathematical conditions for these properties.
• To extend existing theories for fractional dynamical systems.

Main Methods:

• Development of theoretical frameworks for fractional linear systems.
• Application of state-space representations for systems with multiple fractional orders.
• Derivation of sufficient and necessary conditions for controllability and observability.

Main Results:

• Sufficient and necessary conditions for state controllability are established.
• Sufficient and necessary conditions for state observability are established.
• The study provides a generalized framework applicable to systems with distinct fractional orders.

Conclusions:

• The established conditions provide a significant advancement in the analysis of fractional linear systems.
• The results extend prior work on controllability and observability in fractional dynamics.
• This research offers a foundation for further exploration of complex fractional-order control systems.