Explicit stability condition for delta fractional order systems with α∈(0,+∞).

Area of Science:

• Control Systems Engineering
• Fractional Calculus
• Dynamical Systems Analysis

Background:

• Fractional order systems are increasingly important in modeling complex phenomena.
• Conventional stability analysis for fractional systems is often limited to orders between 0 and 1.
• Time-advance delta fractional order systems require specialized stability investigation methods.

Purpose of the Study:

• To investigate the stability of time-advance delta fractional order systems for orders in the range (0,+∞).
• To derive an explicit stability condition for these systems.
• To analyze the unstable region's characteristics.

Main Methods:

• Application of the delta Laplace transform for stability analysis.
• Introduction of a mapping relation ρ=s/(s+1).
• Establishing equivalence between delta and nabla difference operators.
• Eigenvalue distribution analysis of the system matrix.

Main Results:

• An explicit stability condition for time-advance delta fractional order systems is derived.
• The condition is directly related to the eigenvalues of the system matrix.
• Quantitative and qualitative analyses reveal the extent of the unstable region.

Conclusions:

• The developed stability condition is rigorously validated through three illustrative examples.
• The findings extend the understanding of fractional order system stability beyond conventional limits.
• The equivalence between delta and nabla differences aids in validating the results.