A short note concerning simulation of unstable fractional-order transfer functions

Fractional-order dynamic models may allow to capture complex dynamics, as they may be interpreted as the coexistence of extremely slow and fast time constants (Oustaloup (1995) and Trigeassou and Maamri (2025)). Different synthesis methods exist for time-domain simulation of fractional-order systems. One of the most famous methods is the Grünwald-Letnikov method, which relies on a discrete-time approximation of the fractional derivative. This method may be seen as a generalization for the well-known backward Euler method in numerical simulation. Even though a notable advantage of backward Euler’s method for linear system simulation is its unconditional stability for stable system simulation, special care may be required for unstable system simulation. This study is a short note generalizing an already well-known result concerning unstable system simulation for the case of fractional-order systems. A simple mathematical proof as well as some simulation examples will be provided.