Implementation issues in discretization of fractional-order derivative using the Al-Alaoui operator

New Stability Tests for Discretized Fractional‐Order Systems Using the Al‐Alaoui and Tustin Operators

“…The next steps of the proof follow the lines of Theorem 1 of [13]. Note that the characteristic equation of system (4) or (5)…”

“…Firstly, the three discretization schemes lead to infinite complexity of rational, discrete-time counterparts of fractional-order derivative. Therefore, in practical applications various finite-length approximations of the discretization operators have been used, involving the most popular finite fractional difference (FFD) approximation in the Euler approach [1,2] and finite-length implementations of the continuous fraction expansion (CFE) method in the Tustin and Al-Alaoui approaches [2][3][4][5][6]. Also, a number of papers have presented some other approximation/discretization methods for the fractional-order derivative [7][8][9][10].…”

“…The discrete-time fractional-order system(4) or(5), with w z as in(9), is asymptotically stable if and only if all f -poles λ f j of the system, j = 1, … , n, are outside the instability area…”