Digitalisation of control systems

Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements

“…Approximation (22) has an asymptotic behavior of a simple gain in low frequencies and a first-order integrator in high frequencies. It uses a second-order transfer function for C (n ) with two poles and one zero.…”

“…The most common are Euler's, Tustin's, Simpson's, or Al-Alaoui's approximations [8,9,22]. These analogue-to-digital open-loop design methods lead to irrational z-transforms which are then approximated either by a truncated Taylor's series expansion or a continuous fraction expansion.…”

“…Approximations (19), (20), (22), and (23) are compared in Figures 2 and 3. It can be seen that the performance of approximations (19) and (20) depends on the differentiation order n. (19) gives better results than (20) for differentiation orders 0 < n < 0.5 and (20) gives better results than (19) for differentiation orders 0.5 < n < 1.…”

“…This is not restrictive because when {n} > 1, only fractional part of the differentiation or integration is approximated. Consequently, (22) and (23) can be used to approximate any fractional differentiator or integrator.…”

“…The output of one simple eigenmode is represented as shown in Figure 5. I n represents the fractional integrator approximation in a limited frequency band as described by (19), (20), (22) and (23).…”