digital-brain/content/zettels/fractional_order_transfer_functions.md

+++ title = "Fractional Order Transfer Functions" author = ["Dehaeze Thomas"] draft = false +++

Tags

[Digital Filters]({{< relref "digital_filters.md" >}})

Example Using the FOMCON toolbox

The documentation for the toolbox is accessible here.

Here are the parameters that are used to define the wanted properties of the fractional model:

  wb = 2*pi*0.1; % Lowest frequency bound
  wh = 2*pi*1e3; % Highest frequency bound
  n  = 8; % Approximation order
  r = 0.5; % Wanted slope, The corresponding phase will be pi*r

Then, to create an approximation of a fractional-order operator \(s^r\) of order \(n\) which is valid in the frequency range \([\omega_b\, \omega_h]\), the oustafod function can be used:

  G = oustafod(r,n,wb,wh);

G =

  79.27 s^17 + 7.93e05 s^16 + 2.918e09 s^15 + 5.143e12 s^14 + 4.782e15 s^13 + 2.453e18 s^12 + 7.103e20 s^11 + 1.175e23 s^10 + 1.119e25 s^9 + 6.138e26 s^8 + 1.942e28 s^7 + 3.534e29 s^6 + 3.675e30 s^5 + 2.157e31 s^4 + 6.984e31 s^3 + 1.193e32 s^2 + 9.764e31 s + 2.939e31
  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
    s^17 + 1.312e04 s^16 + 6.327e07 s^15 + 1.462e11 s^14 + 1.783e14 s^13 + 1.199e17 s^12 + 4.553e19 s^11 + 9.877e21 s^10 + 1.232e24 s^9 + 8.866e25 s^8 + 3.678e27 s^7 + 8.775e28 s^6 + 1.196e30 s^5 + 9.208e30 s^4 + 3.909e31 s^3 + 8.755e31 s^2 + 9.395e31 s + 3.707e31

Continuous-time transfer function.

Few examples of different slopes are shown in Figure 1.

{{< figure src="/ox-hugo/approximate_deriv_int.png" caption="<span class="figure-number">Figure 1: Example of fractional approximations" >}}

Bibliography