**2. Mathematical preliminaries**

In the paper the following notation will be used. <sup>0</sup> ¼ f g 0, 1, 2, 3, … , *<sup>l</sup>* ¼ f g *l*, *l* þ 1, *l* þ 2, … <sup>þ</sup> ¼ ½ Þ 0, þ∞ . 0*<sup>k</sup>* will denote the zero column vector of dimensions ð Þ� *k* þ 1 1 whereas **0***<sup>k</sup>*,*<sup>k</sup>* is ð Þ� *k* þ 1 ð Þ *k* þ 1 zero matrix. Similarly will be denoted a ð Þ� *k* þ 1 ð Þ *k* þ 1 unit matrix **1***k*.

In general, a fractional-order functions will be denoted by Greek letters *ν*ð Þ� : <sup>0</sup> ! *ν*ð Þ� whereas the integer orders will be denoted by Latin ones *n*∈ þ. In practice, for *l* ∈ 0: 0 <*ν*ð Þ*l* ≤1. For *k*, *l* ∈ <sup>0</sup> and a given order function *ν*ð Þ*l* the function of two discrete variables *<sup>k</sup>*, *<sup>l</sup>* <sup>∈</sup> <sup>0</sup> is defined by the following formula: *<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ*<sup>l</sup>* ð Þ*<sup>k</sup>* as follows:

**Definition 2.1.** For *k*, *l* ∈ <sup>0</sup> and a given order function *ν*ð Þ� one defines the coefficients function of two 13 discrete variables as

$$a^{\left[\nu(l)\right]}(k) = \begin{cases} 1 & \text{for} \quad k = 0\\ \left(-1\right)^{k} \frac{\nu(l)(\nu(l) - 1)\cdots(\nu(l) - k + 1)}{k!} & \text{for} \quad k \in \mathbb{N}\_{1} \end{cases} \tag{1}$$

One should mention that function (1) for *ν*ðÞ¼ *l n l*ðÞ¼ const ∈ <sup>0</sup>

$$a^{[n]}(k) = \begin{cases} 1 & \text{for} \quad k = 0\\ \frac{n(n-1)\cdots(n-k+1)}{\left(-1\right)^{k}!} & \text{for} \quad k \in [1, n] \\\\ 0 & \text{for} \quad k \in \mathbb{N}\_{n+1} \end{cases} \tag{2}$$

The above function will be named as: the "oblivion function" or "decay function".

*Variable, Fractional-Order PID Controller Synthesis Novelty Method DOI: http://dx.doi.org/10.5772/intechopen.95232*
