**2.1 Variable, fractional-order backward difference**

Next one defines the Grünwald–Letnikov variable, fractional-order backward difference (VFOBD). For a discrete-variable bounded real-valued function *f*ð Þ� defined over a discrete interval 0, ½ � *k* the VFOBD is defined as a sum (see for instance [6, 9]).

**Definition 2.2**. The VFOBD with an order function *ν*, with values *ν*ð Þ*k* ∈½ � 0, 1 , is defined as a finite sum, provided that the series is convergent

$$\begin{aligned} \, \_{k\_0} \Delta\_k^{\nu(k)} f(k) &= \sum\_{i=0}^{k-k\_0} a^{\nu(k)}(i) f(k-i) \\ &= \begin{bmatrix} 1 \ a^{\nu(k)}(1) \ a^{\nu(k)}(2) & \cdots & a^{\nu(k)}(k\_0) \end{bmatrix} \begin{bmatrix} f(k) \\ f(k-1) \\ \vdots \\ f(k-k\_0) \end{bmatrix} \end{aligned} \tag{3}$$

Relating to (2) as the first special case of the defined above VFOBD and a constant order function *ν*ð Þ¼ *k ν* ¼ const from (2.1) one gets the fractional-order backward difference (FOBD). The second special case is for a constant integer order function *ν*ð Þ¼ *k ν* ¼ *n* ¼ const where the integer-order backward difference (IOBD) is a classical one.

Equality (3) is valid for *k*, *k* � 1, *k* � 2, … , *k*<sup>0</sup> þ 1, *k*0. Hence, one gets a finite set of equations. Collecting them in a vector matrix form one gets

$$\mathbf{A}\_{k\_0}^{GL} \boldsymbol{\Delta}\_k^{[\boldsymbol{\nu}(k)]} \mathbf{f}(k) = {}\_{k\_0} \mathbf{A}\_k^{[\boldsymbol{\nu}(k)]} \mathbf{f}(k), \tag{4}$$

where

There are numerous continuous and discrete-time PID and FOPID controller synthesis methods [16, 19–31]. One should mention that the optimisation of the closed-loop system in this case is more complicated because of the controller optimization. Apart from the three classical gains there are two additional parameters, namely, a fractional order of differentiation and summation [32]. The FOPID control characterizes by slow achieving the steady state and growing calculation "tail" [12]. In the paper a novelty variable, the fractional-order PID (VFOPID) [6, 28, 33–41]

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

controller synthesis is proposed. It consists of dividing the closed-loop system discrete-transient time division into the finite time intervals over which are defined fractional orders summation and differentiation functions. The main idea is that for the final infinite interval ½ Þ *kL*, þ∞ the difference order equals 0 and the summation is �! preserving quick reaching the zero steady state value. Thus, in the VFOPID control the disadvantages of FOPID are extracted. One should admit that in the FOPID or

Fractional-orders systems are characterized by the so called system "memory". This, in practice, means that in every step the FOPID controller computes its output signals taking into account step-by-step linearly computed number of samples. This causes in practice the micro-controllers realization problems. It is known as "Finite

The paper is organized as follows. In Section 2 the basic information related to the fractional calculus and variable, fractional order Grünwald-Letnikov backward difference is given. The main result of the paper includes Section 3. It contains the proposed VFOPID controller synthesis method with the proposal of the order functions form. The brief description of the controller parameters evaluation algorithm is given. The investigations are supported by a numerical example presented in Chapter 4.

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

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

ð Þ �<sup>1</sup> *<sup>k</sup> <sup>ν</sup>*ð Þ*<sup>l</sup>* ð Þ *<sup>ν</sup>*ðÞ�*<sup>l</sup>* <sup>1</sup> <sup>⋯</sup>ð Þ *<sup>ν</sup>*ðÞ�*<sup>l</sup> <sup>k</sup>* <sup>þ</sup> <sup>1</sup>

*n n*ð Þ � 1 ⋯ð Þ *n* � *k* þ 1 ð Þ �<sup>1</sup> *<sup>k</sup>* !

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

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

1 for *k* ¼ 0

1 for *k* ¼ 0

0 for *k*∈ *<sup>n</sup>*þ<sup>1</sup>

*<sup>k</sup>*! for *<sup>k</sup>*<sup>∈</sup> <sup>1</sup>

for *k*∈ ½ � 1, *n*

(1)

(2)

VFOPID control the microcontrollers are numerically loaded.

memory principle" [12].

**2. Mathematical preliminaries**

*<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ*<sup>l</sup>* ð Þ¼ *<sup>k</sup>*

**2**

denoted a ð Þ� *k* þ 1 ð Þ *k* þ 1 unit matrix **1***k*.

coefficients function of two 13 discrete variables as

8 >>><

>>>:

8 < :

*<sup>a</sup>*½ � *<sup>n</sup>* ð Þ¼ *<sup>k</sup>*

$${}\_{k\_0}\mathbf{A}\_k^{[\nu(k)]} = \begin{bmatrix} 1 & a^{[\nu(k)]}(1) & \cdots & a^{[\nu(k)-1]}(k-k\_0-1) \\ 0 & 1 & \cdots & a^{[\nu(k-1)]}(k-k\_0-1) \\ \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & a^{[\nu(k\_0+1)]}(1) \\ 0 & 0 & \cdots & 1 \end{bmatrix} \tag{5}$$

$$\mathbf{f}(k) = \begin{bmatrix} f(k) \\ f(k-1) \\ \vdots \\ f(k\_0) \end{bmatrix},$$

$${}\_{k\_0}^{GL}\Delta\_k^{[\nu(k)]}\mathbf{f}(k) = \begin{bmatrix} {}\_{k\_0}^{GL}\Delta\_k^{[\nu(k)]}f(k) \\ \vdots \\ \vdots \\ \_{k\_0}^{GL}\Delta\_{k\_0}^{[\nu(k)]}f(k\_0) \end{bmatrix}.$$

**3**
