**1. Introduction**

A continuous-time proportional–integral–derivative controller (PID controller) [1] invented almost 100 years ago is one of the most widely applied controllers in the closed-loop systems [2] with many industrial applications [3–5]. Currently the continuous-time control is successively replaced by discrete-time one in which the integration is replaced by a summation and differentiation by a difference evaluation. So, in the discrete PID controller the classical integral is replaced by a sum and the derivative by a backward difference, [6]. The discrete controller's PID algorithm is mainly realized by micro-controllers [7].

At 70s of the 20-th Century the Fractional Calculus [8] with a great success started a considerable attention in mathematics and engineering [9–12]. Now, the fractional-order backward-difference (FOBD) and the fractional-order backward sum (FOBS) [6, 13] are applied in the dynamical system modeling [14] and discrete control algorithms. The continuous-time FOPID controllers are more difficult in a practical realization [15–18].

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].

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

*Variable, Fractional-Order PID Controller Synthesis Novelty Method*

*k* X�*k*0 *i*¼0

defined as a finite sum, provided that the series is convergent

*<sup>a</sup><sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ*<sup>i</sup> f k*ð Þ � *<sup>i</sup>*

<sup>¼</sup> <sup>1</sup> *<sup>a</sup><sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ<sup>1</sup> *<sup>a</sup><sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ<sup>2</sup> <sup>⋯</sup> *<sup>a</sup><sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>*<sup>0</sup> h i

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

Equality (3) is valid for *k*, *k* � 1, *k* � 2, … , *k*<sup>0</sup> þ 1, *k*0. Hence, one gets a finite set

<sup>1</sup> *<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ<sup>1</sup> <sup>⋯</sup> *<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>* � *<sup>k</sup>*<sup>0</sup>

⋮ ⋮ ⋮

0 0 ⋯ 1

**f**ð Þ¼ *k*

*<sup>k</sup>* **f**ð Þ¼ *k*

0 0 <sup>⋯</sup> *<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ *<sup>k</sup>*0þ<sup>1</sup> ð Þ<sup>1</sup>

*f k*ð Þ

,

*f k*ð Þ � 1

⋮

*f k*ð Þ<sup>0</sup>

⋮

*GL <sup>k</sup>*<sup>0</sup> <sup>Δ</sup>½ � *<sup>ν</sup>*ð Þ*<sup>k</sup> <sup>k</sup> f k*ð Þ

*GL <sup>k</sup>*<sup>0</sup> <sup>Δ</sup>½ � *<sup>ν</sup>*ð Þ *<sup>k</sup>*<sup>0</sup> *<sup>k</sup>*<sup>0</sup> *f k*ð Þ<sup>0</sup>

0 1 <sup>⋯</sup> *<sup>a</sup>*½ � *<sup>ν</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ *<sup>k</sup>* � *<sup>k</sup>*<sup>0</sup> � <sup>1</sup>

*<sup>k</sup>* **<sup>f</sup>**ð Þ¼*<sup>k</sup> <sup>k</sup>*0**A**½ � *<sup>ν</sup>*ð Þ*<sup>k</sup>*

function *ν*ð Þ¼ *k ν* ¼ *n* ¼ const where the integer-order backward difference

of equations. Collecting them in a vector matrix form one gets

*GL <sup>k</sup>*<sup>0</sup> <sup>Δ</sup>½ � *<sup>ν</sup>*ð Þ*<sup>k</sup>*

*GL <sup>k</sup>*<sup>0</sup> <sup>Δ</sup>½ � *<sup>n</sup>*ð Þ*<sup>k</sup>*

instance [6, 9]).

*<sup>k</sup>*0Δ*<sup>ν</sup>*ð Þ*<sup>k</sup>*

(IOBD) is a classical one.

*<sup>k</sup>*0**A**½ � *<sup>ν</sup>*ð Þ*<sup>k</sup> <sup>k</sup>* ¼

where

**3**

*<sup>k</sup> f k*ð Þ¼

*DOI: http://dx.doi.org/10.5772/intechopen.95232*

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

**Definition 2.2**. The VFOBD with an order function *ν*, with values *ν*ð Þ*k* ∈½ � 0, 1 , is

*f k*ð Þ *f k*ð Þ � 1 ⋮ *f k*ð Þ � *k*<sup>0</sup>

*<sup>k</sup>* **f**ð Þ*k* , (4)

(3)

(5)

(6)

In the paper a novelty variable, the fractional-order PID (VFOPID) [6, 28, 33–41] 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 VFOPID control the microcontrollers are numerically loaded.

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 memory principle" [12].

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.
