=Paper=
{{Paper
|id=Vol-3109/paper2
|storemode=property
|title=Information support of controlling influences formation using fractional order controllers
|pdfUrl=https://ceur-ws.org/Vol-3109/paper2.pdf
|volume=Vol-3109
|authors=Yaroslav Marushchak,Bohdan Kopchak
}}
==Information support of controlling influences formation using fractional order controllers==
https://ceur-ws.org/Vol-3109/paper2.pdf
Information support of controlling influences formation using
fractional order controllers
Yaroslav Marushchak1 and Bohdan Kopchak1
1
Lviv Polytechnic National University, 12 Bandera Street, Lviv, 79013, Ukraine
Abstract
Computer research of algorithm and programs realization of differentiating and integrating
fractional order parts, as components of PIλDμ controllers, has shown the application efficiency
of fractional order transfer function approximation. The application of the decomposition
theorem of rational fractions allowed to construct structural schemes from parallel connected
aperiodic parts for the realization of approximated arbitrary order transfer functions. It has been
experimentally proved that the implementation of fractional order controllers based on
approximated transfer functions can work in real time as the integral part of highly dynamic
automatic control systems. Tests of the frequency converter MFC 710 option with the PIλDμ
fractional order controller in the speed control system using the Twerd experimental stand have
confirmed its efficiency in terms of expanding the regulatory capabilities of such automatic
control systems.
Keywords 1
Fractional order, PIλDμ controller, Oustaloup transformation, electromechanical systems
1. Introduction
Construction of automatic control systems (ACS) using fractional order controllers significantly
expands the possibilities compared to conventional controllers. Control influences, which are formed
by controllers, provide the specified indicators of control of electric power and technological processes.
In [1-9] the advantages of ACS for the use of fractional PIλDμ controller with transfer function are
shown
𝑊 (𝑠) = 𝑘 + 𝑘 𝑠 + 𝑘 𝑠 .
In such controller I- and D-fractional order components give a wider range of settings. Naturally, in
addition to the values of proportional, differential and integral components 𝑘 , 𝑘 and 𝑘 , the fractional
order controller has two more parameters: fractional powers λ and μ of the Laplace operator s in the
integrator and differentiator, respectively.
2. Research Analysis
In order to find possible ways to implement fractional order controllers, an analysis of the use of
Riemann, Riemann-Liouville and Grunwald-Letnikov representations for the construction of such
controllers was carried out. To calculate the transition functions for the fractional order integrating
controller in the Riemann representation.
1 (1)
𝐷 𝑓(𝑡) = (𝑡 − 𝜏) 𝑓(𝜏)𝑑𝑡,
Γ(α)
ITEA-2021: 1st Workshop of the 10th International scientific and practical conference Information technologies in energy and agro-industrial
complex, October 6-8, 2021, Lviv, Ukraine
EMAIL: yaroslav.y.marushchak@lpnu.ua (Y. Marushchak); bohdan.l.kopchak@lpnu.ua (B. Kopchak)
ORCID: 0000-0002-7901-3343 (Y. Marushchak); 0000-0002-2705-8240 (B. Kopchak)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
8
and a differential fractional order controller in the Riemann-Liouville representation
1 𝑑 𝑓(𝜏)𝑑𝑡 (2)
𝐷 𝑓(𝑡) = ( ) ,
Γ(n − α) 𝑑𝑡 (𝑡 − 𝜏)
appropriate programs have been developed. Based on the analysis of the obtained results, the following
problems were revealed:
the right integration limit for expressions (1) and (2) gives a division by zero;
the calculation of each subsequent point of the transition process of the integral fractional part
requires the presence of all previous values of the sub integral function (process input signal) starting
from zero, and therefore each subsequent point requires a larger amount of calculations. To calculate
the current value of the function, it is necessary to remember the value of the function at all previous
points in time. Thus, the CPU load increases and thus significantly complicates the work of such
controllers in the ACS, where there are transition processes.
In the literature on the implementation of fractional order controllers, there is a reference to the
integral-differential fractional order controller model with TF s±α in the representation of Grunwald-
Letnikov [10-11]:
Γ(α + 1)f(t − jh)
𝐷 𝑓(𝑡) = lim ℎ (−1) .
→ Γ(j + 1)Γ(α − j + 1)
The advantages of this presentation are:
the formula is easy to use because it is written on the basis of a finite sum, not an integral;
the model for the representation of integrative-differentiating controllers by the Grunwald-
Letnikov formula provides a higher, compared to the above models, the speed of calculations;
the same formula is used to represent the integrating or differentiating fractional order
controller, only the sign of fractional order is changed ("+" for differentiator, "-" for integrator).
The main disadvantages of this model are that the calculation of the transition process of the
integrating or differentiating of fractional order controller is complicated by the presence in the formula
of gamma functions, attempts to increase the accuracy of determining which significantly increases the
calculation time of the transition process.
Thus, the application of these methods for the physical implementation of fractional order
controllers actually makes it impossible to use them in high-speed systems with long-term operating
conditions. Such shortcomings have been partially eliminated in [12-13], where fractional order
controllers are used in indoor climate automation and supercapacitor charge systems. It should be noted
that such systems are characterized by low speed.
If we consider ACS with fractional order controllers that form control effects for various electric
power and electromechanical systems, the main problem to be solved is the operation of such controllers
in real time, when dynamic processes are characterized by high speed.
3. Problem Solving
Given that the implementation of integer-order controllers is well developed in both analog and
digital execution, the problem of technical implementation of synthesized fractional order controllers
in such systems can be solved by equivalent replacement (approximation) of their transfer functions
(TF) to integer order TF. Equivalence implies the provision of the same transition functions and
frequency characteristics in the appropriate frequency range for both TF representations. Such an
approximation can be performed using the known formulas of Oustaloup (Oustaloup A.) transformation
[14]. According to this method, given the lower and upper levels of the frequency range 𝜔 , 𝜔 , for
which the equivalence of the frequency characteristics of fractional controllers both representations, we
can write the following expression of the approximation of the integrating and differentiating fractional
order parts α
9
(3)
𝜔 1 + 𝑠 ⁄𝜔
𝑠± = ,
𝜔 1 + 𝑠 ⁄𝜔
where 𝜔 = 𝜔 𝜔 ; N – order of approximation to be specified; 𝜔 , 𝜔 , − zeros and poles of
equivalent integer order TF, respectively.
The calculation of zeros and poles of the approximate integer order TF is carried out
according to the following expressions:
𝜔 ( , , ⋅ )/( ) (4)
𝜔 =𝜔 ,
𝜔
𝜔 ( , , ⋅ )/( ) (5)
𝜔 =𝜔 .
𝜔
Let us denote 𝑘п = the gain of the approximating TF.
In the general case, the order of approximation is possible at (2N + 1) levels. The idea is that to
replace of the fractional order TF with the whole order TF, the coefficient, zeros and poles of the
expected TF are first calculated. In the next step, using the found zeros and poles, the TF is written in
the form
(𝑠 − 𝜔 )(𝑠 − 𝜔 ) … (𝑠 − 𝜔 ) (6)
𝑊(𝑠) = 𝑘 ,
(𝑠 − 𝜔 )(𝑠 − 𝜔 ) … (𝑠 − 𝜔 )
where 𝜔 , 𝜔 … 𝜔 calculated according to (4) the values of the zeros of the integer order TF;
𝜔 ,𝜔 …𝜔 − calculated according to (5) the poles of the integer order TF.
We present TF (6) as the ratio of polynomials
𝑏 𝑠 + 𝑏 𝑠 + ⋯+ 𝑏 𝑠 + 𝑏 𝑃(𝑠) (7)
𝑊(𝑠) = 𝑘 = .
𝑎 𝑠 + 𝑎 𝑠 + ⋯+ 𝑎 𝑠 + 𝑎 𝑄(𝑠)
To verify the adequacy of the replacement of fractional order TF by the integer order TF, a
comparative analysis of their logarithmic frequency characteristics was performed. For this purpose,
given the order of approximation (2N+1), the frequency characteristics of both TFs were calculated.
The expression 𝑠 ± can be interpreted as the expression of fractional differentiator (+α) or
integrator (-α), by means of which the total TF of fractional PIλDμ controllers is formed. Therefore, the
following analysis is devoted to such TF. To implement it in the MATLAB environment, a program
was developed that implements the Oustaloup method according to (3) to approximate 𝑠 ± by the
integer order TF.
Using the developed program, it is possible to transform differential-integral parts of fractional
order TF with different degrees α in a certain frequency range provided that the order of approximation
changes within N = 1÷5. The frequency range for the differential part is selected within (0.01÷100)s-1,
and for the integral - the range (0.001 ÷ 1000) s-1.
Let us represent the differential and integral fractional order parts 𝑠 ± with powers: α = -1; -
0.75; -0.5; -0.25; 0; 0.25; 0.5; 0.75; 1 by the integer order TF parts.
Using the developed program, we found approximating TF of differentiation and integrating
fractional order parts in the frequency range 0.01-100 s-1 for different orders of approximation N.
Some of them are shown in Table 1.
Logarithmic frequency characteristics were constructed for the TF approximation expressions
obtained in this way.
As shown in [14], the frequency characteristics obtained for the corresponding components of
the integer order controller (α = ± 1.0) based on the Oustaloup A. approximation completely coincide
with the well-known results for integer TFs. This confirms the correctness of using the Oustaloup A.
approximation for the partial case of PIλDμ controllers when λ and μ are integers. In addition, it is shown
that the value of N does not affect these frequency characteristics.
In the case of fractional differential components of the controllers, the logarithmic frequency
characteristics are shown in Figure 1a and Figure 1b.
Figure 1a and Figure 1b show the Bode diagrams of differential (α = 1.0) and integral (α = -1.0)
parts when N = 1 (curve 1), N = 2 (curve 2), N = 3 (curve 3), N = 4 (curve 4).
10
Table 1
Approximating integer order TF obtained using the Oustaloup transformation for N=1 and N=2
α N W(s)
-1.0 N=1 0.01𝑠 + 1.049𝑠 + 4.867𝑠 + 1
𝑠 + 4.867𝑠 + 1.049𝑠 + 0.01
N=2 0.01𝑠 + 1.188𝑠 + 19.31𝑠 + 48.49𝑠 + 18.83𝑠 + 1
𝑠 + 18.83𝑠 + 48.49𝑠 + 19.31𝑠 + 1.188 + 0.01
-0.75 N=1 0.03162𝑠 + 2.259𝑠 + 7.144𝑠 + 1
𝑠 + 7.144𝑠 + 2.259𝑠 + 0.03162
N=2 0.03162𝑠 + 2.985𝑠 + 38.52𝑠 + 76.85𝑠 + 23.71𝑠 + 1
𝑠 + 23.71𝑠 + 76.85𝑠 + 38.52𝑠 + 2.985 + 0.03162
-0.5 N=1 0.1𝑠 + 4.867𝑠 + 10.49𝑠 + 1
𝑠 + 10.49𝑠 + 4.867𝑠 + 0.1
N=2 0.1𝑠 + 7.497𝑠 + 76.85𝑠 + 121.8𝑠 + 29.85𝑠 + 1
𝑠 + 29.85𝑠 + 121.8𝑠 + 76.85𝑠 + 7.497 + 0.1
-0.25 N=1 0.3162𝑠 + 10.49𝑠 + 15.39𝑠 + 1
𝑠 + 15.39𝑠 + 10.49𝑠 + 0.3162
N=2 0.3162𝑠 + 18.83𝑠 + 153.3𝑠 + 193.1𝑠 + 37.57𝑠 + 1
𝑠 + 37.57𝑠 + 193.1𝑠 + 153.3𝑠 + 18.83 + 0.3162
0 N=1 𝑠 + 22.59𝑠 + 22.59𝑠 + 1
𝑠 + 22.59𝑠 + 22.59𝑠 + 1
N=2 𝑠 + 47.3𝑠 + 306𝑠 + 306𝑠 + 47.3𝑠 + 1
𝑠 + 47.3𝑠 + 306𝑠 + 306𝑠 + 47.3𝑠 + 1
0.25 N=1 3.162𝑠 + 48.67𝑠 + 33.16𝑠 + 1
𝑠 + 33.16𝑠 + 48.67𝑠 + 3.162
N=2 3.162𝑠 + 118.8𝑠 + 610.5𝑠 + 484.9𝑠 + 59.55𝑠 + 1
𝑠 + 59.55𝑠 + 484.9𝑠 + 610.5𝑠 + 118.8 + 3.162
0.5 N=1 10𝑠 + 104.9𝑠 + 48.67𝑠 + 1
𝑠 + 48.67𝑠 + 104.9𝑠 + 10
N=2 10𝑠 + 298.5𝑠 + 1218𝑠 + 768.5𝑠 + 74.97𝑠 + 1
𝑠 + 74.97𝑠 + 768.5𝑠 + 1218𝑠 + 298.5 + 10
0.75 N=1 31.62𝑠 + 225.9𝑠 + 71.44𝑠 + 1
𝑠 + 71.44𝑠 + 225.9𝑠 + 31.62
N=2 31.62𝑠 + 749.7𝑠 + 2430𝑠 + 1218𝑠 + 94.38𝑠 + 1
𝑠 + 94.38𝑠 + 1218𝑠 + 2430𝑠 + 749.7𝑠 + 31.62
1.0 N=1 100𝑠 + 486.7𝑠 + 104.9𝑠 + 1
𝑠 + 104.9𝑠 + 486.7𝑠 + 100
N=2 100𝑠 + 1883𝑠 + 4849𝑠 + 1931𝑠 + 118.8𝑠 + 1
𝑠 + 118.8𝑠 + 1931𝑠 + 4849𝑠 + 1883𝑠 + 100
As can be seen from the above characteristics, the phase of each of the parts approaches the value
of 𝜑 = ±𝛼𝜋/2. This is the result obtained analytically for fractional differential parts.
Analysing the obtained results, we can say that the accuracy of the approximation depends on the
value of N, and already at N = 4 the curves almost coincide with the real frequency characteristics of
the respective parts.
Similar results were obtained for other fractional values of α.
Thus, applying the Oustaloup approximation to fractional order controllers of ACS under the
condition N≥4, the approximating TF is described by polynomials P(s) and Q(s) not lower than the 9th
order (n≥9). Reducing the value of N simplifies the expression of the approximating TF and facilitates
its practical implementation. Given the need to ensure the highest adequacy of the approximation, it
seems that there is, at first glance, almost impossible to implement controllers with such high order TF.
11
a b
Figure 1: Bode diagrams of fractional differential (a) and integral (b) (α = 0.75) parts
This problem can be solved by applying to the expression TF (7) the theorem on the
decomposition of a rational fraction into elementary ones. Then expression (6), and hence (7), can
be represented as follows:
1 1 1 (8)
𝑊(𝑠) = 𝐴 +𝐴 + ⋯+ 𝐴 .
𝑠−𝜔 𝑠−𝜔 𝑠−𝜔
The coefficients 𝐴 , 𝐴 … 𝐴 according to the theorem are by expression.
𝑃(𝑠)
𝐴 = 𝑓𝑜𝑟 𝑠 = 𝜔 .
𝑄′(𝑠)
Thus, the integral and differential fractional order parts on the basis of (8) can be represented by a
block diagram, which is shown in Figure 2. In this form, the TF expressions of the components of the
fractional order controller can be easily implemented in any software environment (C, C#, C ++,
Assembler, etc.), or in analog.
Below, as an example, are approximating expressions of TF obtained by applying the Oustaloup
transformation with N = 2 with respect to the differential and integrating fractional order parts with TF
𝑊(𝑠) = 𝑠 ± . [14].
1. Integral fractional order parts 𝑊(𝑠) = 𝑠 .
0.1𝑠 + 7.497𝑠 + 76.85𝑠 + 121.8𝑠 + 29.85𝑠 + 1
𝑠 . = ⇒
𝑠 + 29.85𝑠 + 121.8𝑠 + 76.85𝑠 + 7.497 + 0.1
0.1082 0.1942 0.4678 1.1501 2.5922
⇒ 𝑊(𝑠) = 0.1 + + + + + .
𝑠 + 0.0158 𝑠 + 0.1 𝑠 + 0.6310 𝑠 + 3.9811 𝑠 + 25.1189
2. Differential fractional order parts 𝑊(𝑠) = 𝑠 .
10𝑠 + 298.5𝑠 + 1218𝑠 + 768.5𝑠 + 74.97𝑠 + 1
𝑠 . = ⇒
𝑠 + 74.97𝑠 + 768.5𝑠 + 1218𝑠 + 298.5 + 10
0.0041 0.0726 1.1750 19.4241 430.573
⇒ 𝑊(𝑠) = 10 + + + + + .
𝑠 + 0.0398 𝑠 + 0.2512 𝑠 + 1.5849 𝑠 + 10 𝑠 + 63.0957
Figure 2: Block diagram of the integral or differential fractional order parts in the integer order TF form
12
To implement practical programming of a fractional PIλDμ controller on a microcontroller or
signal processor, a requirement is set that it is not possible to use ready-built functions already built into
MATLAB or another programming language, for example, step, etc. That is, the solution must:
be as simple as possible;
provide a minimum of computational operations (maximum performance of the processor);
provide high accuracy;
provide the ability to choose the calculation step in a wide range;
do not limit the time range of the calculation.
Based on a pre-designed and debugged program in MATLAB, the fractional order controller at
the stage of development and improvement of the algorithm is implemented using the C programming
language and Arduino Mega 2560 and Arduino DUE due to the possibility of such boards working with
a computer. The Arduino Mega 2560 board is built using the Atmel ATMega2560 microcontroller and
has the following main technical characteristics: operating voltage - 5V; clock frequency 16 MHz. The
Arduino DUE board is built using the Atmel ATSAM3X8E ARM microcontroller and its main
differences from the Arduino Mega 2560 board are a higher clock frequency of 84 MHz and the
presence of two 12-bit DACs, i.e. analog outputs.
Physical implementation of fractional order controller is possible in two ways.
The first method involves the possibility of implementing the integrated and differential parts of
the fractional order controller using the complexes "computer - board Arduino Mega 2560" and
"computer - board Arduino DUE". In this case, all calculations are performed by the computer, and
synchronized with the external board "I/O" inputs a hopping input signal x = 1V to the input of the
controller and output the calculated signal "y" to the specified board terminals, which are the output
controller voltage .
The second method involves the possibility of implementing the integrated and differential parts
of the fractional order controller by adapting to the software environment Arduino (programming
language C). After debugging the programs, they were written to the memory of the Arduino Mega
2560 and Arduino DUE, respectively. At that time, their research was conducted autonomous without
the use of computer calculations. In this case, the computer was used only to power the board and to
register the transition processes (output of the calculated analog output signal according to the specified
conversion option). If you provide another power source (battery or original power supply) and
recording devices, you may not use the computer.
In Figure 3a shows the results of the study of transition processes (functions) of integral fractional
order parts with TF 𝑊 (𝑠) = 𝑠 . − curve 1, 𝑊 (𝑠) = 𝑠 . − curve 2, 𝑊 (𝑠) = 𝑠 . − curve 3,
𝑊 (𝑠) = 𝑠 . − curve 4 and 𝑊 (𝑠) = 𝑠 . − curve 5, and Figure 3b − differential fractional units with
TF 𝑊 (𝑠) = 𝑠 . curve 1, 𝑊 (𝑠) = 𝑠 . − curve 2, 𝑊 (𝑠) = 𝑠 . − curve 3, 𝑊 (𝑠) = 𝑠 . − curve 4 and
𝑊 (𝑠) = 𝑠 . − curve 5. These dynamic processes are obtained under the condition of autonomous
operation of the external board Arduino DUE, programmed according to the application of the
Oustaloup transformation. The points of the transition functions of the parts are obtained in the
calculation cycle at the output of the board before writing to the output port.
a b
Figure 3: Transition functions of integral (a) and differential (b) fractional order parts with changes α
within 0.1÷1.0 for the order of approximation N=2
13
Based on a pre-designed and debugged program in the MATLAB environment [14] implemented
fractional order PIλDμ controller using the programming language C and boards Arduino Mega 2560
and Arduino DUE that can work with a computer.
Each of these methods of implementing PIλDμ controller has disadvantages and advantages. There
are advantages to using the MATLAB software environment on a computer with an Arduino Mega
2560 motherboard connected, because in this case the programming language is at a higher level and
provides easy and convenient programming and debugging of the controller. Disadvantages of this
approach include the effect of computer load on the speed of calculations, which sometimes leads to a
twofold increase in calculation time. In addition, the exchange rate between the computer and the board
is limited to 115,500 baud, which also significantly affects the signal delay at the output of the
controller. It should be noted that this version of the implementation of the fractional controller is
appropriate for the use of computer control of the frequency converter.
The use of Arduino Mega 2560 and Arduino DUE boards autonomous using the proposed method
of calculating the instantaneous value of the output voltage of the controller has shown its effectiveness.
It consists in the fact that it is possible to provide a sampling period of calculations at the level of
0.0025s, ie a significant increase in the speed of obtaining the signal of the controller. In addition, there
is the possibility of long-term operation of fractional controllers in stand-alone mode compared to the
option when using a computer. The speed of information exchange between the computer and the
Arduino DUE board in the mode when the computer needs to control it in this case increases
significantly and is 250,000 baud.
Appropriate software has been developed that implements the digital PIλDμ fractional order
controller. As an example, consider a TF of fractional order controller (the question of synthesis of
PIλDμ controller is not considered in this paper), namely:
1 (9)
𝑊 (𝑠) = 3 + + 1.0𝑠 . .
1.0𝑠 .
Experimental studies of fractional order controllers were performed, in particular, according to
expression (9). To build them, we used [15] the frequency converter board MFC1000/10 induction
electric drive. Figure 4a shows the transition process of the of integer order PIλDμ controller, and on
Figure 4b – fractional order (λ = 0.5, μ = 0.5).
a b
λ μ
Figure 4: Transition process of the integer order PI D controller (λ = 1, μ = 1), implemented using the
converter board MFC1000/10 (a) and transition process of the fractional order PIλDμ controller (λ =
0.5, μ = 0.5) implemented using the converter board MFC1000/10 (b)
Of course, of considerable interest are the possibility of implementing fractional controllers in
the ACS. For this purpose, a PIλDμ controller was used as a part of the system “frequency converter -
induction motor” (FC-IM).
Figure 5a shows the transition speed, which corresponds to TF (9).
The green graph corresponds to the signal at the output of the PIλDμ controller, the red graph
corresponds to the signal at the output of the speed sensor, and the blue colour indicates the set speed.
All curves have the appropriate scaling.
The oscillogram clearly shows the effect of fractional Iλ – component on the speed of the FC -
IM system.
14
It is of interest to implement a fractional order controller, if the result of its synthesis is a
fractional and integer component of the TF controller. It turned out that it is possible to implement such
a setting. Figure 5b shows the oscillogram of the dynamic processes of the ACS with such a controller.
Obviously, in such a system it is possible to provide dynamic processes based on the results of
the corresponding synthesis of ACS.
In the above studies, PIλDμ controllers were considered, in which the fractional order is in the
range from 0 to 1. It may be necessary to implement fractional order controllers, where this condition
is not met. Therefore, experimental studies were conducted for this case.
In Figure 5c shows the transition process of the speed, which corresponds to the TF of the
controller with Іλ – component when λ = 1.5 and Dμ – component when μ = 0.5. The oscillogram
demonstrates the possibility of operation of the developed PIλDμ controller if λ> 1.
a b
c
Figure 5: Transition process of speed in the system FC-IM with PIλDμ controller 𝑊 (𝑠) = 3 + 3𝑠 . +
1.0𝑠 . (𝑘 = 3, 𝑘 = 3, 𝑘 = 1, λ = -0,5, μ = 0,5) (a), 𝑊 (𝑠) = 3 + 5𝑠 . + 1.0𝑠 . (𝑘 = 3, 𝑘 = 5, 𝑘 =
1, λ = -1, μ = 0,5) (b), controller 𝑊 (𝑠) = 3 + 3𝑠 . + 1.0𝑠 . (𝑘 = 3, 𝑘 = 1/𝑇 = 3, 𝑇 = 1s, λ = -1,5,
μ = 0,5) (c)
The speed of such ACS decreases with a simultaneous increase in the amount of overshooting. These
parameters of dynamic processes can change due to the implementation of other criteria for the
synthesis of the system.
4. Conclusions
Computer research of algorithm and programs realization of differentiating and integrating
fractional order parts, as components of PIλDμ controllers has shown efficiency of application of
approximation of fractional order TF.
The application of the decomposition theorem of rational fractions allowed to construct structural
schemes from parallel connected aperiodic units for the realization of approximated arbitrary order TFs.
It is experimentally proved that the implementation of fractional order controllers based on
approximated TFs can work in real time as a part of highly dynamic ACS. Tests of the FC MFC 710
option with the PIλDμ - fractional order controller in the speed control system using the Twerd
experimental stand have confirmed its efficiency in terms of expanding the regulatory capabilities of
such ACS.
15
5. References
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