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. Fractional order, PIλDμ controller, Oustaloup transformation, electromechanical systems
+ + .
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 [
addition to the values of proportional, differential and integral components , and , the fractional order controller has two more parameters: fractional powers λ and μ of the Laplace operator s in the integrator and differentiator, respectively.
1 Γ(α) ( ) = ( − ) ( ) , ITEA-2021: 1st Workshop of the 10th International scientific and practical conference Information technologies in energy and agro-industrial and a differential fractional order controller in the Riemann-Liouville representation ( ) =
1 Γ(n − α) ( )
( )
controllers actually makes it impossible to use them in high-speed systems with long-term operating
conditions. Such shortcomings have been partially eliminated in [
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
[
Let us denote п =
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 ± = 1 + ⁄ 1 + ⁄
, ( ( , , , ⋅ )/( , ⋅ )/( ) , ) , …
calculated according to (4) the values of the zeros of the integer order TF; − calculated according to (5) the poles of the integer order TF.
We present TF (6) as the ratio of polynomials ( ) = + + + ⋯ + + + ⋯ + + =
can be interpreted as the expression of fractional differentiator (+α) or was developed that implements the Oustaloup method according to (3) to approximate ± by the integer order TF.
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.
with powers: α = -1; 0.75; -0.5; -0.25; 0; 0.25; 0.5; 0.75; 1 by the integer order TF parts.
fractional order parts in the frequency range 0.01-100 s-1 for different orders of approximation N.
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.
/2. This is the result obtained analytically for fractional differential parts.
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. .
= .
= ⇒ ( ) = 0.1 + ⇒ ( ) = 10 + ( ) =
1 − = + .
(8)
… according to the theorem are by expression.
( ) = ± . [
Integral fractional order parts ( ) = . 0.1 Differential fractional order parts ( ) = .
signal processor, a requirement is set that it is not possible to use ready-built functions already built into
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.
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.
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.
order parts with TF ( ) = . − curve 1, ( ) = . − curve 2, ( ) = . − curve 3, ( ) = . − curve 4 and ( ) = . − curve 5, and Figure 3b − differential fractional units with TF ( ) = . curve 1, ( ) = . − curve 2, ( ) = . − curve 3, ( ) = . − curve 4 and ( ) = . − 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
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.
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.
controller. As an example, consider a TF of fractional order controller (the question of synthesis of
( ) = 3 + 1 1.0 . + 1.0 . . (9)
expression (9). To build them, we used [
the ACS. For this purpose, a PIλDμ controller was used as a part of the system “frequency converter induction motor” (FC-IM).
corresponds to the signal at the output of the speed sensor, and the blue colour indicates the set speed.
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.
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.
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. c Figure 5: Transition process of speed in the system FC-IM with PIλDμ controller ( ) = 3 + 3 . + 1.0 . ( = 3, = 3, = 1, λ = -0,5, μ = 0,5) (a), ( ) = 3 + 5 . + 1.0 . ( = 3, = 5, = 1, λ = -1, μ = 0,5) (b), controller ( ) = 3 + 3 . + 1.0 . ( = 3, = 1/ = 3, = 1s, λ = -1,5, μ = 0,5) (c)
fractional order parts, as components of PIλDμ controllers has shown efficiency of application of approximation of fractional order TF.
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.