Approximation of Fractional Order PI λ D μ -Controller Transfer Function Using Chain Fractions

Yaroslav Marushchak, Damian Mazur, Bogdan Kwiatkowski, Bohdan Kopchak, Tadeusz Kwater and Maciej Koryl
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Yaroslav Marushchak: Institute of Power Engineering and Control Systems, Lviv Polytechnic National University, 12 Bandera Street, 79013 Lviv, Ukraine
Damian Mazur: Department of Electrical and Computer Engineering Fundamentals, Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, 35-959 Rzeszow, Poland
Bogdan Kwiatkowski: Department of Electrical and Computer Engineering Fundamentals, Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, 35-959 Rzeszow, Poland
Bohdan Kopchak: Institute of Power Engineering and Control Systems, Lviv Polytechnic National University, 12 Bandera Street, 79013 Lviv, Ukraine
Tadeusz Kwater: Institute of Technical Engineering, State University of Technology and Economics in Jaroslaw, Czarnieckiego Str. 16, 37-500 Jaroslaw, Poland
Maciej Koryl: Asseco Poland S.A., ul. Olchowa 14, 35-322 Rzeszow, Poland

Energies, 2022, vol. 15, issue 13, 1-12

Abstract: The approximation of a fractional order PI λ D μ -controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of s ± α components of the transfer functions of PI λ D μ -controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of D μ (α = μ = 0.5) and I λ (α = λ = −0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PI λ D μ -controller transfer function by the method of chain fractions with different valuesof λ and μ. For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for α = ±0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the PI λ D μ -controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of PI λ D μ -controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.

Keywords: PI ? D ? -controller; fractional order transfer function; chain fraction; approximation; dynamic characteristics (search for similar items in EconPapers)
JEL-codes: Q Q0 Q4 Q40 Q41 Q42 Q43 Q47 Q48 Q49 (search for similar items in EconPapers)
Date: 2022
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