Research Article
BibTex RIS Cite

Commutativity Conditions of Lame’s Differential Equation

Year 2020, Ejosat Special Issue 2020 (HORA), 211 - 214, 15.08.2020
https://doi.org/10.31590/ejosat.779704

Abstract

The realization of many engineering systems consists of cascade connection of systems of simple orders, which is very important in design of electrical and electronic systems. Although the order of connection of the systems mainly depends on the special design approach, engineering ingenuity, traditional synthetic methods, when the sensitivity, stability, linearity, noise disturbance, robustness effects are considered the change of the order of connection without changing the main function of the total systems (commutativity) may lead positive results. Therefore, the commutativity is very important from the practical point of view. In this study, commutativity conditions of one type of Lame’s differential equations are considered. In the sense of theoretical results for the commutativity of second-order continuous-time linear time-varying systems, it is proved that the system modeled by a Lame’s differential equation has commutative pairs depending on the parameters of the equation. Commutative conjugate of the system modeled by a Lame’s differential equation is constructed. To support the theoretical results, an illustrative example is considered for application. For the illustration, Simulink toolbox of MATLAB 2019b is used. Ode5 (Dormant-Prince) is used as the solver with a fixed step-length. Numerical results are presented. It is observed that the responses computed for x∈[0,120] are identical, which proves the validity of the commutativity results under zero initial conditions. The validity of commutativity with arbitrary initial conditions is also tested. It is observed that the commutativity is spoiled for arbitrarily chosen initial conditions which are not chosen appropriately. Theory of commutativity of the system modeled by Lame’s differential equation with non-zero initial conditions can be conducted in future work using the general formulas in (Koksal, 2019b).

Supporting Institution

The Scientific and Technological Research Council of Turkey

Project Number

115E952

Thanks

This work was supported by the Scientific and Technological Research Council of Turkey under the project no. 115E952.

References

  • Koksal, M. (1982). Commutativity of second order time-varying systems. International Journal of Control. 3, 541-44.
  • Koksal, M. (1985a). A survey on the commutativity of time-varying systems. METU, Technical Report. no: GEEE CAS-85/1.
  • Koksal, M. (1985b). Commutativity of 4th order systems and Euler systems. Presented in National Congress of Electrical Engineers. Paper no: BI-6, Adana, Turkey.
  • Koksal, M. and Koksal, M. E. (2011). Commutativity of linear time-varying differential systems with non-zero initial conditions: A review and some new extensions. Mathematical Problems in Engineering. 2011, 1-25.
  • Koksal, M. E. (2018a). Commutativity and commutative pairs of some well-known differential equations. Communications in Mathematics and Applications. 9 (4), 689-703.
  • Koksal, M. E. (2018b). Commutativity conditions of some time-varying systems. International Conference on Mathematics: “An Istanbul Meeting for World Mathematicians”. 3-6 Jul 2018, Istanbul, Turkey, pp. 109-117.
  • Koksal, M. E. (2019a). Commutativity of systems with their feedback conjugates. Transactions of the Institute of Measurement and Control. 41 (3), 696-700.
  • Koksal, M. E. (2019b). Explicit commutativity conditions for second order linear time-varying systems with non-zero initial conditions. Archives of Control Sciences. 29 (3), 413-432.
  • Marshall, J. E. (1977). Commutativity of time varying systems. Electro Letters. 18, 539-40.
  • Zwillinger, D. (1997). Handbook of Differential Equations. 3rd ed. Boston, MA: Academic Press, p. 127.

Lame Diferansiyel Denkleminin Komütativite Koşulları

Year 2020, Ejosat Special Issue 2020 (HORA), 211 - 214, 15.08.2020
https://doi.org/10.31590/ejosat.779704

Abstract

Çoğu mühendislik sistemlerinin gerçekleştirilmesi, daha basit sistemlerin ardışık bağlantıları ile yapılmaktadır. Bu durum elektrik ve elektronik sistemlerinin tasarımında çok önemlidir. Her ne kadar bu alt sistemlerin bağlantı sırası, kullanılan özel tasarım yöntemlerine, mühendislik tecrübesine, alışılagelmiş sentez yöntemlerine bağlı olmakla beraber, hassasiyet, kararlılık, doğrusallık, gürültüden etkilenme ve dayanıklılık hususları göz önüne alındığında toplam sistemin ana fonksiyonunu değiştirmeden alt sistemlerin bağlantı sırasının değiştirilmesi (komütativite) pozitif sonuçlara yol açabilmektedir. Bu nedenle pratik uygulamalar açısından komütativite çok önemlidir. Bu çalışmada, Lame diferansiyel denkleminin bir türünün komütativite koşulları incelenmiştir. İkinci dereceden doğrusal zamanla değişen sürekli-zaman sistemlerinin komütativitesi için teorik sonuçlar ışığında Lame'nin diferansiyel denklemi ile modellenen sistemin denklemin parametrelerine bağlı olarak komütatif çiftleri olduğu kanıtlanmıştır. Lame diferansiyel denklemi ile modellenen sistemin komütatif eşleniği oluşturulmuştur. Teorik sonuçları desteklemek için açıklayıcı bir örnek ele alınmıştır. Örnek’te MATLAB 2019b'nin Simulink araç kutusu kullanılmıştır. Sabit adım uzunluğuna sahip çözücü olarak Ode45 kullanılmıştır. Sayısal sonuçlar sunulmuştur. x∈[0,120] için hesaplanan cevapların özdeş olduğu, bunun da sıfır başlangıç koşulları altında komütativite sonuçlarının geçerliliğini kanıtladığı görülmüştür. Rasgele seçilen başlangıç koşulları ile komütativitenin geçerli olup olmadığı da test edilmiştir. Komütativitenin uygun şekilde seçilmeyen, keyfi olarak seçilen başlangıç koşulları için bozulduğu gözlenmiştir. Lame diferansiyel denklemi ile modellenen sistemlerin sıfır olmayan başlangıç koşulları ile komütativite teorisi, (Koksal, 2019b) genel formülleri kullanılarak ilerili çalışmalarda incelenebilir.

Project Number

115E952

References

  • Koksal, M. (1982). Commutativity of second order time-varying systems. International Journal of Control. 3, 541-44.
  • Koksal, M. (1985a). A survey on the commutativity of time-varying systems. METU, Technical Report. no: GEEE CAS-85/1.
  • Koksal, M. (1985b). Commutativity of 4th order systems and Euler systems. Presented in National Congress of Electrical Engineers. Paper no: BI-6, Adana, Turkey.
  • Koksal, M. and Koksal, M. E. (2011). Commutativity of linear time-varying differential systems with non-zero initial conditions: A review and some new extensions. Mathematical Problems in Engineering. 2011, 1-25.
  • Koksal, M. E. (2018a). Commutativity and commutative pairs of some well-known differential equations. Communications in Mathematics and Applications. 9 (4), 689-703.
  • Koksal, M. E. (2018b). Commutativity conditions of some time-varying systems. International Conference on Mathematics: “An Istanbul Meeting for World Mathematicians”. 3-6 Jul 2018, Istanbul, Turkey, pp. 109-117.
  • Koksal, M. E. (2019a). Commutativity of systems with their feedback conjugates. Transactions of the Institute of Measurement and Control. 41 (3), 696-700.
  • Koksal, M. E. (2019b). Explicit commutativity conditions for second order linear time-varying systems with non-zero initial conditions. Archives of Control Sciences. 29 (3), 413-432.
  • Marshall, J. E. (1977). Commutativity of time varying systems. Electro Letters. 18, 539-40.
  • Zwillinger, D. (1997). Handbook of Differential Equations. 3rd ed. Boston, MA: Academic Press, p. 127.
There are 10 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Mehmet Emir Köksal This is me 0000-0001-7049-3398

Project Number 115E952
Publication Date August 15, 2020
Published in Issue Year 2020 Ejosat Special Issue 2020 (HORA)

Cite

APA Köksal, M. E. (2020). Commutativity Conditions of Lame’s Differential Equation. Avrupa Bilim Ve Teknoloji Dergisi211-214. https://doi.org/10.31590/ejosat.779704