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Commutativity Conditions of Heine’s Differential Equation

Year 2020, Ejosat Special Issue 2020 (HORA), 239 - 243, 15.08.2020
https://doi.org/10.31590/ejosat.779727

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 system (commutativity) may lead positive results. Therefore, the commutativity is very important from the practical point of view. In this paper, commutativity of Heine differential equation is considered. It is shown that the system modeled by a Heine differential equation has a commutative pair or not depending on the parameters of the equation. The conditions that must be satisfied for the commutativity are set in this contribution. An example is considered for application. The initial conditions are assumed to be zero in this study. For the nonzero initial conditions (or for unrelaxed systems), some further relations should hold in addition to the commutativity conditions for relaxed Heine differential systems derived in this paper. The case of nonzero initial conditions do not gain original results on the paper since they have already been studied in the literature (Koksal, 2019) for general second-order continuous-time linear time-varying systems. The commutative pairs of Heine differential equation are obtained for some parameters satisfying the existence conditions of commutativity and it is seen that they are not of Heine type. So, the commutative pairs of Heine differential systems are not Heine type for the general choice of arbitrary constants relating the commutative pairs.

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. (2019). Explicit commutativity conditions for second order linear time-varying systems with non-zero ınitial 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.

Heine Diferansiyel Denkleminin Komütativite Koşulları

Year 2020, Ejosat Special Issue 2020 (HORA), 239 - 243, 15.08.2020
https://doi.org/10.31590/ejosat.779727

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, Heine diferansiyel denkleminin komütativitesi ele alınmıştır. Bir Heine diferansiyel denklemi ile modellenen sistemin, denklemin parametrelerine bağlı olarak komütatif bir çifti olduğu ya da olmadığı gösterilmiştir. Komütatif olabilmesi için gerekli ve yeterli koşullar belirtilmiştir. Uygulama için bir örnek ele alınmıştır. Bu çalışmada başlangıç koşullarının sıfır olduğu varsayılmıştır. Sıfır olmayan başlangıç koşulları (veya geri beslemeli sistemler) için, bu çalışmada türetilen Heine diferansiyel sistemleri için komütativite koşullarına ek olarak bazı ilave koşullar gereklidir. Sıfır olmayan başlangıç koşulları durumu, genel ikinci dereceden sürekli zamanla değişen doğrusal sistemler için literatürde (Koksal, 2019) daha önce çalışılmış olduklarından bu makalede bunun üzerinde orijinal sonuçlar elde etmemektedir. Komütativite varlığının koşullarını sağlayan bazı parametreler için komütatif Heine diferansiyel denklem çiftleri elde edilmiştir ve bunların Heine tipi olmadığı görülmüştür. Bu nedenle, Heine diferansiyel sistemlerinin komütatif çiftleri, komütatif çiftlerle ilgili keyfi sabitlerin genel seçimi için Heine tipi değildir.

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. (2019). Explicit commutativity conditions for second order linear time-varying systems with non-zero ınitial 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 9 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 Heine’s Differential Equation. Avrupa Bilim Ve Teknoloji Dergisi239-243. https://doi.org/10.31590/ejosat.779727