Search results

Filters

  • Journals
  • Authors
  • Keywords
  • Date
  • Type

Search results

Number of results: 1
items per page: 25 50 75
Sort by:
Download PDF Download RIS Download Bibtex

Abstract

It is shown how a stability test, alternative to the classical Routh test, can profitably be applied to check the presence of polynomial roots inside half-planes or even sectors of the complex plane. This result is obtained by exploiting the peculiar symmetries of the root locus in which the basic recursion of the test can be embedded. As is expected, the suggested approach proves useful for testing the stability of fractional-order systems. A pair of examples show how the method operates. It is believed that the suggested geometric approach can also be of some didactic value in introducing basic control-system tools to engineering students.
Go to article

Bibliography

[1] J.J. Anagnost, C.A. Desoer, and R.J. Minnichelli: Graphical stability robustness tests for linear time-invariant systems: Generalizations of Kharitonov’s stability theorem, Proceedings of the 27th IEEE Conference on Decision and Control (1988), 509–514.
[2] A.T. Azar, A.G. Radwan, and S.Vaidyanathan, Eds.: Mathematical Techniques of Fractional Order Systems, Elsevier, Amsterdam, The Netherlands, 2018.
[3] R. Becker, M. Sagraloff. V. Sharma, J. Xu, and C. Yap: Complexity analysis of root clustering for a complex polynomial, Proceedings of the 41th ACM International Symposium on Symbolic and Algebraic Computation, (2016), 71–78.
[4] T.A. Bickart and E.I. Jury: The Schwarz–Christoffel transformation and polynomial root clustering, IFAC Proceedings 11(1), (1978), 1171–1176.
[5] Y. Bistritz: Optimal fraction–free Routh tests for complex and real integer polynomials, IEEE Transactions on Circuits and Systems I: Regular Papers 60(9), (2013), 2453–2464.
[6] D. Casagrande, W. Krajewski, and U. Viaro: On polynomial zero exclusion from an RHP sector, Proceedings of the 23rd IEEE International Conference on Methods and Models in Automation and Robotics, (2018), 648–653.
[7] D. Casagrande, W. Krajewski, and U. Viaro: Fractional-order system forced-response decomposition and its application, In Mathematical Techniques of Fractional Order Systems, A.T. Azar, A.G. Radwan, and S. Vaidyanathan, Eds., Elsevier, Amsterdam, The Netherlands, 2018.
[8] A. Cohn: Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Mathematische Zeitschrift 14, (1922), 110–148, DOI: 10.1007/BF01215894.
[9] Ph. Delsarte and Y. Genin: The split Levinson algorithm, IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP, 34(3), (1986), 470–478.
[10] Ph. Delsarte and Y. Genin: On the splitting of classical algorithms in linear prediction theory, IEEE Transactions onAcoustics, Speech, and Signal Processing ASSP, 35(5), (1987), 645–653.
[11] A. Doria–Cerezo and M. Bodson: Root locus rules for polynomials with complex coefficients, Proceedings of the 21st Mediterranean Conference on Control and Automation, (2013), 663–670.
[12] A. Doria–Cerezo and M. Bodson: Design of controllers for electrical power systems using a complex root locus method, IEEE Transactions on Industrial Electronics, 63(6), (2016), 3706–3716.
[13] A. Ferrante, A. Lepschy, and U. Viaro: A simple proof of the Routh test, IEEE Transactions on Automatic Control, AC-44(1), (1999), 1306–1309.
[14] A. Hurwitz: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematiche Annalen Band, 46 (1895), 273–284.
[15] R. Imbach and V.Y. Pan: Polynomial root clustering and explicit deflation, arXiv:1906.04920v2.
[16] E.I. Jury and J. Blanchard: A stability test for linear discrete systems in table form, I.R.E. Proceedings, 49(12), (1961), 1947–1948.
[17] T. Kaczorek: Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Sciences, 411, Springer, Berlin, Germany, 2011.
[18] W. Krajewski, A. Lepschy, G.A. Mian, and U. Viaro: A unifying frame for stability-test algorithms for continuous-time systems, IEEE Transactions on Circuits and Systems, CAS-37(2), (1990), 290–296.
[19] W. Krajewski, A. Lepschy, G.A. Mian, and U. Viaro: Common setting for some classical z-domain algorithms in linear system theory, International Journal of Systems Science, 21(4), (1990), 739–747.
[20] W. Krajewski and U. Viaro: Root locus invariance: Exploiting alternative arrival and departure points, IEEE Control Systems Magazine, 27(1), (2007), 36–43.
[21] B.C. Kuo: Automatic Control Systems (second ed.), (1967), Prentice-Hall, Englewood Cliffs, NJ, USA.
[22] P.K.Kythe: Handbook of Conformal Mappings and Applications, Chapman and Hall/CRC Press, London, UK, 2019.
[23] A. Lepschy, G.A. Mian, and U. Viaro: A stability test for continuous systems, Systems and Control Letters, 10(3), (1988), 175–179.
[24] A. Lepschy, G.A. Mian, and U. Viaro: A geometrical interpretation of the Routh test, Journal of the Franklin Institute, 325(6), (1988), 695–703.
[25] A. Lepschy, G.A. Mian, and U. Viaro: Euclid-type algorithm and its applications, International Journal of Systems Science, 20(6), (1989), 945– 956.
[26] A. Lepschy, G.A. Mian, andU. Viaro: Splitting of some s-domain stabilitytest algorithms, International Journal of Control, 50(6), (1989), 2237–2247.
[27] A. Lepschy, G.A. Mian, and U. Viaro: An alternative proof of the Jury- Marden stability criterion, Control and Computers, 18(3), (1990), 70–73.
[28] A. Lepschy, G.A. Mian, and U. Viaro: Efficient split algorithms for continuous-time and discrete-time systems, Journal of the Franklin Institute, 328(1), (1991), 103–121.
[29] A. Lepschy and U. Viaro: On the mechanism of recursive stability-test algorithms, International Journal of Control, 58(2), (1993), 485–493.
[30] A. Lepschy and U. Viaro: Derivation of recursive stability-test procedures, Circuits, Systems, and Signal Processing, 13(5), (1994), 615–623.
[31] S. Liang, S.G. Wang, and Y. Wang: Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees, Journal of the Franklin Institute, 354(1), (2017), 83–104.
[32] A. Lienard and M.H. Chipart: Sur le signe de la partie réelle des racines d’une équation algébrique, Journal of Mathématiques Pures et Appliquée, 10(6), (1914), 291–346.
[33] M. Marden: Geometry of Polynomials [2nd ed.], American Mathematical Society, Providence, RI, USA, 1966.
[34] I. Petras: Stability of fractional-order systems with rational orders: a survey, Fractional Calculus & Applied Analysis, 12(3), (2009), 269–298.
[35] A.G. Radwan, A.M. Soliman, A.S. Elwakil, and A. Sedeek: On the stability of linear systems with fractional order elements, Chaos, Solitons and Fractals, 40(5), (2009), 2317–2328.
[36] E.J. Routh: A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion, Macmillan, London, UK, 1877.
[37] J. Schur: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, Journal für die reine und angewandte Mathematik, 147, (1917) 205– 232, DOI: 10.1515/crll.1917.147.205.
[38] R.Tempo: A Simple Test for Schur Stability of a Diamond of Complex Polynomials, Proceedings of the 28th IEEE Confewrence on Decision and Control (1989), 1892–1895.
[39] U. Viaro: Stability tests revisited, In A Tribute to Antonio Lepschy, G. Picci and M.E. Valcher, Eds., Edizioni Libreria Progetto, Padova, Italy, pp. 189– 199, 2007.
[40] U. Viaro: Twenty–Five Years of Research with Antonio Lepschy, Edizioni Libreria Progetto, Padova, Italy, 2009.
[41] U. Viaro (preface by W. Krajewski): Essays on Stability Analysis and Model Reduction, Polish Academy of Sciences, Warsaw, Poland, 2010.
[42] R.S. Vieira: Polynomials with symmetric zeros, arXiv:1904.01940v1 [math.CV], 2019.
Go to article

Authors and Affiliations

Daniele Casagrande
1
Wiesław Krajewski
2
Umberto Viaro
1

  1. Polytechnic Department of Engineering and Architecture, University of Udine, via delle Scienze 206, 33100 Udine, Italy
  2. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01–447 Warsaw, Poland

This page uses 'cookies'. Learn more