Details

Title

The Lepschy stability test and its application to fractional-order systems

Journal title

Archives of Control Sciences

Yearbook

2021

Volume

vol. 31

Issue

No 1

Authors

Affiliation

Casagrande, Daniele : Polytechnic Department of Engineering and Architecture, University of Udine, via delle Scienze 206, 33100 Udine, Italy ; Krajewski, Wiesław : Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01–447 Warsaw, Poland ; Viaro, Umberto : Polytechnic Department of Engineering and Architecture, University of Udine, via delle Scienze 206, 33100 Udine, Italy

Keywords

fractional-order systems ; D-stability ; recursive algorithms ; complex polynomials ; root locus ; symmetries ; control-theory didactics

Divisions of PAS

Nauki Techniczne

Coverage

145-163

Publisher

Committee of Automatic Control and Robotics PAS

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.

Date

2021.03.30

Type

Article

Identifier

DOI: 10.24425/acs.2021.136884

Source

Archives of Control Sciences; 2021; vol. 31; No 1; 145-163
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