Volume 7, Issue 3, June 2019, Page: 84-91
Synthesis of Hurwitz Polynomial Families Using Root Locus Portraits
Alla Anatolyevna Nesenchuk, Department of the Digital Transformation Technique, United Institute of Informatics Problems of the Belarusian National Academy of Sciences, Minsk, Belarus; Faculty of Information Technologies and Robotics, Belarusian National Technical University, Minsk, Belarus
Received: Aug. 10, 2019;       Accepted: Sep. 16, 2019;       Published: Sep. 27, 2019
DOI: 10.11648/j.acis.20190703.12      View  87      Downloads  17
The paper deals with the problem of synthesis of a stable characteristic polynomial families describing the control systems' dynamics in conditions of the interval uncertainty. Investigation is based on the system mathematical model in the form of its root locus portrait generated by the polynomial free term variation that is named in the paper as the "free root locus portrait". The root loci of the Kharitonov's polynomials family (subfamily) is picked out of the whole polynomial family and is considered for carrying out the investigation. Specific regularities of the interval root locus portrait have been discovered. On the basis of these regularities main properties of the system root locus portrait have been defined. A stability condition has been formulated that allows to calculate the polynomial free term variation interval ensuring the polynomial family hurwitz stability. This stability condition is applicable to the class of polynomials having their free root locus poles lying within the left half-plane of roots or, in other words, being stable when their free term is equal to zero. The stable family is being synthesized by setting up (adjusting) the given initial family that is supposed to be unstable, i.e. the proposed method of synthesis allows to turn stable (hurwitz) the given nonhurwitz interval polynomial family. The setting up criterion is specified in terms of proximity i.e. as the nearest distance from the "unstable" system roots to the "stable" ones as measured along the root trajectories. The stable polynomial could be selected as the nearest to the given unstable one with or without consideration of the system quality requirements. In the course of the setting up procedure new boundaries of only the polynomial free term variation interval (stability interval) are calculated that allows to ensure system stability without modification of its root locus portrait configuration. A numerical example of the polynomial setting up procedure has been given.
Unstable Control System, Parametric Uncertainty, Interval Characteristic Polynomial, Parametric Synthesis, Root Locus, Kharitonov's Polynomials
To cite this article
Alla Anatolyevna Nesenchuk, Synthesis of Hurwitz Polynomial Families Using Root Locus Portraits, Automation, Control and Intelligent Systems. Vol. 7, No. 3, 2019, pp. 84-91. doi: 10.11648/j.acis.20190703.12
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
R. C. Dorf and R. H. Bishop, Modern Control Systems, 12th ed. New York: Prentice Hall, 2011, 1084.
R. Tempo, C. Calafiori, and F. Dabbene, Randomized Algorithms for Analysis and Control of Uncertain Systems With Applications. London: Springer-Verlag, 2013, 357.
V. Kučera, "Polynomial control: past, present, and future," International Journal of Robust and Nonlinear Control, vol. 17, no. 8, pp. 682–705, May 2007.
B. T. Polyak and P. S. Scherbakov, Robust Stability and Control [in Russian]. Moscow: Nauka, 2002, 303.
B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, Linear Systems Control in Presence of External Perturbations [in Russian]. Moscow: LENAND, 2014, 560.
V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations," Differential Equations, vol. 14, pp. 1483–1485, 1979.
V. Dikusar, G. Zelenkov, and N. Zubov, "Criteria of existence of homogeneous classes of equivalence for unstable interval polynomials" [in Russian], Doklady AN, Control theory, vol. 429, pp. 322-324, 2009.
B. T. Polyak, Y. Z. Tsypkin, "Frequency criteria of robust stability and aperiodicity of linear systems,"Automation and Remote Control, vol. 51, no. 9 pt 1, pp. 1192–1201, September 1991.
B. R. Barmish, "Invariance of the strict hurwitz property for polynomials with perturbed coefficients," IEEE Trans Automat Control, vol. 29, pp. 935–936, October 1984.
Y. Soh, "Strict hurwitz property of polynomials under coefficient perturbations," IEEE Trans Automat Control, vol. 34, pp. 629–632, June 1989.
A. Rantzer, "Stability conditions for polytopes of polynomials," IEEE Trans Automat Control, vol. 37, pp. 79–89, Janiary 1992.
X. Li, H. Yu, M. Yuan, and J. Wang, "Design of robust optimal proportional-integral-derivative controller based on new interval polynomial stability criterion and Lyapunov theorem in the multiple parameters' perturbations circumstance," IET Control Theory & Applications, vol. 4, pp. 2427–2440, November 2010.
E. N. Gryazina, B. T. Polyak, "Stability regions in the parameter space: D-decomposition revisited," Automatica, vol. 42, pp. 13–26, January 2006.
A. V. Kraev, А. S. Fursov, "Estimating the instability radii for polynomials of the arbitrary power" [in Russian], in Nonlinear Dynamics and Control, issue 4, S. V. Emelyanov and S. K. Korovin, Eds. Moscow: FIZMATLIT, 2006, pp 127–134.
B. R. Barmish, R. Tempo, "The robust root locus," Automatica, vol. 26, pp. 283–292, February 1990.
A. A. Nesenchuk,"Analysis and Synthesis of Robust Dynamic Systems on the Basis of the Root Locus Approach" [in Russian], Minsk: United Institute of Informatics Problems of the Belarusian National Academy of Sciences, 2005, 234.
A. A. Nesenchuk, "Parametric synthesis of qualitative robust control systems using root locus fields," in Proceedings of the 15th Triennial World Congress of IFAC, Robust Control, vol. E, F. Camacho, L. Basanez, and J. A. de la Puente, Eds. Oxford: Elsevier Science, 2003, pp. 331–335.
A. A. Nesenchuk, "The root locus method of synthesis of stable polynomials by adjustment of all coefficients," Automation and Remote Control, vol. 71, pp. 1515–1525, August 2010.
А. А. Nesenchuk, "Investigation of behavior and synthesis of interval dynamic systems' characteristic polynomials based on the root locus portrait parameter function" [Electronic resource], Proceedings of the 60th American Control Conference (ACC 2018), Milwaukee, USA, pp. 2041–2046, 2018. Mode of access: http://www.proceedings.com/40219.html, date of access: 08.07.2019.
A. A. Nesenchuk, "Investigation and synthesis of robust polynomials in uncertainty on the basis of the Root Locus Theory," in Polynomials – Theory and Applications, Cheon Seoung Ryoo, Ed. London: Intechopen, 2019, ch. 6, pp. 109–130.
Browse journals by subject