Kybernetika 52 no. 3, 461-477, 2016

On stable cones of polynomials via reduced Routh parameters

Ülo Nurges, Juri Belikov and Igor ArtemchukDOI: 10.14736/kyb-2016-3-0461


A problem of inner convex approximation of a stability domain for continuous-time linear systems is addressed in the paper. A constructive procedure for generating stable cones in the polynomial coefficient space is explained. The main idea is based on a construction of so-called Routh stable line segments (half-lines) starting from a given stable point. These lines (Routh rays) represent edges of the corresponding Routh subcones that form (possibly after truncation) a polyhedral (truncated) Routh cone. An algorithm for approximating a stability domain by the Routh cone is presented.


linear systems, Hurwitz stability, convex approximation


93C05, 93D09


  1. J. Ackermann and D. Kaesbauer: Stable polyhedra in parameter space. Automatica 39 (2003), 937-943.   DOI:10.1016/s0005-1098(03)00034-7
  2. I. Artemchuk, Ü. Nurges and J. Belikov: Robust pole assignment via Routh rays of polynomials. In: American Control Conference, Boston 2016, pp. 7031-7036.   CrossRef
  3. I. Artemchuk, Ü. Nurges, J. Belikov and V. Kaparin: Stable cones of polynomials via Routh rays. In: 20th International Conference on Process Control, Štrbské Pleso 2015, pp. 255-260.   DOI:10.1109/pc.2015.7169972
  4. S. P. Bhattacharyya, H. Chapellat and L. H. Keel: Robust Control: The Parametric Approach. Prentice Hall, Upper Saddle River, New Jersy 1995.   CrossRef
  5. G. Calafiore and L. {El Ghaoui}: Ellipsoidal bounds for uncertain linear equations and dynamical systems. Automatica 40 (2004), 773-787.   DOI:10.1016/j.automatica.2004.01.001
  6. H. Chapellat, M. Mansour and S. P. Bhattacharyya: Elementary proofs of some classical stability criteria. Trans. Ed. 33 (1990), 232-239.   DOI:10.1109/13.57067
  7. F. R. Gantmacher: The Theory of Matrices. Chelsea Publishing Company, New York 1959.   DOI:10.1126/science.131.3408.1216-a
  8. R. Greiner: Necessary conditions for Schur-stability of interval polynomials. Trans. Automat. Control 49 (2004), 740-744.   DOI:10.1109/tac.2004.825963
  9. D. Henrion, D. Peaucelle, D. Arzelier and M. Šebek: Ellipsoidal approximation of the stability domain of a polynomial. Trans. Automat. Control 48 (2003), 2255-2259.   DOI:10.1109/tac.2003.820161
  10. D. Hinrichsen and V. L. Kharitonov: Stability of polynomials with conic uncertainty. Math. Control Signals Systems 8 (1995), 97-117.   DOI:10.1007/bf01210203
  11. L. Jetto: Strong stabilization over polytopes. Trans. Automat. Control 44 (1999), 1211-1216.   DOI:10.1109/9.769376
  12. V. L. Kharitonov: Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differ. Equations 14 (1979), 1483-1485.   CrossRef
  13. N. S. Nise: Control Systems Engineering. John Wiley and Sons, Jefferson City 2010.   CrossRef
  14. Ü. Nurges: New stability conditions via reflection coefficients of polynomials. Trans. Automat. Control 50 (2005), 1354-1360.   DOI:10.1109/tac.2005.854614
  15. Ü. Nurges and S. Avanessov: Fixed-order stabilising controller design by a mixed randomised/deterministic method. Int. J. Control 88 (2015), 335-346.   DOI:10.1080/00207179.2014.953208
  16. Ü. Nurges, I. Artemchuk and J. Belikov: Generation of stable polytopes of Hurwitz polynomials via Routh parameters. In: 53rd IEEE Conference on Decision and Control, Los Angeles 2014, pp. 2390-2395.   DOI:10.1109/cdc.2014.7039753
  17. G. Parmar, S. Mukherjee and R. Prasad: System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Model. 31 (2007), 2542-2552.   DOI:10.1016/j.apm.2006.10.004
  18. Q. I. Rahman and G. Schmeisser: Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties. Oxford University Press, London 2002.   CrossRef
  19. A. Rantzer: Stability conditions for polytopes of polynomials. Trans. Autom. Control 37 (1992), 79-89.   DOI:10.1109/9.109640
  20. P. Shcherbakov and F. Dabbene: On the Generation of Random Stable Polynomials. Eur. J. Control 17 (2011), 145-159.   DOI:10.3166/ejc.17.145-159
  21. A. C. Tsoi: Inverse Routh-Hurwitz array solution to the inverse stability problem. Electron. Lett. 15 (1979), 575-576.   DOI:10.1049/el:19790413
  22. E. I. Verriest and W. Michiels: Inverse Routh table construction and stability of delay equations. Systems Control Lett. 55 (2006), 711-718.   DOI:10.1016/j.sysconle.2006.02.002
  23. L. A. Zadeh and C. A. Desoer: Linear System Theory: The State Space Approach. MacGraw-Hill, New York 1963.   CrossRef