Geodesic
On stable cones of polynomials via reduced Routh parameters
Kybernetika, Tome 52 (2016) no. 3, pp. 461-477
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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.
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.
DOI : 10.14736/kyb-2016-3-0461
Classification : 93C05, 93D09
Keywords: linear systems; Hurwitz stability; convex approximation 
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Nurges, Ülo; Belikov, Juri; Artemchuk, Igor. On stable cones of polynomials via reduced Routh parameters. Kybernetika, Tome 52 (2016) no. 3, pp. 461-477. doi: 10.14736/kyb-2016-3-0461

[1] Ackermann, J., Kaesbauer, D.: Stable polyhedra in parameter space. Automatica 39 (2003), 937-943. | DOI | MR | Zbl

[2] Artemchuk, I., Nurges, Ü., Belikov, J.: Robust pole assignment via Routh rays of polynomials. In: American Control Conference, Boston 2016, pp. 7031-7036.

[3] Artemchuk, I., Nurges, Ü., Belikov, J., Kaparin, V.: Stable cones of polynomials via Routh rays. In: 20th International Conference on Process Control, Štrbské Pleso 2015, pp. 255-260. | DOI

[4] Bhattacharyya, S. P., Chapellat, H., Keel, L. H.: Robust Control: The Parametric Approach. Prentice Hall, Upper Saddle River, New Jersy 1995. | Zbl

[5] Calafiore, G., ElGhaoui, L.: Ellipsoidal bounds for uncertain linear equations and dynamical systems. Automatica 40 (2004), 773-787. | DOI | MR

[6] Chapellat, H., Mansour, M., Bhattacharyya, S. P.: Elementary proofs of some classical stability criteria. Trans. Ed. 33 (1990), 232-239. | DOI

[7] Gantmacher, F. R.: The Theory of Matrices. Chelsea Publishing Company, New York 1959. | DOI | MR | Zbl

[8] Greiner, R.: Necessary conditions for Schur-stability of interval polynomials. Trans. Automat. Control 49 (2004), 740-744. | DOI | MR

[9] Henrion, D., Peaucelle, D., Arzelier, D., Šebek, M.: Ellipsoidal approximation of the stability domain of a polynomial. Trans. Automat. Control 48 (2003), 2255-2259. | DOI | MR

[10] Hinrichsen, D., Kharitonov, V. L.: Stability of polynomials with conic uncertainty. Math. Control Signals Systems 8 (1995), 97-117. | DOI | MR | Zbl

[11] Jetto, L.: Strong stabilization over polytopes. Trans. Automat. Control 44 (1999), 1211-1216. | DOI | MR | Zbl

[12] Kharitonov, V. L.: Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differ. Equations 14 (1979), 1483-1485. | MR | Zbl

[13] Nise, N. S.: Control Systems Engineering. John Wiley and Sons, Jefferson City 2010.

[14] Nurges, Ü.: New stability conditions via reflection coefficients of polynomials. Trans. Automat. Control 50 (2005), 1354-1360. | DOI | MR

[15] Nurges, Ü., Avanessov, S.: Fixed-order stabilising controller design by a mixed randomised/deterministic method. Int. J. Control 88 (2015), 335-346. | DOI | MR | Zbl

[16] Nurges, Ü., Artemchuk, I., Belikov, J.: 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

[17] Parmar, G., Mukherjee, S., Prasad, R.: System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Model. 31 (2007), 2542-2552. | DOI | Zbl

[18] Rahman, Q. I., Schmeisser, G.: Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties. Oxford University Press, London 2002. | MR

[19] Rantzer, A.: Stability conditions for polytopes of polynomials. Trans. Autom. Control 37 (1992), 79-89. | DOI | MR | Zbl

[20] Shcherbakov, P., Dabbene, F.: On the Generation of Random Stable Polynomials. Eur. J. Control 17 (2011), 145-159. | DOI | MR | Zbl

[21] Tsoi, A. C.: Inverse Routh-Hurwitz array solution to the inverse stability problem. Electron. Lett. 15 (1979), 575-576. | DOI

[22] Verriest, E. I., Michiels, W.: Inverse Routh table construction and stability of delay equations. Systems Control Lett. 55 (2006), 711-718. | DOI | MR | Zbl

[23] Zadeh, L. A., Desoer, C. A.: Linear System Theory: The State Space Approach. MacGraw-Hill, New York 1963. | Zbl

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