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Editor-in-Chief
Nikiforov
Vladimir O.
D.Sc., Prof.
Partners
doi: 10.17586/2226-1494-2023-23-2-263-270
Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems
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Abstract
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Omorov R.O. Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 2, pp. 263–270. doi: 10.17586/2226-1494-2023-23-2-263-270
Abstract
The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.
Keywords: method of topological roughness, condition number of a matrix, bifurcation of systems, robustness of control systems, interval dynamical systems, multidimensional control systems, frequency-robust method, frequency condition number, synergetic systems, chaos, special points and trajectories, Sylvester matrix equation
References
References
-
Poincaré H. Mémoire sur les courbes définies par une equation différentielle. Journal de Mathématiques Pures et Appliquées. 1881-1886. (in French)
-
Andronov A.A., Pontriagin L.S. Structurally stable systems. Doklady AN SSSR, 1937, vol. 14, no. 5, pp. 247–250. (in Russian)
-
Anosov D.V. Structurally stable systems. Proceedings of the Steklov Institute of Mathematics, 1986, vol. 169, pp. 61–95.
-
Poliak B.T., Tcypkin Ya.Z. Robust stability of linear systems. Itogi nauki i tehniki. Tehnicheskaja kibernetika, 1991, vol. 32, pp. 3–31. (in Russian)
-
Kharitonov V.L. The asymptotic stability of the equilibrium state of a family of systems of linear differential equations. Differential Equations, 1978, vol. 14, no. 11, pp. 2086–2088. (in Russian)
-
Kharitonov V.L. On a generalization of a stability criterion. Izvestija AN Kazahskoj SSR . Serija fiziko-matematicheskaja, 1978, no. 1, pp. 53–57. (in Russian)
-
Neimark Y.I. Robust stability and D-partition. Automation and Remote Control, 1992, vol. 53, no. 7, pp. 957–965.
-
Omorov R.O. Robustness of interval dynamic systems. I. Robustness in continuous linear interval dynamic systems. Journal of Computer and Systems Sciences International, 1996, vol. 34, no. 3, pp. 69–74
-
Omorov R.O. Robustness of interval dynamical systems. II. Robustness of discrete linear interval dynamical systems. Journal of Computer and Systems Sciences International, 1996, vol. 34, no. 4, pp. 1–5.
-
Omorov R.O. Robustness research of interval dynamic systems by algebraic method. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 3, pp. 364–370. (in Russian). https://doi.org/10.17586/2226-1494-2020-20-3-364-370
-
Ushakov A.V., Akunova A., Omorov R.O., Akunov T.A. Robust Multidimensional Control Systems: Frequency and Algebraic Methods. Ed. by R.O. Omorov. Bishkek, Ilim Publ., 2022, 352 p. (in Russian).
-
Omorov R.O. Estimation of roughness of controllable dynamic systems. Russian Electromechanics, 1990, no. 7, pp. 81–87. (in Russian)
-
Omorov R.O. Maximal robustness of dynamical systems. Automation and Remote Control, 1991, vol. 52, no. 8, pp. 1061–1068.
-
Omorov R.O. Measure of roughness of dynamic systems and criteria of emergence of chaotic fluctuations and bifurcations in synergetic systems. Synthesis of algorithms of systems stabilization: department proceedings. Issue 8. Taganrog, 1992, pp. 128–134. (in Russian)
-
Omorov R.O. Dynamical System Quantitative Robustness Measures and Their Applications to Control Systems. Dissertation for the degree of doctor of technical sciences, St. Petersburg, Saint Petersburg Institute of Fine Mechanics and Optics, 1992, 188 p. (in Russian)
-
Omorov R.O. Synergetic systems: problems of roughness, bifurcations and accidents. Nauka i novye tehnologii, 1997, no. 2, pp. 26–36. (in Russian)
-
Omorov R.O. Method of topological roughness: theory and appendices. I. Theory. Izvestija Nacional’noj Akademii nauk Kyrgyzskoj Respubliki, 2009, no. 3, pp. 144–148. (in Russian)
-
Omorov R.O. Topological roughness of synergetic systems. Journal of Automation and Information Sciences, 2012, vol. 44, no. 4, pp. 61–70. https://doi.org/10.1615/JAutomatInfScien.v44.i4.70
-
Omorov R.O. Theory of Topological Roughness of Systems. Applications to Synergetic Systems and Chaos. Bishkek, Ilim Publ., 2019, 288 p. (in Russian)
-
Omorov R.O. Method of topological roughness of dynamic systems: applications to synergetic systems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 2, pp. 257–262. (in Russian ). https://doi.org/10.17586/2226-1494-2020-20-2-257-262
-
Omorov R.O. Sensitivity, Robustness and Roughness of Dynamic Systems. Moscow, LENAND Publ., 2021, 304 p. (in Russian)
-
Omorov R.O. Synergetics and Chaos: Topological Roughness and Bifurcations. Moscow, LENAND Publ., 2022, 160 p. (in Russian)
-
Omorov R.O. The modal sensitivity, robustness and roughness of dynamic systems (review article). Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2021, vol. 21, no. 2, pp. 179–190. (in Russian). https://doi.org/10.17586/2226-1494-2021-21-2-179-190
-
Dorato P.D. A historical review of robust control. IEEE Control Systems Magazine, 1987, vol. 7, no. 2, pp. 44–47. https://doi.org/10.1109/MCS.1987.1105273
-
Jury E.I. Robustness of a discrete system. Automation and Remote Control, 1990, vol. 51, no. 5, pp. 571–592.
-
Discussion on robustness problem in control systems. Avtomatika I Telemkhanika, 1992, no. 1, pp. 165–176. (in Russian)
-
Nikiforov V.O. Robust output control for a linear object. Automation and Remote Control, 1998, vol. 59, no. 9, pp. 1274–1283.
-
Pelevin A.Ye. Robust control law synthesis under uncertainty of model parameters. Giroskopiya i Navigatsiya, 1999, no. 2(25), pp. 63–74. (in Russian)
-
Gusev Yu.M., Yefanov V.N., Krymskiy V.G., Rutkovskiy V.Yu. Analysis and synthesis of linear interval dynamic systems (the state of the problem). I. Analysis which uses interval characteristic polynomials. Soviet journal of computer and systems sciences, 1991, vol. 29, no. 6, pp. 84–103
-
Gusev Yu.M., Yefanov V.N., Krymskiy V.G., Rutkovskiy V.Yu Analysis and synthesis of linear interval dynamical systems (the state of the problem). II. Analysis of the stability of interval matrices and synthesis of robust regulators. Soviet journal of computer and systems sciences, 1992, vol. 30, no. 2, pp. 26–52
-
Polyak B.T., Shcherbakov P.S. Superstable linear control systems. I. Analysis. Automation and Remote Control, 2002, vol. 63, no. 8, pp. 1239–1254. https://doi.org/10.1023/A:1019823208592
-
Kuntsevich V.M. Management under Conditions of Uncertainty: Guaranteed Results in Management and Identification Issues. Kiev, Naukova dumka Publ., 2006, 264 p. (in Russian)
-
Barmish B.R., Hollot C.V. Counter-example to a recent result on the stability of interval matrices by S. Bialas. International Journal of Control, 1984, vol. 39, no. 5, pp. 1103–1104. https://doi.org/10.1080/00207178408933235
-
Barmish B.R., Fu M., Saleh S. Stability of a polytope of matrices: Counterexamples. IEEE Transactions on Automatic Control, 1988, vol. 33, no. 6, pp. 569–572. https://doi.org/10.1109/9.1254
-
Bialas S. A necessary and sufficient condition for the stability of interval matrices. International Journal of Control, 1983, vol. 37, no. 4, pp. 717–722. https://doi.org/10.1080/00207178308933004
-
Kraus F.J., Anderson B.D.O., Jury E.I., Mansour M. On the robustness of low-order Schur polynomials. IEEE Transactions on Circuits and Systems, 1988, vol. 35, no. 5, pp. 570–577. https://doi.org/10.1109/31.1786
-
Mansour M., Kraus F.J. On Robust Stability of Sсhur Polynomials. Report N 87-05, Inst. Autom. Cont. Ind. Electronics, Swiss, Fed. Inst. Tech. (ETH), Züric, 1987, 34 p.
-
Akunov T.A., Slita O.V., Ushakov A.V. Gramians in parametric invariance of continuous systems. Scientific and TechnicalJournal of Information Technologies, Mechanics and Optics, 2005, vol. 5, no. 3, pp. 39–43. (in Russian)
-
Omorov R.O. Robustness of interval dynamic systems I. Robustness in continuous linear interval dynamic systems. Journal of Computer and Systems Sciences International, 1996, vol. 34, no. 3, pp. 69–74.
-
Omorov R.O. Robustness of interval dynamical systems. II. Robustness of discrete linear interval dynamical systems. Journal of Computer and Systems Sciences International, 1996, vol. 34, no. 4, pp. 1–5.
-
Omorov R.O. On discrete analogue of Kharitonov’s theorem. Science and New Technologies, 2002, no. 3, pp. 5–10. (in Russian)
-
Omorov R.O. Robust Stability of Interval Dynamic Systems. Bishkek, Ilim Publ., 2018, 104 p. (in Russian)
-
Omorov R.O. Robustness of Interval Dynamic Systems. II. Radiotehnika i jelektronika, 1995, vol. 40, no. 12, pp. 3–7.
-
Omorov R.O. Robustness of interval dynamic systems. Proc. of the Conference “Mathematical Control Theory and Its Applications”, St.Petersburg, October 7-8, 2020. St. Petersburg, Concern CSRI Elektropribor, 2020, pp. 333–335. (in Russian)
-
Haken H. Advanced Synergetics: Instability Hierarchies of Self- Organizing Systems and Devices. Berlin, Springer-Verlag, 1983.
-
Nikolis G., Prigogine I. Exploring Complexity: An Introduction. W.H. Freeman Publ., 1989, 313 p.
-
Strange Attractors. Moscow, Mir Publ., 1981, 253 p. (in Russian)
-
Zhang W.-B. Synergetic Economics: Time and Change in Nonlinear Economics. Berlin, Springer-Verlag, 1991, 264 p.
-
Kapitca S.P., Kurdiumov S.P., Malinetckii G.G. Synergetics and Future Forecasts. Moscow, URSS Publ., 2001, 288 p. (in Russian)
-
Leonov G.A., Kuznetsov N.V., Kudryashova E.V. Tunisia 2011-2014. Bifurcation, revolution, and controlled stabilization. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika informatika protsessy upravleniya, 2016, no. 4, pp. 92–103. (in Russian). https://doi.org/10.21638/11701/spbu10.2016.409
-
Andrievskii B.R., Fradkov A.L. Control of chaos: Methods and applications. I. Methods. Automation and Remote Control, 2003, vol. 64, no. 5, pp. 673–713. https://doi.org/10.1023/A:1023684619933
-
Kolesnikov A.A. Synergetic Control Methods for Complex Systems: System Synthesis Theory. Moscow, LIBROKOM Publ., 2012, 240 p. (in Russian)
-
Peixoto M.M. On structural stability. Annals of Mathematics, 1959, vol. 69, no. 1, pp. 199–222. https://doi.org/10.2307/1970100
-
Omorov R.O. Topological roughness and bifurcations of synergetic systems. Proc. of the Conference “Mathematical Control Theory and Its Applications”, St. Petersburg, October 7-8, 2020. St. Petersburg, Concern CSRI Elektropribor, 2020, pp. 28–30. (in Russian)
-
Aleksandrov P.S. Poincaré and topology. Poincaré H. Selected Works. Moscow, Nauka Publ., 1972, pp. 808–816. (in Russian)
-
Hopf E. Abzweigungen einer periodischen Losung von elner statlonaren Losung eines differential systems. Wissenschaften, Leipzig, 1942, vol. 94, pp. 1–22.
-
Joss G., Joseph D. Elementary Stability and Bifurcation Theory. Springer New York, 1980, 286 p.
-
Chow S.N., Hale J.K. Methods of Bifurcation Theory. Springer-Verlag, 1982, 515 p.
-
Leontovich E.A., Gordon I.I., Maier A.G., Andronov A.A. Bifurcation Theory of Dynamical Systems on the Plane. Moscow, Nauka Publ., 1967, 488 p. (in Russian)
-
Andronov A.A., Vitt A.A., Khaikin S.E. Oscillation Theory. Moscow, 1981, 568 p. (in Russian)
-
Arnol'd V.I. Catastrophe theory. Itogi Nauki i Tekhniki. Seriya "Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya. V. 5, Moscow, VINITI, 1986, pp. 219–277.(in Russian)