Synthesis of a Control System of a Deterministic Chaotic Process in the Class of Two-Parameter Structurally Stable Mappings
In this paper, the problem of unstable and deterministic chaotic processes in control systems is considered. The synthesis of a control system in the class of two-parameter structurally stable mappings is demonstrated. This is realized via the gradient-velocity method of Lyapunov vector functions. It is shown that the gradient-velocity method of Lyapunov vector functions allows generating an aperiodic robust stable system with the desired characteristics. A simple solution to the problem of synthesis of control systems for unstable and deterministic chaotic processes is obtained. Moreover, it is applicable for complex systems.
[1] A.Yu. Loskutov and A.S. Mikhailov, Osnovyteorii slozhnykh system (Fundamentals of the Theory of Complex Systems), Moscow–Izhevsk: Inst. Komp’yut. Issled., 2007.
[2] W. Brock, Teoriya khaosa (Chaos theory),Moskow: Science, 2011
[3] B.R. Andrievsky, A.L. Fradkov, Control of Chaos: Methods and Applications. II. Applications, Automation and Remote Control, Vol. 65, No. 4, 2004, pp. 505–533.
[4] A. Loskutov, Chaos and Control in Dynamical Systems. Computational Mathematics and Modeling, Vol. 12, No. 4, 2001, pp. 314-352.
[5] V.D. Shalfeev, G.V. Osipov, A.K. Kozlov and A.R. Volkovskii, Chaotic Oscillations: Generation, Synchronization, Control, Zarub. Radioelektron. Usp. Sovr. Radioelektron., 1997, no. 10, pp. 27–49.
[6] F.C. Moon, A.J. Reddy, W.T. Holmes, Experiments in control and anticontrol of chaos in a dry friction oscillirator// J.Vibr.Control. 2003. 9. P. 387-397.
[7] P.A. Meehan, S.F. Asokanthan Control of Chaotic motion in a dual spin spacecraft with nutational damping// J.Guid., Control Dyn. 2002. 25. 2. P. 209-214
[8] M.P. Kennedy, J. Kolumban, Digital communications using chaos. In: Controling chaos and Bifurcations in Engineering Systems/ Ed. G.Chen, CRC Press. 1999. 9 P. 477-500
[9] P.A. Meehan, S.F. Asokanthan, Control of Chaotic instabilities in spinning spacecraft with dissipation using Lyapunov method// Chaos, Solitons and Fractals. 2002. 13. P. 1857-1869.
[10] M.A. Beisenbi, Models and methods of system analysis and control of deterministic chaos in the economy, Astana, 201
[11] M.A. Beisenbi, Investigation of robust stability of automatic control systems by A.M. Lyapunov function method, Astana, 2015.
[12] M.A. Beisenbi, Methods for increasing the robust stability potential of control systems, Astana, 2011.
[13] M.A. Beisenbi, B.A. Erzhanov, Control systems with increased robust stability potential, Astana, 2002.
[14] M.A. Beisenbi, Controlled chaos in the development of the economic system, Nur-Sultan: Master Po LLP, 2019.
[15] B.T. Polyak and P.S. Shcherbakov, Robast nayaustoichivost’ iupravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
[16] P. Dorato and R.K. Yedavalli, Recent Advances in Robust Control, New York: IEE press 1990.
[17] R. Gilmore, Catastrophe Theory, Digital Encyclopedia of Applied Physics, 2007.
[18] Methods of classical and modern theory of automatic control. Textbook 5 vols. Vol.3. Synthesis of automatic control systems regulators, Ed. K.A. Pupkov and N.D. Egupov. - M.: N.E. Bauman MSTU, 2004 .
[19] M.A. Beisenbi, Zh.O. Basheyeva, Solving output control problems using Lyapunov gradient-velocity vector function. International Journal of Electrical and Computer Engineering, Vol. 9, No. 4, 2019, pp. 2874-2879.
[20] Mamyrbek Beisenbi, Aigul Sagymbay, Dana Satybaldina, and Nurgul Kissikova, Velocity Gradient Method of Lyapunov Vector Functions, Proceedings of the 2019 the 5th International Conference on e-Society, e-Learning and e-Technologies, Association for Computing Machinery, New York, NY, USA, 88–92.
[21] Beisenbi M., Uskenbayeva G., Satybaldina D., Martsenyuk V., Shaikhanova F. Robust stability of spacecraft traffic control system using Lyapunov functions. 16th International Conference on Control, Automation and Systems (ICCAS), IEEE, 2016, pp. 743-748
[1] A.Yu. Loskutov and A.S. Mikhailov, Osnovyteorii slozhnykh system (Fundamentals of the Theory of Complex Systems), Moscow–Izhevsk: Inst. Komp’yut. Issled., 2007.
[2] W. Brock, Teoriya khaosa (Chaos theory),Moskow: Science, 2011
[3] B.R. Andrievsky, A.L. Fradkov, Control of Chaos: Methods and Applications. II. Applications, Automation and Remote Control, Vol. 65, No. 4, 2004, pp. 505–533.
[4] A. Loskutov, Chaos and Control in Dynamical Systems. Computational Mathematics and Modeling, Vol. 12, No. 4, 2001, pp. 314-352.
[5] V.D. Shalfeev, G.V. Osipov, A.K. Kozlov and A.R. Volkovskii, Chaotic Oscillations: Generation, Synchronization, Control, Zarub. Radioelektron. Usp. Sovr. Radioelektron., 1997, no. 10, pp. 27–49.
[6] F.C. Moon, A.J. Reddy, W.T. Holmes, Experiments in control and anticontrol of chaos in a dry friction oscillirator// J.Vibr.Control. 2003. 9. P. 387-397.
[7] P.A. Meehan, S.F. Asokanthan Control of Chaotic motion in a dual spin spacecraft with nutational damping// J.Guid., Control Dyn. 2002. 25. 2. P. 209-214
[8] M.P. Kennedy, J. Kolumban, Digital communications using chaos. In: Controling chaos and Bifurcations in Engineering Systems/ Ed. G.Chen, CRC Press. 1999. 9 P. 477-500
[9] P.A. Meehan, S.F. Asokanthan, Control of Chaotic instabilities in spinning spacecraft with dissipation using Lyapunov method// Chaos, Solitons and Fractals. 2002. 13. P. 1857-1869.
[10] M.A. Beisenbi, Models and methods of system analysis and control of deterministic chaos in the economy, Astana, 201
[11] M.A. Beisenbi, Investigation of robust stability of automatic control systems by A.M. Lyapunov function method, Astana, 2015.
[12] M.A. Beisenbi, Methods for increasing the robust stability potential of control systems, Astana, 2011.
[13] M.A. Beisenbi, B.A. Erzhanov, Control systems with increased robust stability potential, Astana, 2002.
[14] M.A. Beisenbi, Controlled chaos in the development of the economic system, Nur-Sultan: Master Po LLP, 2019.
[15] B.T. Polyak and P.S. Shcherbakov, Robast nayaustoichivost’ iupravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
[16] P. Dorato and R.K. Yedavalli, Recent Advances in Robust Control, New York: IEE press 1990.
[17] R. Gilmore, Catastrophe Theory, Digital Encyclopedia of Applied Physics, 2007.
[18] Methods of classical and modern theory of automatic control. Textbook 5 vols. Vol.3. Synthesis of automatic control systems regulators, Ed. K.A. Pupkov and N.D. Egupov. - M.: N.E. Bauman MSTU, 2004 .
[19] M.A. Beisenbi, Zh.O. Basheyeva, Solving output control problems using Lyapunov gradient-velocity vector function. International Journal of Electrical and Computer Engineering, Vol. 9, No. 4, 2019, pp. 2874-2879.
[20] Mamyrbek Beisenbi, Aigul Sagymbay, Dana Satybaldina, and Nurgul Kissikova, Velocity Gradient Method of Lyapunov Vector Functions, Proceedings of the 2019 the 5th International Conference on e-Society, e-Learning and e-Technologies, Association for Computing Machinery, New York, NY, USA, 88–92.
[21] Beisenbi M., Uskenbayeva G., Satybaldina D., Martsenyuk V., Shaikhanova F. Robust stability of spacecraft traffic control system using Lyapunov functions. 16th International Conference on Control, Automation and Systems (ICCAS), IEEE, 2016, pp. 743-748
@article{"International Journal of Information, Control and Computer Sciences:80749", author = "M. Beisenbi and A. Sagymbay and S. Beisembina and A. Satpayeva", title = "Synthesis of a Control System of a Deterministic Chaotic Process in the Class of Two-Parameter Structurally Stable Mappings", abstract = "In this paper, the problem of unstable and deterministic chaotic processes in control systems is considered. The synthesis of a control system in the class of two-parameter structurally stable mappings is demonstrated. This is realized via the gradient-velocity method of Lyapunov vector functions. It is shown that the gradient-velocity method of Lyapunov vector functions allows generating an aperiodic robust stable system with the desired characteristics. A simple solution to the problem of synthesis of control systems for unstable and deterministic chaotic processes is obtained. Moreover, it is applicable for complex systems. ", keywords = "Control system synthesis, deterministic chaotic processes, Lyapunov vector function, robust stability, structurally stable mappings.", volume = "16", number = "3", pages = "98-5", }
