The Second Lyapunov Function Method in Construction of Control Systems with the Increased Potential of Robust Stability in the Class of Catastrophe “Hyperbolic Umbilic”

Abstract:

The current article introduces a new method of construction of control systems for objects with uncertain parameters in the form of three-parameter structurally stable mappings from the theory of catastrophe, allowing to synthesize the highly effective control systems, which demonstrate the extremely wide range of robust stability. The research of robust stability of control systems is based on a new approach to post-rhenium of the A.M Lyapunov’s functions. The method of creation of control system with the increased potential of robust stability is stated. It could be concluded that the application of the suggested approach may assure asymptotically stable invariant states for the system in both negative and positive regions of variations of uncertain parameters.

• Periodical: Applied Mechanics and Materials (Volumes 799-800)
• Pages: 1132-1136
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.799-800.1132
• Online since: October 2015
• Authors: Nurlan Mukataev*, Mamyrbek Beisenbi
• Keywords: Catastrophe Hyperbolic Umbilic, Control Systems, Lyapunov Function Method, Robust Stability, Stability Area, Structurally Stable Mappings, Uncertainty of Parameters

References

[1] B.T. Polyak, P.S. Sherbakov: Robust stability and control, Nauka, Moscow (2002), p.303.

[2] P. Dorato, Vedavalli: Recent Advances in Robust Control, New York (1990), in IEE press.

[3] M.A. Beisenbi: Methods of increasing of potential of robust stability of control system, Astana (2011), p.352.

[4] M.A. Beisenbi: Models and methods of system analysis and management of deterministic chaos in the economy, Astana (2011), p.201.

[5] A.A. Voronov, V.M. Matrosov: The method of vector Lyapunov functions in the theory of stability, Nauka, Moscow (1987), p.312.

[6] T. Poston, I. Stuart: Catastrophe theory and its applications, Nauka, Moscow (2001), #6.

[7] R. Gilmore: Applied theory of catastrophes in 2 part, part 1, Mir, Moscow (1984).

[8] Beisenbi M., Mukataev N., Robust stability of system with one input and one output in a class of accidents hyperbolic umbilic". /CEIT, 14, Tunisia, 2014. Proceedings of IPCO. pp.152-156.

[9] Beisenbi M., Mukataev N., The construction of control systems with an increased potential of the robust stability in form of catastrophe Hyperbolic umbilic, based on method of Lyapunov function. WCICA 2014, Shenyang, China. P. 3046-3052.

DOI: 10.4028/www.scientific.net/amm.799-800.1132

[10] E.A. Barabashin: Introduction to the theory of stability, Nauka, Moscow (1967), p.225.

[11] I.G. Malkin: The theory of stability of motion, Nauka, Moscow (1966), p.540.