Abstract
This paper focuses on finding and the assessment of the variation of the region of robust absolute stability of an impulse control system, with monotonous nonlinearities, as Popov’s parameter varies. A mathematical model of a nonlinear impulsive control system (NICS) is considered. The criterion for absolute stability on the equilibrium position for NICS, with monotonous nonlinearities, can be expressed as a polynomial expression. The robust stability of NICS is tested using Kharitonov’s theorem and a modified root locus method for interval transfer functions. A graphical illustration of roots of the characteristic equation, which have been gotten from the interval transfer function, on the complex plane is used in the assessment of the stability of control system. To evaluate the effect of Popov’s parameter, a specially written program complex Stability is used. An illustrative example is given to demonstrate the effect of varying Popov’s parameter on the region of absolute robust stability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arvanitis, K.G., Soldatos, A.G., Boglou, A.K., Bekiaris-Liberis, N.K.: New simple controller rules for integrating and stable or unstable first order plus dead-time processes. In: Recent Advances in Systems-13th WSEAS International Conference on SYSTEMS, Rhodes, pp. 328–337 (2009)
Leonov, G.A., Kuznetsov, N.V., Seledzhi, S.M., Shumakov, M.M.: Stabilization of unstable control system via design of delayed feedback. In: Recent Researches in Applied and Computational Mathematics-EUROPMENT/WSEAS International Conference on Applied and Computational Mathematics (ICACM ’11), Lanzarote, pp. 18–25 (2011)
Fuh, C.C., Tuhg, P.C.: Robust stability bounds for Lur’e systems with parametric uncertainty. J. Mar. Sci. Technol. 7(2), 73–78 (1999)
Haddad, W.M., Collins, E.G., Jr., Bernstein, D.S.: Robust stability analysis using the small gain, circle positivity, and Popov theorems: A comparative study. IEEE Trans. Control Syst. Technol. 1, 290–293 (1993)
Matusu, R.: Robust stability analysis of discrete-time systems with parametric uncertainty: A graphical approach. Int. J. Math Models Methods Appl. Sci. 8, 95–102 (2014)
Prokop, R., Volkova, N., Prokopova, Z.: Tracking and disturbance attenuation for unstable systems: Algebraic approach. In: Recent Researches in Automatic Control - 13th WSEAS International Conference on Automatic Control, Modeling and Simulation (ACMOS ’11), Lanzarote, pp. 57–62 (2011)
Gazdoš, F., Dostal, P., Marholt, J.: Robust control of unstable systems: Algebraic approach using sensitivity functions. Int. J. Math. Models Methods Appl. Sci. 5(7), 1189–1196 (2011)
Matušu, R., Prokop, R.: Graphical Analysis of Robust Stability for Polynomials with Uncertain Coefficients in Matlab Environment. In: Proceeding of the 16th WSEAS International Conference on Systems, Kos (2012)
Matusu, R., Prokop, R.: Graphical approach to robust stability analysis for discrete-time interval polynomials. In: Proceeding of the 15th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Kuala Lumpur (2013)
Tseligorov, N.A., Tseligorova, E.N., Mafura, G.M.: Using information technology for computer modeling of nonlinear monotonous impulse control system with uncertainties. In: Proceeding of the Computer Modeling and Simulation, Saint Petersburg, pp. 75–80 (2014)
Tseligorov, N.A., Leonov, M.V., Tseligorova, E.N.: Modeling complex “Sustainanability” for the study of robust absolute stability of nonlinear impulsive control systems. In: Proceeding of the Computer Modeling 2012, Saint Petersburg, pp. 9–14 (2012)
Tspkin, Ya.Z., Popkov, Yu.S.: Theory of Nonlinear Impulsive Systems. Nauka, Moscow (1973)
Serkov, V.I., Tseligorov, N.A.: The analysis of absolute stability of nonlinear impulsive automated systems by analytical methods. Autom. Telemech. 9, 60–65 (1975)
Jury, E.I.: Inners and Stability of Dynamic Systems. Wiley, London (1974)
Gantmacher, F.R.: The Theory of Matrices. Taylor & Francis, London (1964)
Rimsky, G.V.: Foundations of the general theory of the root locus for automatic control systems. Science and Technology, Minsk (1972)
Tseligorov, N.A., Tseligorova, E.N.: Using a modified root locus method to study the robust absolute stability of a multivariate control system. System identification and control problem. In: Proceedings of the IV International Conference SICPRO ’07, 29 January–1 February 2007. IPU RAN, 1 DVD Disk, Number: 13034 (2007)
Kharitonov, V.L.: Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differentsial’nye Uravneniya 14, 2086–2088 (1978)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Nikolai, T., Elena, T., Gabriel, M. (2015). An Assessment of the Effect of Varying Popov’s Parameter on the Region of Robust Absolute Stability of Nonlinear Impulsive Control Systems with Parametric Uncertainty. In: Mastorakis, N., Bulucea, A., Tsekouras, G. (eds) Computational Problems in Science and Engineering. Lecture Notes in Electrical Engineering, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-319-15765-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-15765-8_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15764-1
Online ISBN: 978-3-319-15765-8
eBook Packages: EngineeringEngineering (R0)