On the convergence of the attainability sets of a second order linear system
Abstract
The boundary of the attainability sets $D(T)$ of a second order linear differential equation with an external perturbation its determined. It is shown that when the limit $T\to+\infty$ is considered, the bondaury of the attainability sets converge to the only limit cycle $C^*$ that is obtained when the worst external perturbation is considered. The results are illustrated numerically.Downloads
References
Isaac Elishakoff and Makoto Ohsaki, Optimization and anti-optimization of structures under uncertainty, Imperial College Press, (2010), London.
V. I. Blagodatskikh, Introducción al control óptimo: Teoría lineal, Vysshaya Shkola, 2001, Moscú, (En ruso).
D. I. Bugrov, Estimation of the attainability set for a linear system based on a linear matrix inequality, Moscow University Mathematics Bulletin, 2016, 71(6), p. 253-256, DOI 10.3103/S0027132216060061.
D. I. Bugrov and A. M. Formals'kii, Time dependence of the attainability regions of third order systems, Journal of Applied Mathematics and Mechanics, 2017, 81, p. 106--113,
DOI 10.1016/j.jappmathmech.2017.08.004.
A. Kurzhanski and I. Valyi, Ellipsoidal calculus for estimation and control, Systems & Control: Foundations & Applications, Birkhäuser, 1999, Boston.
F. L. Chernousko, Estimación de estados fase de sistemas dinámicos. Método de elipsoides, Nauka,1988, Moscú, (En ruso).
V. N. Zhermolenko, Sobre el problema de Bulgakov referente a la desviación máxima de un sistema oscilatorio de segundo orden, Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., 1980, 2, p. 87--91, (En ruso).
V. N. Zhermolenko, Ciclos límite en el plano de fases, en: El problema de la desviación máxima de Bulgakov y sus aplicaciones, Universidad Estatal de Moscú, 1993, editor V. V. Aleksandrov, p. 35-48, Moscú, (En ruso).
V. N. Zhermolenko, Maximum deviation of oscillating system of the second order with external and parametric disturbances, Journal of Computer and Systems Sciences International, 2007, 46, p. 407-411, DOI 10.1134/S1064230707030094.
V. N. Zhermolenko, On maximal deviation of linear system, Automation and Remote Control, 2012, 73(7), p. 1117-1125, DOI 10.1134/S0005117912070016.
V. V. Aleksandrov, O. V. Alexandrova, I. P. Prikhod'ko and R. Telmotzi-Auila, Synthesis of self-oscillations, Moscow University Mechanics Bulletin, 2007, 62(3), p. 65-68, DOI 10.3103/S0027133007030016.
V. V. Aleksandrov, O. V. Aleksandrova, I.S. Konovalenko and K. V. Tikhonova, Perturbed stable systems on a plane. Part 1, Moscow University Mechanics Bulletin, 2016, 71(5), p. 108-113, DOI 10.3103/S0027133016050022.
