Evaluation of the weighted level of attenuation of external and initial disturbances in nonlinear systems
Abstract
UDC 517.925.51; 681.5.03
We investigate the classes of nonlinear dynamical systems with bounded disturbances and functional uncertainties. Further, we develop the methods aimed at the evaluation of the generalized performance criterion for these systems, which characterizes the weighted levels of damping of external disturbances and the initial disturbances caused by an unknown initial vector. It is proposed to apply these methods in solving the generalized $H_\infty$-control problem for the analyzed classes of systems. An illustrative example of a pseudolinear control system with disturbance is presented.
References
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishman, Linear matrix inequalities in system and control theory, SIAM Stud. Appl. Math., 15 (1994). DOI: https://doi.org/10.1137/1.9781611970777
K. Zhou, J. C. Doyle, K. Glover, Robust and optimal control, Englewood, Prentice-Hall, Inc. (1996).
G. E. Dullerud, F. G. Paganini, A course in robust control theory. A convex approach, Springer-Verlag, Berlin (2000). DOI: https://doi.org/10.1007/978-1-4757-3290-0
А. Г. Мазко, Робастная устойчивость и стабилизация динамических систем. Методы матричных и конусных неравенств, Праці Інституту математики НАН України, 102 (2016).
О. Г. Мазко, Матричнi методи аналiзу та синтезу динамiчних систем, Наук. думка, Київ (2023); https://doi.org/10.37863/6103136622-55. DOI: https://doi.org/10.37863/6103136622-55
D. V. Balandin, M. M. Kogan, Generalized $H_∞$-optimal control as a trade-off between the $H_∞$-optimal and $gamma$-optimal controls, Autom. and Remote Control, 71, № 6, 993–1010 (2010). DOI: https://doi.org/10.1134/S0005117910060020
Z. Feng, J. Lam, S. Xu, S. Zhou, $H_∞$ control with transients for singular systems, Asian J. Control, 18, № 3, 817–827 (2016). DOI: https://doi.org/10.1002/asjc.1163
О. Г. Мазко, С. М. Кусій, Зважене гасіння обмежених збурень у системі керування літака в режимі посадки, Зб. праць Інституту математики НАН України, 15, № 1, 88–99 (2018).
О. Г. Мазко, Синтез статичних регуляторiв для керованих об’єктiв iз екзогенними збуреннями, Нелінійні коливання, 26, № 4, 484–494 (2023). DOI: https://doi.org/10.3842/nosc.v26i4.1425
О. Г. Мазко, Зважена оцiнка i пониження рiвня впливу обмежених збурень у дескрипторних системах керування, Укр. мат. журн., 72, № 11, 1510–1523 (2020). DOI: https://doi.org/10.37863/umzh.v72i11.2389
P. Gahinet, P. Apkarian, A linear matrix inequality approach to $H_∞$ control, Internat. J. Robust and Nonlinear Control, 4, 421–448 (1994). DOI: https://doi.org/10.1002/rnc.4590040403
S. Xu, J. Lam, Y. Zou, New versions of bounded real lemmas for continuous and discrete uncertain systems, Circuits, Systems and Signal Process, 26, 829–838 (2007). DOI: https://doi.org/10.1007/s00034-007-9000-0
I. R. Petersen, R. Tempo, Robust control of uncertain systems: classical results and recent developments, Automatica, 50, № 5, 1315–1335 (2014). DOI: https://doi.org/10.1016/j.automatica.2014.02.042
F. Coloniusa, R. Fabbria, R. Johnson, On non-autonomous $H_∞$ control with infinite horizon, J. Different. Equat., 220, 46–67 (2006). DOI: https://doi.org/10.1016/j.jde.2004.12.009
R. Ravi, K. M. Nagpal, P. P. Khargonekar, $H_∞$ control of linear time-varying systems: a state-space approach, SIAM J. Control and Optim., 29, № 6, 1394–1413 (1991). DOI: https://doi.org/10.1137/0329071
A. J. van der Schaft, $L_2$-Gain analysis of nonlinear systems and nonlinear state feedback $H_∞$ control, IEEE Trans. Automat. Control, 37, № 6, 770–784 (1992). DOI: https://doi.org/10.1109/9.256331
A. J. van der Schaft, $L_2$-Gain and passivity techniques in nonlinear control, third ed., Springer Intern. Publ. AG, Cham, Switzerland (2017). DOI: https://doi.org/10.1007/978-3-319-49992-5
A. Isidori, A. Astolfi, Disturbance attenuation and $H_∞$-control via measurement feedback in nonlinear systems, IEEE Trans. Automat. Control, 37, № 9, 1283–1293 (1992). DOI: https://doi.org/10.1109/9.159566
Xin Wang, E. E. Yaz, S. C. Schneider, Y. I. Yaz, $H_2$–$H_∞$ control of continuous-time nonlinear systems using the state-dependent Riccati equation approach, Systems Science & Control Eng., 5, 224–231 (2017). DOI: https://doi.org/10.1080/21642583.2017.1310636
Wei-Min Lu, J. C. Doyle, $H_∞$ control of nonlinear systems: a convex characterization, IEEE Trans. Automat. Control, 40, № 9, 1668–1675 (1995). DOI: https://doi.org/10.1109/9.412643
D. F. Coutinho, A. Trofino, M. Fu, Nonlinear $H$-infinity control: an LMI approach, IFAC, 15th Triennial World Congress, Barcelona, Spain (2002). DOI: https://doi.org/10.3182/20020721-6-ES-1901.00350
Asep Najmurrokhman, On solvability of output feedback nonlinear $H_infty$-control problem using nonlinear matrix inequalities approach, J. Electr. Eng. and Inform. Technology, 1, № 1, 33–39 (2003).
О. Г. Мазко, Оцiнка та досягнення зважених критерiїв якостi у дескрипторних системах керування, Укр. мат. журн., 74, № 7, 980–990 (2022). DOI: https://doi.org/10.37863/umzh.v74i7.7167
M. S. Berger, M. Berger, Perspective in nonlinearity: an introduction to nonlinear analysis, W. A. Benjamin, New York (1968).
R. Orsi, U. Helmke, J. B. Moore, A Newton-like method for solving rank constrained linear matrix inequalities, Automatica, 42, № 11, 1875–1882 (2006). DOI: https://doi.org/10.1016/j.automatica.2006.05.026
J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan, 284–289 (2004).
D. F. Coutinho, A. Trofino, $H_∞$ Output feedback control for a class of nonlinear systems, Proc. of the 2004 American Control Conference, Boston, Massachusetts, June 30–July 2, 3017–3022 (2004). DOI: https://doi.org/10.23919/ACC.2004.1384371
Copyright (c) 2024 Олексій Григорович Мазко
This work is licensed under a Creative Commons Attribution 4.0 International License.
