CONSTRUCTION OF STABILITY AREAS FOR CONTROLLED SYSTEMS WITH PARAMETRIC AND DYNAMIC UNCERTAINTY
DOI:
https://doi.org/10.30837/ITSSI.2021.17.117Keywords:
asymptotic stability, Lyapunov vector functions, parametric uncertainty, small parameterAbstract
The subject of research in the article is sigularly perturbed controllable systems of differential equations containing terms with a small parameters on the right-hand side, which are not completely known, but only satisfy some constraints. The aim of the work is to expand the study of the behavior of solutions of singularly perturbed systems of differential equations to the case when the system is influenced not only by dynamic (small factor at the derivative) but also parametric (small factor at the right side of equations) uncertainties and to determine conditions under which such systems will be asymptotically resistant to any perturbations, estimate the upper limit of the small parameter, so that for all values of this parameter less than the obtained estimate, the undisturbed solution of the system was asymptotically stable. The following problems are solved in the article: singularly perturbed systems of differential equations with regular perturbations in the form of terms with a small parameter in the right-hand sides, which are not fully known, are investigated; an estimate is made of the areas of asymptotic stability of the unperturbed solution of such systems, that is, the class of systems that can be investigated for stability is expanded, the formulas obtained that allow one to analyze the asymptotic stability of solutions to systems even under conditions of incomplete information about the perturbations acting on them. The following methods are used: mathematical modeling of complex control systems; vector Lyapunov functions investigation of asymptotic stability of solutions of systems of differential equations. The following results were obtained: an estimate was made for the upper bound of a small parameter for sigularly perturbed systems of differential equations with fully known parametric (fully known) and dynamic uncertainties, such that for all values of this parameter less than the obtained estimate, such an unperturbed solution is asymptotically stable; a theorem is proved in which sufficient conditions for the uniform asymptotic stability of such a system are formulated. Conclusions: the method of vector Lyapunov functions extends to the class of singularly perturbed systems of differential equations with a small factor in the right-hand sides, which are not completely known, but only satisfy certain constraints.
References
Tikhonov, A. (1952), Systems of differential equations containing small parameters with derivatives [Sistemy differencial'nyh uravnenij, soderzhashhie malye parametry pri proizvodnyh], Mathematical collection, Vol. 31 (73), No. 3, P. 575–586.
Vasilyeva, A., Butuzov, V. (1990), Asymptotic methods in the theory of singular perturbations [Asimptoticheskie metody v teorii singuljarnyh vozmushhenij], Higher school, Mosсow, 352 p.
Hoppensteadt, F. (1971), "Property of solutions of ordinary differential equations with small parameters", Commun. on pure and applied mathematics, Vol. XXIV, P. 807–840.
Kachalov, V. (2017), "On holomorphic regularization of singularly perturbed systems of differential equations" ["O golomorfnoj reguljarizacii singuljarno vozmushhennyh sistem differencial'nyh uravnenij"], Journal of Computational Mathematics and Mathematical Physics, Vol. 57, No. 4, P. 64–71.
Kachalov, V. (2018), "On a method for solving singularly perturbed systems of Tikhonov type" ["Ob odnom metode reshenija singuljarno vozmushhennyh sistem tihonovskogo tipa"], Proceedings of universities. Mathematics, No. 6, P. 25–30.
Tursunov, D. A., Kozhobekov, K. G. (2017), "Asymptotics of the solution of singularly perturbed differential equations with a fractional turning point" ["Asimptotika resheniya singulyarno vozmushchennyh differencial'nyh uravnenij s drobnoj tochkoj povorota"], Irkutsk State University Bulletin, Series : Mathematics, Vol. 21, P. 108–121. DOI: 10.26516/1997-7670.2017.21.108
Butuzov, V. (2018), "On a certain singularly perturbed system of ordinary differential equations with a multiple root of the degenerate equation", Nonlinear oscillations, Vol. 21, No. 1, P. 6–28.
Binning, H. S., Goodall, D. P. (1997), "Output control for an undefined singularly perturbed system" ["Upravlenie po vyhodu neopredelennoj singuljarno vozmushhennoj sistemoj"], Automation and telemechanics, No. 7, P. 81–97.
Kodra, K., Gajic, Z. (2017), "Optimal control for a new class of singularly perturbed linear systems", Automatica, Vol. 81, P. 203–208. DOI: 10.1016/j.automatica.2017.03.017
Li, Y., Wang, Y., Yao, D. (2018), "A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems", Automatica, No. 97, P. 404–413. DOI: 10.1016/j.automatica.2018.03.066
Liu, H. S., Huang, Y. (2018), "Robust adaptive output feedback tracking control for flexible-joint robot manipulators based on singularly perturbed decoupling", Robotica, Vol. 36, Р. 822–838.
Potapenko, E. M., Savranska, A. V. (1999), "Investigation of robust stability of a combined system with an unexpanded observer" ["Doslidzhennia robastnoi stiikosti kombinovanoi systemy z nerozshyrenym sposterihachem"], Bulletin of Zaporizhia University, No. 1, P. 108–113.
Potapenko, E. M., Savranska, A. V. (1999), "Advancement of the robustness of the control system with the help of the vector of non-value", ["Doslidzhennya robastnoyi stijkosti sy`stemy` upravlinnya zi sposterigachem vektora nevy`znachennosti"], Bulletin of Zaporizhia University, No. 2, P. 108–11.
Martynyuk, A. (1986), "Uniform asymptotic stability of a singularly perturbed system based on a matrix, the Lyapunov function" ["Ravnomernaja asimptoticheskaja ustojchivost' singuljarno vozmushhennoj sistemy na osnove matricy - funkcii Ljapunova"], Reports of the USSR Academy of Sciences, Vol. 287, No. 4, P. 786–789.
Xingwen Liu, Yongbin Yu, Hao Chen (2017), "Stability of perturbed switched nonlinear systems with delays", Nonlinear Analysis: Hybrid Systems, Vol. 25, P. 114–125. DOI: org/10.1016/j.nahs.2017.03.003
Potapenko, E., Savranskaya, A. (1998), "Generalization of Tikhonov's theorem for a singularly excited system" ["Uzagal`nennya teoremy` Ty`xonova dlya sy`ngulyarno-zbudzhenoyi sy`stemy`"], Bulletin of Zaporizhia University, No. 1, P. 61–65.
Potapenko, E., Savranskaya A. (1999), "Uniform asymptotic stability of a singularly excited system under constant excitations", ["Rivnomirna asy`mptoty`chna stijkist` sy`ngulyarno-zbudzhenoyi sy`stemy` pry` postijno diyuchy`x zbudzhennyax"], Bulletin of the University of Kiev, No. 4, P. 55–59.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Our journal abides by the Creative Commons copyright rights and permissions for open access journals.
Authors who publish with this journal agree to the following terms:
Authors hold the copyright without restrictions and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-commercial and non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their published work online (e.g., in institutional repositories or on their website) as it can lead to productive exchanges, as well as earlier and greater citation of published work.
