Abstract
We use difference inclusions to describe the dynamics of a family of nonlinear discrete systems subject to bounded disturbances. For a family of linear discrete systems, we get an analytic solution of the problem of finding the invariant set, and for families of nonlinear systems, we propose an iterative process that finds their invariant set and converges with the speed of a geometric progression. We also provide illustrative examples.
Similar content being viewed by others
References
Bulgakov, B.V., On Collecting Disturbances in Linear Oscillatory Systems with Constant Parameters, Dokl. Akad. Nauk USSR, 1946, vol. 51, no. 5, pp. 7–15.
Gnoenskii, L.S., Bulgakov’s Problem on Collecting Disturbances, in Zadacha Bulgakova o maksimal’nom otklonenii i ee primenenie (Bulgakov’s Maximal Deviation Problem and Its Applications), Aleksandrov, V.V., Ed., Moscow: Mosk. Gos. Univ., 1993.
Kuntsevich, V.M. and Pshenichnyi, B.N., Minimal Invariant Sets of Dynamic Systems with Bounded Disturbances, Kibern. Sist. Anal., 1996, no. 1, pp. 74–81.
Kuntsevich, V.M. and Pshenichny, B.N., Analysis of Some Classes of Nonlinear Discrete Systems under Bounded Disturbances, Preprint 4th Sympos. IFAC Nonlinear Control Systems Design, 1998.
Schweppe, F., Uncertain Dynamic Systems, New Jersey: Prentice Hall, 1973.
Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezul’tata (Controlling a Dynamic System. The Minimum Guaranteed Result Problem), Moscow: Nauka, 1985.
Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertainty), Moscow: Nauka, 1977.
Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem (Estimating Phase States of Dynamical Systems), Moscow: Nauka, 1988.
Glover, D. and Schweppe, F., Control of Linear Dynamic Systems with Set Constrained Disturbances, IEEE Trans. Automat. Control, 1971, vol. 16, pp. 411–423.
Kuntsevich, V.M., Upravlenie v usloviyakh neopredelennosti: garantirovannye rezul’taty v zadachakh upravleniya i identifikatsii (Control under Uncertainty: Guaranteed Results in Control and Identification Problems), Kiev: Naukova Dumka, 2006.
Nazin, S.A., Polyak, B.T., and Topunov, M.V., Rejection of Bounded Exogenous Disturbances by the Method of Invariant Ellipsoids, Autom. Remote Control, 2007, vol. 68, no. 3, pp. 467–486.
Polyak, B.T., Shcherbakov, P.S., and Topunov, M.V., Invariant Ellipsoids Approach to Robust Rejection of Persistent Disturbances, Proc. 17th World IFAC Congr., Seoul, July 6–11, 2008, pp. 3976–3981.
Blanchini, F. and Miani, S., Set-theoretic Methods in Control, Boston: Birkhauser, 2008.
Boyd, S., Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 2008.
Kuntsevich, V.M. and Polyak, B.T., Invariant Sets of Nonlinear Discrete Systems wih Bounded Perturbances and Control Problems, Probl. Upravlen. Informat., 2009, no. 6, pp. 6–21.
Barbashin, E.A, Funktsii Lyapunova (Lyapunov Functions), Moscow: Nauka, 1978.
Kuntsevich, A.V. and Kuntsevich, V.M., Stability in the Region of Nonlinear Difference Inclusions, Kibern. Sist. Anal., 2010, no. 5, pp. 11–17.
Kappel, F. and Kuntsevich, A.V., An Implementation of Shor’s r -algorithm, Comput. Optim. Appl., 2000, vol. 15, no. 2, pp. 193–205.
Lur’e, A.I. and Postnikov, V.N., On Stability Theory of Controllable Systems, Prikl. Mat. Mekh., 1944, vol. 8, no. 3, pp. 246–248.
Aizerman, M.A. and Gantmakher, F.R., Absolyutnaya ustoichivost’ reguliruemykh sistem (Absolute Stability in Controllable Systems), Moscow: Akad. Nauk SSSR, 1963.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.V. Kuntsevich, V.M. Kuntsevich, 2012, published in Avtomatika i Telemekhanika, 2012, No. 1, pp. 92–106.
Rights and permissions
About this article
Cite this article
Kuntsevich, A.V., Kuntsevich, V.M. Invariant sets for families of linear and nonlinear discrete systems with bounded disturbances. Autom Remote Control 73, 83–96 (2012). https://doi.org/10.1134/S0005117912010067
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117912010067