Abstract
This paper introduces a new class of controls that ensure an effect similar to that produced by conventional matching conditions between control and disturbance inputs in a linear system, but now for a broader class of such inputs. Namely, this is due to an application of piecewise-constant control functions with varying amplitudes, generated by approximations of “ideal controls,” which are linear combinations of delta functions and their higher order derivatives. Such a class allows to calculate feedback control solutions by solving problems of open-loop control, thus reducing the overall computation burden.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Symbol \(\mathop {\dot{-}}\) denotes the geometric (Minkowski) difference of the sets: .
References
Başar, T., Bernhard, P.: \(H^\infty \)-Optimal Control and Related Minimax Design Problems. Birkhäuser, Boston (1995)
Bensoussan, A., Lions, J.L.: Contrôle impulsionnel et inéquations quasi variationnelles. Dunod, Paris (1982)
Dar’in, A.N., Kurzhanski, A.B.: Control synthesis in a class of higher-order distributions. Diff. Equ. 43(11), 1479–1489 (2007)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions. In: Properties and Operations, vol. 1. Academic Press, NY (1964)
Krasovski, N.N.: Rendezvous Game Problems. National Technical Information Service, Springfield (1971)
Krasovskii, N.N., Subbotin, A.I.: Game-theoretical control problems. (1988)
Kurzhanski, A.B.: Pontryagin’s alternated integral in the theory of control synthesis. Proc. Steklov Inst. Math. 224, 212–225 (1999)
Kurzhanski, A.B., Daryin, A.N.: Dynamic programming for impulse controls. Annu. Rev. Control 32(2), 213–227 (2008)
Kurzhanski, A.B., Osipov, Y.S.: On controlling linear systems through generalized controls. Diff. Equ. 5(8), 1360–1370 (1969) (In Russian)
Leitmann, G.: Optimality and reachability with feedback control. In: Avez, A., Blaquière, A., Marzollo, A. (eds.) Dynamical Systems and Microphysics: Geometry and Mechanics, pp. 119–141. Elsevier (1982)
Polovinkin, E.S., Balashov, M.V.: Elements of Convex and Strongly Convex Analysis. Fizmatlit, Moscow (2004) (In Russian)
Rockafellar, R.T., Wets, R.J.B.: Variational analysis. In: Grundlehren der mathematischen Wissenschaften, vol. 317. Springer (1998)
Schwartz, L.: Théorie des distributions. Hermann, Paris (1950)
Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow (1979)
Vostrikov, I.V., Dar’in, A.N., Kurzhanski, A.B.: On the damping of a ladder-type vibration system subjected to uncertain perturbations. Diff. Equ. 42(11), 1524–1535 (2006)
Acknowledgments
This work is supported by the Russian Foundation for Basic Research (grants 12-01-00261-a, 12-01-31416-mol-a) and by the program State Support of the Leading Scientific Schools (grant NS-2239.2012.1).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kurzhanski, A.B., Daryin, A.N. (2016). Attenuation of Uncertain Disturbances Through Fast Control Inputs. In: Dimirovski, G. (eds) Complex Systems. Studies in Systems, Decision and Control, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-319-28860-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-28860-4_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28858-1
Online ISBN: 978-3-319-28860-4
eBook Packages: EngineeringEngineering (R0)