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JOURNALS // Avtomatika i Telemekhanika // Archive

Avtomat. i Telemekh., 2019 Issue 9, Pages 112–121 (Mi at15344)

This article is cited in 10 papers

Transient response in matrix discrete-time linear systems

B. T. Polyaka, G. V. Smirnovb

a Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
b University of Minho, Braga, Portugal

Abstract: The behavior of trajectories of multidimensional linear discrete-time systems with nonzero initial conditions is considered in two cases as follows. The first case is the systems with infinite degree of stability (the processes of a finite duration); the second case is the stable systems with a spectral radius close to 1. It is demonstrated that in both cases, large deviations of the trajectories from the equilibrium may occur. These results are applied to accelerated unconstrained optimization methods (such as the Heavy-ball method) for explaining the nonmonotonic behavior of the methods, which is observed in practice.

Keywords: discrete-time systems, transient response, stability, large deviations, infinite degree of stability, multidimensional systems, Heavy-ball method.


Received: 22.06.2018
Revised: 06.09.2018
Accepted: 08.11.2018

DOI: 10.1134/S0005231019090083


 English version:
Automation and Remote Control, 2019, 80:9, 1645–1652

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© Steklov Math. Inst. of RAS, 2025