Automation and Remote Control

, Volume 80, Issue 9, pp 1645–1652 | Cite as

Transient Response in Matrix Discrete-Time Linear Systems

  • B. T. PolyakEmail author
  • G. V. SmirnovEmail author
Topical Issue


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.


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


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The work of B.T. Polyak was supported by the Russian Science Foundation, project no. 16-11-10015. We are grateful to P.S. Shcherbakov, M. Danilova and A. Kulakova for careful reading of the manuscript and helpful remarks.


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.University of MinhoBragaPortugal

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