Automation and Remote Control

, Volume 76, Issue 6, pp 957–976 | Cite as

Large deviations in linear control systems with nonzero initial conditions

  • B. T. PolyakEmail author
  • A. A. Tremba
  • M. V. Khlebnikov
  • P. S. Shcherbakov
  • G. V. Smirnov
Linear Systems


Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper [2] by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane.

In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.


Remote Control Linear Matrix Inequality Lyapunov Equation Companion Form Root Location 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • B. T. Polyak
    • 1
    • 2
    Email author
  • A. A. Tremba
    • 1
  • M. V. Khlebnikov
    • 1
  • P. S. Shcherbakov
    • 1
  • G. V. Smirnov
    • 3
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of ScienceMoscowRussia
  2. 2.ITMO UniversitySt. PetertsburgRussia
  3. 3.University of MinhoBragaPortugal

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