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Robust Stability Conditions for a Family of Linear Discrete-Time Systems Subjected to Uncertainties

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Abstract

Robust stability conditions are established for a family of linear discrete-time systems subjected to uncertainties. The traditional approach, which involves the construction of a common quadratic Lyapunov function for the entire family of systems with uncertainty, often leads to the problem of conservatism. In this connection, constructing parametric quadratic Lyapunov functions seems promising. The main tools of the proposed approach are the apparatus of linear matrix inequalities and a modification, presented here, of the well-known Petersen’s lemma. A simple approach to finding the robust quadratic stability radius of the family in question is proposed in the paper as well. The corresponding optimization problems have the form of semidefinite programming and one-dimensional minimization, which can be easily solved numerically. The efficiency of our approach is demonstrated via a numerical example. The results obtained can be generalized to design problems for linear discrete-time systems subjected to uncertainties, to other robust statements, and to the case of exogenous disturbances.

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Funding

This work was partly financially supported by the Russian Foundation for Basic Research, project no. 18-08-00140.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    M. V. Khlebnikov & Ya. I. Kvinto

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  1. M. V. Khlebnikov
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  2. Ya. I. Kvinto
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Correspondence to M. V. Khlebnikov or Ya. I. Kvinto.

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Translated by V. Potapchouck

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Khlebnikov, M.V., Kvinto, Y.I. Robust Stability Conditions for a Family of Linear Discrete-Time Systems Subjected to Uncertainties. Autom Remote Control 82, 1456–1462 (2021). https://doi.org/10.1134/S0005117921080117

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  • DOI: https://doi.org/10.1134/S0005117921080117

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