2D system analysis via dual problems and polynomial matrix inequalities

Vladimir  Pozdyayev

doi:10.3934/naco.2016022

Article Contents

2016, Volume 6, Issue 4: 491-504. Doi: 10.3934/naco.2016022

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2D system analysis via dual problems and polynomial matrix inequalities

Vladimir Pozdyayev^1,

1.
Arzamas Polytechnic Institute, Alekseev Nizhny Novgorod State Technical University, 607220, Arzamas

Received: September 2015

Revised: November 2016

Published: November 2016

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Abstract

Application of the Lyapunov method to 2D system stability and performance analysis yields algebraic systems that can be interpreted as either sum-of-squares problems for nontrivial matrix polynomials, or parameterized linear matrix inequalities that need to be satisfied for certain ranges of parameter values. In this paper we show that dualizing core inequalities in the latter forms allows converting these systems to conventional optimization problems on sets described by polynomial matrix inequalities. Methods for solving these problems include moment-based methods or the “atomic optimization” method proposed earlier by the author. As a result, we obtain necessary conditions for 2D system stability and lower bounds on system performance. In particular, we demonstrate respective results for discrete-discrete system stability and mixed continuous-discrete system $\mathcal{H}_\infty$ performance. A numerical example is provided.

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Mathematics Subject Classification: Primary: 93D05, 90C26; Secondary: 49M29.

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