The Problem of Generalized D -Stability in Unbounded LMI Regions and Its Computational Aspects
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The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects Olga Y. Kushel1 · Raffaella Pavani2
Received: 28 April 2020 / Revised: 28 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (socalled LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known particular cases of unbounded LMI regions (namely, conic sectors and shifted halfplanes) are considered. A new D-stability criterion as well as sufficient conditions for generalized D-stability are analyzed. Several applications of the developed theory to dynamical systems are shown. Keywords D-stability · LMI regions · Hurwitz stability · Relative stability · Eigenvalue clustering Mathematics Subject Classification 15A18 · 15A12 · 34D10
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stability and D-stability with Respect to LMI Regions . . . . . . . . . 2.1 LMI Regions and Matrix Spectra Localization . . . . . . . . . . . 2.2 Links Between Matrix Stability and D-stability . . . . . . . . . . 2.3 Robust D-stability and Generalization of D-stability . . . . . . . . 2.4 Links Between D-stability and (D, D)-stability . . . . . . . . . . 3 “Forbidden Boundary” Approach . . . . . . . . . . . . . . . . . . . . 3.1 General “Forbidden Boundary” Approach . . . . . . . . . . . . . 3.2 “Forbidden Boundary” Conditions for Conic Regions . . . . . . . 3.3 “Forbidden Boundary” Conditions for the Classical Case D = C−
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Raffaella Pavani [email protected] Olga Y. Kushel [email protected]
1
Department of Mathematics, Shanghai University, Shangda Road 99, 200444 Shanghai, China
2
Department of Mathematics, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italia
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Journal of Dynamics and Differential Equations 4 Generalized D-stability Criteria Using Additive Compound Matrices . . . 4.1 Stability and D-stability Criteria Using Additive Compound Matrices 4.2 D-stability Criterion Using Additive Compound Matrices . . . . . . . 4.3 (D, D)-stability Criteria Using Additive Compound Matrices . . . . . 5 Computational Aspects and Numerical Examples . . . . . . . . . . . . . . 6 Applications to Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 6.1 Relative Stability of Mechanical Systems . . . . . . . . . . . . . . . . 6.2 Stability of Fractional-order Systems
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