On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Yadykin, Igor B.; Galyaev, A. A.

doi:10.1134/s0005117913020045

Cited by 16 publications

(4 citation statements)

References 11 publications

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“…In the monograph [25], there appeared formulas for computing the elements of the controllability Gramian for a system in which the equations of state are represented in diagonal canonical form. A similar result was previously obtained, by a different method in [32], for computing the controllability and observability Gramians in a broader setting, taking into account the multiplicity of the eigenvalues of the dynamics matrix. It is shown that Gramian matrices are pseudo-Hankel matrices-Xiao matrices.…”

Section: Discussionsupporting

confidence: 80%

“…Proof. The proof of the general formulas is based on the results of [32], taking into account the separability properties of spectral expansion ns of Gramians. The validity of the formulas of the modal Lyapunov Equations ( 16)-( 19) is established by substituting the formulas into the original Lyapunov equations and taking into account the equalities…”

Section: Resultsmentioning

confidence: 99%

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Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

Yadykin

2023

Mathematics

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New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

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Section: Discussionsupporting

confidence: 80%

Section: Resultsmentioning

confidence: 99%

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

Yadykin

2023

Mathematics

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“…Approach Nr. 6 In order to improve the static stability, the quality of transient processes and the oscillation damping problems of the system synthesis of automatic control are solved. It is obvious, that the problem solving quality of the mode management and the oscillation damping depends on the problem solving quality of dynamic quality analysis.…”

Section: Approach Nr 5: Inverse Iterationmentioning

confidence: 99%

“…Artificial neuron networks (ANN) [8]; 3. Methods for computing the Gramians [6]; 4. The pseudo-spectral analysis [9] 5.…”

Section: A Brief Description Of Approaches Used Up-to-date While mentioning

confidence: 99%

Localization of the roots of the characteristic polynomial (CP) and isolines to analyze static stability

Zicmane

Kovaļenko

Sobolevskis

et al. 2014

2014 55th International Scientific Conference on Power and Electrical Engineering of Riga Technical University (RTUCON)

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The market economy dictates the terms of unification of the energy systems of the countries that leads to a change in number of interconnections, the increase of the transmitted power, and therefore, it is necessary to reconfigure the networks and, consequently, the recalculation of their normal modes, as well as the project assessment of static stability is required. Stability is an important constraint in power system operation and control. Power system stability is understood as the ability to regain an equilibrium state after being subjected to a physical disturbance. A key method among those of static stability analysis is the determination of system stability on the consideration of eigenvalues in steady state condition. This provides appropriate strategy to have a quick decision on the system's steady state behavior. The configuration of the electrical systems should be often changed in terms of market relations. The research is aimed to overcome the difficulties while dealing with the characteristic polynomial of a higher degree.Latvian power systems static stability is evaluated applying the methods of localization of the roots of the CP and isolines.

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Characterization of power systems near their stability boundary using the sub-Gramian method

Yadykin

Kataev

Iskakov

et al. 2016

Control Engineering Practice

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On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Cited by 16 publications

References 11 publications

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

Localization of the roots of the characteristic polynomial (CP) and isolines to analyze static stability

Characterization of power systems near their stability boundary using the sub-Gramian method

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