Structural Spectral Methods for Solving Continuous Lyapunov Equations

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For linear multivariable continuous stationary stable control systems with a simple spectrum, presented in the form of a canonical diagonal form, controllability and observability forms, a method was developed and analytical formulas for spectral decompositions of gramians in the form of various Xiao matrices were obtained. A method and algorithm for calculatin  generalized Xiao matrices in the form of the Hadamard product for multiply connected continuous
linear systems with many inputs and many outputs have been developed. This allows us to calculate the elements of the corresponding controllability and observability gramians in the
form of products of the corresponding elements of the multiplier matrices and a matrix that is the sum of all possible products of the numerator matrices of the matrix transfer function of the system. New results are obtained in the form of spectral and singular decompositions of the inverse gramians of controllability and observability. This makes it possible to obtain invariant decompositions of energy functionals and formulate new criteria for the stability of linear systems taking into account the nonlinear effects of mode interaction.

作者简介

I. Yadykin

Trapeznikov Institute of Control Sciences, Russian Academy of Sciences

Email: jad@ipu.ru
Moscow, Russia

I. Galyaev

Trapeznikov Institute of Control Sciences, Russian Academy of Sciences

编辑信件的主要联系方式.
Email: ivan.galyaev@yandex.ru
Moscow, Russia

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