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Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products
Version 1
: Received: 28 November 2023 / Approved: 29 November 2023 / Online: 29 November 2023 (10:20:27 CET)
A peer-reviewed article of this Preprint also exists.
Yadykin, I. Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products. Mathematics 2024, 12, 36. Yadykin, I. Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products. Mathematics 2024, 12, 36.
Abstract
New possibilities of Gramian computation by using canonical transformations into diagonal, controllable and observable canonical forms are shown. With the help of such a technique the Gramian matrices can be represented in the form of products of Hadamard matrices of multipliers and matrices of the transformed right-hand side of Lyapunov equations. It is shown that the 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 element-by-element computation of Gramian matrices for stable continuous MIMO LTI systems are developed. For continuous SISO LTI systems given by equations of state in controllable and observable canonical forms, new algorithms for the computation of controllability Gramians and their traces in the form of Hadamard products in the form of Xiao matrices are developed. The application of transformations to the canonical forms of controllability and observability allowed to simplify the formulas of spectral decompositions in the form of Xiao matrices. In the paper new spectral decompositions in the form of Hadamard’s products for solutions of Sylvester algebraic and differential equations of MIMO LTI systems, including spectral decompositions of finite and infinite cross-Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.
Keywords
spectral decompositions; linear continuous systems, Gramians; Sylvester and Lyapunov equations; Xiao matrices; Hadamard product.
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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