Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products
View through CrossRef
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.
Title: Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products
Description:
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.
Related Results
Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms
Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms
The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation ...
Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms
Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms
The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation ...
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
New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the ...
The Application of S‐transform Spectrum Decomposition Technique in Extraction of Weak Seismic Signals
The Application of S‐transform Spectrum Decomposition Technique in Extraction of Weak Seismic Signals
AbstractIn processing of deep seismic reflection data, when the frequency band difference between the weak useful signal and noise both from the deep subsurface is very small and h...
Element-wise Multiplicative Operations in Neural Architectures: A Comprehensive Survey of the Hadamard Product
Element-wise Multiplicative Operations in Neural Architectures: A Comprehensive Survey of the Hadamard Product
The Hadamard product, also known as the element-wise multiplication operation, has increasingly emerged as a fundamental primitive in the design and analysis of modern deep learnin...
On Empirical System Gramians
On Empirical System Gramians
AbstractState‐space realizations of input‐output systems or control systems are a widely used class of models in engineering, physics, chemistry and biology. For the qualitative an...
On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems
On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems
The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also show...
Structural Spectral Methods for Solving Continuous Lyapunov Equations
Structural Spectral Methods for Solving Continuous Lyapunov Equations
For linear multivariable continuous stationary stable control systems with a simple spectrum, presented in the form of a canonical diagonal form, controllability and observability ...