Continuous Algebraic Lyapunov Equation Research Articles

A unified approach of the measurement of solution bounds of the continuous and discrete algebraic Lyapunov equations

In this paper, an integrated study for measuring the solutions of the continuous and discrete Lyapunov equations is developed. A unified algebraic Lyapunov equation (UALE) will be given first. Then, by using a bilinear transformation, the mentioned equation can be transformed into the quasi-standard form of the discrete algebraic Lyapunov equation (DALE). Thus, those approaches appeared in the literature for solving the estimation problem of the solution of the Lyapunov equations may be used to solve the same problem for the UALE. By extending mainly the approach proposed in [20] for the continuous algebraic Lyapunov equation (CALE) associated with linear algebraic techniques, several upper and lower matrix bounds of the solution of the UALE are developed. Then, matrix bounds of the solutions of the CALE and DALE can be obtained directly by these obtained matrix bounds. Algorithms for getting better bounds of the solutions of the CALE and DALE are also given. By comparisons, the obtained solution bounds of the CALE and DALE are less restrictive or easier to be calculated than those parallel results appeared in the literature. Finally, numerical examples have given to demonstrate the merits of the proposed approach.

Experimental Study of Lyapunov Equation Solution Bounds for Power Systems Models

Abstract This research work investigates the applicability of some lower and upper, matrix and scalar bounds for the solution of the Continuous Algebraic Lyapunov Equation (CALE), when the coefficient matrices are the state matrices of real data models. The bounds are illustrated for two different models describing the dynamic behavior of power systems - a two-area power system and an interconnected power system. Some important conclusions referring to the accuracy of the respective estimates are made, as well.

On the robust stability of interval time-delay systems – An application of the upper solution bounds of the Lyapunov equation

In this paper, the robust stability test problem of the interval time-delay systems is discussed. We first study the upper solution bounds of the continuous algebraic Lyapunov equation (the CALE). By using linear algebraic techniques, a simple approach is proposed to derive new upper bounds of the CALE. Compared to existing works on this topic, these newly obtained bounds are less restrictive and easier to calculate. Then, we apply them to solve the mentioned problem. By using Lyapunov equation approach associated with these upper bounds, new concise criteria for the robust stability and decay rate test problem of the mentioned systems are presented. Comparing to some well-known results of the time-delay systems, the obtained criteria are sharper. An interesting consequence of these results is that it is not necessary to solve any Lyapunov equation although the Lyapunov equation approach is used.

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

By using a bilinear transformation and some linear algebraic techniques, new matrix bounds of the solution of the continuous algebraic Lyapunov equation (CALE) are derived in this paper. Comparing to existing works, these obtained matrix bounds are less restrictive and are easy to be calculated. A numerical example is also given to demonstrate the merits of the present results.

New lower matrix bounds for the solution of the continuous algebraic lyapunov equation

AbstractNew lower matrix bounds are derived for the solution of the continuous algebraic Lyapunov equation (CALE). Following each bound derivation, an iterative algorithm is proposed to derive tighter matrix bounds. In comparison to existing results, the presented results are more concise and are always valid when the CALE has a non‐negative definite solution. We finally give numerical examples to show the effectiveness of the derived bounds and make comparisons with existing results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

New Upper Estimates for the Solution of the Continuous Algebraic Lyapunov Equation

A new result for bounding the summations for solution eigenvalues of the algebraic Lyapunov equation is presented. This makes possible to generate the best known upper scalar solution estimates.

On the upper and lower bounds of the solution for the continuous Riccati matrix equation

In this paper, concise upper and lower matrix bounds for the solution of the continuous Riccati equation are presented. These results are new and the lower matrix bound is better than the previously reported one. According to these estimates, all the bounds for eigenvalues, trace,determinant, etc, can then be defined. A lower matrix bound and eigenvalue bounds of the solution of the continuous algebraic Lyapunov equation are also determined. The main feature of this paper is thatallthe presentresults are new and are sharper than the majority ofthose reported in the literature. Finally, we give an illustrative example to make comparisons for the present schemes and existing bounds.

New results for the bounds of the solution for the continuous Riccati and Lyapunov equations

This paper presents upper and lower matrix bounds for the solution of the continuous algebraic matrix Riccati equation. Furthermore, a new lower matrix bound for the solution of the continuous algebraic Lyapunov equation is also developed. These are new results.

A Finite Series Solution of the Matrix Equation $AX - XB = C$

A Finite Series Solution of the Matrix Equation $AX - XB = C$