Abstract
In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.
Highlights
It is known that the Lyapunov equation is widely used in various control systems
Robust stability analysis for time-delay systems, robust root clustering for linear systems, determination of the size of the estimation error for multiplicative systems, and others can be solved by the mentioned solution bounds
Gajic and Qureshi [1] explained one motive for studying the solution bounds of the Lyapunov equation: sometimes we are interested in the general behavior of the underlying system, and this behavior can be determined by examining certain bounds on the parameters of the solution, rather than the full solution
Summary
It is known that the Lyapunov equation is widely used in various control systems. solution bounds of the above equation can treat many control problems. During the past few decades, research on deriving solution bounds of the Lyapunov equation has become an attractive research topic, and a number of research approaches have been proposed to this problem [2]-[9]. Among those results, they focus on the evaluation for the bounds of single eigenvalues including the extreme ones, the trace, the determinant, as well as the bounds of solution matrix. It seems that most of these approaches for the matrix bounds contain points of weakness Those results proposed in [5]-[7] must assume that the matrix Q is positive definite. In comparison with existing literature on the subject, the proposed results are less restrictive and more calculated
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