On the estimation of solution bounds of the generalized Lyapunov equations and the robust root clustering for the linear perturbed systems

Abstract

On the estimation of solution bounds of the generalized Lyapunov equations and the robust root clustering for the linear perturbed systems CHIEN-HUA LEE{ and SU-TSUNG LEE{This paper measures the solution bounds for the generalized Lyapunov equations (GLE). By making use of linear algebraic techniques, we estimate the upper and lower matrix bounds for the solutions of the above equations. All the proposed bounds are new, and it is also shown that the majority of existing bounds are the special cases of these results. Furthermore, according to these bounds, the problem of robust root clustering in sub-regions of the complex plane for linear time-invariant systems subjected to parameter perturbations is solved. The tolerance perturbation bounds for robust clustering in the given sub-regions are estimated. Compared to previous results, the feature of these tolerance bounds is that they are independent of the solution of the GLE.

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

“…Therefore, during the past two decades, research for deriving solution bounds including matrix and eigenvalue bounds of the CALE (1) has become an attractive topic. A number of research approaches for this topic have been proposed in the literature [2,3,[5][6][7][8][9][11][12][13][14][15][16][20][21][22]24]. Among those solution bounds, the matrix bounds can define all eigenvalue bounds such as bounds of the extreme eigenvalues, the summation of eigenvalues, the trace, the product of eigenvalues, and the determinant; hence they are the most general findings.…”

“…Davies et al [3] have also pointed out that instead of solving the CALE (1) for the solution matrix P, one can use solution bounds in place of the exact solution P to solve the optimization problem for a linear system. Besides, it is found that they can be applied to treat many control problems such as robust stability analysis for time-delay systems [17,25], robust root clustering [16,26], determination of the size of the estimation error for multiplicative systems [10], and so on [19]. In the literature, Gajic and Qureshi [4] also explained a motivation for studying the solution bounds but not the exact solution of the CALE (1).…”

The open question of the relation between square matrix’s eigenvalues and its similarity matrix’s singular values in linear discrete system

“…However, unfortunately, most of the bounds are based on the assumption 1 ( ) 1 T AA λ < [8][9][10][11][12][13][14]. To remove the strong assumptions, Dong-Gi in [7] have firstly utilized similarity transformation and provided some upper bounds without the assumption that the system is asymptotically stable.…”

“…For the discrete Lyapunov matrix equation, many researchers have interest in estimating the solution bounds during the past three decades [7][8][9][10][11][12][13][14]. However, unfortunately, most of the bounds are based on the assumption 1 ( ) 1 T AA λ < [8][9][10][11][12][13][14].…”

Further results on the measurement of solution bounds of the generalized Lyapunov equations

“…However, it might be troublesome to solve the GLE especially when the dimensions of a system become large. Recently, by extending the methods developed by Lee [5][6][7] and making use of linear algebraic techniques, the estimation problem of the solution bounds of the GLE was first treated in [17]. Some matrix bounds and several eigenvalue bounds for the solutions of the GLE were developed.…”