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

Abstract

New 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.

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.…”

“…Fortunately, solution bounds of the CALE (1) can be utilized to solve problems [17]. 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].…”

“…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. Surveying existing works, matrix bounds of the solution of the CALE (1) have been proposed in [2,3,6,[12][13][14][15]20,22]. However, all of those estimations have some limitations.…”

“…However, all of those estimations have some limitations. In [2,5,[12][13][14], it must be assumed that the matrix Q is positive definite and results in [3,12,13] are nonlinear functions of some free variables. Besides, bounds proposed in [14] possess a free positive matrix.…”

Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

“…In [19][20][21], upper matrix bounds for the solution of the CCARE have been presented and iterative algorithms have been proposed to derive tighter upper matrix bounds. And there are many other works for studying the solutions of the CCARE, such as matrix bounds and properties [22][23][24][25][26][27], matrix eigenvalue bounds [28][29][30], numerical solution [31][32][33], and the explicit solution [34,35].…”