Upper and lower bounds of the solutions of the discrete algebraic Riccati and Lyapunov matrix equations

Abstract

Upper and lower bounds of the solutions of the discrete algebraic Riccati and Lyapunov matrix equationsCHIEN-HUA LEE² Matrix bounds, upper and lower, for the solution of the discrete algebraic Riccati and Lyapunov equations respectively, are proposed in this paper. They are new or sharper than the majority of existing results. By making use of these new matrix bounds, the corresponding bounds for each eigenvalue, the trace and the determinant of these solution matrices are also presented. Comparisons are made between these bounds and the majority of those appearing in the literature. It is shown that the obtained results are tighter. Finally, we give illustrative examples to show that these estimates appear to be considerably stronger than previously available results in many cases. Notation real number ® eld A Â transpose of a matrix A Î n´m |a| the absolute value of a complex number a i ( A) the i th eigenvalue of A Î n´n , all of |¸i ( A) | are arranged in nonincreasing order, i.e. |¸1 ( A) | |¸2 ( A) | ´´´ |¸n ( A) | If A is symmetric, then all of¸i( A) are arranged in non-increasing order, i.e.¸1 ( A) ¸2( A) ´´´ ¸n( A) s i ( A) the i th singular value of A Î n´m , the values of s i (X) are arranged in non-increasing order, i.e. s 1 ( A) s 2 (A) ´´´ s n ( A) tr ( A) the trace of matrix A Î n´n det ( A) the determinant A Î n´n

New upper solution bounds of the discrete algebraic Riccati matrix equation

“…Furthermore, we have, for this case, that the bound P U 1 only works when 2 1 (A) < 1, whilst the bound P U 2 only works when 1 + 2 n (B) > 2 1 (A) and Q is nonsingular. For the remaining upper matrix bounds existing in the literature [9,14,16,19,20,2,3,17], one can see that the presented bounds cannot be compared with the existing ones by any mathematical method. However, comparison via a numerical example is always possible.…”

“…Of these bounds, the matrix bounds are the most general, because they can directly offer all other types of bounds mentioned. However, viewing the literature [9,2,3,5,[14][15][16][17]19,20], it appears that all proposed upper matrix bounds for the DARE have been developed under assumptions additional to the fundamental existence conditions for the DARE solution. Therefore, this note develops two upper matrix bounds, of which the bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists.…”

“…In practice, analytical solution of this equation is complicated, particularly when the dimensions of the system matrices are high. As such, a number of works have been presented over the past three decades for deriving solution bounds of this equation [2][3][4][5][7][8][9][10][11][12][13][14][15][16][17][18][19][20]22,[24][25][26], to reduce the computational burdens required to solve it analytically. Not only do solution bounds provide estimates for the solution of this equation, but they can also be applied to deal with practical situations involving the solution of this equation.…”

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

“…For the discrete Riccati matrix equation, numerous jobs have been devoted to the estimation of the extent or size of the solution during the past three decades [15][16][17][18][19][20][21][22][23]. However, viewing the literatures [15][16][17][18][19][20][21][22], it seems that all proposed bounds have been developed under additional fundamental existing assumptions.…”

“…However, viewing the literatures [15][16][17][18][19][20][21][22], it seems that all proposed bounds have been developed under additional fundamental existing assumptions. To remove the restrictions, Davies et al [23] have developed the following matrix bound by using similarity transformation, which is always calculated if the stabilizing solution exists.…”