Abstract:Matrix bounds of the solutions of the continuous and discrete Riccati equations-a unified approach
CHIEN-HUA LEE{In this paper, a new scheme is introduced to measure the matrix bounds of the continuous and discrete Riccati equations. By estimating upper and lower matrix bounds of the solution of the unified algebraic Riccati equation (UARE), the same measurements for the solutions of the continuous and discrete Riccati equations, respectively, can be obtained in limiting cases. According to these obtained matr… Show more
“…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.…”
Section: Remark 22mentioning
confidence: 83%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.
“…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.…”
Section: Remark 22mentioning
confidence: 83%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.
“…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.…”
Section: Introductionmentioning
confidence: 98%
“…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.…”
In linear discrete system, we often need to utilize similarity transformation to extend the solution ranges of its corresponding discrete matrix equations. Consequently, how to choose the similarity transformation matrix remains an open question in recent years. In this paper, by applying the theory of matrix's Jordan canonical form and the related properties of nonnegative matrices, we construct the similarity transformation matrix of some special similarity transformation, then present a necessary and sufficient condition and a corresponding algorithm, thus solve the open question totally.
Redundant control inputs play an important part in engineering and are often used in
control problems, dynamic control allocation, quadratic performance optimal control, and many uncertain systems. In this paper, by the equivalent form of the discrete algebraic Riccati equation (DARE), we propose new upper and lower bounds of the solution for the equivalent DARE. Compared with some existing work on this topic, the new bounds are more tighter. Next, when increasing the columns of the input matrix, we give the applications of these new upper and lower solution bounds to obtain a sufficient condition for strictly decreasing feedback controller gain. Finally, corresponding numerical examples illustrate the effectiveness of our results.
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