Matrix bounds of the discrete ARE solution

This paper provides new lower and upper matrix bounds of the solution to the discrete algebraic Riccati equation. The lower bound always works if the solution exists. The upper bounds are presented in terms of the solution of the discrete Lyapunov equation and its upper matrix bound. The upper bounds are always calculated if the solution of the Lyapunov equation exists. A numerical example shows that the new bounds are tighter than previous results in many cases.

[1]  T. Mori,et al.  On the discrete Riccati equation , 1987 .

[2]  I. Troch.  Improved bounds for the eigenvalues of solutions of Lyapunov equations , 1987 .

[3]  T. Mori,et al.  A brief summary of the bounds on the solution of the algebraic matrix equations in control theory , 1984 .

[4]  Chien-Hua Lee.  Upper matrix bound of the solution for the discrete Riccati equation , 1997, IEEE Trans. Autom. Control..

[5]  K. Hirai,et al.  Upper and lower bounds on the solution of the algebraic Riccati equation , 1979 .

[6]  A. Laub,et al.  On the numerical solution of the discrete-time algebraic Riccati equation , 1980 .

[7]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[8]  N. Komaroff,et al.  Lower summation bounds for the discrete Riccati and Lyapunov equations , 1992 .

[9]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[10]  Young Soo Moon,et al.  Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results , 1996 .

[11]  Myung-Joong Youn,et al.  On bounds of the Riccati and Lyapunov matrix equations , 1985 .

[12]  M. T. Tran,et al.  A note on the discrete Lyapunov and Riccati matrix equations , 1984 .

[13]  N. Komaroff,et al.  Iterative matrix bounds and computational solutions to the discrete algebraic Riccati equation , 1994, IEEE Trans. Autom. Control..

[14]  J. Garloff.  Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equations and the continuous Lyapunov equation , 1986 .

[15]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.