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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 9, 2014

Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L

  • Maria Adam and Nicholas Assimakis EMAIL logo
From the journal Open Mathematics

Abstract

In this paper, we present two new algebraic algorithms for the solution of the discrete algebraic Riccati equation. The first algorithm requires the nonsingularity of the transition matrix and is based on the solution of a standard eigenvalue problem for a new symplectic matrix; the proposed algorithm computes the extreme solutions of the discrete algebraic Riccati equation. The second algorithm solves the Riccati equation without the assumption of the nonsingularity of the transition matrix; the proposed algorithm is based on the solution of the matrix equation X + A*X-1A=L, where A is a singular matrix and L a positive definite matrix.

References

[1] Adam M., Assimakis N., Matrix equations solutions using Riccati Equation, Lambert, Academic Publishing, 2012 10.5402/2012/625897Search in Google Scholar

[2] Adam M., Assimakis N., Sanida F., Algebraic Solutions of the Matrix Equations X ATX-1A=Q and X-ATX-1A=Q, International Journal of Algebra, 2008, 2(11), 501–518 Search in Google Scholar

[3] Adam M., Sanida F., Assimakis N., Voliotis S., Riccati Equation Solution Method for the computation of the extreme solutions of X A*X-1A=Q and X-A*X-1A=Q, IWSSIP 2009, Proceedings 2009 IEEE, 978-1-4244-4530-1/09, 41–44 10.1109/IWSSIP.2009.5367796Search in Google Scholar

[4] Anderson B.D.O., Moore J.B., Optimal Filtering, Dover Publications, New York, 2005 Search in Google Scholar

[5] Assimakis N., Sanida F., Adam M., Recursive Solutions of the Matrix Equations X+ATX-1A=Q and X-ATX-1A=Q Q; Applied Mathematical Sciences, 2008, 2(38), 1855–1872 Search in Google Scholar

[6] Assimakis N.D., Lainiotis D.G., Katsikas S.K., Sanida F.L., A survey of recursive algorithms for the solution of the discrete time Riccati equation, Nonlinear Analysis, Theory, Methods & Applications, 1997, 30, 2409–2420 10.1016/S0362-546X(97)00062-XSearch in Google Scholar

[7] Engwerda J.C., On the existence of a positive definite solution of the matrix equation X+ATX-1A=I, Linear Algebra and Its Applications, 1993, 194, 91–108 10.1016/0024-3795(93)90115-5Search in Google Scholar

[8] Engwerda J.C., Ran A.C.M., Rijkeboer A.L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q, Linear Algebra and Its Applications, 1993, 186, 255–275 10.1016/0024-3795(93)90295-YSearch in Google Scholar

[9] Gaalman G.J., Comments on "A Nonrecursive Algebraic Solution for the Discrete Riccati Equation", IEEE Transactions on Automatic Control, June 1980, 25(3), 610–612 10.1109/TAC.1980.1102386Search in Google Scholar

[10] Horn R.A., Johnson C.R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991 10.1017/CBO9780511840371Search in Google Scholar

[11] Hwang T.-M., Chu E.K.-W., Lin W.-W., A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations, International Journal of Control, 2005, 78(14), 1063—1075 10.1080/00207170500155827Search in Google Scholar

[12] Ionescu V., Weiss M., On Computing the Stabilizing Solution of the Diecrete-Time Riccati Equation, Linear Algebra and Its Applications, 1992, 174, 229–238 10.1016/0024-3795(92)90053-DSearch in Google Scholar

[13] Hasanov V.I., Ivanov I.G., On two perturbation estimates of the extreme solutions to the equations X±A*X-1A=Q; Linear Algebra and Its Applications, 2006, 413(1), 81–92 10.1016/j.laa.2005.08.013Search in Google Scholar

[14] Hasanov V.I., Ivanov I.G., Uhlig F., Improved perturbation estimates for the matrix equations X±A*X-1A=Q; Linear Algebra and Its Applications, 2004, 379, 113–135 10.1016/S0024-3795(03)00424-5Search in Google Scholar

[15] Kalman R.E., A new approach to linear filtering and prediction problems, Transactions of the ASME -Journal of Basic Engineering, 1960, 82(Series D), 34–45 10.1115/1.3662552Search in Google Scholar

[16] Lancaster P., Rodman L., Algebraic Riccati Equations, Clarendon Press, Oxford, 1995 Search in Google Scholar

[17] Laub A., A Schur method for solving algebraic Riccati equations, IEEE Transactions on Automatic Control, 1979, 24, 913–921 10.1109/TAC.1979.1102178Search in Google Scholar

[18] Lin W.-W., Xu S.-F., Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations, SIAM Journal on Matrix Analysis and Applications, 2006, 28(1), 26—39 10.1137/040617650Search in Google Scholar

[19] Pappas T., Laub A., Sandell, N.Jr., On the numerical solution of the discrete-time algebraic Riccati equation, IEEE Transactions on Automatic Control, 1980, 25, 631–641 10.1109/TAC.1980.1102434Search in Google Scholar

[20] Ran A.C.M., Rodman L., Stable Hermitian Solutions of Discrete Algebraic Riccati Equations, Math. Control Signals Systems, 1992, 5, 165–193 10.1007/BF01215844Search in Google Scholar

[21] Salah M. El-Sayed, Petkov M.G., Iterative methods for nonlinear matrix equations X+A*X-aA=I; Linear Algebra and Its Applications, 2005, 403, 45–52 10.1016/j.laa.2005.01.010Search in Google Scholar

[22] Vaughan D.R., A Nonrecursive Algebraic Solution for the Discrete Riccati Equation, IEEE Transactions on Automatic Control, October 1970, 597–59910.1109/TAC.1970.1099549Search in Google Scholar

Received: 2013-8-24
Accepted: 2014-7-29
Published Online: 2014-10-9
Published in Print: 2015-1-1

© 2015 Maria Adam, Nicholas Assimakis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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