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2015 | 13 | 1 |
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
wydano
2015-01-01
otrzymano
2013-08-24
zaakceptowano
2014-07-29
online
2014-10-09
Twórcy
autor
  • Department of Computer Science and Biomedical Informatics, University of Thessaly,
    2-4 Papasiopoulou str., P.O. 35100 Lamia, Greece, madam@dib.uth.gr
  • Department of Electronic Engineering, Technological Educational Institute of Central Greece,
    3rd km Old National Road Lamia-Athens, P.O. 35100 Lamia, Greece, assimakis@mail.teiste.gr
Bibliografia
  • [1] Adam M., Assimakis N., Matrix equations solutions using Riccati Equation, Lambert, Academic Publishing, 2012
  • [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
  • [3] Adam M., Sanida F., Assimakis N., Voliotis S., Riccati Equation Solution Method for the computation of the extreme solutions ofX A*X-1A=Q and X-A*X-1A=Q, IWSSIP 2009, Proceedings 2009 IEEE, 978-1-4244-4530-1/09, 41–44
  • [4] Anderson B.D.O., Moore J.B., Optimal Filtering, Dover Publications, New York, 2005
  • [5] Assimakis N., Sanida F., Adam M., Recursive Solutions of the Matrix Equations X+ATX-1A=Q and X-ATX-1A=QQ; Applied Mathematical Sciences, 2008, 2(38), 1855–1872
  • [6] Assimakis N.D., Lainiotis D.G., Katsikas S.K., Sanida F.L., A survey of recursive algorithms for the solution of the discrete timeRiccati equation, Nonlinear Analysis, Theory, Methods & Applications, 1997, 30, 2409–2420
  • [7] Engwerda J.C., On the existence of a positive definite solution of the matrix equation X+ATX-1A=I, Linear Algebra andIts Applications, 1993, 194, 91–108
  • [8] Engwerda J.C., Ran A.C.M., Rijkeboer A.L., Necessary and sufficient conditions for the existence of a positive definite solution ofthe matrix equation X+A*X-1A=Q, Linear Algebra and Its Applications, 1993, 186, 255–275
  • [9] Gaalman G.J., Comments on "A Nonrecursive Algebraic Solution for the Discrete Riccati Equation", IEEE Transactions on AutomaticControl, June 1980, 25(3), 610–612[Crossref]
  • [10] Horn R.A., Johnson C.R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991
  • [11] Hwang T.-M., Chu E.K.-W., Lin W.-W., A generalized structure-preserving doubling algorithm for generalized discrete-time algebraicRiccati equations, International Journal of Control, 2005, 78(14), 1063-1075[Crossref]
  • [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
  • [13] Hasanov V.I., Ivanov I.G., On two perturbation estimates of the extreme solutions to the equations X±A*X-1A=Q; LinearAlgebra and Its Applications, 2006, 413(1), 81–92[WoS]
  • [14] Hasanov V.I., Ivanov I.G., Uhlig F., Improved perturbation estimates for the matrix equations X±A*X-1A=Q; LinearAlgebra and Its Applications, 2004, 379, 113–135
  • [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[Crossref]
  • [16] Lancaster P., Rodman L., Algebraic Riccati Equations, Clarendon Press, Oxford, 1995
  • [17] Laub A., A Schur method for solving algebraic Riccati equations, IEEE Transactions on Automatic Control, 1979, 24, 913–921[WoS][Crossref]
  • [18] Lin W.-W., Xu S.-F., Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations, SIAMJournal on Matrix Analysis and Applications, 2006, 28(1), 26-39
  • [19] Pappas T., Laub A., Sandell, N.Jr., On the numerical solution of the discrete-time algebraic Riccati equation, IEEE Transactionson Automatic Control, 1980, 25, 631–641[Crossref]
  • [20] Ran A.C.M., Rodman L., Stable Hermitian Solutions of Discrete Algebraic Riccati Equations, Math. Control Signals Systems,1992, 5, 165–193
  • [21] Salah M. El-Sayed, Petkov M.G., Iterative methods for nonlinear matrix equations X+A*X-aA=I; Linear Algebra and ItsApplications, 2005, 403, 45–52[WoS]
  • [22] Vaughan D.R., A Nonrecursive Algebraic Solution for the Discrete Riccati Equation, IEEE Transactions on Automatic Control,October 1970, 597–599[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_math-2015-0006
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