Kimura, Morishige Convergence of the doubling algorithm for the discrete-time algebraic Riccati equation. (English) Zbl 0645.93012 Int. J. Syst. Sci. 19, No. 5, 701-711 (1988). Summary: If the doubling algorithm (DA) for the discrete-time algebraic Riccati equation converges, the speed of convergence is high. However, its convergence has not yet been examined exactly. Firstly, a certain matrix appearing in the DA is shown to be non-singular and therefore the algorithm is well defined. Secondly, it is found that the loss of significant digits hardly occurs in the DA. Finally, it is proved that if the time-invariant discrete-time linear system, whose state is estimated by the steady-state Kalman filter, is reachable and detectable, or stabilizable and observable, then all three matrix sequences in the DA converge. Cited in 1 ReviewCited in 9 Documents MSC: 93B40 Computational methods in systems theory (MSC2010) 15A24 Matrix equations and identities 65F30 Other matrix algorithms (MSC2010) 93E11 Filtering in stochastic control theory 93C05 Linear systems in control theory 93C55 Discrete-time control/observation systems Keywords:doubling algorithm; discrete-time algebraic Riccati equation; convergence; time-invariant PDFBibTeX XMLCite \textit{M. Kimura}, Int. J. Syst. Sci. 19, No. 5, 701--711 (1988; Zbl 0645.93012) Full Text: DOI References: [1] DOI: 10.1080/00207177808922455 · Zbl 0385.49017 · doi:10.1080/00207177808922455 [2] ANDERSON B. D. O., Optimal Filtering (1979) [3] BIERMAN G. J., Factorization Methods for Discrete Sequential Estimation (1977) · Zbl 0372.93001 [4] DOI: 10.1080/00207177008931892 · Zbl 0205.15902 · doi:10.1080/00207177008931892 [5] DOI: 10.1109/TAC.1971.1099832 · doi:10.1109/TAC.1971.1099832 [6] DOI: 10.1016/0022-247X(73)90135-2 · Zbl 0256.93028 · doi:10.1016/0022-247X(73)90135-2 [7] ISAACSON , E. , and KELLER , H. B. , 1966 ,Analysis of Numerical Methods( New York Wiley ), p. 11 . · Zbl 0168.13101 [8] KALMAN R. E., Trans. Am. Soc. mech. Engrs, J. bas. Engng 82 pp 35– (1960) [9] KATAYAMA T., Applied Kalman Filter (1983) [10] KUČERA V., Kybernetika 8 pp 430– (1972) [11] KUO B. C., Digital Control Systems (1980) [12] KWAKERNAAK H., Linear Optimal Control Systems (1972) · Zbl 0276.93001 [13] POTTER , J. E. , 1964 ,Astronautical Guidance, edited by R. H. Battin ( New York McGraw-Hill ), pp. 338 – 340 . [14] RAO , C. R. , and MITRA , S. K. , 1971 ,Generalized Inverse of Matrices and its Applications( New York Wiley ), pp. 123 – 124 . [15] THORNTON , C. L. , and BIERMAN , G. J. , 1980 ,Control and Dynamic Systems, edited by C. T. Leondes , Vol. 16 ( New York Academic Press ), pp. 177 – 248 . [16] DOI: 10.1109/TAC.1970.1099549 · doi:10.1109/TAC.1970.1099549 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.