On the phase portrait of the fast filtering algorithms

Yishao Zhou

ESAIM: Control, Optimisation and Calculus of Variations (1999)

  • Volume: 4, page 609-630
  • ISSN: 1292-8119

How to cite

top

Zhou, Yishao. "On the phase portrait of the fast filtering algorithms." ESAIM: Control, Optimisation and Calculus of Variations 4 (1999): 609-630. <http://eudml.org/doc/90558>.

@article{Zhou1999,
author = {Zhou, Yishao},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Kalman filtering; fast filtering algorithms; nonlinear dynamical systems; Riccati equations; positivity conditions},
language = {eng},
pages = {609-630},
publisher = {EDP Sciences},
title = {On the phase portrait of the fast filtering algorithms},
url = {http://eudml.org/doc/90558},
volume = {4},
year = {1999},
}

TY - JOUR
AU - Zhou, Yishao
TI - On the phase portrait of the fast filtering algorithms
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1999
PB - EDP Sciences
VL - 4
SP - 609
EP - 630
LA - eng
KW - Kalman filtering; fast filtering algorithms; nonlinear dynamical systems; Riccati equations; positivity conditions
UR - http://eudml.org/doc/90558
ER -

References

top
  1. [1] G. Ammar and C. Martin, The geometry of matrix eigenvalue methods. Acta Appl. Math. 5 ( 1986) 239-279. Zbl0639.34046MR868890
  2. [2] F.A. Badawi and A. Lindquis, A Hamiltonian approach to the factorization of the matrix Riccati equation. Math. Programming Stud. 18 ( 198227-38. Zbl0484.49014
  3. [3] C.I. Byrnes, A. Lindquist, S.V. Gusev and S. Matee, A complete parametrization of all positive rational extensions of a covariance sequence. IEEE Trans. Automat. Control AC-40 ( 1995) 1841-1857. Zbl0847.93008MR1358002
  4. [4] C.I. Byrnes and A. Lindquist, On the partial stochastic realization problem. IEEE Trans. Automat. Control AC-42 ( 1997) 1049-1070. Zbl0885.93012MR1469067
  5. [5] C.I. Byrnes, A. Lindquist and T. McGregor, Predictability and unpredictability in Kalman filtering. IEEE Trans. Automat. Control 36 ( 1991) 563-579. Zbl0738.93071MR1101711
  6. [6] C.I. Byrnes, A. Lindquist and Y. Zhou, Stable, unstable and center manifolds for fast filtering algorithms. Modeling, Estimation and Control of Systems with Uncertainty, G.B. Di Masi, A. Gombani and A. Kurzhanski, Eds., Birkhäuser Boston Inc. ( 1991). Zbl0749.93076MR1132263
  7. [7] C.I. Byrness, A. Lindquist and Y. Zhou, On the nonlinear dynamics of fast filtering algorithms. SIAM J. Control Optim. 32 ( 1994) 744-789. Zbl0825.93906MR1269991
  8. [8] J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge ( 1956). Zbl0077.04801MR87708
  9. [9] H.J. Landau, C.I. Byrnes and A. Lindquist, On the well-posedness of the rational covariance extension problem, Tech. Report TRITA/MAT-96-OS5, Department of Mathematics, KTH, Royal Institute of Technology, Stockholm, Sweden ( 1996). Zbl0932.93015MR1445098
  10. [10] S.V. Gusev, C.I. Byrnes and A. Lindquist, A convex optimization approach to the rational covariance extension problem, Tech. Report TRITA/MAT-97-OS9, Department of Mathematics, KTH, Royal Institute of Technology, Stockholm, Sweden ( 1997). Zbl0947.30027
  11. [11] P. Faurre, M. Clerget and F. Germain, Opérateurs Rationnels Positifs, Dunod ( 1979). Zbl0424.93002MR525380
  12. [12] G.H. Hardy and J.E. Littlewood, Some problems of Diophantine approximation. Acta Math. 37 ( 1914) 155-239. Zbl45.0305.03MR1555098JFM45.0305.03
  13. [13] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford at the Clarendon Press ( 1954). Zbl0058.03301MR67125
  14. [14] R. Hermann and C. Martin, Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations, I: General Lie-theoretic methods. Math. Systems Theory 15 ( 1982) 277-284. Zbl0508.58034MR667827
  15. [15] R. Hermann and C. Martin, Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations. II. Math. Systems Theory 16 ( 1983) 297-306. Zbl0527.58033MR721102
  16. [16] J.F. Koksma, Diophantische Approximationen, Chelsea Publishing Company, New York ( 1936). Zbl0012.39602JFM62.0173.01
  17. [17] A.J. Laub and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices. Celestial Mech. 9 ( 1974) 213-238. Zbl0316.15005MR339273
  18. [18] A. Lindquist, A new algorithm for optimal filtering of discrete-time stationary processes. SIAM J. Control 12 ( 1974) 736-746. Zbl0296.93037MR408968
  19. [19] A. Lindquist, Some reduced-order non-Riccati equations for linear least-squares estimation: the stationary, single-output case. Int. J. Control 24 ( 1976) 821-842. Zbl0342.93054MR421808
  20. [20] C. Martin, Grassmannian manifolds, Riccati equations, and feedback invariants of linear systems, Geometrical Methods for the Theory of Linear Systems, C.I. Byrnes and C. Martin, Eds., Reidel Publishing Company ( 1980) 195-211. Zbl0477.93030MR608994
  21. [21] I. Niven, Diophantine Approximations, Interscience Publishers, New York, London ( 1956). Zbl0115.04402MR148613
  22. [22] I.R. Shafarevitch, Basic Algebraic Geometry, Springer-Verlag, Heidelberg ( 1974). Zbl0284.14001MR366917
  23. [23] M. Shayman, Phase portrait of the matrix Riccati equations. SIAM J. Control Optim. 24 ( 1986) 1-65. Zbl0594.34044MR818936
  24. [24] Y. Zhou, Monotonicity and finite escape time of solutions of the discrete-time riccati equation, to appear in proceedings of European Control Conference, Karlsruhe ( 1999). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.