# On the Phase Portrait of the Fast Filtering Algorithms

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topZhou, Yishao. "On the Phase Portrait of the Fast Filtering Algorithms." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 609-630. <http://eudml.org/doc/197372>.

@article{Zhou2010,

abstract = {
Fast filtering algorithms arising from linear filtering and
estimation are nonlinear dynamical systems whose initial values
are the statistics of the observation process. In this paper, we
give a fairly complete description of the phase portrait for such
nonlinear dynamical systems, as well as a special type of naturally
related matrix Riccati equation.
},

author = {Zhou, Yishao},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Fast filtering algorithms; Riccati equations; Kalman
filtering; nonlinear dynamical systems.; Kalman filtering; fast filtering algorithms; nonlinear dynamical systems; positivity conditions},

language = {eng},

month = {3},

pages = {609-630},

publisher = {EDP Sciences},

title = {On the Phase Portrait of the Fast Filtering Algorithms},

url = {http://eudml.org/doc/197372},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Zhou, Yishao

TI - On the Phase Portrait of the Fast Filtering Algorithms

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 609

EP - 630

AB -
Fast filtering algorithms arising from linear filtering and
estimation are nonlinear dynamical systems whose initial values
are the statistics of the observation process. In this paper, we
give a fairly complete description of the phase portrait for such
nonlinear dynamical systems, as well as a special type of naturally
related matrix Riccati equation.

LA - eng

KW - Fast filtering algorithms; Riccati equations; Kalman
filtering; nonlinear dynamical systems.; Kalman filtering; fast filtering algorithms; nonlinear dynamical systems; positivity conditions

UR - http://eudml.org/doc/197372

ER -

## References

top- G. Ammar and C. Martin, The geometry of matrix eigenvalue methods. Acta Appl. Math.5 (1986) 239-279.
- F.A. Badawi and A. Lindquist, A Hamiltonian approach to the factorization of the matrix Riccati equation. Math. Programming Stud.18 (1982) 27-38.
- 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. ControlAC-40 (1995) 1841-1857.
- C.I. Byrnes and A. Lindquist, On the partial stochastic realization problem. IEEE Trans. Automat. ControlAC-42 (1997) 1049-1070.
- C.I. Byrnes, A. Lindquist and T. McGregor, Predictability and unpredictability in Kalman filtering. IEEE Trans. Automat. Control36 (1991) 563-579.
- 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).
- , On the nonlinear dynamics of fast filtering algorithms. SIAM J. Control Optim.32 (1994) 744-789.
- J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge (1956).
- 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).
- 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).
- P. Faurre, M. Clerget and F. Germain, Opérateurs Rationnels Positifs, Dunod (1979).
- G.H. Hardy and J.E. Littlewood, Some problems of Diophantine approximation. Acta Math.37 (1914) 155-239.
- G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford at the Clarendon Press (1954).
- 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 Theory15 (1982) 277-284.
- , Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations. II. Math. Systems Theory16 (1983) 297-306.
- J.F. Koksma, Diophantische Approximationen, Chelsea Publishing Company, New York (1936).
- A.J. Laub and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices. Celestial Mech.9 (1974) 213-238.
- A. Lindquist, A new algorithm for optimal filtering of discrete-time stationary processes. SIAM J. Control12 (1974) 736-746.
- , Some reduced-order non-Riccati equations for linear least-squares estimation: the stationary, single-output case. Int. J. Control24 (1976) 821-842.
- 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.
- I. Niven, Diophantine Approximations, Interscience Publishers, New York, London (1956).
- I.R. Shafarevitch, Basic Algebraic Geometry, Springer-Verlag, Heidelberg (1974).
- M. Shayman, Phase portrait of the matrix Riccati equations. SIAM J. Control Optim.24 (1986) 1-65.
- 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).

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