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The Kalman-Bucy filter revisited
1. | Dipartimento di Sistemi e Informatica, Università di Firenze, Facolta' di Ingegneria, Via di Santa Marta 3, 50139 Firenze |
2. | Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid |
References:
[1] |
B. Anderson and J. Moore, The Kalman-Bucy filter as a true time-varying Wiener filter, IEEE T. Syst. MAN Cyb., SMC-1 (1971), 119-128. |
[2] |
B. Anderson and J. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Opt., 19 (1981), 20-32.
doi: 10.1137/0319002. |
[3] |
V. I. Arnold, On a characteristic class entering into conditions of quantization, Funk. Anal. Appl., 1 (1967), 1-13.
doi: 10.1007/BF01075861. |
[4] |
B. Bell, J. Burke and G. Pillonetto, An inequality constrained nonlinear Kalman-Bucy smoother by interior point likelihood maximization, Automatica, 45 (2009), 25-33.
doi: 10.1016/j.automatica.2008.05.029. |
[5] |
A. Benavoli and L. Chisci, Robust stochastic control based on imprecise probabilities, Proc. 18th IFAC World Congress, (2011), 4606-4613. |
[6] |
P. Bougerol, Filtre de Kalman-Bucy et exposants de Lyapounov, Oberwolfach, 1990, Lecture Notes in Math., Springer, Berlin, 1486 (1991), 112-122,.
doi: 10.1007/BFb0086662. |
[7] |
P. Bougerol, Some results on the filtering Riccati equation with random parameters, Applied Stochastic Analysis (New Brunswick, NJ, 1991), Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992, 30-37.
doi: 10.1007/BFb0007046. |
[8] |
P. Bougerol, Kalman filtering with random coefficients and contractions, SIAM Jour. Control Optim., 31 (1993), 942-959.
doi: 10.1137/0331041. |
[9] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[10] |
R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties, Z. angew. Math. Phys., 54 (2002), 484-502.
doi: 10.1007/s00033-003-1068-1. |
[11] |
R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems II: The Floquet coefficient, Z. angew. Math. Phys., 54 (2003), 652-676.
doi: 10.1007/s00033-003-1057-4. |
[12] |
F. Fagnani and J. Willems, Deterministic Kalman filtering in a behavioral framework, Sys. Cont. Letters, 32 (1997), 301-312.
doi: 10.1016/S0167-6911(97)00086-8. |
[13] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Heidelberg, Berlin, 1975. |
[14] |
R. Johnson, $m$-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl., 147 (1987), 211-248.
doi: 10.1007/BF01762419. |
[15] |
R. Johnson and M. Nerurkar, On null controllability of linear systems with recurrent coefficients and constrained controls, J. Dynam. Differ. Equations, 4 (1992), 259-273.
doi: 10.1007/BF01049388. |
[16] |
R. Johnson and M. Nerurkar, Exponential dichotomy and rotation number for linear Hamiltonian systems, J. Differential Equations, 108 (1994), 201-216.
doi: 10.1006/jdeq.1994.1033. |
[17] |
R. Johnson and M. Nerurkar, Stabilization and random linear regulator problem for random linear control processes, J. Math. Anal. Appl., 197 (1996), 608-629.
doi: 10.1006/jmaa.1996.0042. |
[18] |
R. Kalman and R. Bucy, New results in linear filtering and prediction theory, Trans. ASME, Basic Eng. Ser. D, 83 (1961), 95-108.
doi: 10.1115/1.3658902. |
[19] |
Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972. |
[20] |
A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer-Verlag, Berlin, Heidelberg, New York, 1990.
doi: 10.1007/978-3-642-61259-6. |
[21] |
V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960. |
[22] |
S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems, J. Differential Equations, 148 (1998), 148-185.
doi: 10.1006/jdeq.1998.3469. |
[23] |
R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems III, J. Differential Equations, 22 (1976), 497-522.
doi: 10.1016/0022-0396(76)90043-7. |
[24] |
R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.
doi: 10.1016/0022-0396(78)90057-8. |
[25] |
M. P. Wojtkowski, Measure theoretic entropy of the system of hard spheres, Ergodic Theory Dynam. Systems, 8 (1988), 133-153.
doi: 10.1017/S0143385700004363. |
[26] |
Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187.
doi: 10.1006/jdeq.1993.1026. |
show all references
References:
[1] |
B. Anderson and J. Moore, The Kalman-Bucy filter as a true time-varying Wiener filter, IEEE T. Syst. MAN Cyb., SMC-1 (1971), 119-128. |
[2] |
B. Anderson and J. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Opt., 19 (1981), 20-32.
doi: 10.1137/0319002. |
[3] |
V. I. Arnold, On a characteristic class entering into conditions of quantization, Funk. Anal. Appl., 1 (1967), 1-13.
doi: 10.1007/BF01075861. |
[4] |
B. Bell, J. Burke and G. Pillonetto, An inequality constrained nonlinear Kalman-Bucy smoother by interior point likelihood maximization, Automatica, 45 (2009), 25-33.
doi: 10.1016/j.automatica.2008.05.029. |
[5] |
A. Benavoli and L. Chisci, Robust stochastic control based on imprecise probabilities, Proc. 18th IFAC World Congress, (2011), 4606-4613. |
[6] |
P. Bougerol, Filtre de Kalman-Bucy et exposants de Lyapounov, Oberwolfach, 1990, Lecture Notes in Math., Springer, Berlin, 1486 (1991), 112-122,.
doi: 10.1007/BFb0086662. |
[7] |
P. Bougerol, Some results on the filtering Riccati equation with random parameters, Applied Stochastic Analysis (New Brunswick, NJ, 1991), Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992, 30-37.
doi: 10.1007/BFb0007046. |
[8] |
P. Bougerol, Kalman filtering with random coefficients and contractions, SIAM Jour. Control Optim., 31 (1993), 942-959.
doi: 10.1137/0331041. |
[9] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[10] |
R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties, Z. angew. Math. Phys., 54 (2002), 484-502.
doi: 10.1007/s00033-003-1068-1. |
[11] |
R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems II: The Floquet coefficient, Z. angew. Math. Phys., 54 (2003), 652-676.
doi: 10.1007/s00033-003-1057-4. |
[12] |
F. Fagnani and J. Willems, Deterministic Kalman filtering in a behavioral framework, Sys. Cont. Letters, 32 (1997), 301-312.
doi: 10.1016/S0167-6911(97)00086-8. |
[13] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Heidelberg, Berlin, 1975. |
[14] |
R. Johnson, $m$-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl., 147 (1987), 211-248.
doi: 10.1007/BF01762419. |
[15] |
R. Johnson and M. Nerurkar, On null controllability of linear systems with recurrent coefficients and constrained controls, J. Dynam. Differ. Equations, 4 (1992), 259-273.
doi: 10.1007/BF01049388. |
[16] |
R. Johnson and M. Nerurkar, Exponential dichotomy and rotation number for linear Hamiltonian systems, J. Differential Equations, 108 (1994), 201-216.
doi: 10.1006/jdeq.1994.1033. |
[17] |
R. Johnson and M. Nerurkar, Stabilization and random linear regulator problem for random linear control processes, J. Math. Anal. Appl., 197 (1996), 608-629.
doi: 10.1006/jmaa.1996.0042. |
[18] |
R. Kalman and R. Bucy, New results in linear filtering and prediction theory, Trans. ASME, Basic Eng. Ser. D, 83 (1961), 95-108.
doi: 10.1115/1.3658902. |
[19] |
Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972. |
[20] |
A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer-Verlag, Berlin, Heidelberg, New York, 1990.
doi: 10.1007/978-3-642-61259-6. |
[21] |
V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960. |
[22] |
S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems, J. Differential Equations, 148 (1998), 148-185.
doi: 10.1006/jdeq.1998.3469. |
[23] |
R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems III, J. Differential Equations, 22 (1976), 497-522.
doi: 10.1016/0022-0396(76)90043-7. |
[24] |
R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.
doi: 10.1016/0022-0396(78)90057-8. |
[25] |
M. P. Wojtkowski, Measure theoretic entropy of the system of hard spheres, Ergodic Theory Dynam. Systems, 8 (1988), 133-153.
doi: 10.1017/S0143385700004363. |
[26] |
Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187.
doi: 10.1006/jdeq.1993.1026. |
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