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October  2014, 34(10): 4139-4153. doi: 10.3934/dcds.2014.34.4139

The Kalman-Bucy filter revisited

Russell Johnson 1, and  Carmen Núñez 2,

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

Received  November 2012 Published  April 2014

We study the properties of the error covariance matrix and the asymptotic error covariance matrix of the Kalman-Bucy filter model with time-varying coefficients. We make use of such techniques of the theory of nonautonomous differential systems as the exponential dichotomy concept and the rotation number.
Keywords: rotation number., Nonautonomous control, Kalman-Bucy filter, exponential dichotomy, error covariance matrix.
Mathematics Subject Classification: 37B55, 93C05, 94E1.
Citation: Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139
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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