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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, On a characteristic class entering into conditions of quantization, Funk. Anal. Appl., 1 (1967), 1-13. doi: 10.1007/BF01075861.  Google Scholar

[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.  Google Scholar

[5]

A. Benavoli and L. Chisci, Robust stochastic control based on imprecise probabilities, Proc. 18th IFAC World Congress, (2011), 4606-4613. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Bougerol, Kalman filtering with random coefficients and contractions, SIAM Jour. Control Optim., 31 (1993), 942-959. doi: 10.1137/0331041.  Google Scholar

[9]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Heidelberg, Berlin, 1975.  Google Scholar

[14]

R. Johnson, $m$-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl., 147 (1987), 211-248. doi: 10.1007/BF01762419.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972.  Google Scholar

[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.  Google Scholar

[21]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, On a characteristic class entering into conditions of quantization, Funk. Anal. Appl., 1 (1967), 1-13. doi: 10.1007/BF01075861.  Google Scholar

[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.  Google Scholar

[5]

A. Benavoli and L. Chisci, Robust stochastic control based on imprecise probabilities, Proc. 18th IFAC World Congress, (2011), 4606-4613. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Bougerol, Kalman filtering with random coefficients and contractions, SIAM Jour. Control Optim., 31 (1993), 942-959. doi: 10.1137/0331041.  Google Scholar

[9]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Heidelberg, Berlin, 1975.  Google Scholar

[14]

R. Johnson, $m$-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl., 147 (1987), 211-248. doi: 10.1007/BF01762419.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972.  Google Scholar

[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.  Google Scholar

[21]

V. Nemytskii and V. Stepanoff, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ, 1960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

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