In this work, we seek exact formulations of the optimal estimator and filter for a non-linear framework, as the Kalman filter is for a linear framework. The solution is well established with the Mortensen filter in a continuous-time setting, but we seek here its counterpart in a discrete-time context. We demonstrate that it is possible to pursue at the discrete-time level an exact dynamic programming strategy and we find an optimal estimator combining a prediction step using the model and a correction step using the data. This optimal estimator reduces to the discrete-time Kalman estimator when the operators are in fact linear. Furthermore, the strategy that consists of discretizing the least square criterion and then finding the exact estimator at the discrete level allows to determine a new time-scheme for the Mortensen filter which is proven to be consistent and unconditionally stable, with also a consistent and stable discretization of the underlying Hamilton-Jacobi-Bellman equation.
Accepted:
DOI: 10.1051/cocv/2017077
Keywords: Deterministic observer, optimal filtering, time-discrete non-linear systems, Hamilton-Jacobi-Bellman
@article{COCV_2018__24_4_1815_0, author = {Moireau, P.}, title = {A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the {Mortensen} observer}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1815--1847}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017077}, zbl = {1414.93192}, mrnumber = {3922444}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017077/} }
TY - JOUR AU - Moireau, P. TI - A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1815 EP - 1847 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017077/ DO - 10.1051/cocv/2017077 LA - en ID - COCV_2018__24_4_1815_0 ER -
%0 Journal Article %A Moireau, P. %T A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1815-1847 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017077/ %R 10.1051/cocv/2017077 %G en %F COCV_2018__24_4_1815_0
Moireau, P. A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1815-1847. doi : 10.1051/cocv/2017077. http://archive.numdam.org/articles/10.1051/cocv/2017077/
[1] Detectability and stabilizability of time-varying discrete-time linear systems. SIAM J. Control Optim. 19 (1981) 20–32. | DOI | MR | Zbl
and ,[2] On Observer Problems for Systems Governed by Partial Differential Equations. Technical Report. Maryland Univ., College Park (1987). | DOI
and ,[3] Dynamic observers as asymptotic limits of recursive filters: special cases. SIAM J. Appl. Math. 48 (1988) 1147–1158. | DOI | MR | Zbl
, and ,[4] Nonlinear Filtering: The Set-Membership (Bounding) and the H8 Techniques. Technical Report TR 1995-40, ISR (1995).
and ,[5] Dynamic Programming. Princeton University Press (1957). | MR | Zbl
,[6] Filtrage Optimal des Systèmes Linéaires. Dunod (1971). | Zbl
,[7] Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992). | DOI | MR | Zbl
,[8] II of Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl
, , and , Representation and Control of Infinite-Dimensional Systems. Vol.[9] Dynamics Programming and Optimal Control. 3rd edn. Athena Scientific, Vol. 1 (2005). | MR | Zbl
,[10] Data assimilation for geophysical fluids. Comput. Methods Atmos. Ocean 14 (2009) 385–441. | MR
, and ,[11] An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equations. J. Sci. Comput. 55 (2013) 575–605. | DOI | MR | Zbl
, , and ,[12] Sparse grids. Acta Numer. 13 (2004) 147–269. | MR | Zbl
and ,[13] Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. II. Existence theorems for weak solutions. Trans. Am. Math. Soc. 124 (1966) 413–430. | DOI | MR | Zbl
,[14] Fundamental principles of data assimilation underlying the verdandi library: applications to biophysical model personalization within euheart. Med. Biol. Eng. Comput. 51 (2012) 1221–1233. | DOI
, , and ,[15] A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – analysis, assessments and applications to parameter estimation. ESAIM: M2AN 47 (2013) 1821–1843. | DOI | Numdam | MR | Zbl
, , and ,[16] Nonlinear Least Squares for Inverse Problems. Springer (2010). | DOI | MR | Zbl
,[17] Bayesian filtering: From Kalman filters to particle filters, and beyond. Statistics 182 (2003) 1–69.
,[18] Data assimilation of time under-sampled measurements using observers, the wave-like equation example. ESAIM: COCV 21 (2015) 635–669. | Numdam | MR | Zbl
, and ,[19] On the estimation of state variables and parameters for noisy dynamic systems. IEEE Trans. Autom. Control (1964). | MR
,[20] Discrete approximations in optimal control, in Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. Springer, New York, NY (1996) 59–80. | DOI | MR | Zbl
,[21] Deterministic nonlinear filtering. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997) 435–454. | Numdam | MR | Zbl
,[22] A max-plus-based algorithm for a Hamilton–Jacobi–Bellman equation of nonlinear filtering. SIAM J. Control Optim. 38 (2000) 683–710. | DOI | MR | Zbl
and ,[23] Deterministic and Stochastic Optimal Control. Springer-Verlag (1975). | DOI | MR | Zbl
and ,[24] Asymptotic nonlinear filtering and large deviations. Adv. Filter. Optim. Stoch. Control (1982) 170–176. | DOI | MR | Zbl
,[25] Nonlinear filtering and large deviations: a PDE-control theoretic approach. Stochastics 23 (1988) 391–412. | DOI | MR | Zbl
and ,[26] New extension of the Kalman filter to nonlinear systems. Proc. SPIE 3068 (1997) 182–193. | DOI
and ,[27] Linear Estimation. Prentice Hall, New Jersey, Vol. 1 (2000).
, and ,[28] Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5 (1960) 102–119. | MR | Zbl
,[29] A new approach to linear filtering and prediction problems. J. Basic Eng. 82 (1960) 35–45. | DOI | MR
,[30] Mathematical description of linear dynamical systems. J. SIAM Control Ser. A 1 (1963) 152–192. | MR | Zbl
,[31] New results in linear filtering and prediction theory. Trans. ASME J. Basic Eng. 83 (1961) 95–108. | DOI | MR
and ,[32] A Lyapunov theory of nonlinear observers, in and eds. Stochastic Analysis, Control, Optimization and Applications. Springer (1998) 409–420. | MR | Zbl
,[33] The convergence of the minimum energy estimator, in New Trends in Nonlinear Dynamics and Control, and their Applications. Springer, Berlin (2003). | MR | Zbl
,[34] A hybrid computational approach to nonlinear estimation, in Proceedings of the 35th IEEE Decision and Control, 1996 (1996) 1815–1819. | DOI
and ,[35] HJB-POD-based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3 (2004) 701–722. | DOI | MR | Zbl
, and ,[36] Dynamical equations for optimal nonlinear filtering. J. Differ. Equ. 3 (1967) 179–190. | DOI | MR | Zbl
,[37] Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38 (2010) 97–110. | DOI
and ,[38] Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2011) 380–405. | Numdam | MR | Zbl
and ,[39] Maximum-likelihood recursive nonlinear filtering. J. Optim. Theory Appl. 2 (1968) 386–394. | DOI | MR | Zbl
,[40] Data assimilation for numerical weather prediction: a review, in and eds. Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications. Springer, Berlin, Heidelberg (2009). | DOI
,[41] Nonlinear dynamic systems and optimal control problems on time scales. ESAIM: COCV 17 (2010) 654–681. | Numdam | MR | Zbl
, and ,[42] A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst. 16 (1998) 323–340. | DOI
, and ,[43] Optimal State Estimation: Kalman, H^{∞}, and Nonlinear Approaches. Wiley-Interscience (2006). | DOI
,[44] Nonlinear Systems Analysis. Prentice-Hall Internaltional Editions, Englewood Cliffs, NJ (1993). | Zbl
,Cited by Sources: