Nonlinear filters for chaotic oscillatory systems

This paper examines and contrasts the feasibility of joint state and parameter estimation of noise-driven chaotic systems using the extended Kalman filter (EKF), ensemble Kalman filter (EnKF), and particle filter (PF). In particular, we consider the chaotic vibration of a noisy Duffing oscillator perturbed by combined harmonic and random inputs ensuing a transition probability density function (pdf) of motion which displays strongly non-Gaussian features. This system offers computational simplicity while exhibiting a kaleidoscope of dynamical behavior with a slight change of input and system parameters. An extensive numerical study is undertaken to contrast the performance of various nonlinear filtering algorithms with respect to sparsity of observational data and strength of model and measurement noise. In general, the performance of EnKF is better than PF for smaller ensemble size, while for larger ensembles PF outperforms EnKF. For moderate measurement noise and frequent measurement data, EKF is able to correctly track the dynamics of the system. However, EKF performance is unsatisfactory in the presence of sparse observational data or strong measurement noise.

[1]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[2]  R. Kalman,et al.  New results in linear prediction and filtering theory Trans. AMSE , 1961 .

[3]  J. E. Handschin.  Monte Carlo techniques for prediction and filtering of non-linear stochastic processes , 1970 .

[4]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[5]  H. Akashi,et al.  Application of Monte Carlo method to optimal control for linear systems under measurement noise with Markov dependent statistical property , 1975 .

[6]  Yu-Kweng Michael Lin.  Probabilistic Theory of Structural Dynamics , 1976 .

[7]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[8]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[9]  M. Hoshiya,et al.  Structural Identification by Extended Kalman Filter , 1984 .

[10]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[11]  F. L. Dimet,et al.  Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects , 1986 .

[12]  Guanrong Chen,et al.  Kalman Filtering with Real-time Applications , 1987 .

[13]  G. Evensen.  Using the Extended Kalman Filter with a Multilayer Quasi-Geostrophic Ocean Model , 1992 .

[14]  Pierre Gauthier,et al.  Chaos and quadri-dimensional data assimilation: a study based on the Lorenz model , 1992 .

[15]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[16]  Hisashi Tanizaki,et al.  Nonlinear Filters: Estimation and Applications , 1993 .

[17]  P. Courtier,et al.  Assimilation of Simulated Wind Lidar Data with a Kalman Filter , 1993 .

[18]  Jun S. Liu,et al.  Sequential Imputations and Bayesian Missing Data Problems , 1994 .

[19]  Chan Ghee Koh,et al.  Identification and Uncertainty Estimation of Structural Parameters , 1994 .

[20]  Christian Soize,et al.  The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions , 1994, Series on Advances in Mathematics for Applied Sciences.

[21]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[22]  Michael Ghil,et al.  Advanced data assimilation in strongly nonlinear dynamical systems , 1994 .

[23]  Chin-Hsiung Loh,et al.  A system identification approach to the detection of changes in both linear and non‐linear structural parameters , 1995 .

[24]  Stephen E. Cohn,et al.  An Introduction to Estimation Theory (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[25]  R. Errico.  What is an adjoint model , 1997 .

[26]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[27]  S. Cohn,et al.  An Introduction to Estimation Theory , 1997 .

[28]  Pierre Del Moral,et al.  Discrete Filtering Using Branching and Interacting Particle Systems , 1998 .

[29]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[30]  A. Olsson,et al.  On Latin Hypercube Sampling for Stochastic Finite Element Analysis , 1999 .

[31]  P. Fearnhead,et al.  Improved particle filter for nonlinear problems , 1999 .

[32]  Robert N. Miller,et al.  Data assimilation into nonlinear stochastic models , 1999 .

[33]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[34]  Peter Jan van Leeuwen,et al.  Comment on ''Data Assimilation Using an Ensemble Kalman Filter Technique'' , 1999 .

[35]  M. Allen,et al.  Sensitivity analysis of the climate of a chaotic system , 2000 .

[36]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[37]  Guanrong Chen,et al.  Controlling the Duffing oscillator to the Lorenz system and generalizations , 2000, 2000 2nd International Conference. Control of Oscillations and Chaos. Proceedings (Cat. No.00TH8521).

[38]  Jinqiao Duan.  Probabilistic structural dynamics of protein folding , 2000, Appl. Math. Comput..

[39]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[40]  Neil J. Gordon,et al.  Editors: Sequential Monte Carlo Methods in Practice , 2001 .

[41]  D. Pham.  Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems , 2001 .

[42]  S. Haykin.  Kalman Filtering and Neural Networks , 2001 .

[43]  Alberto Corigliano,et al.  Parameter identification of a time-dependent elastic-damage interface model for the simulation of debonding in composites ☆ , 2001 .

[44]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[45]  J Kurths,et al.  Estimation of parameters and unobserved components for nonlinear systems from noisy time series. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  A. Bennett.  Inverse Modeling of the Ocean and Atmosphere: The ocean and the atmosphere , 2002 .

[47]  J. Whitaker,et al.  Ensemble Data Assimilation without Perturbed Observations , 2002 .

[48]  Rudolph van der Merwe,et al.  The Unscented Kalman Filter , 2002 .

[49]  Armin Köhl,et al.  An adjoint method for the assimilation of statistical characteristics into eddy-resolving ocean models , 2002 .

[50]  G. Kivman,et al.  Sequential parameter estimation for stochastic systems , 2003 .

[51]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[52]  Andrew C. Lorenc,et al.  The potential of the ensemble Kalman filter for NWP—a comparison with 4D‐Var , 2003 .

[53]  Chris Snyder,et al.  Toward a nonlinear ensemble filter for high‐dimensional systems , 2003 .

[54]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

[55]  G. Eyink,et al.  Ensemble Filtering for Nonlinear Dynamics , 2003 .

[56]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[57]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[58]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[59]  A. Corigliano,et al.  Parameter identification in explicit structural dynamics: performance of the extended Kalman filter , 2004 .

[60]  Matthew Fisher,et al.  On the equivalence between Kalman smoothing and weak‐constraint four‐dimensional variational data assimilation , 2005, Quarterly Journal of the Royal Meteorological Society.

[61]  Julia C. Hargreaves,et al.  Parameter estimation in an atmospheric GCM using the Ensemble , 2005 .

[62]  James D. Annan,et al.  Parameter estimation using chaotic time series , 2005 .

[63]  H. Dijkstra,et al.  Nonlinear Processes in Geophysics Nonlinear data-assimilation using implicit models , 2005 .

[64]  Stefano Mariani,et al.  Unscented Kalman filtering for nonlinear structural dynamics , 2007 .

[65]  Emil M. Constantinescu,et al.  Ensemble-based chemical data assimilation II: Real observations , 2006 .

[66]  J. Beck,et al.  Bayesian State and Parameter Estimation of Uncertain Dynamical Systems , 2006 .

[67]  Jonathan D. Beezley,et al.  PREDICTOR-CORRECTOR AND MORPHING ENSEMBLE FILTERS FOR THE ASSIMILATION OF SPARSE DATA INTO HIGH-DIMENSIONAL NONLINEAR SYSTEMS , 2006 .

[68]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[69]  C. S. Manohar,et al.  Monte Carlo filters for identification of nonlinear structural dynamical systems , 2006 .

[70]  Roger Ghanem,et al.  Health monitoring for strongly non‐linear systems using the Ensemble Kalman filter , 2006 .

[71]  Emil M. Constantinescu,et al.  Ensemble‐based chemical data assimilation. I: General approach , 2007 .

[72]  C. S. Manohar,et al.  Nonlinear structural dynamical system identification using adaptive particle filters , 2007 .

[73]  C. S. Manohar,et al.  A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[74]  J. P.,et al.  A Variance-Minimizing Filter for Large-Scale Applications , 2009 .