Monte Carlo filters for identification of nonlinear structural dynamical systems

The problem of identification of parameters of nonlinear structures using dynamic state estimation techniques is considered. The process equations are derived based on principles of mechanics and are augmented by mathematical models that relate a set of noisy observations to state variables of the system. The set of structural parameters to be identified is declared as an additional set of state variables. Both the process equation and the measurement equations are taken to be nonlinear in the state variables and contaminated by additive and (or) multiplicative Gaussian white noise processes. The problem of determining the posterior probability density function of the state variables conditioned on all available information is considered. The utility of three recursive Monte Carlo simulation-based filters, namely, a probability density function-based Monte Carlo filter, a Bayesian bootstrap filter and a filter based on sequential importance sampling, to solve this problem is explored. The state equations are discretized using certain variations of stochastic Taylor expansions enabling the incorporation of a class of non-smooth functions within the process equations. Illustrative examples on identification of the nonlinear stiffness parameter of a Duffing oscillator and the friction parameter in a Coulomb oscillator are presented.

[1]  D. Talay.  Numerical solution of stochastic differential equations , 1994 .

[2]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[3]  B. Peeters,et al.  Stochastic System Identification for Operational Modal Analysis: A Review , 2001 .

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

[5]  Brian Schwarz,et al.  Experimental modal analysis , 1999 .

[6]  R. Ghanem,et al.  Structural-System Identification. I: Theory , 1995 .

[7]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

[8]  Yukito IBA,et al.  Population Monte Carlo algorithms , 2000, cond-mat/0008226.

[9]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

[10]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[11]  Simon J. Godsill,et al.  On sequential simulation-based methods for Bayesian filtering , 1998 .

[12]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

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

[14]  S. Zacks,et al.  Introduction to stochastic differential equations , 1988 .

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

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

[17]  T. Kailath.  Lectures on Wiener and Kalman Filtering , 2003 .

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

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

[20]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice Using MATLAB , 2001 .

[21]  Paul Sas,et al.  Modal Analysis Theory and Testing , 2005 .

[22]  J. Schoukens,et al.  Parametric and nonparametric identification of linear systems in the presence of nonlinear distortions-a frequency domain approach , 1998, IEEE Trans. Autom. Control..

[23]  Richard A. Brown,et al.  Introduction to random signals and applied kalman filtering (3rd ed , 2012 .

[24]  J. K. Hedrick,et al.  Disturbance Adaptive Discrete-Time Sliding Control With Application to Engine Speed Control , 2001 .

[25]  Roger Ghanem,et al.  Structural System Identification. II: Experimental Verification , 1995 .

[26]  Alan E. Gelfand,et al.  Bayesian statistics without tears: A sampling-resampling perspective , 1992 .

[27]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[28]  Masanobu Shinozuka,et al.  Fundamentals of system identification in structural dynamics , 1989 .

[29]  G. R. Tomlinson,et al.  Nonlinearity in experimental modal analysis , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[30]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .