System identification of nonlinear state-space models

This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution.

[1]  A. Rollett,et al.  The Monte Carlo Method , 2004 .

[2]  S. Duncan,et al.  Using the EM algorithm to estimate the disease parameters for smallpox in 17th century London , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[3]  Sten Bay Jørgensen,et al.  Parameter estimation in stochastic grey-box models , 2004, Autom..

[4]  Lennart Ljung,et al.  Guest Editorial: Special Issue on System Identification , 2005, IEEE Trans. Autom. Control..

[5]  Thomas B. Schön,et al.  Parameter Estimation for Discrete-Time Nonlinear Systems Using EM , 2008 .

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  A. Doucet,et al.  Smoothing algorithms for state–space models , 2010 .

[8]  A. Doucet,et al.  Parameter estimation in general state-space models using particle methods , 2003 .

[9]  Torsten P. Bohlin,et al.  Practical Grey-box Process Identification: Theory and Applications , 2006 .

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

[11]  A. Rama Kalyan,et al.  Systems and control engineering , 1999 .

[12]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

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

[14]  I. J. Leontaritis,et al.  Input-output parametric models for non-linear systems Part II: stochastic non-linear systems , 1985 .

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

[16]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[17]  P. Fearnhead,et al.  A sequential smoothing algorithm with linear computational cost. , 2010 .

[18]  Kunio Takezawa.  Whiley Series in Probability and Statistics , 2005 .

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

[20]  Y. Bresler.  Two-filter formulae for discrete-time non-linear bayesian smoothing , 1986 .

[21]  Richard D. Braatz,et al.  On the "Identification and control of dynamical systems using neural networks" , 1997, IEEE Trans. Neural Networks.

[22]  Thomas Bo Schön,et al.  Maximum Likelihood Nonlinear System Estimation , 2006 .

[23]  R. B. Gopaluni.  Identification of Nonlinear Processes with known Model Structure Under Missing Observations , 2008 .

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

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

[26]  G.C. Goodwin,et al.  Approximate EM Algorithms for Parameter and State Estimation in Nonlinear Stochastic Models , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[27]  Brett Ninness,et al.  Robust maximum-likelihood estimation of multivariable dynamic systems , 2005, Autom..

[28]  Margaret H. Wright,et al.  Direct search methods: Once scorned, now respectable , 1996 .

[29]  Gianluigi Pillonetto,et al.  Optimal smoothing of non-linear dynamic systems via Monte Carlo Markov chains , 2008, Autom..

[30]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[31]  K. Poolla,et al.  New results for Hammerstein system identification , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[32]  A. Doucet,et al.  Monte Carlo Smoothing for Nonlinear Time Series , 2004, Journal of the American Statistical Association.

[33]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[34]  Zoubin Ghahramani,et al.  Learning Nonlinear Dynamical Systems Using an EM Algorithm , 1998, NIPS.

[35]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[36]  Christophe Andrieu,et al.  Particle methods for change detection, system identification, and control , 2004, Proceedings of the IEEE.

[37]  D. Stoffer,et al.  Fitting Stochastic Volatility Models in the Presence of Irregular Sampling via Particle Methods and the EM Algorithm , 2008 .

[38]  G. Kitagawa.  A self-organizing state-space model , 1998 .

[39]  T. Westerlund,et al.  Remarks on "Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems" , 1980 .

[40]  Brett Ninness,et al.  Strong laws of large numbers under weak assumptions with application , 2000, IEEE Trans. Autom. Control..

[41]  Peter Salamon,et al.  Facts, Conjectures, and Improvements for Simulated Annealing , 1987 .

[42]  Xiao-Li Hu,et al.  A Basic Convergence Result for Particle Filtering , 2008, IEEE Transactions on Signal Processing.

[43]  R. B. Gopaluni.  A particle filter approach to identification of nonlinear processes under missing observations , 2008 .

[44]  Brett Ninness,et al.  Maximum-likelihood parameter estimation of bilinear systems , 2005, IEEE Transactions on Automatic Control.

[45]  Petre Stoica,et al.  Decentralized Control , 2018, The Control Systems Handbook.

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

[47]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[48]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[49]  Lennart Ljung,et al.  Perspectives on system identification , 2010, Annu. Rev. Control..

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

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

[52]  Winson Taam,et al.  Nonlinear System Analysis and Identification From Random Data , 1991 .

[53]  S. Roweis,et al.  Learning Nonlinear Dynamical Systems Using the Expectation–Maximization Algorithm , 2001 .

[54]  K. Lange.  A gradient algorithm locally equivalent to the EM algorithm , 1995 .

[55]  Thomas Bo Schön,et al.  Particle Filters for System Identification of State-Space Models Linear in Either Parameters or States , 2003 .