Cramér-Rao-type bound for state estimation in linear discrete-time system with unknown system parameters

Tracking problems are usually investigated using the Bayesian approach. Many practical tracking problems involve some unknown deterministic nuisance parameters such as the system parameters or noise statistical parameters. This paper addresses the problem of state estimation in linear discrete-time dynamic systems in the presence of unknown deterministic system parameters. A Cramér-Rao-type bound on the mean-sqaure-error (MSE) of the state estimation is introduced. The bound is based on the concept of risk-unbiasedness and can be computed recursively. It allows evaluating the optimality of the estimation procedure. Some sequential estimators for this problem are proposed such that the estimation procedure can be considered an on-line technique. Simulation results show that the proposed bound is asymptotically achieved by the considered estimators.

[1]  Ehud Weinstein.  The extended least-squares and the joint maximum-a-posteriori maximum-likelihood estimation criteria , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[2]  Hagit Messer,et al.  Notes on the Tightness of the Hybrid CramÉr–Rao Lower Bound , 2009, IEEE Transactions on Signal Processing.

[3]  K. Åström,et al.  Numerical Identification of Linear Dynamic Systems from Normal Operating Records , 1966 .

[4]  M. Levin.  Estimation of a system pulse transfer function in the presence of noise , 1964 .

[5]  M. Aoki,et al.  On Certain Convergence Questions in System Identification , 1970 .

[6]  H. Messer,et al.  The hybrid Cramér-Rao bound and the generalized Gaussian linear estimation problem , 2008, 2008 5th IEEE Sensor Array and Multichannel Signal Processing Workshop.

[7]  T. Nayak,et al.  The concept of risk unbiasedness in statistical prediction , 2010 .

[8]  D. G. Lainiotis,et al.  Optimal adaptive filter realizations for sample stochastic processes with an unknown parameter , 1969 .

[9]  Biao Huang,et al.  System Identification , 2000, Control Theory for Physicists.

[10]  Bernard Widrow,et al.  Least-Mean-Square Adaptive Filters: Haykin/Least-Mean-Square Adaptive Filters , 2005 .

[11]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[12]  D. Spain,et al.  Identification and modelling of discrete stochastic linear systems , 1972, CDC 1972.

[13]  R. Kashyap.  Maximum likelihood identification of stochastic linear systems , 1970 .

[14]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[15]  R. Mehra.  On the identification of variances and adaptive Kalman filtering , 1970 .

[16]  R. K. Mehra,et al.  Identification of stochastic linear dynamic systems using Kalman filter representation , 1971 .

[17]  Henry Cox,et al.  On the estimation of state variables and parameters for noisy dynamic systems , 1964 .

[18]  Joseph Tabrikian,et al.  Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part I: Performance Bounds , 2015, IEEE Transactions on Signal Processing.

[19]  Arie Yeredor,et al.  The joint MAP-ML criterion and its relation to ML and to extended least-squares , 2000, IEEE Trans. Signal Process..

[20]  Karl Johan Åström,et al.  Numerical Identification of Linear Dynamic Systems from Normal Operating Records , 1965 .

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

[22]  H. Akaike,et al.  Comment on "An innovations approach to least-squares estimation, part I: Linear filtering in additive white noise" , 1970 .

[23]  Bernard Widrow,et al.  Least-mean-square adaptive filters , 2003 .

[24]  R. Mehra.  On-line identification of linear dynamic systems with applications to Kalman filtering , 1971 .

[25]  Yaakov Bar-Shalom,et al.  Optimal simultaneous state estimation and parameter identification in linear discrete-time systems , 1970 .

[26]  K. Steiglitz,et al.  Maximum likelihood estimation of rational transfer function parameters , 1967, IEEE Transactions on Automatic Control.

[27]  Demetrios G. Lainiotis,et al.  Optimal Estimation in the Presence of Unknown Parameters , 1969, IEEE Trans. Syst. Sci. Cybern..

[28]  H. Messer.  The Hybrid Cramer-Rao Lower Bound - From Practice to Theory , 2006, Fourth IEEE Workshop on Sensor Array and Multichannel Processing, 2006..

[29]  Joseph Tabrikian,et al.  Bayesian cramér-rao type bound for risk-unbiased estimation with deterministic nuisance parameters , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[30]  Hans-Andrea Loeliger,et al.  AR model parameter estimation: from factor graphs to algorithms , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.