Accuracy of linear multiple-input multiple-output (MIMO) models obtained by maximum likelihood estimation

In this paper, we study the accuracy of linear multiple-input multiple-output (MIMO) models obtained by maximum likelihood estimation. We present a frequency-domain representation for the information matrix for general linear MIMO models. We show that the variance of estimated parametric models for linear MIMO systems satisfies a fundamental integral trade-off. This trade-off is expressed as a multivariable 'water-bed' effect. An extension to spectral estimation is also discussed.

[1]  E. Hannan,et al.  The Statistical Theory of Linear Systems. , 1990 .

[2]  Lennart Ljung,et al.  Optimal experiment designs with respect to the intended model application , 1986, Autom..

[3]  Yucai Zhu,et al.  System identification for process control: recent experience and outlook , 2009, Int. J. Model. Identif. Control..

[4]  Cristian R. Rojas,et al.  Fundamental Limitations on the Variance of Estimated Parametric Models , 2009, IEEE Transactions on Automatic Control.

[5]  André Klein,et al.  A generalization of Whittle's formula for the information matrix of vector-mixed time series , 2000 .

[6]  P. Whittle.  The Analysis of Multiple Stationary Time Series , 1953 .

[7]  G. Goodwin,et al.  EXPERIMENT DESIGN FOR TIME SERIES ANALYSIS: THE MULTIVARIATE CASE* , 1979 .

[8]  Graham C. Goodwin,et al.  Choosing Between Open- and Closed-Loop Experiments in Linear System Identification , 2007, IEEE Transactions on Automatic Control.

[9]  Graham C. Goodwin,et al.  On the equivalence of time and frequency domain maximum likelihood estimation , 2010, Autom..

[10]  P. Caines,et al.  Asymptotic normality of prediction error estimators for approximate system models , 1980 .

[11]  Håkan Hjalmarsson,et al.  Variance error quantifications that are exact for finite-model order , 2004, IEEE Transactions on Automatic Control.

[12]  Håkan Hjalmarsson,et al.  The fundamental role of general orthonormal bases in system identification , 1999, IEEE Trans. Autom. Control..

[13]  Graham C. Goodwin,et al.  Control System Design , 2000 .

[14]  Håkan Hjalmarsson,et al.  On the frequency domain accuracy of closed loop estimates , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[15]  W. E. Larimore,et al.  A Survey of Some Recent Developments in System Parameter Identification , 1982 .

[16]  D.G. Dudley,et al.  Dynamic system identification experiment design and data analysis , 1979, Proceedings of the IEEE.

[17]  P. Spreij,et al.  Matrix differential calculus applied to multiple stationary time series and an extended Whittle formula for information matrices , 2009 .

[18]  Graham C. Goodwin,et al.  Identifiability of errors in variables dynamic systems , 2008, Autom..

[19]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[20]  Graham C. Goodwin,et al.  Fundamental limitations on the accuracy of MIMO linear models obtained by PEM for systems operating in open loop , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[21]  Lennart Ljung,et al.  Unprejudiced optimal open loop input design for identification of transfer functions , 1985, Autom..

[22]  L. Ljung,et al.  Asymptotic normality of prediction error estimators for approximate system models , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[23]  Yucai Zhu.  Black-box identification of mimo transfer functions: Asymptotic properties of prediction error models , 1989 .

[24]  Torsten Söderström,et al.  Errors-in-variables methods in system identification , 2018, Autom..

[25]  Torsten Söderström,et al.  Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification , 2010, Autom..

[26]  Håkan Hjalmarsson,et al.  Least-squares estimation of a class of frequency functions: A finite sample variance expression , 2006, Autom..

[27]  L. Ljung,et al.  Black-box identification of multivariable transfer functions—asymptotic properties and optimal input design , 1984 .

[28]  L. Ljung.  Asymptotic variance expressions for identified black-box transfer function models , 1984 .

[29]  Cristian R. Rojas,et al.  Asymptotic statistical analysis for model-based control design strategies , 2011, Autom..

[30]  Guy Melard,et al.  The information matrix of multiple-input single-output time series models , 1994 .

[31]  Karl Johan Åström,et al.  On the Achievable Accuracy in Identification Problems , 1967 .

[32]  Benjamin Friedlander,et al.  A general lower bound for parametric spectrum estimation , 1984 .

[33]  Erik Weyer,et al.  Guaranteed non-asymptotic confidence regions in system identification , 2005, Autom..

[34]  George A. F. Seber,et al.  A matrix handbook for statisticians , 2007 .

[35]  Alex Simpkins,et al.  System Identification: Theory for the User, 2nd Edition (Ljung, L.; 1999) [On the Shelf] , 2012, IEEE Robotics & Automation Magazine.

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

[37]  Håkan Hjalmarsson,et al.  A Geometric Approach to Variance Analysis in System Identification , 2011, IEEE Transactions on Automatic Control.

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

[39]  B. Ninness,et al.  The Waterbed Effect in Spectral Estimation , 2004, IEEE Signal Processing Magazine.

[40]  Erik Weyer,et al.  Finite sample properties of system identification methods , 2002, IEEE Trans. Autom. Control..

[41]  Håkan Hjalmarsson,et al.  Input design via LMIs admitting frequency-wise model specifications in confidence regions , 2005, IEEE Transactions on Automatic Control.

[42]  Graham C. Goodwin,et al.  Robustness in Experiment Design , 2012, IEEE Transactions on Automatic Control.

[43]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[44]  Torsten Söderström,et al.  The Cramér-Rao lower bound for noisy input-output systems , 2000, Signal Process..