Consistent dynamic PCA based on errors-in-variables subspace identification

Abstract In this paper, we make a comparison between dynamic principal component analysis (PCA) and errors-in-variables (EIV) subspace model identification (SMI) and establish consistency conditions for the two approaches. We first demonstrate the relationship between dynamic PCA and SMI. Then we show that when process variables are corrupted by measurement noise dynamic PCA fails to give a consistent estimate of the process model in general whether or not process noise is present. We then propose an indirect dynamic PCA approach for the consistent estimate of the process model resorting to EIV SMI algorithms. Consistent dynamic PCA models are obtained with and without process disturbances. Additional features of the indirect approach include (i) easy determination of the number of lagged variables in the model; (ii) determination of the number of significant process disturbances; and (iii) consistent estimate of the dynamic PCA models with and without process disturbances. We conduct two simulation examples and an industrial case study to support our theoretical results, where the relationship between dynamic PCA and EIV SMI is numerically verified.

[1]  Thomas Kailath,et al.  Detection of signals by information theoretic criteria , 1985, IEEE Trans. Acoust. Speech Signal Process..

[2]  Wallace E. Larimore,et al.  Canonical variate analysis in identification, filtering, and adaptive control , 1990, 29th IEEE Conference on Decision and Control.

[3]  Bart De Moor,et al.  A unifying theorem for three subspace system identification algorithms , 1995, Autom..

[4]  Sabine Van Huffel,et al.  Recent advances in total least squares techniques and errors-in-variables modeling , 1997 .

[5]  B. Moor,et al.  A geometrical strategy for the identification of state space models of linear multivariable systems with singular value decomposition , 1987 .

[6]  Patrick Dewilde,et al.  Subspace model identification Part 1. The output-error state-space model identification class of algorithms , 1992 .

[7]  Lei Guo,et al.  Consistent estimation of the order of stochastic control systems , 1987 .

[8]  Michel Verhaegen,et al.  A Novel Non-Iterative Mimo State Space Model Identification Technique , 1991 .

[9]  I. Jolliffe.  Principal Component Analysis , 2002 .

[10]  Ilan Ziskind,et al.  Detection of the number of coherent signals by the MDL principle , 1989, IEEE Trans. Acoust. Speech Signal Process..

[11]  C. Heij,et al.  Global total least squares modeling of multivariable time series , 1995, IEEE Trans. Autom. Control..

[12]  T. W. Anderson.  ASYMPTOTIC THEORY FOR PRINCIPAL COMPONENT ANALYSIS , 1963 .

[13]  Bo Wahlberg,et al.  On Consistency of Subspace Methods for System Identification , 1998, Autom..

[14]  Thomas Kailath,et al.  On smoothed rank profile tests in eigenstructure methods for directions-of-arrival estimation , 1987, IEEE Trans. Acoust. Speech Signal Process..

[15]  Bart De Moor,et al.  N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems , 1994, Autom..

[16]  Barry M. Wise,et al.  The process chemometrics approach to process monitoring and fault detection , 1995 .

[17]  Thomas Kailath,et al.  Linear Systems , 1980 .

[18]  Dietmar Bauer,et al.  Consistency and asymptotic normality of some subspace algorithms for systems without observed inputs , 1999, Autom..

[19]  Mats Viberg,et al.  Subspace Methods in System Identification , 1994 .

[20]  J. Gertler,et al.  Optimal residual decoupling for robust fault diagnosis , 1995 .

[21]  H. Akaike.  A new look at the statistical model identification , 1974 .

[22]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[23]  P. Holmes,et al.  Suppression of bursting , 1997, Autom..

[24]  K. A. Kosanovich,et al.  Monitoring Process Performance in Real-Time , 1992, 1992 American Control Conference.

[25]  Hong Wang,et al.  On the performance of signal-subspace processing- Part I: Narrow-band systems , 1986, IEEE Trans. Acoust. Speech Signal Process..

[26]  S. Wold.  Cross-Validatory Estimation of the Number of Components in Factor and Principal Components Models , 1978 .

[27]  W. Larimore.  System Identification, Reduced-Order Filtering and Modeling via Canonical Variate Analysis , 1983, 1983 American Control Conference.

[28]  Michel Verhaegen,et al.  Identification of the deterministic part of MIMO state space models given in innovations form from input-output data , 1994, Autom..

[29]  Z. Bai,et al.  On detection of the number of signals when the noise covariance matrix is arbitrary , 1986 .

[30]  G. C. Tiao,et al.  Modeling Multiple Time Series with Applications , 1981 .

[31]  Michael J. Piovoso,et al.  Process data chemometrics , 1991 .

[32]  Thomas F. Edgar,et al.  Identification of faulty sensors using principal component analysis , 1996 .

[33]  Z. Bai,et al.  On detection of the number of signals in presence of white noise , 1985 .

[34]  Michel Verhaegen,et al.  Subspace Algorithms for the Identification of Multivariable Dynamic Errors-in-Variables Models , 1997, Autom..

[35]  Christos Georgakis,et al.  Disturbance detection and isolation by dynamic principal component analysis , 1995 .

[36]  J. Baillieul,et al.  Identification and filtering of nonlinear systems using canonical variate analysis , 1990, 29th IEEE Conference on Decision and Control.

[37]  Weihua Li,et al.  Isolation enhanced principal component analysis , 1999 .

[38]  Thomas E. Marlin,et al.  Multivariate statistical monitoring of process operating performance , 1991 .

[39]  Bart De Moor,et al.  Subspace algorithms for the stochastic identification problem, , 1993, Autom..

[40]  S. Joe Qin,et al.  Subspace approach to multidimensional fault identification and reconstruction , 1998 .

[41]  T. Kailath,et al.  Fast Estimation of Principal Eigenspace Using LanczosAlgorithm , 1994 .

[42]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

[44]  Manfred Deistler,et al.  Consistency and relative efficiency of subspace methods , 1994, Autom..

[45]  M. Moonen,et al.  On- and off-line identification of linear state-space models , 1989 .

[46]  Jie Chen,et al.  Robust fault detection of jet engine sensor systems using eigenstructure assignment , 1991 .

[47]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[48]  Consistency of system identification by global total least squares , 1999, Autom..