Elsevier

Automatica

Volume 42, Issue 2, February 2006, Pages 315-320
Automatica

Brief paper
Closed-loop subspace identification using the parity space

https://doi.org/10.1016/j.automatica.2005.09.012Get rights and content

Abstract

It is known that many subspace algorithms give biased estimates for closed-loop data due to the existence of feedback. In this paper we present a new subspace identification method using the parity space employed in fault detection in the past. The basic algorithm, known as subspace identification method via principal component analysis (SIMPCA), gives consistent estimation of the deterministic part and stochastic part of the system under closed loop. Column weighting for SIMPCA is introduced which shows improved efficiency/accuracy. A simulation example is given to illustrate the performance of the proposed algorithm in closed-loop identification and the effect of column weighting.

Introduction

Subspace identification methods (SIMs) have been one of the main streams of research in system identification (Gevers, 2003). However, due to the correlation between the input and the unmeasured disturbance under feedback control, many subspace algorithms do not work on closed-loop data (Forssell and Ljung, 1999, Ljung and McKelvey, 1996), even though the data satisfy identifiability conditions for prediction error methods. This problem comes from a step in most SIMs that projects the future output on the orthogonal complement of the future input, which requires the future input to be uncorrelated to the past noise. To address this aspect, several algorithms for closed-loop identification have been developed in the last few years. Ljung and McKelvey (1996) presented a subspace identification method through the classical realization path using estimated predictors based on ARX models. Chou and Verhaegen (1997) developed an instrumental variable method which can be applied to errors-in-variables and closed-loop identification problem, but it handles white-noise input differently from correlated input. Qin et al. (2005) and Qin and Ljung (2003b) pointed out the non-causal projection in traditional SIMs and proposed a closed-loop identification method through innovation estimation (Qin & Ljung, 2003a). In the meantime Jansson (2003) proposed to use high order ARX to pre-estimate the system Markov parameters and then estimate the observability matrix, thus avoiding the non-causal projection. Chiuso and Picci (2005) provided consistency analysis of the methods in Qin and Ljung (2003a) and Jansson (2003), and considered them as significant progresses in closed-loop SIM.

In parallel, Wang and Qin, 2001, Wang and Qin, 2002 proposed the use of parity space and principal component analysis (SIMPCA) for errors-in-variables identification with colored input excitation, which can also be applied to closed-loop identification. Instead of pre-estimating the Markov parameters or eliminating them via non-causal projections, SIMPCA reformulates the SIM problem in parity space. Huang et al. (2005) developed a new closed-loop subspace identification algorithm (SOPIM) by adopting the EIV model structure of SIMPCA, and proposed a revised instrumental variable method to avoid identifying the parity space of the feedback controller.

In this paper we present a new SIMPCA algorithm that identifies the process model and noise model in the same framework with appropriate column weighting, referred to as SIMPCA-Wc. Then we discuss the persistent excitation condition for SIMPCA for closed-loop data. We further investigate the relationship between SOPIM (Huang et al., 2005) and SIMPCA. It is shown that the column space of the observability matrix extracted from SOPIM is equivalent to that from SIMPCA-Wc.

The remaining parts of the paper are organized as follows. Section 2 gives the problem formulation and assumptions. Section 3 briefly reviews the original SIMPCA algorithm (deterministic part) and presents the estimation of stochastic part. Section 4 presents the SIMPCA with column weighting (SIMPCA-Wc). Section 5 gives a simulation example. The final section concludes the paper.

Section snippets

Problem formulation and assumptions

Consider the linear time-invariant system in its innovation representation:x(k+1)=Ax(k)+Bu(k)+Ke(k),y(k)=Cx(k)+Du(k)+e(k).Here, x(k)Rn is the state vector, u(k)Rl and y(k)Rm are the measured input and output signals. e(k)Rm is the innovation process. We introduce the following assumptions:

  • A1:

    (A, C) is observable.

  • A2:

    (A, [BK]) is controllable.

  • A3:

    The input u and innovation e are jointly stationary and one-way uncorrelated, i.e.,

E¯[e(k)e(l)T]=Reδkl,E¯[e(k)u(l)T]=0,k>l,where E¯ is defined as in Ljung

SIMPCA for estimating the deterministic and stochastic parts

In this section, we briefly review the original SIMPCA algorithm and discuss the persistent excitation condition. Then we use the SIMPCA model structure to estimate the stochastic model part, namely, the Kalman gain and innovation covariance under closed-loop conditions.

SIMPCA with column weighting

Like other subspace identification methods, SIMPCA also uses SVD or PCA to obtain the observability matrix first. It is thus of great interest to optimize the estimate of the observability matrix. One direction is to apply a weighting matrix to improve the estimate. Several contributions in the literature have appeared in this area (Gustafsson, 2002, Jansson and Wahlberg, 1996, Viberg et al., 1997). Gustafsson (2002) pointed out that although different approaches of analysis were adopted, the

Simulation study

In this section a simulation study is presented to demonstrate the performance of SIMPCA algorithm and the effect of column weighting in closed-loop identification. Results from SIMPCA, N4SID with CVA weighting from the Matlab System Identification Toolbox (version 6.5) (denoted as N4SID-WCVA), MOESP-PO and SIMPCA-Wc are presented for comparison. The simulation example is a first order SISO system under closed-loop operation,y(k)-0.9y(k-1)=u(k-1)+e(k)+0.9e(k-1).The feedback has the following

Conclusions

In this paper, a complete closed-loop subspace identification algorithm is developed using the parity space with appropriate column weighting. Persistent excitation condition of SIMPCA is given. Because SIMPCA makes use of the parity space to estimate the system model and avoids projecting out the future input, it is applicable to closed-loop identification. By scaling the instrumental variables to unit variance, SIMPCA-Wc significantly improves the estimate accuracy. The equivalence between

Acknowledgements

Financial support from National Science Foundation under CTS-9985074 and an Overseas Young Investigator Award from NSF China (60228001) is gratefully acknowledged.

Jin Wang obtained the BS degree in Chemical Engineering from Tsinghua University in Beijing, China, in 1994. She received the MS and PhD degrees in Chemical Engineering from the University of Texas at Austin in 2001 and 2004, respectively. She is currently a Senior Development Engineer with Advanced Process Control at Advanced Micro Devices, Inc. Her research interests include system identification, control performance monitoring, semiconductor process modeling and control, and fault detection

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    The authors also determined the reason that SIMPCA and SOPIM are not effective in closed-loop conditions is due to the influence of feedback. In 2006, Wang and Qin proposed SIMPCA with column weighting (SIMPCA-Wc) [7], which is based on the SIMPCA method and uses column weighting to improve estimation accuracy in closed-loop conditions. They also proved that the performance of SIMPCA-Wc is equivalent to that of SOPIM.

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Jin Wang obtained the BS degree in Chemical Engineering from Tsinghua University in Beijing, China, in 1994. She received the MS and PhD degrees in Chemical Engineering from the University of Texas at Austin in 2001 and 2004, respectively. She is currently a Senior Development Engineer with Advanced Process Control at Advanced Micro Devices, Inc. Her research interests include system identification, control performance monitoring, semiconductor process modeling and control, and fault detection and classification.

S. Joe Qin holds Paul D. and Betty Robertson Meek and American Petrofina Foundation Centennial Professorship in Chemical Engineering at University of Texas at Austin. He obtained his BS and MS degrees in Automatic Control from Tsinghua University in Beijing, China, in 1984 and 1987, respectively. He received his PhD degree in Chemical Engineering from University of Maryland in 1992. He worked as a Principal Engineer at Fisher-Rosemount from 1992 to 1995 and then joined University of Texas as a professor. His research interests include system identification, process monitoring and fault diagnosis, model predictive control, run-to-run control, microelectronics process control, and control performance monitoring. He is a recipient of the NSF CAREER Award, DuPont Young Professor Award, NSF-China Outstanding Young Investigator Award. He was an Editor for Control Engineering Practice from 1999 to 2005 and is currently a Member of the Editorial Board for Journal of Chemometrics.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor Torsten Soederstroem.

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