Elsevier

Automatica

Volume 36, Issue 4, April 2000, Pages 497-509
Automatica

Analysis of the asymptotic properties of the MOESP type of subspace algorithms

https://doi.org/10.1016/S0005-1098(99)00174-0Get rights and content

Abstract

The MOESP type of subspace algorithms are used for the identification of linear, discrete time, finite-dimensional state-space systems. They are based on the geometric structure of covariance matrices and exploit the properties of the state vector extensively. In this paper the asymptotic properties of the algorithms are examined. The main results include consistency and asymptotic normality for the estimates of the system matrices, under suitable assumptions on the noise sequence, the input process and the underlying true system.

Introduction

Subspace algorithms are used for the estimation of linear, time-invariant, finite-dimensional, discrete time, state-space systems. They are an alternative to the more classical maximum likelihood and prediction error methods. The main advantages of subspace algorithms are their conceptual simplicity and their numerical properties. The main idea of these algorithms lies in the observation that the predictions of a time series from the whole past of the outputs and possibly the whole series of observed exogenous inputs for different time horizons are a function of the state vector and the future of the exogenous inputs: Every optimal (in the least-squares sense) predictor of the future of the process based on the entire past of the output process and the whole input process is a linear function of the state and the future of the exogenous inputs under appropriate assumptions on the noise and the data generating process. This fact can be used for estimation of the state (cf. Larimore, 1983; Peternell, Scherrer & Deistler, 1996) or the estimation of the linear mapping attaching the predictions to the state vectors and the future of the exogenous inputs (cf. Van Overschee and De Moor 1994, Van Overschee and De Moor 1996; Verhaegen, 1994). The statistical properties of the first type of algorithms are clarified to a large extent by Deistler, Peternell and Scherrer (1995), Peternell et al. (1996), Bauer, Deistler and Scherrer (1999) and Bauer (1998). Within the second type of algorithms, the MOESP class of algorithms is very popular. MOESP has been developed by Verhaegen and coworkers in a series of papers (Verhaegen and Dewilde 1992a, Verhaegen and Dewilde 1992b; Verhaegen & Dewilde, 1993; Verhaegen, 1994). The numerical properties of the latter algorithms have been investigated thoroughly in these papers. The consistency of this approach has been investigated in Jansson and Wahlberg 1997, Jansson and Wahlberg 1998. The main conclusion from these papers is that, in general, it is not enough to impose persistence of excitation type of conditions on the exogenous inputs in order to guarantee consistency. However, there are some special cases (see Jansson & Wahlberg, 1998). Asymptotic normality of the estimates of the poles of the transfer function has been established in Viberg, Ottersten, Wahlberg and Ljung (1993). In the current paper the asymptotic properties of the subspace estimates using various conditions on the exogenous inputs are considered. The analysis will center on conditions ensuring consistency of the approach in generic situations, and on asymptotic normality of the system matrix estimates.

The paper is organized as follows: Section 2 introduces the model class used for identification and presents some standard assumptions. Section 3 presents the class of algorithms considered. Section 4 then contains the main results of this paper, namely consistency and asymptotic normality of the system matrix estimates. Section 5 presents some numerical examples and finally Section 6 concludes the paper.

Throughout the paper the following notation will be used: Bold face symbols are used for matrices and vectors, lower case latin and greek symbols are used for scalars. As usual → will denote convergence for deterministic quantities and → a.s. stands for almost sure convergence of stochastic quantities. d will denote convergence in distribution. Also the notation at,bt〉=(1/T)∑t=1TatbtT, where T denotes the sample size, is introduced. Here the initial conditions are such that at,bt〉=〈at+j,bt+j holds for |j|≤α+β, where α and β are integers to be specified in the following section. Finally fn=o(gn) means limn→∞fn/gn=0.

Section snippets

Model set

In this paper the model class is restricted to linear, finite-dimensional, discrete time, time-invariant, state-space systems of the formxt+1=Axt+But+Kεt,yt=Cxt+Dut+Eεt,where t∈Z, ytRs is the s-dimensional observed output, εtRs, denotes the s-dimensional white noise with zero mean and covariance matrix equal to unity. utRm denotes the m-dimensional exogenous input series, which is assumed to be independent of the noise εt in an appropriate sense to be defined below. Finally, xtRn denotes

The algorithms

In this section a brief presentation of the algorithms considered in this paper will be given. The main fact that is used by subspace algorithms can be formulated as follows: Let Yt,β=[yt−1T,yt−2T,…,yt−βT]T be the vector of the stacked (finite) past of the process and let Yt,α=[ytT,yt+1T,…,yt+α−1T]T be the vector of the stacked (finite) future of the output process. Define Ut,β and Ut,α analogously from ut, and let Pt,β=[Yt,βT,Ut,βT]T. In what follows, it is assumed that α>n and βn.

Asymptotic properties

The first part of this section will focus on the question of consistency of the estimates. There will be two different concepts concerning the consistency, depending on whether the estimate of the transfer function is concerned, or whether the convergence of the system matrix estimates is investigated. From the description of the algorithm it can be seen that the system matrix estimates are a nonlinear function of the sample covariances of the joint process zt=[ytT,utT]T up to lag α+β−1. Up to

Numerical examples

In the previous section, the asymptotic normality of the MOESP algorithm has been derived. In Theorem 13 the variance of the limiting normal distribution has been denoted with V. As has been stated already, V depends on the covariance sequence of the inputs, the choice of the weighting matrices and the choice of the indices α,β. The theorem also shows that V can be calculated from the knowledge of the covariances of the covariance estimates of the joint process zt=[ytT,utT]T. This merely

Conclusions

In this paper the asymptotic performance of a special class of subspace algorithms has been investigated. The estimate of the transfer function from the exogenous inputs to the outputs has been shown to be a.s. consistent for a generic set of linear systems. The results in Jansson and Wahlberg (1997) show that this actually is the best result that can be expected. Furthermore, for a smaller generic set also the consistency for the system matrices has been shown, as well as asymptotic normality

Acknowledgements

Support by the Austrian ‘Fonds zur Förderung der wissenschaftlichen Forschung’ Projekt P11213-MAT, the foundation BLANCEFLOR Boncompagni-Ludovisi, née Bildt, and the Swedish Foundation for International Cooperation in Research and Higher Education is gratefully acknowledged.

Dietmar Bauer was born in St. Pölten, Austria, in 1972. He received his masters and Ph.D. degrees in Applied Mathematics from the Technical University of Vienna in 1995 and 1998 respectively. From 1995 until 1998 he was with the Institute for Econometrics, Operations Research and System Theory, Technical University of Vienna. Currently he is visiting the Department of Electrical and Computer Engineering, University of Newcastle, Australia. His research interests include system identification in

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    Dietmar Bauer was born in St. Pölten, Austria, in 1972. He received his masters and Ph.D. degrees in Applied Mathematics from the Technical University of Vienna in 1995 and 1998 respectively. From 1995 until 1998 he was with the Institute for Econometrics, Operations Research and System Theory, Technical University of Vienna. Currently he is visiting the Department of Electrical and Computer Engineering, University of Newcastle, Australia. His research interests include system identification in particular subspace algorithms and parametrisation of linear systems, and economic applications of time series analysis. For a recent photograph of Dietmar Bauer please refer to Automatica 35(7) 1243–1254.

    Magnus Jansson was born in Enköping, Sweden, in 1968. He received the Master of Science, Technical Licentiate, and Ph.D. degrees in electrical engineering from the Royal Institute of Technology (KTH), Stockholm, Sweden, in 1992, 1995 and 1997, respectively. From September 1998 he spent one year at the Department of Electrical and Computer Engineering, University of Minnesota, USA. He is currently a Research Associate at the Department of Signals, Sensors and Systems, Royal Institute of Technology.

    His research interests include sensor array signal processing, time series analysis, and system identification.

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

    1

    On leave from S3-Automatic Control, Royal Institute of Technology (KTH), Stockholm, Sweden.

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