Subspace identification of closed loop systems by the orthogonal decomposition method

doi:10.1016/j.automatica.2004.11.026

Automatica

Volume 41, Issue 5, May 2005, Pages 863-872

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Abstract

In this paper, we consider a problem of identifying the deterministic part of a closed loop system by applying the stochastic realization technique of (Signal Process. 52 (2) (1996) 145) in the framework of the joint input–output approach. Using a preliminary orthogonal decomposition, the problem is reduced to that of identifying the plant and controller based on the deterministic component of the joint input–output process. We discuss the role of input signals in closed loop identification and the realization method based on a finite data, and then sketch a subspace method for identifying state space models of the plant and controller. Since the obtained models are of higher order, a model reduction procedure should be applied for deriving lower order models. Some numerical results are included to show the applicability of the present technique.

Introduction

The identification problem for linear systems operating in closed loop has received much attention in the literature (Söderström and Stoica, 1989, Van den Hof, 1997, Forssell and Ljung, 1999). Also, the identification of multivariable systems operating in closed loop by subspace methods has been object of active research in the past decade; among early references using the joint input–output approach, we quote papers (Van der Klauw et al., 1991, Verhaegen, 1993).

Since the joint input–output approach requires assumptions of linearity of the feedback channel and is generally computationally demanding, several attempts have been made to identify directly the plant dynamics without estimating the feedback channel. For example modifying the N4SID method (Van Overschee & De Moor, 1996), a closed loop subspace identification method has been derived in Van Overschee and De Moor (1997). However this approach requires that a finite number of Markov parameters of the controller are known. Several other subspace-based closed loop identification have been proposed, e.g. (Chou and Verhaegen, 1999, Ljung and McKelvey, 1996). In general, these methods need some side information on the system and their statistical properties (e.g. consistency) have not been assessed; see e.g. Bauer (2004) and Chiuso and Picci (2004) for an up-to-date discussion of these aspects.

In this paper, we study a joint input–output subspace method for identifying the plant and controller operating in closed loop, by extending the orthogonal decomposition based technique in Katayama et al., 2001, Katayama et al., 2002. It is not assumed that all the input signals are purely non-deterministic (and hence persistently exciting of arbitrary order) as in the references above, but one of the exogenous inputs is allowed to be a purely deterministic (p.d.) (or “linearly singular” (Rozanov, 1967)) signal. First we compute the deterministic component of the joint input–output process, that is linearly related to the exogenous inputs, by means of an orthogonal decomposition as in Picci and Katayama (1996). Thereby, the identification of the closed loop system is reduced to that of obtaining the plant and controller based on the deterministic component of the joint input–output process. Based on the realization of the deterministic component, a subspace method for identifying the plant and controller is then derived by adapting some standard subspace method from the literature. In general the obtained models will be of higher order, so that a model reduction procedure should be applied for deriving lower order models.

The organization of the paper is as follows. In Section 2, the problem is stated along with the underlying assumptions. In Section 3, we formulate a joint input–output approach to the closed loop identification problem and derive the deterministic component of the joint input–output process by the preliminary orthogonal decomposition. Section 4 considers the state space realization of the deterministic component, with special emphasis on the role of input signals in closed loop identification and on the realization from a finite data. Section 5 derives some formulas for computing the plant and controller from overall transfer matrices. In Section 6, a subspace method of identifying the plant and controller is briefly sketched, together with a model reduction procedure. Section 7 includes some numerical results and Section 8 concludes the paper.

Section snippets

Problem statement

We consider the problem of identifying a closed loop system shown in Fig. 1, where $y \in R^{p}$ is the output vector of the plant, and $u \in R^{m}$ the input vector. The effect of stochastic unmeasurable disturbances, modeling errors, etc. is described by stationary error (or disturbance) processes $H (z) ξ$ and $F (z) η$ , acting additively on the outputs of the plant and controller, respectively. The transfer matrices of noise filters $H (z)$ and $F (z)$ will be assumed square rational, minimum phase with $H (\infty) = I_{p}$ and $F (\infty) = I$

The joint input–output approach

In this section, we shall be concerned with the case where infinite data are available.

Realization of closed-loop system

In this section, we shall construct a state space realization for the joint “deterministic” input–output process described in the previous section. The technique will be based on stochastic realization ideas described in Picci and Katayama (1996), Katayama and Picci (1999) and Verhaegen (1994).

Extracting plant and controller models

Assume now that we have estimated the joint model (7). Then by simple manipulations of the joint state space equations (7), we can derive state space models for the plant and controller. The formulas are collected in the following proposition.

Proposition 3

A (non-minimal) state space representations of the plant and controller are respectively given by $\begin{matrix} \end{matrix}$ and $\begin{matrix} \end{matrix}$

Proof

It follows from (7) that the closed loop transfer matrices are expressed as $\begin{matrix} \end{matrix}$ We see from (4) that the plant and controller are computed from $P (z) = T_{yd} (z) T_{ud}^{- 1}$

Closed-loop subspace identification algorithm

We briefly discuss a subspace identification method based on the data measured on a finite interval. Suppose that the input–output data⁴ ${d (t), r (t), u (t), y (t), t = 0, 1, \dots, N + 2 k - 2}$ , with $k > n$ and N very large, be sample values from the jointly stationary “true” input–output

Simulation results

Some simulation results are included to show the applicability of the present technique. Suppose that the plant, controller and two noise models are given by Van den Hof and Schrama (1993) $P (z) = \frac{z^{- 1}}{1 - 1.6 z^{- 1} + 0.89 z^{- 2}}, C (z) = 1 - 0.8 z^{- 1}$ and $F (z) = 1, H (z) = \frac{1 - 1.56 z^{- 1} + 1.045 z^{- 2} - 0.3338 z^{- 3}}{1 - 2.35 z^{- 1} + 2.09 z^{- 2} - 0.6675 z^{- 3}} .$ The configuration of the feedback system is the same as the one shown in Fig. 1, where d, $η$ and $ξ$ are Gaussian white noises with variances $σ_{d}^{2} = 0.2$ , $σ_{η}^{2} = 0.01$ and $σ_{ξ}^{2} = \frac{1}{9}$ , respectively. The reference

Conclusions

In this paper we have developed a subspace method for identifying the deterministic part, i.e. the plant and controller, of closed loop systems in the joint input–output framework. It is assumed that one of the exogenous inputs is purely deterministic and the other is purely non-deterministic. We have discussed the realization method based on a finite data and the role of input signals in closed loop system identification, and derived a subspace method to identify the plant and controller.

Acknowledgements

The authors wish to thank H. Tanaka for his assistance in performing simulations.

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Tohru Katayama received the B.E., M.E. and Ph.D. degrees all in applied mathematics and physics from Kyoto University, Kyoto, in 1964, 1966 and 1969, respectively. Since 1986, he has been in the Department of Applied Mathematics and Physics, Kyoto University. He had visiting positions at UCLA from 1974–1975, and at University of Padova in 1997. He was an Associate Editor of IEEE Transactions on Automatic Control from 1996 to 1998, and is now a Subject Editor of International Journal of Robust and Nonlinear Control, and the Chair of IFAC Technical Committee of Stochastic Systems for 1999–2002, and is now the Chair of IFAC Coordinating Committee of Signals and Systems for 2002–2005. His research interests includes estimation theory, stochastic realization, subspace method of identification, blind identification, and control of industrial processes.

Hidetoshi Kawauchi received the B.E. and M.E. degrees both in applied mathematics and physics from Kyoto University, Kyoto, in 1999 and 2001, respectively. Since 2001, he has been an engineer of development at the Sony Corporation, Shinagawa, Japan. His research interests includes theory of subspace system identification of closed loop systems.

Giorgio Picci holds a full professorship with the University of Padova, Italy, Department of Information Engineering, since 1980. He has held several long-term visiting appointments with various American and European universities among which Brown University, M.I.T., the University of Kentucky, Arizona State University, the Center for Mathematics and Computer Sciences (C.W.I.) in Amsterdam, the Royal Institute of Technology, Stockholm, Sweden, Kyoto University and Washington University, St. Louis, MO. He has been contributing to Systems and Control theory mostly in the area of modeling, estimation and identification of stochastic systems and published over 100 papers and edited three books in this area. He has been involved in various joint research projects with industry and state agencies. He is currently general coordinator of the Commission of European Communities IST project RECSYS, of the fifth Framework Program. Giorgio Picci is a Fellow of the IEEE, past chairman of the IFAC Technical Committee on Stochastic Systems and a member of the EUCA council.

^☆: 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 T. Söderström.

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Brief PaperSubspace identification of closed loop systems by the orthogonal decomposition method☆

Abstract

Introduction

Section snippets

Problem statement

The joint input–output approach

Realization of closed-loop system

Extracting plant and controller models

Closed-loop subspace identification algorithm

Simulation results

Conclusions

Acknowledgements

Automatica

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Signal Processing

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Automatica

Some algorithmic aspects of subspace identification with inputs

International Journal of Applied Mathematics & Computer Science

Brief Paper
Subspace identification of closed loop systems by the orthogonal decomposition method☆