Elsevier

Automatica

Volume 50, Issue 6, June 2014, Pages 1632-1640
Automatica

Brief paper
Realization and identification of autonomous linear periodically time-varying systems

https://doi.org/10.1016/j.automatica.2014.04.003Get rights and content

Abstract

The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as “lifting”, is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data.

Introduction

Periodically time-varying systems, i.e., systems with periodic coefficients, appear in many applications and have been studied from both theoretical and practical perspectives. The source of the time-variation can be rotating parts in mechanical systems Bittanti and Colaneri (2008); hearth beat and/or breathing in biomedical applications Ionescu, Kosinski, and De Keyser (2010); Sanchez et al. (2013); and seasonality in econometrics (Ghysels, 1996, Osborn, 2001). Linear periodically time-varying systems also appear when a nonlinear system is linearized about a periodic trajectory Sracic and Allen (2011).

In this paper, we restrict our attention to the subclass of discrete-time autonomous linear periodically time-varying systems. A specific application of autonomous linear periodically time-varying system identification in mechanical engineering is vibration analysis, also known as operational modal analysis; see, e.g., Allen and Ginsberg (2006) and Allen, Sracic, Chauhan, and Hansen (2011). The problems considered in the paper are exact (Section  2, Problem 1) and approximate (Section  5, Problem 2) identifications. The exact identification of an autonomous linear periodically time-varying system is equivalent to realization of an input–output linear periodically time-varying system from impulse response measurement. The approximate identification problem yields a maximum-likelihood estimator in the output error model.

Input–output identification methods for linear periodically time-varying systems are proposed in Hench (1995); Liu (1997); Mehr and Chen (2002); Verhaegen and Yu (1995); Xu, Shi, and You (2012) and Yin and Mehr (2010). Less attention is devoted to the autonomous identification problem. A method for exact identification, based on polynomial algebra, is proposed in Kuijper (1999) and a frequency domain method for output-only identification is developed in Allen (2009) and Allen and Ginsberg (2006). Both the method of Kuijper (1999); Kuijper and Willems (1997) and the method of Allen (2009) are based on a lifting approach, i.e., the time-varying system is represented equivalently as a multivariable time-invariant system. The number of outputs p of the lifted system is equal to the number of outputs p of the original periodic system times the number of samples P in a period of the parameter variation.

Most methods proposed in the literature consist of the following main steps (see also Fig. 1):

  • (1)

    preprocessing—lifting of the data,

  • (2)

    main computation—derivation of a linear time-invariant model for the lifted data,

  • (3)

    postprocessing—derivation of an equivalent linear periodically time-varying model.

The key in solving the linear periodic time-varying realization and identification problem is the lifting operation, which converts the time-varying dynamics into time-invariant dynamics of a system with p=pP outputs. From a computational point of view, the realization of the lifted dynamics is a rank-revealing factorization of a block-Hankel matrix. A numerically stable way of doing this operation is the singular value decomposition of a pL×(TL) matrix, where L is an upper bound on the order, p is the number of outputs, and T is the number of time samples. Its computational complexity is O(L2p2PT) operations.

Once the linear time-invariant dynamics of the lifted model is obtained, it is transformed back to a linear periodically time-varying model in a postprocessing step. In the subspace identification literature, see, e.g., Hench (1995), this operation is done indirectly by computing shifted versions of the state sequence of the model and solving linear systems of equations for the model parameters. This method, referred to as the “indirect method” is Algorithm 1 in the paper, has computational complexity O(L2p2P2T).

The main shortcoming of the indirect method is that it requires extra computations for the derivation of the shifted state sequences and the solution of the systems of equations for the model parameters. This increases the computational complexity by a factor of P compared with the complexity of the realization of the lifted system. We show in Section  4 that the linear periodically time-varying model’s parameters can be obtained directly from the lifted model’s parameters without extra computations. The resulting method, referred to as the “direct method,” is Algorithm 2 in the paper. Its computational complexity is O(L2p2PT). A further improvement of the indirect method (Algorithm 3) operates on a L×p(TL) Hankel matrix and requires O(L2pT) operations.

The maximum-likelihood estimation problem is considered in Section  5. Using the results relating the realization problem to rank revealing factorization of a Hankel matrix constructed from the data, we show that the maximum-likelihood identification problem is equivalent to Hankel structured low-rank approximation. Subsequently, we use existing efficient local optimization algorithms Usevich and Markovsky (2013) for solving the problem.

The motivation for reformulating the maximum likelihood identification problem as structured low-rank approximation is the possibility to use readily available solution methods. Structured-low-rank approximation is an active area of research that offers a variety of solution methods, e.g., convex relaxation methods, based on the nuclear norm heuristic. There are also methods for solving problems with missing data Markovsky and Usevich (2013). Identification with missing data is a challenging problem; however, using the link between system identification and low-rank approximation, the identification of autonomous periodically time-varying systems with missing data becomes merely an application of existing methods.

The main contributions of the paper are summarized next.

  • (1)

    Reduction of the computational cost of linear periodically time-varying system realization from O(L2p2P2T) to O(L2pT).

  • (2)

    Maximum-likelihood method for linear periodically time-varying system identification with computational complexity per iteration that is linear in the number of data points. In addition, the maximum-likelihood method can deal with missing data.

Section snippets

Preliminaries, problem formulation, and notation

An autonomous discrete-time linear time-varying system can be represented by a state space model =(A,C){yx(t+1)=A(t)x(t),y(t)=C(t)x(t),for all  t, with  x(1)=xiniRn}, where A(t)Rn×n and C(t)Rp×n are the model coefficient functions—A is the state transition matrix and C is the output matrix. A state space representation (A,C) of the model is not unique due to a change of basis, i.e., =(A,C)=(Â,Ĉ), where, for all tÂ(t)=V(t+1)A(t)V1(t)andĈ(t)=C(t)V1(t), with a nonsingular

Realization of the lifted system

As shown in Bittanti and Colaneri (2008, Section 6.2.3), the lifted system liftP((A,C)) admits an nth-order linear time-invariant representation (Φ̂,Ψ̂)=liftP((A,C)),with  Φ̂Rn×n  and  Ψ̂Rp×n. The problem of obtaining the parameters Φ̂ and Ψ̂ from the lifted trajectory y of the periodically time-varying system is a classical linear time-invariant realization problem. We use Kung’s method Kung (1978), which is based on the Hankel matrix L(y). The number of block-rows L must be such that

Indirect method

Define the matrices

X̂iVi[x(i)x(i+P)x(i+2P)x(i+(TL1)P)],for  i=1,,P, constructed from the state sequence (x(1),x(2),) in a state-space basis, defined by Vi. The derivation of X̂1 is a by-product of the realization of the lifted system (Φ̂,Ψ̂); see (7). The shifted state sequences X̂2,,X̂P can also be computed from (4) by using the i-steps shifting data (y(i),,y(T)) instead of (y(1),,y(T)). Note that the computation of X̂i through (4) results in general in a basis Vi that is

Maximum likelihood identification

As commented in Note 3, Note 4, the realization Algorithms 1–3 can be used in the case of noisy data as estimation methods. Using instrumental variables, the basic algorithms presented can be extended to different noise assumptions, resulting in a class of the non-iterative identification methods, such as the MOESP methods Verhaegen and Dewilde (1992). Despite many advantages, however, non-iterative methods do not estimate optimal (in an a priori specified sense) models. Therefore, the problem

Numerical examples

The estimation accuracy of Algorithms 1–4, measured by the prediction error, is compared on a test example from Allen et al. (2011). In Section  6.2, the computational advantages of the modified method (Algorithm 3) over the classical method (Algorithm 1) is illustrated on a marginally stable linear periodically time-varying system. Identification with missing data is shown in Section  6.3 and statistical properties of the maximum likelihood estimator (Algorithm 4) are shown in Section  6.4.

Conclusions

In this paper, we developed realization and maximum likelihood identification algorithms for autonomous linear periodically time-varying systems. The algorithms are based on (1) lifting of the original time series, (2) modeling of the lifted time-series by a linear time-invariant system, (3) the transition from the time-invariant system’s parameters to the ones of the periodic time-varying system. It is shown that the derivation of the periodic time-varying system’s state space parameters in

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant agreement number 258581 “Structured low-rank approximation: Theory, algorithms, and applications”, the Research Foundation Flanders (FWO-Vlaanderen), the Flemish Government (Methusalem Fund, METH1), and the Belgian Federal Government (Interuniversity Attraction Poles programme VII, Dynamical Systems, Control, and

Ivan Markovsky obtained M.S. degree in Control and Systems Engineering from the Technical University of Sofia in July 1998 and Ph.D. degree in Electrical Engineering from the Katholieke Universiteit Leuven in February 2005. From January 2007 to September 2012 he was a lecturer at the School of Electronics and Computer Science of the University of Southampton. Since October 2012 he is with the department ELEC of the Vrije Universiteit Brussel. His current research interests are structured

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    Ivan Markovsky obtained M.S. degree in Control and Systems Engineering from the Technical University of Sofia in July 1998 and Ph.D. degree in Electrical Engineering from the Katholieke Universiteit Leuven in February 2005. From January 2007 to September 2012 he was a lecturer at the School of Electronics and Computer Science of the University of Southampton. Since October 2012 he is with the department ELEC of the Vrije Universiteit Brussel. His current research interests are structured low-rank approximation, system identification, and data-driven control.

    Jan Goos was born in Geel (Belgium) in 1986. He graduated from the Katholieke Universiteit Leuven as an engineer in Computer sciences (Artificial Intelligence) in 2009 and in Mathematical Engineering in 2011. In October 2011 he joined the department of ELEC as a Ph.D. student. His main interests are the measurement and modeling of Linear Parameter Varying (LPV) systems, but he also loves non-linear dynamics.

    Konstantin Usevich obtained a specialist degree in 2007 and Ph.D. degree in 2011, both from the Department of Statistical Modeling, St.Petersburg State University, Russia. From September 2011 to September 2012 he was working as a postdoctoral research fellow at the University of Southampton, United Kingdom. Since October 2012, he is a postdoctoral researcher at the Department ELEC of the Vrije Universiteit Brussel, Belgium. His research interests are in low-rank approximation of structured matrices, system identification, signal and image processing, and approximate polynomial computations.

    Rik Pintelon was born in Gent, Belgium, on December 4, 1959. He received a master’s degree in electrical engineering in 1982, a doctorate (Ph.D.) in engineering in 1988, and the qualification to teach at university level (geaggregeerde voor het hoger onderwijs) in 1994, all from the Vrije Universiteit Brussel (VUB), Brussels, Belgium.

    From 1982 to 1984 and 1986–2000, Dr. Pintelon was a researcher with the Belgian National Fund for Scientific Research (FWO-Vlaanderen) at the Electrical Engineering (ELEC) Department of the VUB. From 1984 to 1986 he did his military service overseas in Tunesia at the Institut National Agronomique de Tunis. From 1991 to 2000 he was a part-time lecturer at the department ELEC of the VUB, and since 2000 he is a full-time professor in electrical engineering at the same department. Since 2009 he is a visiting professor at the department of Computer Sciences of the Katholieke Universiteit Leuven, and since 2013 he is a honorary professor in the School of Engineering of the University of Warwick. His main research interests include system identification, signal processing, and measurement techniques. Dr. Pintelon is the coauthor of 4 books on System Identification and the coauthor of more than 200 articles in refereed international journals. He has been a Fellow of IEEE since 1998. Dr. Pintelon was the recipient of the 2012 IEEE Joseph F. Keithley Award in Instrumentation and Measurement (IEEE Technical Field Award).

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Alessandro Chiuso under the direction of Editor Torsten Söderström.

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