Brief paperRealization and identification of autonomous linear periodically time-varying systems☆
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 of the lifted system is equal to the number of outputs of the original periodic system times the number of samples in a period of the parameter variation.
Most methods proposed in the literature consist of the following main steps (see also Fig. 1):
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preprocessing—lifting of the data,
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main computation—derivation of a linear time-invariant model for the lifted data,
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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 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 matrix, where is an upper bound on the order, is the number of outputs, and is the number of time samples. Its computational complexity is 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 .
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 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 . A further improvement of the indirect method (Algorithm 3) operates on a Hankel matrix and requires 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.
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Reduction of the computational cost of linear periodically time-varying system realization from to .
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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 where and are the model coefficient functions— is the state transition matrix and is the output matrix. A state space representation of the model is not unique due to a change of basis, i.e., where, for all with a nonsingular
Realization of the lifted system
As shown in Bittanti and Colaneri (2008, Section 6.2.3), the lifted system admits an th-order linear time-invariant representation The problem of obtaining the parameters and from the lifted trajectory 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 . The number of block-rows must be such that
Indirect method
Define the matrices
constructed from the state sequence in a state-space basis, defined by . The derivation of is a by-product of the realization of the lifted system ; see (7). The shifted state sequences can also be computed from (4) by using the -steps shifting data instead of . Note that the computation of through (4) results in general in a basis 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).
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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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