Time-Varying Eigensystem Realization Algorithm Manoranjan Majji * Texas A&M University, College Station, Texas 77843-3141 Jer-Nan Juang † National Cheng-Kung University, Tainan City, Taiwan 700 Republic of China and John L. Junkins ‡ Texas A&M University, College Station, Texas 77843-3141 DOI: 10.2514/1.45722 An identification algorithm called the time-varying eigensystem realization algorithm is proposed to realize discrete-time-varying plant models from input and output experimental data. It is shown that this singular value decomposition based method is a generalization of the eigensystem realization algorithm developed to realize time invariant models from pulse response sequences. Using the results from discrete-time identification theory, the generalized Markov parameter and the generalized Hankel matrix sequences are computed via a least squares problem associated with the input–output map. The computational procedure presented in the paper outlines a methodology to extract a state space model from the generalized Hankel matrix sequence in different time-varying coordinate systems. The concept of free response experiments is suggested to identify the subspace of the unforced system response. For the special case of systems with fixed state space dimension, the free response subspace is used to construct a uniform coordinate system for the realized models at different time steps. Numerical simulation results on general systems discuss the details and effectiveness of the algorithms. I. Introduction T HE eigensystem realization algorithm (ERA) [1–3] has occupied the center stage in the current system identification theory and practice owing to its ease, efficiency, and robustness of implementation in several spheres of engineering. Connections of ERA with modal and principal component analyses made the algorithm an invaluable tool for the analysis of mechanical systems. As a consequence, the associated algorithms have contributed to several successful applications in design, control, and model order reduction of mechanical systems. ERA is the member of a class of algorithms derived from system realization theory based on the now classical Ho-Kalman method [4]. Because both left and right singular vector matrices of the singular value decomposition are used, ERA yields state space realizations that are not only minimal but also balanced [1]. The key utility of ERA has been in the development of discrete-time invariant models from input and output experimental data. Owing to the one-to-one mapping of linear time invariant dynamical system models between the continuous and discrete-time domains, the ERA identified discrete-time model is tantamount to the identification of a continuous-time model (with the standard assumptions on the sampling theorem). Furthermore, the physical parameters of a mechanical system (natural frequencies, normal modes, and damping) can be derived from the identified plant models by using ERA. A variety of system identification methods for such time invariant systems are available, the fundamental unifying features of which are now well understood [5–7] and can be shown to be related (and/or equivalent) to the corresponding features of ERA. Several efforts were undertaken in the past to develop a holistic approach for the identification of time-varying systems. Specifically, it has been desired for some time to generalize ERA to the case of time-varying systems. Earliest efforts in the development of methods for time-varying systems involved recursive and fast imple- mentations of the time invariant methods by exploring structural properties of the input–output realizations. The classic paper by Chu et al., [8] exploring the displacement structure in the Hankel matrices is representative of the efforts of this nature. Subsequently, significant results were obtained by Shokoohi and Silverman [9] and Dewilde and Van der Veen [10], that generalized several concepts in the classical linear time invariant system theory consistently. Verhaegen and Yu [11] and Verhaegen [12] subsequently introduced the idea of repeated experiments (termed ensemble input/output data), rendering practical methods to realize the conceptual identification strategies presented earlier. These methods are referred to as ensemble state space model identification problems in the literature. This class of generalized system realization methods was applied to complex problems such as the modeling the dynamics of human joints, with much success. Liu [13] developed a methodology for developing time-varying models from free response data (for systems with an asymptotically stable origin) and made initial contributions to the development of time-varying modal parameters and their identification [14]. Although the effects of time-varying coordinate systems are shown to exist by these classical developments, it is not clear if the identified plant models (more generally identified model sequence sets) are useful in state propagation. This is because no guarantees are given as to whether the system matrices identified are, in fact, all realized in the same coordinate system. This limits the utility of the classical solutions because model sequences identified by different procedures cannot be merged as the sequences would lose compatibility at the time instance at which the algorithm is switched. In other words, most classical results developed thus far have realized models that are topologically equivalent (defined mathe- matically in subsequent sections) from an input and output standpoint. However, this does not imply that they are in coordinate systems consistent in time for state propagation purposes. It is Presented at the AAS/AIAA Spaceflight Mechanics Meeting, Galveston, TX, Jan. 2008; received 29 May 2009; revision received 16 Oct. 2009; accepted for publication 19 Oct. 2009. Copyright © 2009 by M. Majji, J. N. Juang, J. L. Junkins. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in correspondence with the CCC. * Post Doctoral Research Associate, Aerospace Engineering Department, Room 616 – D, TAMU 3141; majji@tamu.edu. Member AIAA. † Professor; currently Adjunct Professor, Aerospace Engineering Depart- ment, TAMU 3141; jjuang@cox.net. Fellow AIAA. ‡ Distinguished Professor, Regents Professor, Royce E. Wisenbaker Chair in Engineering, Aerospace Engineering Department, Room 722 B, TAMU 3141; junkins@aeromail.tamu.edu. Fellow AIAA. JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 33, No. 1, January–February 2010 13