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Subspace-Based Identification for linear and nonlinear systems

# Subspace-Based Identification for linear and nonlinear systems

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Subspace-Based Identification for Linear and Nonlinear Systems Harish J. Palanthandalam-Madapusi, Seth Lacy, Jesse B. Hoagg and Dennis S. Bernstein 1. I NTRODUCTION Mathematical models describe the dynamic behavior of a system as a function of time, and arise in all scientific disciplines. These mathematical models are used for simula- tion, operator training, analysis, monitoring, fault detection, prediction, optimization, control system designs and quality control. System identification is the process of constructing mathe- matical models of dynamical systems using measured data. The identified models can be used for output prediction, system analysis and diagnostics, system design, and control. Identification of linear models has been well studied and numerous frequency-domain and time-domain methods are available [21]. Among the available time-domain methods, subspace algorithms are an important class of algorithms for identifying linear state-space models. Subspace methods are naturally applicable to multi-input multi-output (MIMO) systems and identify the system in the state-space form. These methods are computationally tractable and robust since they use QR factorization and singular value decompositions. Another advantage of sub- space algorithms is that no a priori information about the system is needed, and thus they are widely applicable. Since the number of user choices is small, the software is user- friendly. All these factors make subspace identification a valuable tool. Overviews of linear subspace identification are presented in detail in [5, 34, 45]. Detailed examination of subspace- based identification of bilinear and linear parameter-varying (LPV) systems is done in [36]. Subspace algorithms have also been applied to numerous industrial and real-life ap- plications ranging from glass tube manufacturing processes to bridge dynamics. A few application-based papers are [9], Sections 6.3 and 6.4 of [34] and [31]. Section 6.4 of [34] references several other interesting applications. In this paper, we introduce the basic ideas behind subspace algorithms to the uninitiated reader and provide an overview of linear, time-varying, nonlinear and closed-loop methods. The basic idea behind subspace is that the Kalman filter state sequence can directly be estimated from the input/output observations. Once the state estimates are avail- able, the state space matrices are estimated using a linear least squares fit. This is contrary to traditional identification S. Lacy is with the Air Force Base, Albuquerque, New Mexico H. J. Palanth, J. B. Hoagg and D. S. Bernstein are with the department of Aerospace Engineering at the University of Michigan, Ann Arbor, MI 48109-2140. { hpalanth,hoagg,dsbaero } @umich.edu methods, where the model is identified first and then the states are generated using the identified model.

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