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IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32. NO. 2, FEBRUARY 1987 115 Computation of System Balancing Transformations - and Other Applications Simultaneous Diagonalization Algorithms ALAN J. LAUB, FELLOW, IEEE, MICHAEL T. HEATH, CHRIS C. PAIGE, AND RO3ERT c. WARD Abstract--An algorithm is presented in this paper for computing state- space balancing transformations directly from a state-space realization. The algorithm requires no “squaring up” or unnecessary matrix prod- ucts. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken. involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems. I. IKTRODUCTION T HE notion of balancing in linear systems and control theory has become a cornerstone for important new theoretical developments with far-reaching implications. Some early ideas were discussed by Moore in [I91 and further refinements, extensions. and applications have appeared in. for example. [4], [6]-[8], [13], [17], [22], 1251-[27], [29] and the references therein. The literature on the subject is now voluminous. One of the main reasons why balancing is of interest in control and one of its principal applications is in model reduction. For state-space models, a methodology for deriving reduced-order models is provided in terms of a system‘s realization in balanced coordinates. The key computational problem there is the calcula- tion of a balancing transformation and the matrices of the balanced realization. Other applications include a solution to the Hankel- norm approximation problem and other frequency domain and frequency respmse approximation problems. The key paper by Glover [8] provides many further details andreferences.Some important new advances are described in [8] and much of the previous work is tied together in averyreadable and coherent presentation. Other applications of balancing transformations (although not by that name) can be found in signal processing; see, for example. [20]. In this paper we shall concentrate on describing a numerical algorithm for computing a state-space balancing transformation. Other algorithms, particularly some early ideas in [I61 which motivated this paper, and numerous related aspects will also be Manuscript received January 13.
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This note was uploaded on 07/21/2011 for the course MAD 5932 taught by Professor Gallivan during the Spring '06 term at FSU.

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