Abstract
A new numerical scheme for computing balancing coordinate transformations for signature symmetric realizations in linear systems theory is presented. The method is closely related to the Jacobi method for diagonalizing symmetric matrices. Here the minimization of the sum of traces of the Gramians by orthogonal and hyperbolic Jacobi-type rotations is considered. Local quadratic convergence of the algorithm is shown.
Similar content being viewed by others
References
De Abreu-García, J.A. and Fairman, F.W. 1987, Balanced realization of orthogonally symmetric transfer function matrices. IEEE Transactions on Circuits and Systems, 34(9): 997-1010
Anderson, B.D.O. and Bitmead, R.R. 1977, The matrix Cauchy index: Properties and applications. SIAM J. Appl. Math. 33: 655-672.
Benner, P., Quintana-Ortí, E.S. and Quintana-Ortí, G. 1999, Balanced truncation model reduction of large-scale dense systems on parallel computers. Technical report, University of Bremen, Berichte aus der Technomathematik 99–07.
Byrnes, C.I. and Duncan, T.W. 1982, On certain topological invariants arising in system theory. In P.J. Hilton and G.S. Young, eds., New Directions in Applied Mathematics, pp. 29-72. Springer, New York.
Fortuna, L., Nunnari, G., and Gallo, A. 1992, Model Order Reduction Techniques with Applications in Electrical Engineering. Springer, London.
Gawronski, W. 1996, Balanced Control of Flexible Structures. Springer, London.
Glover, K. 1984, All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds. Internat. J. Control, 39(6): 1115-1193.
Helmke, U. and Hüper, K. 2000, A Jacobi-type method for computing balanced realizations. Systems & Control Letters, 39: 19-30.
Helmke, U. and Moore, J.B. 1994, Optimization and Dynamical Systems. CCES. Springer, London.
Hüper, K. 1996, Structure and convergence of Jacobi-type methods for matrix computations. PhD thesis, Technical University of Munich.
Laub, A.J., Heath, M.T., Paige, C.C. and Ward, R.C. 1987, Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Transactions on Automatic Control, 32(2): 115-122.
Liu, W.Q., Sreeram, V., and Teo. K.L. 1998, Model reduction for state-space symmetric systems. Systems & Control Letters, 34(4): 209-215.
Ober, R.J. 1987, Balanced realizations: canonical form, parametrization, model reduction. Internat. J. Control, 46(2): 643-670.
Safonov, M.G. and Chiang, R.Y. (1989), A Schur method for balanced-truncation model reduction. IEEE Transactions on Automatic Control, 34(7): 729-733.
Yan, W., Moore, J.B. and Helmke, U. 1994, Recursive algorithms for solving a class of nonlinear matrix equations with applications to certain sensitivity optimization problems. SIAM J. Control and Optimization, 32(6): 1559-1576.
Youla, D.C. and Tissi, P. 1966, N-port synthesis via reactance extraction-Part I. IEEE Intern. Convention Record, 183-205.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Helmke, U., Hüper, K. & Moore, J. Computation of Signature Symmetric Balanced Realizations. Journal of Global Optimization 27, 135–148 (2003). https://doi.org/10.1023/A:1024822603531
Issue Date:
DOI: https://doi.org/10.1023/A:1024822603531