The Jacobi Method: A Tool for Computation and Control

The interaction between numerical linear algebra and control theory has crucially influenced the development of numerical algorithms for linear systems in the past. Since the performance of a control system can often be measured in terms of eigenvalues or singular values, matrix eigenvalue methods have become an important tool for the implementation of control algorithms. Standard numerical methods for eigenvalue or singular value computations are based on the QR-algorithm. However, a number of computational problems in control and signal processing are not amenable to standard numerical theory or cannot be easily solved using current numerical software packages. Various examples can be found in the digital filter design area. For instance, the task of finding sensitivity optimal realizations for finite word length implementations requires the solution of highly nonlinear optimization problems for which no standard numerical solution of algorithms exist.

[1]  Raimund J. Ober,et al.  Balanced realizations: canonical form, parametrization, model reduction , 1987 .

[2]  G. Kempf,et al.  The length of vectors in representation spaces , 1979 .

[3]  M. Safonov,et al.  A Schur method for balanced-truncation model reduction , 1989 .

[4]  Lothar Thiele,et al.  On the sensitivity of linear state-space systems , 1986 .

[5]  R. Brockett.  Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[6]  Clifford T. Mullis,et al.  Synthesis of minimum roundoff noise fixed point digital filters , 1976 .

[7]  K. Huper,et al.  Structure and convergence of conventional Jacobi-type methods minimizing the off-norm function , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[8]  Michael Atiyah,et al.  Convexity and Commuting Hamiltonians , 1982 .

[9]  U. Helmke.  Balanced realizations for linear systems: a variational approach , 1993 .

[10]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[11]  Wei-Yong Yan,et al.  On L/sup 2/-sensitivity minimization of linear state-space systems , 1992 .

[12]  M. Gevers,et al.  Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects , 1993 .

[13]  John B. Moore,et al.  Singular-Value Decomposition via Gradient and Self-Equivalent Flows , 1992 .

[14]  S. Sternberg,et al.  Convexity properties of the moment mapping , 1982 .

[15]  A. Laub,et al.  Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms , 1987 .

[16]  C. Jacobi,et al.  C. G. J. Jacobi's Gesammelte Werke: Über ein leichtes Verfahren, die in der Theorie der Sacularstorungen vorkommenden Gleichungen numerisch aufzulosen , 1846 .